EFFECT OF TEMPERATURE MODULATION ON RAYLEIGH …repository.christuniversity.in/1840/1/Rashmi_-_M... · 2.1 Rayleigh-Bénard Convection in Fluids 7 2.2 Convection in the Presence of

EFFECT OF TEMPERATURE MODULATION

ON RAYLEIGH-BÈNARD CONVECTION

IN A ROTATING LAYER OF A

FERROMAGNETIC FLUID

Dissertation submitted in partial fulfillment of the requirements for the award of the degree of

MASTER OF PHILOSOPHY IN MATHEMATICS

By

RASHMI VENKATESH MURTHY Register No. 0935309

Supervisor Dr. S. MARUTHAMANIKANDAN

Department of Mathematics Christ University

Bangalore-560 029

HOSUR ROAD

BANGALORE-560 029

2010

DEDICATED TO

MY BELOVED PARENTS

VENKATESH MURTHY & VIJAYA

DECLARATION

I hereby declare that the dissertation entitled “Effect of Temperature

Modulation on Rayleigh-Bènard Convection in a Rotating Layer of

a Ferromagnetic Fluid” has been undertaken by me for the award of

M.Phil. degree in Mathematics. I have completed this under the guidance of

Dr. S. MARUTHAMANIKANDAN, Assistant Professor, Department of

Mathematics, Christ University, Bangalore-560 029. I also declare that this

dissertation has not been submitted for the award of any Degree, Diploma,

Associateship, Fellowship or other title.

Place:

Date:

RASHMI VENKATESH MURTHY

Candidate

Dr. S. MARUTHAMANIKANDAN Assistant Professor Department of Mathematics Christ University Bangalore - 560 029.

CERTIFICATE This is to certify that the dissertation submitted by RASHMI

VENKATESH MURTHY on the title “Effect of Temperature

Modulation on Rayleigh-Bènard Convection in a Rotating Layer of

a Ferromagnetic Fluid” is a record of research work done by her during the

academic year 2009 – 2010 under my guidance and supervision in partial

fulfillment of the requirements for the award of the degree of Master of

Philosophy in Mathematics. This dissertation has not been submitted for the

award of any Degree, Diploma, Associateship, Fellowship or other title.

Place:

Date:

Dr. S. MARUTHAMANIKANDAN

Supervisor

ACKNOWLEDGEMENT

First and foremost, I would like to express my sincere thanks and appreciation to

my supervisor, Dr. S. Maruthamanikandan, for his phenomenal guidance and

support all through the work. I am appreciative of his constant encouragement and

unwavering support. I am immensely grateful to him for helping me improve my

scientific skills.

I am thankful to Dr. S. Pranesh, Coordinator, Postgraduate Department of

Mathematics, Christ University for his genuine concern and constant support.

I gratefully acknowledge Prof. T.V. Joseph, HOD, Department of Mathematics,

Christ University and Mrs. Sangeetha George, Assistant Professor, Department of

Mathematics, Christ University for their whole-hearted support.

I extend my gratitude to Prof. K.A. Chandrasekharan, the General Research

Coordinator and Prof. Dr. Nanje Gowda, The Dean of Science of Christ

University for their valuable advice and constant support.

Special thanks to the Vice–Chancellor, Dr. (Fr.) Thomas C. Mathew,

Pro-Vice-Chancellor and Director of Centre for Research and Consultancy,

Dr. (Fr.) Abraham V. M. of Christ University for the opportunity provided to do

this course.

I really need to acknowledge my friends who provided a stimulating and

fascinating environment. I am especially thankful to Chitra, Aparna, Shibiraj

Singh and Deepika for sharing the good time.

My deep appreciation goes to my brother Srinivasa and my sister Sushma for

their remarkable support and love.

RASHMI VENKATESH MURTHY

ABSTRACT

The stability of a rotating horizontal layer of ferromagnetic fluid heated

from below is examined when, in addition to a steady temperature

difference between the walls of the layer, a time-dependent sinusoidal

perturbation is applied to the wall temperatures. Only infinitesimal

disturbances are considered. The effects of the oscillating temperature field

are treated by a perturbation expansion in powers of the amplitude of the

applied field. The onset criterion is derived when the condition for the

principle of exchange of stabilities is valid. The shift in the critical Rayleigh

number is calculated as a function of the frequency of modulation,

magnetic parameters, Taylor number and Prandtl number. The effect of

various parameters is found to be significant for moderate values of the

frequency of modulation. It is shown that, when the thermal excitation is

symmetric, supercritical motion is more pronounced for low Prandtl

number ferro fluids. Further, for the case in which only the bottom wall

temperature is modulated, the effect of rotation is to stabilize the system at

low frequencies and the opposite is true for moderately large frequencies.

The problem throws light on external means of controlling convection in

ferromagnetic fluid applications.

CONTENTS

Page No.

CHAPTER I Introduction 1 1.1 Objective and Scope 1 CHAPTER II Literature Review 7

2.1 Rayleigh-Bénard Convection in Fluids 7 2.2 Convection in the Presence of Rotation 16 2.3 Convection with Temperature Modulation 28 2.4 Convection in Ferromagnetic Fluids 41 2.5 Plan of Work 54 CHAPTER III Basic Equations, Boundary Conditions and

Dimensionless Parameters 55

3.1 Basic Equations 57 3.2 Boundary Conditions 61 3.3 Dimensionless Parameters 65 CHAPTER IV

Effect of Temperature Modulation on Rayleigh-Bènard Convection in a Rotating Layer of a Ferromagnetic Fluid

68

4.1 Introduction 68 4.2 Mathematical Formulation 72 4.3 Basic State 74 4.4 Linear Stability Analysis 75 4.5 Method of Solution 79 CHAPTER V

Results, Discussion and Concluding Remarks 84

5.1 Results and Discussion 84 5.2 Concluding Remarks 86 BIBLIOGRAPHY

100

1

CHAPTER I

INTRODUCTION

1.1 Objective and Scope

In most part of the last century the engineering applications of fluid mechanics

were restricted to systems in which electric and magnetic fields played no role.

Recently, the study of the interaction of electromagnetic fields with fluids started

gaining attention with the promise of applications to areas like nuclear fusion,

chemical engineering, medicine and high-speed noiseless printing. This study could

be divided into three main categories.

Magnetohydrodynamics – the study of the interaction between magnetic

fields and electrically conducting fluids.

Ferrohydrodynamics – the study of the mechanics of fluid motion influenced

by strong forces of magnetic polarization and

Electrohydrodynamics – the branch of fluid mechanics concerned with

electric force effects.

The investigation of convective heat transfer together with the aforementioned

electrical and magnetic forces in Newtonian fluids is of practical importance. A

systematic study through a proper theory is essential so as to understand the physics

of the complex flow behaviour of these fluids and to obtain invaluable scaled-up

information for industrial applications as well. The emerging areas of applications

of magnetic fluids have brought to light new thoughts and ideas for advanced level

research. The variety of situations related to these application-oriented problems

makes the modelling of the same intricate and complex. Since Ferro fluids are

basically suspensions which respond thermally and to magnetic fields, the problems

concerning magnetic fluids are both mathematically and physically challenging.

2

Ferromagnetic Fluids

Ferromagnetism is a property of iron, nickel, cobalt and some compounds and

alloys of these elements. It was thought that to create a magnetic fluid one might

heat the metal until it becomes molten, but this strategy did not work as

ferromagnetism disappears above a certain temperature called the Curie point,

which is invariably well below the melting point of the material. A magnetic fluid,

better known as ferrofluid, consists of kinetically stabilized ultramicroscopic ferro-

or ferrimagnetic particles coated with a monomolecular layer of surfactant and

colloidally dispersed in a magnetically passive liquid. Under the influence of an

external magnetic field, such a fluid exhibits a large magnetization and as soon as

the field is removed, the fluid attains its zero magnetization state at once. As each

particle possesses a giant magnetic moment when compared with paramagnetic

particles, such a medium is called superparamagnetic i.e. having zero remanence

and coercivity. Ferrofluids have almost the same magnetic characteristics as a solid,

but in many respects behave as liquid continua. Magnetic liquids can be controlled

by magnetic forces.

Composition of Magnetic Fluids

The development of many innovative applications warrants the unique

combination of magnetic and fluidic property. A magnetic fluid is a two phase

matter consisting of solid and liquid and a three component system comprising

magnetic particles, carrier liquid and surfactant. Since randomizing Brownian

energy may not be sufficient to counteract attractions owing to van der Waal and

dipole-dipole forces, aggregation and sedimentation are prevented by providing

suitable repulsive forces either by Coulomb or by steric repulsion. In the former

case particles are either positively or negatively charged and the fluid is called ionic

ferrofluid while in the latter case each particle is coated with an appropriate

surfactant and the resulting fluid is known as surfacted ferrofluid (Figure 1.1).

3

Figure 1.1: (a) Surfacted ferrofluid and (b) ionic ferrofluid. (Upadhyay, 2000) On the other hand, the magnetic fluid should remain stable in the presence of a

magnetic field, that is, there should be no agglomeration and/or phase separation.

To meet this requirement, each of the three components should satisfy certain

conditions. Non-isothermal application situations are relevant to the theme of the

dissertation. For most non-isothermal applications, the most sought-after properties

of a ferrofluid liquid are the following (Fertman, 1990; Berkovskii et. al., 1993;

Upadhyay, 2000):

Long-term stability within the operating temperature range of the device and

within the range of magnetic field strengths.

High saturation magnetization and large initial susceptibility.

Low viscosity and low vapour pressure.

Stability in gravitational fields and the magnetic field gradient.

Absence of significant aggregation in the presence of a uniform magnetic field.

Good thermal conductivity.

4

The investigation of heat transfer in ferrofluids is of practical importance. Their

commercial applications include ink jet printing, sealing, pumping, semiconductor

and computer industries, and medicine and high speed noiseless printing

(Rosensweig, 1986 and Fertman, 1990).

Rayleigh-Benard and Marangoni Convection Rayleigh-Benard convection (RBC) is the instability of a fluid layer which is

confined between two thermally conducting plates, and is heated from below to

produce a fixed temperature difference. Since liquids typically have positive

thermal expansion coefficient, the hot liquid at the bottom of the cell expands and

produces an unstable density gradient in the fluid layer. If the density gradient is

sufficiently strong, the hot fluid will rise, causing a convective flow which results

in enhanced transport of heat between the two plates. In order for convection to

occur, a small plume of hot fluid which begins to rise toward the top of the cell

must grow in strength, rather than fizzle out.

There are two processes that oppose this amplification. First, viscous damping in

the fluid directly opposes the fluid flow. In addition, thermal diffusion will suppress

the temperature fluctuation by causing the rising plume of hot fluid to equilibrate

with surrounding fluid, destroying the buoyant force. Convection occurs if the

amplifying effect exceeds the disippative effect of thermal diffusion and buoyancy.

This competition of forces is parameterized by the Rayleigh number, which is the

temperature difference, but appropriately normalized to take into account the

geometry of the convection cell and the physical properties of the fluid. If the

temperature difference is very large, then the fluid rises very quickly, and a

turbulent flow may be created. If the temperature difference is not far above the

onset, an organized flow resembling overturning of cylinders is formed. It is the

patterns created by these convection "rolls" that most people study.

5

Cross-sectional view of cell illustrating convection rolls

The first intensive experiments were carried out by Benard in 1900. He

experimented with a fluid of thin layer and observed appearance of hexagonal cells

when the instability in the form of convection developed. Rayleigh in 1916

developed the theory which found the condition for the instability with two free

surface. He showed that the instability would manifest if the temperature gradient

was large enough so that the so-called Rayleigh number exceeds a certain value

(critical value).

Experiments in the early stage were carried out with fluid heated from bottom

and the top surface is open to atmosphere. Thus the top surface is free to move and

deform. It was later (around 1960) realized that this could lead to another instability

mechanism (thermocapillary convection) due to gradient in surface tension. This

mechanism coexists with the Rayleigh's mechanism but dominates in thin layer.

Most of the findings reported by Benard were actually due to this latter instability

mechanism. The instability driven by surface tension decreases as the layer

becomes thicker. Experiments on thermal convection (with or without free upper

surface) have exhibited convective cells of many forms such as rolls, square and

hexagons.

When it comes to Rayleigh-Benard convection, only buoyancy force is

responsible for the appearance of convection cells. The initial movement is the

upwelling of warmer liquid from the heated bottom layer. In case of a free liquid

surface in contact with air, surface tension effect will play a role besides buoyancy.

It is known that liquids flow from places of lower surface tension to places of

higher surface tension. This is called the Marangoni effect. When applying heat

from below, the temperature at the top layer will show temperature fluctuations.

6

With increasing temperature, surface tension decreases. Thus a lateral flow of

liquid at the surface will take place, from warmer areas to cooler areas. In order to

preserve a horizontal (or nearly horizontal) liquid surface, liquid from the cooler

places on the surface have to go down into the liquid. Thus the driving force of the

convection cells is the downwelling of liquid. Marangoni convection plays an

important role in Benard convection in shallow fluid layers, in chemical

engineering as well as in crystal growth and other materials processing

technologies.

Assorted Constraints In quite a few heat transfer problems, suppressing or augmenting the convection

plays a vital role. There are several mechanisms that can be used effectively to

either delay or advance the convection, namely, by applying a magnetic/electric

field externally or by Coriolis force due to rotation or by maintaining non-uniform

temperature gradient across the porous layer. A non-uniform temperature gradient

can arise in various ways notably due to (i) transient heating or cooling at a

boundary (ii) volumetric distribution of heat sources (iii) radiative heat transfer

(iv) thermal modulation (v) vertical throughflow and (vi) chemical reaction.

Goal

The problem of convection in a rotating layer of a ferromagnetic fluid has been

extensively studied. However, attention has not been given to the study of effect of

temperature modulation on Rayleigh–Benard convection in a rotating layer of a

ferromagnetic fluid. Therefore, the objective of the work is to investigate

theoretically the influence of temperature modulation on Rayleigh-Benard

convection in a ferromagnetic fluid in the presence of rotation with emphasis on

how the stability criterion for the onset of convection is modified in the presence of

both temperature modulation and rotation.

7

CHAPTER II

LITERATURE REVIEW

The main objective of the dissertation is to deal with Rayleigh-Bènard

convection in a rotating layer of a ferromagnetic fluid with temperature modulation.

Literature pertinent to this is classified as follows.

• Rayleigh-Bénard convection (RBC) in fluids

• Convection in the presence of rotation.

• Convection with temperature modulation.

• Convection in ferromagnetic fluids.

The relevant literature on the problem at hand is briefly discussed below in

keeping with the above classifications.

2.1 Rayleigh-Bénard Convection in Fluids Natural convection in a horizontal layer of fluid heated from below and cooled

from above has been the subject of investigation for many decades owing to its

implications for the control and exploitation of many physical, chemical and

biological processes. We now make a brief review of the RBC problem keeping in

mind the objective and scope of the thesis.

The earliest experiment which called attention to the thermal instability was

briefly reported by Thompson (1882). Benard (1901) later presented a much more

complete description of the development of the convective flow. Lord Rayleigh

(1916) was the first to study the problem theoretically and aimed at determining the

conditions delineating the breakdown of the quiescent state. As a result, the thermal

instability situation described in the foregoing paragraph is referred to as Rayleigh-

Bénard convection (RBC). The Rayleigh theory was generalized and extended to

consider several boundary combinations by Jeffreys (1926), Low (1929) and

8

Sparrow et al. (1964). Chandra (1938) examined the RBC problem experimentally

for a gas. The most complete theory of the thermal instability problem was

presented by Pellew and Southwell (1940).

Malkus and Veronis (1958) investigated finite amplitude cellular convection

and determined the form and amplitude of convection by expanding the nonlinear

equations describing the fields of motion and temperature in a sequence of

inhomogeneous linear equations. Veronis (1959) studied finite amplitude cellular

convection in a rotating fluid and showed that the fluid becomes unstable to finite

amplitude disturbances before it becomes unstable to infinitesimal perturbations.

Palm (1960) showed that for a certain type of temperature-dependence of

viscosity, the critical Rayleigh number and the critical wavenumber are smaller

than those for constant viscosity and explained the observed fact that steady

hexagonal cells are formed frequently at the onset of convection.

Lorenz (1963) solved a simple system of deterministic ordinary nonlinear

differential equations representing cellular convection numerically. For those

systems with bounded solutions, it is found that non-periodic solutions are unstable

with respect to small modifications and that slightly differing initial states can

evolve into considerably different states.

Veronis (1966) analyzed the two-dimensional problem of finite amplitude

convection in a rotating layer of fluid by considering the boundaries to be free.

Using a minimal representation of Fourier series, he showed that, for a restricted

range of Taylor number, steady finite amplitude motions can exist for values of the

Rayleigh number smaller than the critical value required for overstability. Veronis

(1968) also examined the effect of a stabilizing gradient of solute on thermal

convection using both linear and finite amplitude analysis. It is found that the onset

of instability may occur as an oscillatory motion because of the stabilizing effect of

the solute in the case of linear theory and that finite amplitude instability may occur

first for fluids with a Prandtl number somewhat smaller than unity.

9

Krishnamurthy (1968a, b) presented a nonlinear theory of RBC problem and

discussed the formation of hexagonal cells and the existence of subcritical

instabilities. Torrance and Turcotte (1971) investigated the influence of large

variations of viscosity on convection in a layer of fluid heated from below.

Solutions for the flow and temperature fields were obtained numerically assuming

infinite Prandtl number, free-surface boundary conditions and two-dimensional

motion. The effect of temperature-dependent and depth-dependent viscosity was

studied motivated by the convective heat transport in earth’s mantle.

Busse (1975) considered the interaction between convection in a horizontal fluid

layer heated from below and an ambient vertical magnetic field. It is found that

finite amplitude onset of steady convection becomes possible at Rayleigh numbers

considerably below the values predicted by linear theory.

Booker (1976) investigated experimentally the heat transport and structure of

convection in a high Prandtl number fluid whose viscosity varies by up to a factor

of 300 between the boundary temperatures. Horne and Sullivan (1978) examined

the effect of temperature-dependent viscosity and thermal expansion coefficient on

the natural convection of water through permeable formations. They found that the

convective motion is unstable at even moderate values of the Rayleigh number and

exhibits a fluctuating convective state analogous to the case of a fluid with constant

viscosity and coefficient of thermal expansion.

Carey and Mollendorf (1980) presented a regular perturbation analysis for

several laminar natural convection flows in liquids with temperature-dependent

viscosity. Several interesting variable viscosity trends on flow and transport are

suggested by the results obtained. Stengel et al. (1982) obtained, using a linear

stability theory, the viscosity-ratio dependences of the critical Rayleigh number and

critical wave number for several types of temperature-dependence of viscosity.

Richter et al. (1983) showed, by an experiment with temperature-dependent

viscosity ratio as large as 106, the existence of subcritical convection of finite

10

amplitude near the critical Rayleigh number. Busse and Frick (1985) analyzed the

problem of RBC with linear variation of viscosity and showed an appearance of

square pattern for a viscosity ratio larger than 2.

White (1988) made an experiment for the fluid with Prandtl number of o(105)

and studied convective instability with several planforms for the Rayleigh number

up to 63000 and the temperature-dependent viscosity ratio up to 1000. He found

that if the viscosity ratio is 50 or 100 and the Rayleigh number is less than 25000,

stable hexagonal and square patterns are formed in a certain range of wavenumber

and that their wavenumbers increase with viscosity ratio. The possibility of multi-

valued solution in the thermal convection problem with temperature-dependent

viscosity has been examined numerically by Hirayama and Takaki (1993).

Tong and Shen (1992) studied high Rayleigh number turbulent convection using

the technique of photon-correlation homodyne spectroscopy to measure velocity

differences at various length scales. The measured power-law exponents are found

to be in excellent agreement with the theoretical predictions.

Massaioli et al. (1993) investigated the probability density function (pdf) of the

temperature field by numerical simulations of Rayleigh-Bénard convection in two

spatial dimensions. The Pdf of the temperature has been shown to have exponential

tails, consistently with previous laboratory experiments and numerical simulations.

They also offered a new theoretical explanation for the exponential tail of the Pdf.

Xi and Gunton (1993) presented a numerical study of the spontaneous formation

of spiral patterns in Rayleigh-Benard convection in non-Boussinesq fluids. They

solved a generalized two-dimensional Swift-Hohenberg equation that includes a

quadratic nonlinearity and coupling to mean flow. They showed that this model

predicts in quantitative detail many of the features observed experimentally in

studies of Rayleigh-Benard convection in CO2 gas. In particular, they studied the

appearance and stability of a rotating spiral state obtained during the transition from

an ordered hexagonal state to a roll state.

11

Mukutmoni and Yang (1994) reviewed the broad area of flow transitions of

Rayleigh-Benard convection in rectangular enclosures with sidewalls. They looked

into pattern selection for both small and intermediate enclosures.

Kafoussias and Williams (1995) studied, using an efficient numerical

technique, the effect of a temperature-dependent viscosity on an incompressible

fluid in steady, laminar, free-forced convective boundary layer flow over an

isothermal vertical semi-infinite flat plate. It is concluded that the flow field and

other quantities of physical interest are significantly influenced by the viscosity-

temperature parameter. Kafoussias et al. (1998) studied the combined free-forced

convective laminar boundary layer flow past a vertical isothermal flat plate with

temperature-dependent viscosity. The obtained results showed that the flow field is

appreciably influenced by the viscosity variation.

Severin and Herwig (1999) investigated the variable viscosity effect on the

onset of instability in the RBC problem. An asymptotic approach is considered

which provides results that are independent of specific property laws.

Kozhhoukharova et al. (1999) examined the influence of a temperature-dependent

viscosity on the axisymmetric steady thermocapillary flow and its stability with

respect to non-axisymmetric perturbations by means of a linear stability analysis.

The onset of oscillatory convection is studied numerically by a mixed Chebyshev-

collocation finite-difference method.

Rogers and Schatz (2000) reported the first observations of superlattices in

thermal convection. The superlattices are selected by a four-mode resonance

mechanism that is qualitatively different from the three-mode resonance

responsible for complex-ordered patterns observed previously in other

nonequilibrium systems. Numerical simulations quantitatively describe both the

pattern structure and the stability boundaries of superlattices observed in laboratory

experiments. It is found that, in the presence of inversion symmetry, superlattices

numerically bifurcate supercritically directly from conduction or from a striped

base state.

12

Rogers et al. (2000) reported on the quantitative observations of convection in a

fluid layer driven by both heating from below and vertical sinusoidal oscillation.

Just above onset, convection patterns are modulated either harmonically or

subharmonically to the drive frequency. It is found that single frequency patterns

exhibit nearly solid-body rotations with harmonic and subharmonic states always

rotating in opposite directions. Further, flows with both harmonic and subharmonic

responses have been found near a co-dimension two point, yielding novel

coexisting patterns with symmetries not found in either single-frequency states.

You (2001) presented a simple method which can be applied to estimate the

onset of natural convection in a fluid with a temperature-dependent viscosity.

Straughan (2002) developed an unconditional nonlinear energy stability analysis

for thermal convection with temperature-dependent viscosity. The nonlinear

stability boundaries are shown to be sharp when compared with the instability

thresholds of linear theory.

Hossain et al. (2002) analyzed the effect of temperature-dependent viscosity on

natural convection flow from a vertical wavy surface using an implicit finite

difference method. They have focused their attention on the evaluation of local

skin-friction and the local Nusselt number. Chakraborty and Borkakati (2002)

studied the flow of a viscous incompressible electrically conducting fluid on a

continuously moving flat plate in the presence of uniform transverse magnetic field.

Assuming the fluid viscosity to be an inverse linear function of temperature, the

nature of fluid velocity and temperature is analyzed.

Getling and Brausch (2003) studied numerically the evolution of three-

dimensional, cellular convective flows in a plane horizontal layer of Boussinesq

fluid heated from below. It is found that the flow can undergo a sequence of

transitions between various cell types. In particular, two-vortex polygonal cells may

form at some evolution stages, with an annular planform of the upflow region and

downflows localized in both central and peripheral regions of the cells. They also

13

showed that, if short-wave hexagons are stable, they exhibit a specific, stellate fine

structure.

Rudiger and Knobloch (2003) described the results of direct numerical

simulations of convection in a uniformly rotating vertical cylinder with no-slip

boundary conditions. They used these results to study the dynamics associated with

transitions between states with adjacent azimuthal wave numbers far from onset. In

certain regimes a novel burst-like state is identified and described.

Ma and Wang (2004) studied the bifurcation and stability of the solutions of the

Boussinesq equations, and the onset of the Rayleigh-Benard convection. A

nonlinear theory for this problem is established using a new notion of bifurcation

called attractor bifurcation and its corresponding theorem developed recently. This

theory includes the following three aspects. First, the problem bifurcates from the

trivial solution an attractor AR when the Rayleigh number R crosses the first critical

Rayleigh number Rc for all physically sound boundary conditions, regardless of the

multiplicity of the eigenvalue Rc for the linear problem. Second, the bifurcated

attractor AR is asymptotically stable. Third, when the spatial dimension is two, the

bifurcated solutions are also structurally stable and are classified as well. In

addition, the technical method developed provides a recipe, which can be used for

many other problems related to bifurcation and pattern formation.

Sprague et al. (2005) investigated pattern formation in a rotating Rayleigh-

Benard configuration for moderate and rapid rotation in moderate aspect-ration

cavities. While the existence of the Kuppers-Lortz rolls is predicted by the theory at

the onset of convection, square patterns have been observed in physical and

numerical experiments at relatively high rotation rates. In addition to presenting

numerical results produced from the direct numerical simulation of the full

Boussinesq equations, they derived a reduced system of nonlinear PDEs valid for

convection in a cylinder in the rapidly rotating limit.

14

Yanagisawa and Yamagishi (2005) carried out simulations of the Rayleigh-

Benard convection with infinite Prandtl number and high Rayleigh numbers in the

spherical shell geometry to understand the thermal structure of the mantle and the

evolution of the earth. The analysis reveals that the structural scale of convection

differs between the boundary region and the isothermal core region. The structure

near the boundary region is characterized by the cell type structure constructed by

the sheet-shaped downwelling and upwelling flows, and that of the core region by

the plume type structure which consists of the cylindrical flows.

Ma and Wang (2007) attempted at linking the dynamics of fluid flows with the

structure of these fluid flows in physical space and the transitions of this structure.

The two-dimensional Rayleigh-Bénard convection, which serves as a prototype

problem has been given attention and the analysis is based on two recently

developed nonlinear theories: geometric theory for incompressible flows and

bifurcation and stability theory for nonlinear dynamical systems (both finite and

infinite dimensional). They have shown that the Rayleigh-Bénard problem

bifurcates from the basic state to an attractor AR when the Rayleigh number R

crosses the first critical Rayleigh number Rc for all physically sound boundary

conditions, regardless of the multiplicity of the eigenvalue Rc for the linear

problem. In addition to a classification of the bifurcated attractor AR, the structure

of the solutions in physical space and the transitions of this structure are classified,

leading to the existence and stability of two different flows structures: pure rolls

and rolls separated by a cross the channel flow.

Zhou et al. (2007) presented an experimental study of the morphological

evolution of thermal plumes in turbulent thermal convection. They noted that as the

sheet-like plumes move across the plate, they collide and convolute into spiraling

swirls and that these swirls then spiral away from the plates to become

mushroomlike plumes which are accompanied by strong vertical vorticity. The

fluctuating vorticity is found to have the same exponential distribution and scaling

behaviour as the fluctuating temperature.

15

Barletta and Nield (2009) revisited the classical Rayleigh–Bénard problem in an

infinitely wide horizontal fluid layer with isothermal boundaries heated from

below. The effects of pressure work and viscous dissipation are taken into account

in the energy balance. A linear analysis is performed in order to obtain the

conditions of marginal stability and the critical values of the wave number and of

the Rayleigh number for the onset of convective rolls. Mechanical boundary

conditions are considered such that the boundaries are both rigid, or both stress-

free, or the upper stress-free and the lower rigid. It is shown that the critical value

of Ra may be significantly affected by the contribution of pressure work, mainly

through the functional dependence on the Gebhart number and on a thermodynamic

Rayleigh number. While the pressure work term affects the critical conditions

determined through the linear analysis, the viscous dissipation term plays no role in

this analysis being a higher order effect.

Song and Tang (2010) carried out a systematic study of turbulent Rayleigh-

Bénard convection in two horizontal cylindrical cells of different lengths filled with

water. Global heat transport and local temperature and velocity measurements are

made over varying Rayleigh numbers Ra. The scaling behavior of the measured

Nusselt number and the Reynolds number associated with the large-scale

circulation remains the same as that in the upright cylinders. The scaling exponent

for the rms value of local temperature fluctuations, however, is strongly influenced

by the aspect ratio and shape of the convection cell. The experiment clearly reveals

the important roles played by the cell geometry in determining the scaling

properties of convective turbulence.

For detailed descriptions of linear and nonlinear problems of both RBC, one may

refer to the books of Chandrasekhar (1961), Gershuni and Zhukhovitsky (1976),

Kays and Crawford (1980), Ziener and Oertel (1982), Platten and Legros (1984),

Gebhart et al. (1988), Getling (1998), Colinet et al. (2001) and Straughan (2004).

Chapters on thermal convection are included in the books by Turner (1973), Joseph

(1976), Tritton (1979) and Drazin and Reid (1981). Reviews of research on

16

convective instability have been given by Normand et al. (1977), Davis (1987) and

Bodenschatz et al. (2000).

We have so far reviewed the literature pertaining to Rayleigh-Benard

convection. In what follows we review the literature on convective instabilities in

the presence of rotation.

2.2 Convection in the Presence of Rotation Linear stability theory of Bénard convection in a rotating fluid has shown that

fluids with large Prandtl number, σ , exhibit behaviour markedly different from

that of fluids with 1σ ≤ . This difference in behaviour extends also into the finite-

amplitude range. Veronis (1968) reported a numerical study of two-dimensional

Be´nard convection in a rotating fluid confined between free boundaries. A study of

the resultant velocity and temperature fields shows how rotation controls the

system, with the principal behaviour reflected by the thermal wind balance; i.e. the

horizontal temperature gradient is largely balanced by the vertical shear of the

velocity component normal to the temperature gradient. A fluid with a small

Prandtl number becomes unstable to finite-amplitude disturbances at values of the

Rayleigh number significantly below the critical value of linear stability theory.

The subsequent steady vorticity and temperature fields exhibit a structure which is

quite different from that of fluids with large σ. The rotational constraint is balanced

primarily by non-linear processes in a limited range of Taylor number. For larger

values of Taylor number the system first becomes unstable to infinitesimal

oscillatory disturbances but a steady, finite-amplitude flow is established at

supercritical values of R which are none the less smaller than the values that one

would expect from linear theory.

Rossby (1969) presented an experimental study of the response of a thin

uniformly heated rotating layer of fluid. It is shown that the stability of the fluid

depends strongly upon the three parameters that described its state, namely the

Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl

17

numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear

theory is insufficient to fully describe their stability properties. For water,

subcritical instability will occur for all Taylor numbers greater than 5 × 104,

whereas mercury exhibits a subcritical instability only for finite Taylor numbers

less than 105. At all other Taylor numbers there is good agreement between linear

theory and experiment.

Eltayeb (1972) examined the linear stability of a rotating, electrically conducting

viscous layer, heated from below and cooled from above, and lying in a uniform

magnetic field using the Boussinesq approximation. Several orientations of the

magnetic field and rotation axes are considered under a variety of different surface

conditions. The analysis is, however, limited to large Taylor numbers, T, and large

Hartmann numbers, M. (These are non-dimensional measures of the rotation rate

and magnetic field strength, respectively.) Except when field and rotation are both

vertical, the most unstable mode at marginal stability has the form of a horizontal

roll whose orientation depends in a complex way on the directions and strengths of

the field and angular velocity. Also, in this case the mean applied temperature

gradient and the wavelength of the tesselated convection pattern are both

independent of viscosity when the layer is marginally stable. Furthermore, the

Taylor-Proudman theorem and its extension to the hydromagnetic case are no

longer applicable even qualitatively. Over the interior of the layer, however, the

Coriolis forces to which the convective motions are subjected are, to leading order,

balanced by the Lorentz forces. The results obtained in this paper have a bearing on

the possibility of a thermally driven steady hydromagnetic dynamo.

Roberts and Stewartson (1974) examined a particular M.A.C. - wave model

originally proposed by Braginsky. It consists of a horizontal layer containing a

uniform horizontal magnetic field, B0, and rotated about the vertical, an adverse

temperature gradient being maintained on the horizontal boundaries to provide the

unstable density stratification. In the rotationally dominant case of large A, a

measure of the relative importance of Coriolis and magnetic forces, the principle of

18

the exchange of stabilities holds, and the motions that arise in the marginal state are

steady. A theory is developed for the weakly nonlinear convection that arises when

R exceeds only slightly the critical value Rc at which marginal convection occurs.

It is concluded that, starting from an arbitrary initial perturbation, the convection

that arises when R exceeds Re will ultimately become a completely regular

tesselated pattern filling the horizontal plane. The relevance of the theory to

sunspot formation is discussed.

Daniels (1978) considered the effect of rotation on two-dimensional Benard

convection between horizontal stress-free boundaries which are maintained at

different constant temperatures. The fluid is confined laterally by rigid sidewalls

which are assumed only approximately insulating, the possibility of small lateral

heat losses, which are observed experimentally, being incorporated in the theory. A

weakly nonlinear theory based on the method of multiple scales is developed to

describe the motion for slightly supercritical Rayleigh numbers R, and large aspect

ratios (L > 1), although the results are also valid for finite values of L if the speed

of rotation is large (T > 1). In the exchange case a steady finite amplitude solution

evolves if the Prandtl number (Pr) of the fluid is greater than 0.577, but subcritical

instability and bursting can occur for a certain range of Taylor numbers if a <

0.577. In the overstable case disturbances propagate between the sidewalls, and

ultimately either decay or, for Rayleigh numbers greater than a critical value

depending on both Pr and T, attain an equilibrium state controlled by reflexion at

the sidewalls.

Galdi and Straughan (1985) studied the stabilizing effect of rotation in the

Benard problem using a novel generalized energy. It is found that the nonlinearity

boundary is in very close agreement with the experiments of Rossby, who predicted

sub-critical instabilities for high Taylor numbers for fluids with Prandtl number

greater than or equal to 1, such as water.

Magnan and Reiss (1988) considered the secondary and cascading bifurcation of

two-dimensional steady and period thermal convection states in a rotating box.

19

Previously developed asymptotic and perturbation methods that rely on the

coalescence of two, steady convection, primary bifurcation points of the conduction

state as the Taylor number approaches a critical value are employed. A multitime

analysis is employed to construct asymptotic expansions of the solutions of the

initial-boundary value problem for the Boussinesq theory. The small parameter in

the expansion is proportional to the deviation of the Taylor number from its critical

value. To leading order, the asymptotic expansion of the solution involves the mode

amplitudes of the two interacting steady convection states. The asymptotic analysis

yields a first-order system of two coupled ordinary differential equations for the

slow-time evolution of these amplitudes.

Zhonga et al. (1991) investigated Rayleigh-Bénard convection with rotation

about a vertical axis for small dimensionless rotation rates 0 < Ω < 50. The

convection cell is cylindrical with aspect ratio Γ = 10 and the convecting fluid is

water with a Prandtl number of 6.8 at T = 23.8°C. Comparisons are made between

experimental data and linear stability theory for the onset Rayleigh number and for

the wavenumber dependence of the convective pattern. The nonlinear Küppers-

Lortz transition is found to occur significantly below the theoretically expected

rotation rate Ωc and to be nucleated by defects created at the lateral cell.

Riahi (1992) studied the problem of weakly nonlinear two- and three-

dimensional oscillatory convection in the form of standing waves for a horizontal

layer of fluid heated from below and rotating about a vertical axis. The solutions to

the nonlinear problem are determined by a perturbation technique and the stability

of all the base flow solutions is investigated with respect to both standing wave and

travelling wave disturbances. The results of the stability and the nonlinear analyses

for various values of the rotation parameter T and the Prandtl number P(0 < P <

0.677) indicate that there is no subcritical instability and that all the base flow

solutions are unstable. The dependence on P and r of the nonlinear effect on the

frequency and of the heat flux are also discussed.

20

Zhong et al. (1993) presented optical shadowgraph flow visualization and heat

transport measurements of Rayleigh-Bénard convection with rotation about a

vertical axis. The fluid, water with Prandtl number 6.4, is confined in a cylindrical

convection cell with radius-to-height ratio Γ = 1. For dimensionless rotation rates

150 < Ω < 8800, the onset of convection occurs at critical Rayleigh numbers Rc(Ω)

much less than those predicted by linear stability analysis for a laterally infinite

system.

Riahi (1994) studied nonlinear convection in a porous medium and rotating

about vertical axis. An upper bound to the heat flux is calculated by the method

initiated first by Howard for the case of infinite Prandtl number.

Julien et al. (1996) studied turbulent Boussinesq convection under the influence

of rapid rotation (i.e. with comparable characteristic rotation and convection

timescales). The transition to turbulence proceeds through a relatively simple

bifurcation sequence, starting with unstable convection rolls at moderate Rayleigh

(Ra) and Taylor numbers (Ta) and culminating in a state dominated by coherent

plume structures at high Ra and Ta. Like non-rotating turbulent convection, the

rapidly rotating state exhibits a simple power-law dependence on Ra for all

statistical properties of the flow. When the fluid layer is bounded by no-slip

surfaces, the convective heat transport (Nu-1, where Nu is the Nusselt number)

exhibits scaling with Ra2/7 similar to non-rotating laboratory experiments. When

the boundaries are stress free, the heat transport obeys 'classical' scaling (Ra1/3) for

a limited range in Ra, then appears to undergo a transition to a different law at Ra

4 x 107.

Cox (1998a) examined thermal convection in a horizontal layer of Boussinesq

fluid. The fluid layer rotates about a given axis with constant angular velocity, and

a constant mean shear is maintained in the fluid by uniform differential motion of

the horizontal boundaries. The problem is motivated by convection in geophysical

and astrophysical contexts, for which the rotation of the system is often significant,

and where there is often a significant background shear flow. The imposed shear

21

flow and the rotation of the layer individually tend to align convective rolls parallel

to some preferred direction. This analysis reveals certain parameter values near

which the orientation of the preferred rolls depends very sensitively upon the

parameters, and many modes may simultaneously be close to marginal stability.

Numerical simulations of the fully nonlinear governing equations near such points

of sensitivity reveal nonlinear interactions between these modes. One interaction

between three resonant modes results in a stable limit cycle that at different times

has a planform approximating rolls, hexagons and rhombs. This oscillatory form of

convection in a three-dimensional phase space is analyzed and the various changes

in its symmetry that take place as the Rayleigh number varies is explained.

Cox (1998b) examined the onset of thermal convection in a horizontal layer of

fluid rotating about a vertical axis by means of a nonlocal model partial differential

equation (PDE). This PDE is obtained asymptotically from the Navier-Stokes and

heat equations in the limit of small conductivity of the horizontal boundaries. The

model describes the onset of convection near a steady bifurcation from the

conduction state and is valid provided the Prandtl number of the fluid is not too

small and the rotation rate of the layer is not too great. It is known that a restricted

version of our model PDE for convection in a nonrotating fluid layer predicts a

preference for convection in a square planform rather than two-dimensional roll

motions. It is found that this preference carries over to the rotating layer. The

instability of rolls in a nonrotating layer is compounded by the Kiippers-Lortz

instability when rotation is introduced. The stability of weakly nonlinear rolls and

square planforms is analyzed and the analysis is supplemented with numerical

simulations of the model PDE. The most notable feature of the numerical

simulations in square periodic domains of moderate size is the strong preference for

convection in a square planform.

Plapp et al. (1998) reported, for Rayleigh-Bénard convection of a fluid with

Prandtl number σ = 1.4, experimental and theoretical results on a pattern selection

mechanism for cell-filling, giant, rotating spirals. They have shown that the pattern

selection in a certain limit can be explained quantitatively by a phase-diffusion

22

mechanism. This mechanism for pattern selection is very different from that for

spirals in excitable media.

Govender (2003a) used the linear stability theory to investigate analytically the

effects of gravity on centrifugally driven convection in a rotating porous layer

offset from the axis of rotation. The stability of a basic solution is analysed with

respect to the onset of stationary and oscillatory convection. It is also demonstrated

that the stationary mode is the critical mode of convection thereby resulting in the

convection rolls being aligned parallel to the axis of rotation. Besides providing a

non-motionless basic solution and dictating the direction of the wave number,

gravity plays a passive role and does not affect the stability results.

Govender (2003b) investigated analytically the Coriolis effect on centrifugally

driven convection in a rotating porous layer using the linear stability theory. The

problem corresponding to a layer placed far away from the axis of rotation was

identified as a distinct case and therefore justifying special attention. The stability

of the basic centrifugally driven convection is analysed. The marginal stability

criterion is established as a characteristic centrifugal Rayleigh number in terms of

the wavenumber and the Taylor number.

Sharma and Monica Sharma (2004) considered the thermal instability of a

couple-stress fluid with suspended particles. Following the linear stability analysis

and normal mode analysis, the dispersion relation is obtained. For the case of

stationary convection, couple-stress is found to postpone the onset of convection,

whereas suspended particles hasten it. It is found that the principle of exchange of

stabilities is valid. The thermal instability of a couple-stress fluid with suspended

particles, in the presence of rotation and magnetic field, is also considered. The

magnetic field and rotation are found to have stabilizing effects on the stationary

convection and introduce oscillatory modes in the system. A sufficient condition

for the nonexistence of ovestabllity is also obtained.

23

Sharma and Mehta (2005) paid attention to a layer of compressible, rotating,

couple-stress fluid heated and soluted from below. For the case of stationary

convection, the compressibility, stable solute gradient and rotation postpone the

onset of convection, whereas the couple-stress viscosity postpones as well as

hastens the onset of convection depending on rotation parameter. The case of

overstability is also studied wherein a sufficient condition for the non-existence of

overstability is found.

Sharma et al. (2006) considered combined effect of magnetic field and rotation

on the stability of stratified viscoelastic Walters’ (Model B′) fluid in porous

medium. In contrast to the Newtonian fluids, the system is found to be unstable at

stable stratification for low values of permeability or high values of kinematic

viscoelasticity. Magnetic field is found to stabilize the small wavelength

perturbations for unstable stratification. It has been found that the growth rate

increases with the increase in kinematic viscosity and permeability, whereas it

decreases with the increase in kinematic viscoelasticity.

Pardeep Kumar et al. (2006) considered the thermal instability of a rotating

Rivlin-Ericksen viscoelastic fluid in the presence of uniform vertical magnetic

field. For the case of stationary convection, Rivlin-Ericksen viscoelastic fluid

behaves like a Newtonian fluid. It is found that rotation has a stabilizing effect,

whereas the magnetic field has both stabilizing and destabilizing effects. The

rotation and magnetic field are found to introduce oscillatory modes in the system,

which were nonexistent in their absence.

Govender and Vadas (2007) investigated Rayleigh–Benard convection in a

porous layer subjected to gravitational and Coriolis body forces, when the fluid and

solid phases are not in local thermodynamic equilibrium. The Darcy model

(extended to include Coriolis effects and anisotropic permeability) is used to

describe the flow, whilst the two-equation model is used for the energy equation

(for the solid and fluid phases separately). The linear stability theory is used to

evaluate the critical Rayleigh number for the onset of convection and the effect of

24

both thermal and mechanical anisotropy on the critical Rayleigh number is

discussed.

Malashetty and Heera (2008) studied double diffusive convection in a fluid-

saturated rotating porous layer heated from below and cooled from above, when the

fluid and solid phases are not in local thermal equilibrium, using both linear and

non-linear stability analyses. A two-field model that represents the fluid and solid

phase temperature fields separately is used for energy equation. The onset criterion

for stationary, oscillatory and finite amplitude convection is derived analytically. It

is found that small inter-phase heat transfer coefficient has significant effect on the

stability of the system. There is a competition between the processes of thermal and

solute diffusions that causes the convection to set in through either oscillatory or

finite amplitude mode rather than stationary. The effect of solute Rayleigh number,

porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz

number and Taylor number on the stability of the system is investigated. The non-

linear theory, based on the truncated representation of Fourier series method,

predicts the occurrence of subcritical instability in the form of finite amplitude

motions. The effect of thermal non-equilibrium on heat and mass transfer is also

brought out.

Siri and Hashim (2008) analyzed the problem of the effect of feedback control

on the onset of steady and oscillatory Marangoni convection in a rotating horizontal

fluid layer. The role of the controller gain parameter on the Pr - Ta parameter space

is determined.

Sharma et al. (2008) studied the problem of convection of micropolar fluids

heated from below in the presence of suspended particles (fine dust) and uniform

vertical rotation. It is found that the presence of coupling between thermal and

micropolar effects, rotation parameter and suspended particles may introduce

overstability in the system. It is found that Rayleigh number for the case of

overstability and stationary convection increases with increase in rotation

parameters and decreases with increase in micropolar coefficients, for a fixed wave

25

number, showing thereby the stabilizing effect of rotation parameters and

destabilizing effect of micropolar coefficients on the thermal convection of

micropolar fluids. It is also found from the graphs that the Rayleigh number for the

case of overstability is always smaller than the Rayleigh number for the case of

stationary convection, for a fixed wave number.

Recent experiments in rotating convection have shown that the spatio-temporal

bulk convective state with Küppers–Lortz dynamics can be suppressed by small-

amplitude modulations of the rotation rate. The resultant axisymmetric pulsed

target patterns were observed to develop into axisymmetric travelling target

patterns as the modulation amplitude and Rayleigh number were increased. Using

the Navier–Stokes–Boussinesq equations with physical boundary conditions, Rubio

et al. (2008) were able to numerically reproduce the experimental results and gain

physical insight into the responsible mechanism, relating the onset of the travelling

target patterns to a symmetry-restoring saddle-node on an invariant circle

bifurcation. Movies are available with the online version of the paper.

Aggarwal and Suman Makhija (2009) examined theoretically the thermal

stability of a couple-stress fluid in the presence of magnetic field and rotation.

Following the linear stability theory and normal mode analysis, the dispersion

relation is obtained. For stationary convection, rotation has stabilizing effect

whereas couple stresses in fluid and magnetic field have stabilizing effect under

certain conditions. It is found that principle of exchange of stabilities is satisfied in

the absence of magnetic field and rotation. The sufficient conditions for the non

existence of overstability are also obtained.

Om et al. (2009) studied the effect of rotation speed modulation on the onset of

centrifugally driven convection using linear stability analysis. Darcy flow model

with zero-gravity is used to describe the flow. The perturbation method is applied

to find the correction in the critical Rayleigh number. It is found that by applying

modulation of proper frequency to the rotation speed, it is possible to delay or

advance the onset of centrifugal convection.

26

Hashim and Siri (2009) applied the linear stability theory to investigate the

effects of rotation and feedback control on the onset of steady and oscillatory

thermocapillary convection in a horizontal fluid layer heated from below with a

free-slip bottom. The thresholds and codimension-2 points for the onset of steady

and oscillatory convection are determined.

Pardeep Kumar and Mahinder Singh. (2009) considered the thermosolutal

instability of couple-stress fluid in the presence of uniform vertical rotation.

Following the linear stability theory and normal mode analysis, the dispersion is

obtained. For the case of stationary convection, the stable solute gradient and

rotation have stabilizing effects on the system, whereas the couple-stress has both

stabilizing and destabilizing effects. The dispersion relation is also analyzed

numerically. The stable solute gradient and the rotation introduce oscillatory modes

in the system, which did not occur in their absence. The sufficient conditions for

the non-existence of overstability are also obtained.

Chauhan and Rastogi (2010) investigated the unsteady natural convection MHD

flow of a rotating viscous electrically conducting fluid in a vertical channel

partially filled by a porous medium with high porosity in the presence of radiation

effects. It is assumed that the conducting fluid is gray, emitting-absorbing radiation,

and non-scattering medium. The two infinite vertical porous plates of the channel

are subjected to a constant injection velocity at the one plate and the same constant

suction velocity at the other plate. The entire system rotates about the axis normal

to the plates with a uniform angular velocity. The analytic expressions for velocity

and temperature field are obtained and effects of the radiation-conduction

parameter (Stark number), Prandtl number, Grashof number, rotation parameter,

magnetic field, and permeability of the porous medium on the velocity field,

temperature field and Nusselt number have been discussed in the analysis.

Falsaperla et al. (2010) considered the problem of thermal convection in a

rotating horizontal layer of porous medium. The porous medium is described by the

equations of Darcy. A novel aspect of this work is to consider boundary conditions

27

for the temperature of Newton–Robin type with heat flux prescribed as a limiting

case. The effect of rotation is found to be crucial. For the Taylor number small

enough the critical wave number is zero, but it is found that a threshold such that

for Taylor numbers beyond this non-zero critical wave numbers are found. The

threshold is verified via a weakly nonlinear analysis. Finally, a sharp global

nonlinear stability analysis is given.

Paul et al. (2010) investigated two-dimensional Rayleigh-Bénard convection

using direct numerical simulation in Boussinesq fluids with Prandtl number P = 6.8

confined between thermally conducting plates. They have shown through the

simulation that in a small range of reduced Rayleigh number r (770 < r < 890) the

2D rolls move chaotically in a direction normal to the roll axis. The lateral shift of

the rolls may lead to a global flow reversal of the convective motion. The chaotic

travelling rolls are observed in simulations with free-slip as well as no-slip

boundary conditions on the velocity field. Further, they have shown the travelling

rolls and the flow reversal are due to an interplay between the real and imaginary

parts of the critical modes.

Vanishree and Siddheshwar (2010) performed a linear stability analysis for

mono-diffusive convection in an anisotropic rotating porous medium with

temperature-dependent viscosity. The Galerkin variant of the weighted residual

technique is used to obtain the eigenvalue of the problem. The effect of Taylor–

Vadasz number and the other parameters of the problem are considered for

stationary convection in the absence or presence of rotation. Oscillatory convection

seems highly improbable. Some new results on the parameters’ influence on

convection in the presence of rotation, for both high and low rotation rates, are

presented.

In what follows we review the literature on convective instability problems with

temperature modulation.

28

2.3 Convection with Temperature Modulation Donnelly (1964) studied the behavior of disturbances in circular Couette flow

between coaxial cylinders, when the motion of the inner cylinder consists of a small

oscillation about a steady rotation, while the outer cylinder is at rest. It is found

that when the critical Taylor number is increased in the presence of the periodic

motion, the magnitude of the enhancement is a function of the oscillation frequency

and amplitude.

Venezian (1969) studied the stability of a horizontal layer of fluid heated from

below when, in addition to a steady temperature difference between the walls of the

layer, a time-dependent sinusoidal perturbation is applied to the wall temperatures.

The object is to determine the modulating effect of the oscillation on the stability

characteristics of the mean gradient. He showed that at low frequencies the

equilibrium state becomes unstable, because at that frequency the disturbances

grow to a sufficient size so that the inertia effects becomes important.

Rosenblat and Herbert (1970) investigated the low-frequency modulation of

thermal instability of a Boussinesq fluid heated from below. They considered the

applied temperature gradient as the sum of a steady component and a low-

frequency sinusoidal component. An asymptotic solution is obtained which

describes the behavior of infinitesimal disturbances. The solution is discussed from

the viewpoint of the stability or otherwise of the basic state and possible stability

criteria are analyzed. Some comparison is made with known experimental results.

Rosenblat and Tanaka (1971) studied linear stability problem for a fluid in a

classical Bénard geometry, when the temperature gradient has both a steady and a

time-periodic component. The modulating effect of the oscillatory gradient on the

stability characteristics of the basic configuration was examined. It is found that

there is enhancement of the critical value of a suitably defined Rayleigh number.

Srivastava (1976) investigated the thermal convective instability of a plane fluid

layer heated from below when the temperature has both a steady and time periodic

29

component in the presence of a uniform magnetic field parallel to the gravity and

normal to the boundaries. It is found that the maximum modulation occurs as the

frequency of the oscillating temperature tends to zero and that the modulation

remains finite for large values of the magnetic field parameter.

Banerjee and Bhattacharjee (1984) considered the onset of convection in 3He-4He mixtures with the temperature difference between the plates in the Rayleigh-

Benard geometry modulated sinusoidally. It is found that the system can show

stabilization or destabilization depending on the mean temperature. For oscillatory

instability in the system the modulation leads to parametric resonances.

Bhattacharjee (1989) considered rotating Rayleigh-Benard convection with

modulated rotation speed. Galerkin truncation under realistic, i.e. experimentally

realisable, boundary conditions is carried out. The threshold of convection can be

raised or lowered depending on the Prandtl number and rotation speed.

Bhattacharjee (1990) studied the onset of convection in a rotating fluid layer

heated from below is studied when the rotation speed is sinusoidally modulated.

We study the instability that, in the absence of modulation, leads to stationary

convection. We show that under modulation the conduction state can be stabilized

or destabilized depending on the Taylor number, that the instability can be forward

or backward for a range of Taylor numbers and modulation amplitudes, and that the

critical Taylor number for the Küppers-Lortz instability is lowered. The

calculations are carried out in the limit of large Prandtl number.

Ramaswamy (1993) studied sinusoidal gravity modulation fields imposed on

two-dimensional Rayleigh-Benard convection flow to understand the effects of

periodic source (g-jitter) on fluids system and heat transfer mechanism. The

transient Navier-Stokes and energy equations are solved by semi-implicit operator

splitting finite element method. Results include two sets. One is considered at

normal terrestrial condition and the other one is related to low-gravity condition.

Under low-gravity condition the research focuses on the effects of modulation

30

frequency and direction in order to find out the critical frequency for heat transfer

mechanism transferring from conduction to convection.

Malashetty and Wadi (1999) investigated the stability of a Boussinesq fluid-

saturated horizontal porous layer heated from below for the case of a time-

dependent wall temperature. Forchheimer flow model with effective viscosity

larger than the viscosity of the fluid is considered to give a more general theoretical

result. A method based on small amplitude of the modulation proposed by

Venezian is used to compute the critical values of the Rayleigh number and wave

number. The shift in the critical Rayleigh number is calculated as a function of

frequency of modulation, Prandtl number, porous parameter, and viscosity ratio. It

is shown that the system is most stable when the boundary temperature is

modulated out of phase. It is also found that the low-frequency thermal modulation

can have a significant effect on the stability of the system. The effect of the

viscosity ratio and Prandtl number on the stability of the system is also brought out.

Siddheshwar and Pranesh (1999) studied the effect of time-periodic

temperature/gravity modulation on the onset of magneto-convection in weak

electrically conducting fluids with internal angular momentum using linear stability

analysis. The Venezian approach is adopted in arriving at the critical Rayleigh and

wave numbers for small amplitude temperature/gravity modulation. The

temperature modulation is shown to give rise to sub-critical motion and gravity

modulation leads to delayed convection. An asymptotic analysis is also presented

for small and large frequencies.

Siddheshwar and Pranesh (2000) studied the effect of time-periodic

temperature/gravity modulation on the onset of magneto-convection in electrically

conducting fluids with internal angular momentum using a linear stability analysis.

The results of the are presented against the background of the results of weak

electrically conducting fluids.

31

Aniss et al. (2001) investigated the effect of a time-sinusoidal magnetic field on

the onset of convection in a horizontal magnetic fluid layer heated from above and

bounded by isothermal non magnetic boundaries. The analysis is restricted to static

and linear laws of magnetization. A first order Galerkin method is performed to

reduce the governing linear system to the Mathieu equation with damping term.

Therefore, the Floquet theory is used to determine the convective threshold for the

free-free and rigid-rigid cases. With an appropriate choice of the ratio of the

magnetic and gravitational forces, it is shown the possibility to produce a

competition between the harmonic and subharmonic modes at the onset of

convection.

Malashetty and Basavaraja (2002) investigated the effect of time-periodic

temperature/gravity modulation at the onset of convection in a Boussinesq fluid-

saturated anisotropic porous medium using a linear stability analysis. Brinkman

flow model with effective viscosity larger than the viscosity of the fluid is

considered to give a more general theoretical result. The perturbation method is

applied for computing the critical Rayleigh and wave numbers for small amplitude

temperature/gravity modulation. The shift in the critical Rayleigh number is

calculated as a function of frequency of the modulation, viscosity ratio, anisotropy

parameter and porous parameter. It is shown that the small anisotropy parameter

has a strong influence on the stability of the system. The effect of viscosity ratio,

anisotropy parameter, the porous parameter and the Prandtl number is discussed.

Kelly and Or (2002) investigated the effects of thermal modulation with time on

the thermocapillary instability of a thin horizontal fluid layer with a deformable

free surface on the basis of linear stability theory. First, a sinusoidal heating with a

mean component is applied at the lower wall corresponding to the boundary

conditions either in the form of prescribed temperature or heat flux. Thermal

modulation with moderate modulation amplitude tends to stabilize the mean basic

state, and optimal values of frequency and amplitude of modulation are determined

to yield maximum stabilization. However, large-amplitude modulation can be

destabilizing. A basic state with zero mean is then considered and the critical

32

Marangoni number is obtained as a function of frequency. The effects of

modulation are also investigated in the long-wavelength limit. For the case of

prescribed temperature, the modulation does not affect the onset of the long-

wavelength mode associated with the mean basic state and a purely oscillating

basic state is always stable with respect to long-wavelength disturbances. For the

case of prescribed heat flux both at the wall and free surface, by contrast, thermal

modulation exerts a significant effect on the onset of convection from a mean basic

state and long-wavelength convection can occur even for a purely oscillating basic

state.

Bhadauria (2002a) studied thermal convection in a fluid layer confined between

two horizontal rigid boundaries with the help of the Floquet theory. The

temperature distribution consists of a steady part and an oscillatory time-dependent

part. Disturbances are assumed to be infinitesimal. Numerical results for the critical

Rayleigh numbers and wave numbers are obtained. It is found that the disturbances

are either synchronous with the primary temperature field or have half its

frequency.

Bhadauria (2002b) considered the linear stability of a horizontal fluid layer,

confined between two rigid walls, heated from below and cooled from above. The

temperature gradient between the walls consists of a steady part and a periodic part

that oscillates with time. Only infinitesimal disturbances are considered. Numerical

results for the critical Rayleigh number are obtained for various Prandtl numbers

and for various values of the frequency.

Siddheshwar and Abraham (2003) investigated the thermal instability in a layer

of a ferromagnetic liquid when the boundaries of the layer are subjected to

synchronous/asynchronous imposed time- periodic boundary temperature (ITBT)

and time-periodic body force (TBF). The Venezian approach is adopted in arriving

at the critical Rayleigh and wave numbers for small amplitudes of ITBT. When the

ITBT at the two walls are synchronized then, for moderate frequency values, the

role of magnetization in inducing sub-critical instabilities is delineated. A similar

33

role has been shown by the Prandtl number. They showed that the magnetization

parameters and Prandtl number have opposite effect at large frequencies. Low

Prandtl number liquids are shown to be more easily vulnerable to destabilization by

TBF compared to very large Prandtl number liquids.

Govender (2004) used the linear stability theory to investigate analytically the

effects of gravity modulation on convection in a homogenous porous layer heated

from below. The gravitational field consists of a constant part and a sinusoidally

varying part, which is tantamount to a vertically oscillating porous layer subjected

to constant gravity. The linear stability results are presented for the specific case of

low amplitude vibration for which it is shown that increasing the frequency of

vibration stabilises the convection.

Bhadauria and Lokenath (2004) studied the linear stability of a horizontal layer

of fluid heated from below and above with thermal modulation. In addition to a

steady temperature difference between the walls of the fluid layer, a time-

dependent periodic perturbation is applied to the wall temperatures. Only

infinitesimal disturbances are considered. Numerical results for the critical

Rayleigh number are obtained at various Prandtl numbers and for various values of

the frequency.

Malashetty and Basavaraja (2004) studied the effect of time-periodic boundary

temperatures on the onset of double diffusive convection in a fluid-saturated

anisotropic porous medium using a linear stability analysis. The correction thermal

Rayleigh number is calculated as a function of frequency of modulation, viscosity

ratio, anisotropy parameter, porous parameter, Prandtl number, diffusivity ratio and

solute Rayleigh number. The effect of various physical parameters is found to be

significant at moderate values of the frequency. They found that it is possible to

advance or delay the onset of double diffusive convection by proper tuning of the

frequency of modulation of the wall temperature.

34

Liu (2004) examined the stability of a horizontally extended second-grade liquid

layer heated from below when a steady temperature difference between the walls is

superimposed on sinusoidal temperature perturbations. He found that the onset of

convection can be delayed or advanced by parameters such as modulation

frequency, the Prandtl number and the viscoelastic parameter.

Bhadauria (2006) investigated the convective instability in a horizontal layer of

electrically conducting fluid heated from below in the presence of an applied

vertical magnetic field. The temperature gradient between the walls of the fluid

layer consists of a steady part and a time dependent oscillatory part. The

temperature of both walls is modulated. The combined effect of the vertical

magnetic field and modulation of walls temperature is studied using Floquet theory.

It is found that the effect of magnetic field has a stabilizing influence on the onset

of thermal instability. Further, it is also found that it is possible to advance or delay

the onset of convection by proper tuning of the frequency of modulation of the

walls temperature.

Candel (2006) studied the effect of the temperature modulation, applied at the

horizontal boundaries, on the onset of convection of a horizontal Maxwellian liquid

layer. The Floquet theory and a technique of converting a boundary value problem

to an initial value problem are used to solve the system of equations corresponding

to the onset of convection. Results obtained have been used to characterize the

influence of modulation effects and that of the viscoelastic nature of liquid on the

critical Rayleigh number.

Mahabaleswar (2007) studied the combined effect of time-periodic boundary

temperature and time-periodic body force of small amplitude on magneto-

convection in a micro polar liquid using a linear stability analysis. A regular

perturbation method is used to arrive at an expression for the correction Rayleigh

number that throws light on the possibility of sub-critical motions. The Venezian

approach is adopted in arriving at the critical Rayleigh and wave numbers for small

35

amplitudes. Comparison is made between the effects of temperature, gravity and

combined modulations.

Bhadauria (2007a) investigated thermal instability in an electrically conducting

fluid saturated porous medium, confined between two horizontal walls in the

presence of an applied vertical magnetic field and rotation using the Brinkman

model. The combined effect of permeability, rotation, vertical magnetic field, and

temperature modulation has been investigated using Galerkin's method and Floquet

theory. The value of the critical Rayleigh number is calculated as function of

amplitude and frequency of modulation, Chandrasekhar number, Taylor number,

porous parameter, Prandtl number, and magnetic Prandtl number. It is found that

rotation, magnetic field, and porous medium all have stabilizing influence on the

onset of thermal instability. Further, it is also found that it is possible to advance or

delay the onset of convection by proper tuning of the frequency of modulation of

the walls' temperature. In addition the results corresponding to the Brinkman model

and Darcy model have been compared for neutral instability.

Bhadauria (2007b) studied the linear stability of thermal convection in a rotating

horizontal layer of fluid-saturated porous medium, confined between two rigid

boundaries, considering temperature modulation using Brinkman’s model. In

addition to a steady temperature difference between the walls of the porous layer, a

time-dependent periodic perturbation is applied to the wall temperatures. The

combined effect of rotation, permeability and modulation of wall temperature on

the stability of flow through porous medium has been investigated using Galerkin

method and Floquet theory. The critical Rayleigh number is calculated as function

of amplitude and frequency of modulation, Taylor number, porous parameter and

Prandtl number. It is found that both rotation and permeability are having

stabilizing influence on the onset of thermal instability. Further it is also found that

it is possible to advance or delay the onset of convection by proper tuning of the

frequency of modulation of the wall temperatures.

36

Bhadauria (2007c) performed linear stability analysis to find the effect of

temperature modulation on the onset of double diffusive convection in a rotating

horizontal layer of a fluid-saturated porous medium using Darcy-Lapwood-

Brinkman's model. The combined effect of rotation, permeability, and temperature

modulation on the onset of double diffusive convection in a porous medium has

been investigated using Galerkin method. The value of the critical Rayleigh number

is calculated as a function of frequency and amplitude of modulation, Prandtl

number, Taylor number, Darcy number, diffusivity ratio, and solute Rayleigh

number. Stabilizing and destabilizing effects of modulation on the onset of double

diffusive convection have been obtained. Furthermore, it is found that both rotation

and the porous medium have stabilizing influences on the system.

Malashetty and Swamy (2007) investigated the stability of a fluid-saturated

horizontal rotating porous layer subjected to time-periodic temperature modulation,

when the condition for the principle of exchange of stabilities is valid. A regular

perturbation method based on small amplitude of applied temperature field is used

to compute the critical values of Darcy–Rayleigh number and wavenumber. The

shift in critical Darcy–Rayleigh number is calculated as a function of frequency of

modulation, Taylor number, and Darcy–Prandtl number. It is established that the

convection can be advanced by the low frequency in-phase and lower-wall

temperature modulation, where as delayed by the out-of-phase modulation. The

effect of Taylor number and Darcy–Prandtl number on the stability of the system is

also discussed. It is found that by proper tuning of modulation frequency, Taylor

number, and Darcy–Prandtl number it is possible to advance or delay the onset of

convection.

Malashetty and Swamy (2007) investigated the effect of temperature modulation

on the onset of stationary convection in a Boussinesq fluid-saturated sparsely

packed porous layer, when the porous layer is subjected to rotation about an axis

parallel to the gravity vector. The linear stability analysis is used to study the effect

of infinitesimal disturbances. The onset criterion is derived using perturbation

expansion in powers of the small amplitude of the applied temperature field. The

37

critical correction Rayleigh number is found to be a function of the frequency of

modulation, the Taylor number, the Darcy number, the Prandtl number, and the

viscosity ratio. It is found that the rotation reinforces the effect of thermal

modulation. The large-frequency symmetric modulation, asymmetric modulation,

and bottom-wall temperature modulation stabilizes the system compared with the

unmodulated system, while small-frequency symmetric modulation destabilizes it.

The effect of the Darcy number, the Prandtl number, and the viscosity ratio on the

stability of the system is also discussed.

Natalia (2008) examined the effect of vertical harmonic vibration on the onset of

convection in an infinite horizontal layer of fluid saturating a porous medium. The

mathematical model is described by equations of filtration convection in the

Darcy–Oberbeck–Boussinesq approximation. The linear stability analysis for the

quasi-equilibrium solution is performed using Floquet theory. The neutral curves of

the Rayleigh number Ra versus horizontal wave number α for the synchronous and

sub harmonic resonant modes are constructed for different values of frequency Ω

and amplitude A of vibration. It is shown that, at some finite frequencies of

vibration, there exist regions of parametric instability.

Singh and Bajaj (2008) investigated the stability characteristics of an infinite

horizontal fluid layer excited by a time-periodic, sinusoidally varying free-

boundary temperature using the Floquet theory. It has been found that the

modulation of the temperature gradient across the fluid layer affects the onset of the

Rayleigh–Bénard convection. Modulation can give rise to instability in the

subcritical conditions and it can also suppress the instability in the supercritical

cases. The instability in the fluid layer manifests itself in the form of either a

harmonic or subharmonic flow, controlled by thermal modulation.

Bhadauria (2008a) studied the effect of temperature modulation on the onset of

thermal convection in an electrically conducting fluid-saturated-porous medium

heated from below using linear stability analysis. The porous medium is confined

between two horizontal walls and subjected to a vertical magnetic field; flow in

38

porous medium is characterized by Brinkman–Darcy model. The correction in the

critical Rayleigh number is calculated as a function of frequency of modulation,

Darcy number, Darcy Chandrasekhar number, magnetic Prandtl number, and a

nondimensional group number. The influence of the magnetic field is found to be

stabilizing. The results of the present model have been compared with that of Darcy

model.

Bhadauria (2008b) studied the effect of temperature modulation on the onset of

thermal instability in a horizontal layer of a fluid-saturated porous medium heated

from below and subjected to constant rotation. An extended Darcy model, which

includes the time derivative term, has been considered, and a time-dependent

periodic temperature field is applied to modulate the surface temperatures. A

perturbation procedure based on the small amplitude of imposed temperature

modulation is used to study the combined effect of rotation, permeability, and

temperature modulation on the stability of flow through a porous medium. The

correction in the critical Darcy Rayleigh number is calculated as a function of

amplitude and frequency of modulation, the Darcy Taylor number, and the Vadasz

number. It is found that both rotation and permeability suppress the onset of

thermal instability. Furthermore, we find that temperature modulation can advance

or delay the onset of convection.

Das and Kumar (2008) investigated the effects of time-periodic forcing in a few-

mode model for zero-Prandtl number convection with rigid body rotation. The

time-periodic modulation of the rotation rate about the vertical axis and gravity

modulation are considered separately. In the presence of periodic variation of the

rotation rate, the model shows modulated waves with a band of frequencies. The

increase in the external forcing amplitude widens the frequency band of the

modulated waves, which ultimately leads to temporally chaotic waves. The gravity

modulation, on the other hand, with small frequencies, destroys the quasiperiodic

waves at the onset and leads to chaos through intermittency. The spectral power

density shows more power to a band of frequencies in the case of periodic

modulation of the rotation rate. In the case of externally imposed vertical vibration,

39

the spectral density has more power at lower frequencies. The two types of forcing

show different routes to chaos.

Recent experiments in rotating convection have shown that the spatio-temporal

bulk convective state with Küppers–Lortz dynamics can be suppressed by small-

amplitude modulations of the rotation rate. The resultant axisymmetric pulsed

target patterns were observed to develop into axisymmetric travelling target

patterns as the modulation amplitude and Rayleigh number were increased. Using

the Navier–Stokes–Boussinesq equations with physical boundary conditions, Rubio

et al. (2008) were able to numerically reproduce the experimental results and gain

physical insight into the responsible mechanism, relating the onset of the travelling

target patterns to a symmetry-restoring saddle-node on an invariant circle

bifurcation. Movies are available with the online version of the paper.

Malashetty and Swamy (2008) examined the stability of a rotating horizontal

fluid layer heated from below when the walls of the layer are subjected to time-

periodic temperature modulation. A regular perturbation method based on small

amplitude of applied temperature field is used to compute the critical values of

Rayleigh number and wave number. The shift in critical Rayleigh number is

calculated as a function of frequency of modulation, Taylor number and Prandtl

number. It is established that the instability can be enhanced by the rotation at low

frequency symmetric modulation and with moderate to high frequency lower wall

temperature modulation, whereas the stability can be enhanced by the rotation in

case of asymmetric modulation.

Bhadauria et al. (2009) examined the stability of a horizontal layer of fluid

heated from below as well as from above. The temperature gradient between the

walls of the fluid layer consists of a steady part and a time-dependent part, which is

oscillatory. By considering the weakly non-linear analysis, it is shown that the

modulation produces a range of stable hexagons near the critical Rayleigh number.

40

Raju and Bhattacharyya (2009) studied thermal instability in a horizontal layer

of fluid with the boundary temperatures modulated sinusoidally in time. The

amplitude of modulation is assumed small and is used as an expansion parameter. It

is shown that an exact solution can be obtained, even when the boundaries are

considered to be rigid. When only the lower boundary temperature is modulated,

for small values of the Prandtl number modulation is always stabilizing, while for

large values it can be stabilizing or destabilizing depending on the modulation

frequency. When both boundary temperatures are modulated in phase, modulation

is destabilizing for low modulation frequency, but for higher modulation frequency

stabilization occurs for low values of the Prandtl number. When the two boundary

temperatures are modulated out of phase the modulation always has a stabilizing

effect.

Siddheshwar and Abraham (2009) discussed the thermal instability in a layer of

dielectric fluid when the boundaries of the layer are subjected to

synchronous/asynchronous time-periodic temperatures. Perturbation solution in

powers of the amplitude of the applied temperature field is obtained. In the case

when the Imposed Time-periodic Boundary Temperatures (ITBT) at the two walls

are synchronized, then for moderate values of frequency the role of the electric

Rayleigh Number in inducing subcritical instabilities is delineated. A similar role is

shown to be played by the Prandtl number. The dielectric parameters and Prandtl

number have the opposite effect at large frequencies. The system is most stable

when the ITBT is asynchronous. The problem has relevance in many dielectric

fluid applications wherein regulation of thermal convection is called for.

Bhadauria and Srivastava (2010) investigated thermal instability in an

electrically conducting two component fluid-saturated-porous medium, considering

temperature modulation of the boundaries. The porous medium is confined between

two horizontal surfaces subjected to a vertical magnetic field; flow in the porous

medium is characterized by Brinkman–Darcy model. Making linear stability

analysis and applying perturbation procedure, the correction in the critical Darcy

Rayleigh number is calculated. It is found that the correction in the critical Darcy

41

Rayleigh number is a function of frequency of modulation, solute Rayleigh

number, diffusivities ratio, Darcy number, Darcy Chandrasekhar number, magnetic

Prandtl number and the non-dimensional group number χ. Also the effects of

various parameters on thermal instability have been studied; we found that these

parameters may have stabilizing or destabilizing effects, thus may advance or delay

the onset of convection. A comparison between the results from the present model

has been made with that of Darcy model.

We next review the literature pertaining to convective instability problems of

ferromagnetic fluids.

2.4 Convection in Ferromagnetic Fluids

The problem of convection in a ferromagnetic fluid is different from

magnetoconvection even though the influence of the magnetic field exists in both

the problems. In the case of magnetoconvection, the fluid is electrically conducting

and we see the influence of a body force, known as, Lorentz force. Magnetic fluids

are not electrically conducting and hence the Lorentz force does not appear. As a

result of the magnetization of the micron-sized suspended ferrite particles a

pondermotive force, analogous to the Lorentz force, appears and gives rise to a

dynamically different situation than the type that occurs in the magnetoconvection

problem.

Finlayson (1970) made a detailed study of convective instability in a

ferromagnetic fluid. He showed that convection is caused by a spatial variation in

the magnetization which is induced when the magnetization is a function of

temperature and a temperature gradient is established across the fluid layer. He also

predicted the critical temperature gradient for the onset of convection when only

the magnetic mechanism is important as well as when both the magnetic and

buoyancy mechanisms are operative. The magnetic mechanism is shown to

predominate over the buoyancy mechanism in fluid layers which are about 1 mm

thick. For fluid layers contained between two free boundaries, which are

42

constrained flat, the exact solution has been obtained for some parameter values

and oscillatory stability is ruled out. For rigid boundaries, an approximate solution

for stationary instability using a higher order Galerkin method is obtained. It is

shown that, the Galerkin method yields an eigenvalue which is stationary to small

changes in the trial functions because the Galerkin method is equivalent to an

adjoint variational principle.

Lalas and Carmi (1971) investigated a nonlinear analysis of the convective

stability problem in magnetic fluids using the energy method. They showed that the

linear and energy theories give identical results for stationary ferromagnetic flow

under the assumption that the magnetization is independent of the magnetic field

intensity. Subcritical instabilities were ruled out.

Berkovskii and Bashtovoi (1971) investigated the problem of gravitational

convection in an incompressible non-conducting ferromagnetic fluid resulting from

the magnetocaloric effect. This problem is shown to be equivalent to the problem of

natural convection with a vertical temperature gradient. Closed form solutions for

both velocity and temperature are obtained in this study and numerical estimates of

the critical magnetic field gradients are given. Kamiyama et al. (1988) investigated

an analogous problem both numerically and analytically using a perturbation

procedure. Elaborate comments on Oberbeck convection in magnetic fluids have

been made.

Shilomis (1973) studied the conditions under which instability arises in the

equilibrium of a non-uniformly heated ferrofluid in a gravitational field and a non-

uniform magnetic field. Shulman et al. (1976) experimentally investigated the

effect of a constant magnetic field on the heat transfer process in ferromagnetic

suspensions by varying the type and concentration of the disperse phase, the

strength of the magnetic field and the orientation of the field relative to the

direction of the temperature gradient. They observed that the thermal resistance of

disperse systems depends on the size, shape, nature and surface purity of the

43

particles of the disperse phase. The effective thermal conductivity of ferromagnetic

suspensions has been shown to be anisotropic in character.

Berkovsky et al. (1976) presented numerical and experimental study of

convective heat transfer in a vertical layer of a ferromagnetic fluid. A critical

relationship is given between heat transfer and characteristic parameters.

Nogotov and Polevikov (1977) studied Oberbeck convection in a vertical layer

of a magnetic liquid in a magnetic field of current carrying sheet. The dependence

of heat transfer on Rayleigh number, Prandtl number and aspect ratio were clearly

exhibited. The convective stability of a vertical layer of magnetic fluid in a uniform

longitudinal magnetic field was studied by Bashtovoi and Pavlinov (1978).

Rosensweig et al. (1978) established experimentally the penetration of ferrofluids

in the Heleshaw cell.

Gupta and Gupta (1979) investigated thermal instability in a layer of

ferromagnetic fluid subject to Coriolis force and permeated by a vertical magnetic

field. It is substantiated that overstability cannot occur if the Prandtl number is

greater than unity. Gotoh and Yamada (1982) investigated the linear convective

instability of a ferromagnetic fluid layer heated from below and confined between

two horizontal ferromagnetic boundaries. The Galerkin technique is used and the

Legendre polynomials are taken as the trial functions. It is shown that the

magnetization of the boundaries and the nonlinearity of fluid magnetization reduce

the critical Rayleigh number and the effects of magnetization and buoyancy forces

are shown to compensate each other.

Schwab et al. (1983) performed an experiment to examine the influence of a

homogeneous vertical magnetic field on the Rayleigh-Bénard convection in a

ferrofluid layer. The results agreed with theoretical predictions. Schwab and

Stierstadt (1987) demonstrated the preparation and visualization of distinct

wavevectors for thermal convection in ferrofluids.

44

Blums (1987) examined the possibility of having convection in ferromagnetic

fluids as a result of magneto-diffusion of colloidal particles which give rise to non-

uniform magnetization. Kamiyama et al. (1988) studied both analytically and

numerically the effect of combined forced and free steady convection in a vertical

slot of ferromagnetic fluid in the presence of a transverse magnetic field taking into

account the magnetocaloric effect. The relative magnitudes of the magnetization

parameter and thermal Rayleigh number along with the uniform pressure gradient

are shown to significantly influence the dynamics of the ferrofluid in a vertical slot.

Ageev et al. (1990) studied magnetic fluid convection in a non-uniform

magnetic field. Results from both numerical and experimental studies are

presented. Nakatsuka et al. (1990) studied the effect of thermomagnetic convection,

which arises when a temperature sensitive magnetic fluid is heated in a vessel

under a non-uniform magnetic field.

Stiles and Kagan (1990) examined the thermoconvective instability of a

horizontal layer of ferrofluid in a strong vertical magnetic field. Their paper also

questioned the satisfactory agreement claimed to exist between the experiments and

the theoretical model of Finlayson which has been generalized by them. Schwab

(1990) investigated the stability of flat layers of ferrofluid subject to a vertical

temperature gradient and a vertical magnetic field experimentally. It is shown that

magnetostatic stresses reinforce the surface deformation of Marangoni convection

but they work against the surface deformation of Rayleigh-Bénard convection.

Abdullah and Lindsay (1991) examined convection in a nonlinear magnetic

fluid under the influence of a non-vertical magnetic field. It is found that both

stationary and overstable instabilities can be expected to be realizable possibilities.

Sekhar and Rudraiah (1991) studied convective instability in magnetic fluids

bounded by isothermal non-magnetic boundaries with internal heat generation.

Oscillatory convection is ruled out by proving the validity of the principle of

exchange of stabilities. The solutions are obtained using a higher order Galerkin

expansion technique.

45

Blennerhassett et al. (1991) analyzed the linear and weakly nonlinear

thermoconvective stability of a ferrofluid, confined between rigid horizontal plates

at different temperatures and subjected to a strong uniform external magnetostatic

field in the vertical direction. When the ferrofluid is heated from above and when

convection is due to magnetic forces, the Nusselt numbers for a given supercritical

temperature gradient are significantly higher than when the ferrofluid s heated from

below. Following the analysis of Blennerhassett et al. (1991), Stiles et al. (1992)

analyzed linear and weakly nonlinear thermoconvective stability in weakly

magnetized ferrofluids. They showed that if the ferrofluid is heated from above, the

magnitudes of the critical horizontal wavenumbers are substantially higher than

those when the ferrofluid is heated from below.

Rudraiah and Sekhar (1992) analyzed the thermohaline convection in a

Boussinesq-ferrofluid layer confined between rigid-rigid boundaries using the

Galerkin method. The conditions for direct and oscillatory modes are established. It

is shown that the concentration gradient and the diffusivity ratio significantly

influence the stability of the system.

Siddheshwar (1993) investigated the RBC problem of a Newtonian

ferromagnetic fluid with second sound. It is shown that oscillatory convection is

possible for heating from above. He further showed that the critical eigenvalue for

stationary convection, when heated from below, is significantly influenced by

second sound effects. Aniss et al. (1993) made an experimental investigation of the

RBC problem in a magnetic fluid contained in an annular Hele-Shaw cell.

Qin and Kaloni (1994) developed a nonlinear stability analysis based on energy

method to study the effects of buoyancy and surface tension in a ferromagnetic

fluid layer which is heated from below. The free surface is assumed to be flat and

non-deformable. The possibility of the existence of subcritical instabilities is

pointed out. Venkatasubramanian and Kaloni (1994) studied the effects of rotation

46

on the thermoconvective instability in a horizontal layer of ferrofluid heated from

below in the presence of a uniform vertical magnetic field.

Aniss et al. (1995) made a theoretical investigation of Rayleigh-Bénard

convection in a magnetic liquid enclosed in a Hele-Shaw cell. It is shown that the

Hele-Shaw approximation leads to two nonlinear problems; each one depending on

the order of magnitude of the Prandtl number. Results of linear and weakly

nonlinear analysis of stability near the onset of convection are presented.

Odenbach (1995a) investigated the convective flow generated by the interaction

of a magnetic field gradient with a gradient in magnetization in a magnetic fluid.

This gradient was caused by the diffusion of the magnetic particles in the field

gradient. Odenbach (1995b) investigated the onset and the flow profile of

thermomagnetic convection in a cylindrical fluid layer experimentally. Under

microgravity conditions and with periodic boundary conditions, he established

counter-rotating vortices.

Russell et al. (1995) examined heat transfer in strongly magnetized ferrofluids

in the case of strong heating from above. The convective patterns at critical

conditions have a large wave number and this is used to derive simplified equations

for the temperature field in the ferrofluid. The results show that the heat transfer

depends nonlinearly on the temperature difference.

Siddheshwar (1995) studied convective instability of a ferromagnetic fluid in

the Rayleigh-Bénard situation between fluid-permeable, magnetic boundaries and

subject to a uniform, transverse magnetic field. The Galerkin method is used to

predict the critical eigenvalue for free-free and rigid-rigid boundaries. This paper

reaffirmed the qualitative findings of earlier investigations which are in fact

limiting cases of the present study.

Weilepp and Brand (1996) presented a linear stability analysis of a layer of a

magnetic fluid with a deformable free surface, which is heated from below and

exposed to a uniform, vertically applied magnetic field. In this configuration the

47

temperature dependence of the surface tension, the buoyancy and the focusing of

the magnetic field due to surface fluctuations act as destabilizing effects. It is

demonstrated that there is no oscillatory instability in the regions of the parameter

space considered in this problem.

Odenbach (1996) investigated the behaviour of a magnetic fluid under the

influence of an inhomogeneous magnetic field gradient. The onset of the

convective flow is described by a model based on a time-dependent dimensionless

parameter. Zebib (1996) performed a theoretical study of the character and stability

of thermomagnetic flow in a microgravity environment. Convection is driven

owing to imposed radial magnetic and temperature gradients in a cylindrical shell

containing a ferrofluid. It is shown that convection sets in as a stable supercritical

bifurcation.

Bajaj and Malik (1997) have investigated a nonlinear convective instability in a

layer of magnetic fluid in the presence of an applied magnetic field and temperature

gradient. The stability of steady state patterns resulting from the convective

instability has been discussed using bifurcation theory. Rolls are found to be stable

on both the square and hexagonal lattices.

Morimoto et al. (1998) investigated the dissipative structure of thermomagnetic

convection by microgravity experiments through linear and nonlinear numerical

simulations. The effect of the aspect ratio of the magnetic fluid layer on the pattern

formations is investigated. In the case of linear theory, the critical magnetic

Rayleigh number and the critical wave number have been obtained by solving the

eigenvalue equations using harmonic analysis and the finite difference method.

Linear stability theory results agree with the microgravity experiments. The

nonlinear equations have been solved by the control volume finite difference

method. The flow patterns obtained by the nonlinear calculation coincide with

those obtained by the microgravity experiments. It is found that the critical

magnetic Rayleigh number obtained by the nonlinear analysis agrees with that

48

obtained by the linear stability analysis and the bifurcations from one pattern to

another are clearly demonstrated as a problem of probability.

Bajaj and Malik (1998) studied pattern formation due to double-diffusive

convection in ferrofluids in the presence of an externally applied transverse

magnetic field. The critical value of the Rayleigh number for steady state

bifurcation is found to be different from that for Hopf-bifurcation in contrast to

ordinary fluids where the two critical values are the same.

Siddheshwar and Abraham (1998) considered the problem of convection in

ferromagnetic fluids occupying a rectangular vertical slot with uniform heat flux

along the vertical walls. A closed form solution based on the Oseen-linearization

technique is obtained. It is found that the effect of the magnetization is to increase

the Nusselt number. Rudraiah et al. (1998) examined the effect of non-uniform

concentration distribution on double diffusive convection in a Boussinesq-magnetic

fluid layer confined between two rigid boundaries analytically using the Galerkin

method. The conditions for direct and oscillatory modes for different nonlinear

basic concentration distributions have been established.

Russell et al. (1999) examined the structure of two-dimensional vortices in a

thin layer of magnetized ferrofluid heated from above in the limit as the critical

wave number of the roll cells become large. They present a nonlinear asymptotic

description of the vortex pattern that occurs directly above the critical point in the

parameter space where instability first sets in. Tangthieng et al. (1999) investigated

heat transfer enhancement in ferrofluids subjected to steady magnetic fields. Luo et

al. (1999) examined novel convective instabilities in a magnetic fluid.

Yamaguchi et al. (1999) studied experimentally and numerically the natural

convection of a magnetic fluid in a two dimensional cell whose aspect ratio is one.

Results obtained reveal that the vertically imposed magnetic field has a

destabilizing influence and at the supercritical state the flow mode becomes quite

different from that without the magnetic field.

49

Sekar et al. (2000) studied the effect of ferrothermohaline convection in a

rotating medium heated from below and salted from above. The effect of salinity is

included in the magnetization and density of the ferrofluid. The conditions for both

stationary and oscillatory modes have been obtained using linear stability analysis

and it is found that the stationary mode is favored in comparison with oscillatory

mode. Auernhammer and Brand (2000) investigated the effect of rotation on RBC

in a ferrofluid using both a linear and a weakly nonlinear analysis of the governing

hydrodynamic equations in the Boussinesq approximation.

Aniss et al. (2001) investigated the effect of a time-sinusoidal magnetic field on

the onset of convection in a horizontal magnetic fluid layer heated from above. The

Floquet theory is used to determine the convective threshold for free-free and rigid-

rigid cases. The possibility to produce a competition between the harmonic and

sub-harmonic modes at the onset of convection is discussed.

Rudraiah et al. (2002) and Shivakumara et al. (2002) investigated the effect of

different basic temperature gradients on the onset of MC, and on the onset of the

combined RBC and MC in ferrofluids respectively in the presence of a vertical

uniform magnetic field. The mechanism of suppressing or augmenting the

ferroconvection is discussed.

Abraham (2002a) investigated the RBC problem in a micropolar ferromagnetic

fluid layer in the presence of a vertical uniform magnetic field analytically. It is

shown that the micropolar ferromagnetic fluid layer heated from below is more

stable as compared with the classical Newtonian ferromagnetic fluid. Lange (2002)

studied the thermomagnetic convection of magnetic fluids in a cylindrical geometry

subject to a homogeneous magnetic field. The general condition for the existence of

a potentially unstable stratification in the magnetic fluid is derived.

Siddheshwar and Abraham (2003) examined the thermal instability in a layer of

a ferromagnetic fluid when the boundaries of the layer are subjected to

synchronous/asynchronous imposed time-periodic boundary temperatures (ITBT)

50

and time-periodic body force (TBF). It is shown that the stability or instability of

ferrofluids can be controlled with the help of ITBT and TBF.

Sunil et al. (2004) considered the thermosolutal convection in a ferromagnetic

fluid for a fluid layer heated and soluted from below in the presence of uniform

vertical magnetic field. For the case of two free boundaries, an exact solution is

obtained using a linear stability analysis. For the case of stationary convection,

magnetization has a destabilizing effect, whereas stable solute gradient has a

stabilizing effect on the onset of instability. The principle of exchange of stabilities

is found to hold true for the ferromagnetic fluid heated from below in porous

medium in the absence of stable solute gradient. The oscillatory modes are

introduced due to the presence of the stable solute gradient, which were non-

existent in its absence. A sufficient condition for non-existence of the overstability

is also obtained.

Kaloni and Lou (2004) carried out a theoretical investigation of the convective

instability problem in the thin horizontal layer of a magnetic fluid heated from

below. The effects of the relaxation time and the vortex (rotational) viscosity are

considered and discussed. The Chebyshev pseudospectral method is employed to

solve the eigenvalue problems and numerical calculations are carried out for a

number of magnetic fluids and in full range of the magnetic field. A variety of

results under gravity-free conditions are also presented and the critical temperature

gradients are determined for a variety of situations.

Kaloni and Lou (2005) presented linear and weakly nonlinear analysis of

thermal instability in a layer of ferromagnetic fluid rotating about a vertical axis

and permeated by a vertical magnetic field. The amplitude equation is developed by

multiscale perturbation method and it is found that the ratio of heat transfer by

convection to that by conduction decreases as magnetic field increases.

Bajaj (2005) considered thermosolutal convection in magnetic fluids in the

presence of a vertical magnetic field and bifrequency vertical vibrations. The

51

regions of parametric instability have been obtained using the Floquet theory.

Vaidyanathan et al. (2005) obtained the condition for the onset of

thermoconevctive instability in ferrofluids due to the Soret effect. Both stationary

and oscillatory instabilities have been investigated.

Ramanathan and Muchikel (2006) investigated the effect of temperature-

dependent viscosity on ferroconvective instability in a porous medium. It is found

that the stationary mode of instability if preferred to oscillatory mode and that the

effect of temperature-dependent viscosity has a destabilizing effect on the onset of

convection.

Martinez-Mardones et al. (2006) reported theoretical and numerical results on

for a binary magnetic mixture subject to rotation. The effect of magnetophoresis

and Kelvin force have been emphasized. They also analyzed the stabilizing effect

of rotation on instability thresholds for aqueous suspensions. Sekar et al. (2006)

studied the Soret effect on multicomponent ferrofluid saturating a porous medium

with large variation in permeability. It is found that stationary instability is

preferred irrespective of values of permeability of pores.

Sunil et al. (2007) dealt with the theoretical investigation of the double-diffusive

convection in a micropolar ferromagnetic fluid layer heated and soluted from below

subjected to a transverse uniform magnetic field. The principle of exchange of

stabilities is found to hold true for the micropolar ferromagnetic fluid heated from

below in the absence of micropolar viscous effect, microinertia and solute gradient.

The oscillatory modes are introduced due to the presence of the micropolar viscous

effect, microinertia and solute gradient, which were non-existent in their absence.

Sunil et al. (2008) made a theoretical investigation of the effect of magnetic

field dependent (MFD) viscosity on thermal convection in a ferromagnetic fluid in

the presence of dust particles. As for stationary convection, dust particles always

have a destabilizing effect, whereas the MFD viscosity has a stabilizing effect on

the onset of convection. It is observed that the critical magnetic thermal Rayleigh

52

number is reduced solely because the heat capacity of clean fluid is supplemented

by that of the dust particles. The principle of exchange of stabilities is found to hold

true for the ferromagnetic fluid heated from below in the absence of dust particles.

The oscillatory modes are introduced due to the presence of the dust particles,

which were non-existent in their absence.

Sunil et al. (2008) performed a linear stability analysis for a micropolar

ferrofluid layer, heated from below subjected to a transverse uniform magnetic field

in the presence of uniform vertical rotation. The principle of exchange of stabilities

is found to hold true for the micropolar ferrofluid heated from below in the absence

of micropolar viscous effect, microinertia and rotation. The oscillatory modes are

introduced due to the presence of the micropolar viscous effect, microinertia and

rotation, which were non-existent in their absence. Sunil et al. (2008) developed

generalized energy method, which gives sufficient condition for the stability, for

convection problem in a magnetized ferrofluid with magnetic field dependent

(MFD) viscosity heated from below. Both linear and nonlinear analyses are carried

out and comparison of results shows a marked difference in the stability boundaries

and thus indicates that the sub-critical instabilities are possible.

Saravanan (2009) made a theoretical investigation to study the influence of

magnetic field on the onset of convection induced by centrifugal acceleration in a

magnetic fluid filled porous medium. The layer is assumed to exhibit anisotropy in

mechanical as well as thermal sense. Numerical solutions are obtained using the

Galerkin method. It is found that the magnetic field has a destabilizing effect and

can be suitably adjusted depending on the anisotropy parameters to enhance

convection. The effect of anisotropies of magnetic fluid filled porous media is

shown to be qualitatively different from that of ordinary fluid filled porous media.

Sunil and Mahajan (2009a) developed the generalized energy method to study

the nonlinear stability analysis for a magnetized ferrofluid layer heated from below

saturating a porous medium in the stress-free boundary case. It is found that the

nonlinear critical stability magnetic thermal Rayleigh number does not coincide

53

with that of linear instability analysis, and thus indicates that the subcritical

instabilities are possible. However, it is noted that, in the case of non-ferrofluid,

global nonlinear stability Rayleigh number is exactly the same as that for linear

instability. For lower values of magnetic parameters, this coincidence is

immediately lost. It is shown that with the increase of magnetic parameter (M3) and

Darcy number (Da), the subcritical instability region between the two theories

decreases quickly.

Sunil and Mahajan (2009b) performed a nonlinear energy stability analysis for a

rotating magnetized ferrofluid layer heated from below in the stress-free boundary

case. It is shown that with the increase of magnetic parameter, M3, the subcritical

instability region between the two theories decreases quickly while with the

increase of Taylor number, the subcritical region expands a little for small values

and expands significantly for large values.

Suresh and Vasanthakumari (2009) studied ferroconvection induced by

magnetic field dependent viscosity in an anisotropic porous medium using Darcy

model with computational methods. Galerkin method is applied. Linear stability

analysis is carried out for both stationary and oscillatory modes. The critical

magnetic Rayleigh number is computed for various values of the parameters which

characterize the flow. It is found that the increase in magneto viscosity stabilizes

the system through stationary mode.

Singh and Bajaj (2009) investigated numerically the effect of frequency of

modulation, applied magnetic field, and Prandtl number on the onset of a periodic

flow in the ferrofluid layer using the Floquet theory. Some theoretical results have

also been obtained to discuss the limiting behavior of the underlying instability

with the temperature modulation. Depending upon the parameters, the flow patterns

at the onset of instability have been found to consist of time-periodically oscillating

vertical magnetoconvective rolls.

54

Belyaev and Smorodin (2010) used the Langevin law of magnetization to study

the linear stability of a convective flow in a flat vertical layer of ferrofluid subject

to a transverse temperature gradient and a uniform magnetic field. The stability of

the flow against planar, spiral and three-dimensional perturbations is examined, and

the stability boundaries and characteristics of critical disturbances are determined.

The competition between the monotonic mode and two types of wave modes is

analyzed taking into account the properties of the fluid (magnetic susceptibility and

Prandtl number) and the magnetic field strength. The domain of parameters where

the oscillatory thermomagnetic wave instability exists is found.

2.5 Plan of Work

The Third Chapter consists of basic equations, approximations, boundary

conditions and a discussion of the dimensionless parameters. In Chapter IV, the

effect of temperature modulation on Rayleigh-Bènard ferroconvection in the

presence of rotation is studied. The Venezian approach will be adopted to find the

criterion for the onset of convection for small amplitudes of time periodic boundary

temperatures. A perturbation solution in powers of the amplitude of the applied

temperature field will be obtained. The results are discussed in Chapter V with the

help of figures. An exhaustive bibliography follows this last chapter.

55

CHAPTER III

BASIC EQUATIONS, BOUNDARY CONDITIONS AND DIMENSIONLESS PARAMETERS

In this chapter, we discuss the mathematical modelling of convective

phenomena involving a ferromagnetic liquid. The relevant boundary conditions and

dimensionless parameters arising in the problem are explained in a general manner.

It is now well known that ferrofluids represent a class of magnetizable liquids

with interesting properties capable of having a substantial impact on technology. In

many commercial applications, ferrofluid is an essential component of the system

or is an addition, which enhances the performance. Since the force exerted by a

magnetic field gradient on the fluid is proportional to its susceptibility, even weak

magnetic fields can exert reasonable forces to magnetic fluids.

It should be remarked that, upon application of a magnetic field, the entropy

associated with the magnetic degree of freedom in magnetic fluids is changed due

to the field-induced ordering. If performed adiabatically, this leads to a temperature

change in the fluid (Resler and Rosensweig, 1964; Parekh et al., 2000). The

magnitude of this effect depends on the physical and magnetic properties such as,

size, temperature dependence of magnetization, heat capacity of the material and

carrier liquid. We note, in view of this, that the energy conservation equation

should account for heat sources (sinks), which have implications for

magnetocaloric pumping.

On the other hand, if the magnetic force is to have any engineering application

to the control of fluid motion, there must be an interface or temperature gradients.

In what follows we elucidate briefly the development of some of the classical

instabilities that arise in ferromagnetic fluids.

56

The interfacial phenomena provide an area where the fluid mechanics of a

ferromagnetic liquid differs from that of a non-magnetic material. It is shown both

theoretically and experimentally that when a vertical magnetic field is applied on a

magnetic fluid having a flat surface with air above, the flat surface becomes

unstable when the applied magnetic field exceeds the critical value of the magnetic

field (Rosensweig and Cowley, 1967). This normal field instability (also known as

Rosensweig instability) is a direct consequence of the interaction of nonlinear

instabilities in magnetic fluids (Bajaj and Malik, 1996) and thanks to which a

pattern of spikes appear on the fluid surface.

It is well known that parametric stabilization can also be observed in fluid

dynamics, the most impressive example being the inhibition of the Rayleigh-Taylor

instability: a horizontal fluid layer placed above another one of smaller density

could be stabilized by vertically vibrating their container (Racca and Annett, 1985).

However, this requires a container with a rather small horizontal extension because

modes with a large enough wavelength are not parametrically stabilized. It should

be remarked that the parametric excitation of surface waves, the so-called Faraday

instability, can also be achieved in magnetic fluids by temporal modulation of an

external field (Mahr and Rehberg, 1998).

As has been discussed in Chapter I, dissipative instabilities, such as Rayleigh-

Bénard instability arising due to density variation and Marangoni instability arising

owing to surface-tension variation in ferromagnetic fluids in the presence of a

temperature gradient, have been studied by many researchers (Finlayson, 1970;

Siddheshwar, 1995; 2005; Zebib, 1996; Auernhammer and Brand, 2000; Abraham,

2003). It is worth noting that, in contrast to the dissipative Marangoni instability in

magnetic fluids, the Rosensweig instability is static whose critical wavelength is

nearly independent of the layer thickness (Weilepp and Brand, 1996). It has been

predicted recently that the Rosensweig instability could be inhibited by vertical

vibrations with an appropriate choice of the fluid and vibration parameters (Muller,

1998; Petrelis et al., 2000).

57

We now discuss the basic equations pertaining to the problem of

ferroconvection.

3.1 Basic Equations

To derive the basic equations, we make the following approximations: a) The ferromagnetic fluid is a homogeneous, incompressible medium and the

total magnetic moment of the particles is equally distributed throughout any

elementary fluid volume. Since the carrier fluids are good insulators, forces due

to interaction of magnetic fields with currents of free charge, such as found in

magnetohydrodynamics, are negligible (Cowley and Rosensweig, 1967). The

particles are prevented from agglomerating in the presence of a magnetic field

as they are surrounded by a surfactant such as oleic acid. The combination of

the short-range repulsion due to the surfactant and the thermal agitation yields a

material which behaves as a continuum (Papell and Faber, 1966).

b) Since we are considering small particle concentrations dipole-dipole

interactions are negligible and hence the applied magnetic field is not distorted

by the presence of the ferromagnetic fluid (Bean, 1955). Hysteresis is unlikely

in ferromagnetic fluids as the applied magnetic field is not rapidly changing

(Cowley and Rosensweig, 1967).

c) Maxwell’s equations are considered for non-conducting liquids with no

displacement currents.

d) The Boussinesq approximation is assumed to be valid, i.e.,

(1 )( ) .ρ Dρ Dt q→<< ∇ . As a result, the equation of continuity, viz.,

( ) ( ) 0.Dρ Dt ρ q→+ ∇ = , reduces to 0. q→∇ = . In other words, Boussinesq

fluids behave as incompressible fluids. This assumption also allows the fluid

density to vary only in the buoyancy force term in the momentum equation and

elsewhere it is treated as a constant. This is valid provided the velocity of the

58

fluid is much less than that of sound, i.e., Mach number << 1. The basic idea of

this approximation is to filter out high frequency phenomena such as sound

waves as they do not play an important role in transport processes (Spiegel and

Veronis, 1960).

e) Other fluid properties such as viscosity, thermal conductivity and heat capacity

are assumed to be constants.

f) The heating due to magnetocaloric effect of the magnetic substance in the

presence of a magnetic field is assumed negligible.

g) The viscous dissipation and radiation effects are neglected.

h) The temperature range of operation is below the Curie point.

i) Magnetization induced by temperature variations is small compared to that

induced by the external magnetic field, i.e., ( )1 o1K T Hχ∆ << + .

j) The magnetization is assumed to get aligned with the magnetic field.

The governing equations for ferrofluids (Neuringer and Rosensweig, 1964;

Finlayson, 1970) are the following:

Conservation of Mass (Continuity Equation)

The general form of the continuity equation is

( ) 0D qDtρ ρ •+ ∇ = , (3.1)

where DDt

denotes the material or substantial derivative ( )( )t q •= ∂ ∂ + ∇ ,

( )wvuq ,,= is the fluid velocity, t is the time, ρ is the fluid density and ∇ is the

vector differential operator. Eq. (3.1), for a fluid with Boussinesq approximation,

reduces to

59

0q•∇ = . (3.2) Conservation of Momentum (Momentum Equation) The momentum equation for a ferromagnetic fluid under the Boussinesq

approximation is

( ) ( ) 2R

q q q p g H B qt

ρ ρ µ• •

∂+ ∇ = −∇ + + ∇ + ∇ ∂

, (3.3)

where Rρ is a reference density, p is the pressure, g is the acceleration due to

gravity, H is the magnetic field, B is the magnetic induction and µ is the

coefficient of viscosity. The left side of Eq. (3.3) represents the rate of change of

momentum per unit volume. The first, second, third and fourth terms on the right

side represent respectively the pressure force due to normal stress, body force due

to gravity, pondermotive force arising due to the magnetization of the fluid (called

the Maxwell’s stress) and the viscous force arising due to shear.

Conservation of Energy (Heat conduction equation)

The heat transport equation for the considered ferromagnetic fluid which obeys

modified Fourier law is

2R 1

, ,o oVH

V H V H

M DT M DHC H T K TT Dt T Dt

ρ µ µ• •

∂ ∂ − + = ∇ ∂ ∂ ,

(3.4) where HVC is the specific heat at constant volume and constant magnetic field,

oµ is the magnetic permeability, T is the temperature and 1K is the thermal

conductivity.

Equation of State

The equation of state for a single component fluid is

60

( ) ( )R 1 RT T Tρ ρ α= − − , (3.5)

where α is the coefficient of thermal expansion and RT is a reference temperature.

The equation above is derived by expanding the density ( )Tρ using a Taylor’s

series about RT T= and neglecting the second and higher terms.

Maxwell’s Equations

Maxwell’s equations, simplified for a non-conducting ferromagnetic fluid with

no displacement currents, become

0B•∇ = and 0H∇× = , (3.6)

where B is the magnetic induction. The magnetic induction B , in terms of the

magnetization M and magnetic field H , is expressed as

( )oB M Hµ= + . (3.7)

Since the magnetization M is aligned with the magnetic field and is a function

of temperature and magnetic field, we have

( ),HM M H TH

= . (3. 8)

The magnetic equation of state is linearized about the magnetic field oH and

the reference temperature RT to become

( ) ( )oo m RM M χ H H K T T= + − − − , (3. 9) where χ is the magnetic susceptibility and mK is the pyromagnetic coefficient. In what follows we discuss various boundary conditions arising in the

convective instability problems of ferromagnetic liquids.

61

3.2 Boundary Conditions

(i) Velocity Boundary Conditions

The boundary conditions on velocity are obtained from conservation of mass,

the no-slip condition and the Cauchy’s stress principle depending on the nature of

the bounding surfaces of the fluid. The following combinations of boundary

surfaces are considered in the convective instability problems:

(i) Both lower and upper boundary surfaces are rigid.

(ii) Both lower and upper boundary surfaces are free.

(iii) Lower surface is rigid and upper surface is free.

a) Rigid surfaces

If the fluid layer is bounded above and below by rigid surfaces, then the viscous

fluid adheres to its bounding surface; hence the velocity of the fluid at a rigid

boundary surface is that of the boundary. This is known as the no-slip condition

and it indicates that the tangential components of velocity in the x and y directions

are zero, i.e. u = 0, v = 0. If the boundary surface is fixed or stationary, then in

addition to u = 0, v = 0, the normal component of velocity .q n∧→ is also zero, i.e.,

w = 0. Hence at the rigid boundary we have

u = v = w = 0. (3.10)

Since u = v = 0 for all values of x and y at the boundary, we have 0ux∂

=∂

and

0vx

∂=

∂, and hence from the continuity equation subject to the Boussinesq

approximation, it follows that

0wz

∂=

∂

62

at the boundaries. Thus, in the case of rigid boundaries, the boundary conditions for

the z-component of velocity are

0wwz

∂= =∂

. (3. 11)

b) Free surfaces

In the case of a free surface the boundary conditions for velocity depend on

whether we consider the surface-tension or not. If there is no surface-tension at the

boundary, i.e., the free surface does not deform in the direction normal to itself, we

must require that

w = 0. (3.12)

We have taken the z-axis perpendicular to the xy plane, therefore w does not

vary with respect to x and y, i.e.

0wx

∂=

∂ and 0w

y∂

=∂

. (3.13)

In the absence of surface tension, the non-deformable free surface (assumed

flat) is free from shear stresses so that

0u vz z

∂ ∂= =

∂ ∂. (3.14)

From the equation of continuity subject to the Boussinesq approximation, we have

0u v wx y z

∂ ∂ ∂+ + =

∂ ∂ ∂. (3.15)

Differentiating this equation with respect to ‘z’ and using Eq. (3.14) yields

2

2 0wz

∂=

∂. (3.16)

63

Thus, in the absence of surface-tension, the conditions for the z-component of

velocity at the free surfaces are

2

2 0wwz

∂= =∂

. (3.17)

This condition is the stress-free condition.

(ii) Thermal Boundary Conditions

The thermal boundary conditions depend on the nature of the boundaries

(Sparrow et al., 1964). Four different types of thermal boundary conditions are

discussed below.

(a) Fixed surface temperature

If the bounding wall of the fluid layer has high heat conductivity and large heat

capacity, the temperature in this case would be spatially uniform and independent

of time, i.e. the boundary temperature would be unperturbed by any flow or

temperature perturbation in the fluid. Thus

T = 0 (3.18)

at the boundaries. The effect is to maintain the temperature and this boundary

condition is known as isothermal boundary condition or boundary condition of the

first kind which is the Dirichlet type boundary condition.

(b) Fixed surface heat flux Heat exchange between the free surface and the environment takes place in the

case of free surfaces. According to Fourier’s law, the heat flux TQ passing through

the boundary per unit time and area is

1TTQ kz

∂= −

∂ (3.19)

64

where Tz

∂∂

is the temperature gradient of the fluid at the boundary. If TQ is

unperturbed by thermal or flow perturbations in the fluid, it follows that

Tz

∂∂

= 0 (3.20)

at the boundaries. This thermal boundary condition is known as adiabatic boundary

condition or insulating boundary condition or boundary condition of the second

kind which is the Neumann type boundary condition.

(c) Boundary condition of the third kind

This is a general type of boundary condition on temperature which is given by

T Bi Tz

∂= −

∂. (3.21)

When Bi→ ∞ , we are led to the isothermal boundary condition T = 0 and when

0Bi → , we obtain the adiabatic boundary condition 0Tz

∂=

∂.

(iii) Magnetic Potential Boundary Conditions

The general boundary conditions for the perturbed magnetic potential Φ are

0 at 0,1

0 at 1,1

aΦDΦ T zχ

aΦDΦ T zχ

+ − = = + − − = =+

(3.22)

where dDd z

= and a is the dimensionless wave number. If we take a → ∞ in

(3.22), we obtain the boundary condition of the first kind, i.e., Φ = 0 at both the

boundaries. This type of boundary condition has been used by Gotoh and Yamada

(1982) for a liquid layer confined between two ferromagnetic boundaries. In this

65

case the magnetic permeability of the boundary is much higher than that of the

fluid. If we consider isothermal boundary conditions for temperature and take

χ → ∞ in (3.22) at both boundaries, we obtain the boundary condition of the

second kind, i.e., 0DΦ = . Finlayson (1970) used this type of boundary condition

in order to obtain exact solution to the convective instability problem of

ferromagnetic fluids for free-free, isothermal boundaries.

3.3 Dimensionless Parameters

Exact solutions are rare in many branches of fluid mechanics because of

nonlinearities and general boundary conditions. Hence to determine approximate

solutions of the problem, numerical techniques or analytical techniques or a

combination of both are used. The key to tackle modern problems is mathematical

modelling. This process involves keeping certain elements, neglecting some, and

approximating yet others. To accomplish this important step one needs to decide

the order of magnitude, i.e., smallness or largeness of the different elements of the

system by comparing them with one another as well as with the basic elements of

the system. This process is called non-dimensionalization or making the variables

dimensionless. Expressing the equations in dimensionless form brings out the

important dimensionless parameters that govern the behaviour of the system. The

first method used to make the equations dimensionless is by introducing the

characteristic quantities and the other is by comparing similar terms. We use the

former method of introducing characteristic quantities. The following are the

important dimensionless parameters arising in the present study.

(i) Raleigh number The Raleigh number R is defined to be

3R g T hR ρ α

µκ∆

= .

66

The thermal Rayleigh number plays a significant role in fluid layers where the

buoyancy forces are predominant. Physically it represents the balance of energy

released by the buoyancy force and the energy dissipation by viscous and thermal

effects. We observe from the expression for R that the terms in the numerator drive

the motion and the terms in the denominator oppose the motion. Mathematically,

this number denotes the eigenvalue in the study of stability of thermal convection.

The critical thermal Rayleigh number is the value of the eigenvalue at which the

conduction state breaks down and convection sets in.

(ii) Prandtl number

The Prandtl number Pr , which is a property of a particular fluid, is defined to be

RPr µ

ρ κ= .

Pr is the ratio between diffusivity of momentum and vorticity to diffusivity of

heat. High Pr liquids are very viscous ones and low Pr ones have high thermal

diffusivities. When Pr is large the velocity boundary layer is thick compared with

the temperature boundary layer. The Prandtl number is very high for non-

Newtonian fluids.

(iii) Buoyancy-magnetization parameter This parameter 1M is defined to be

( )

2o

1R

∆1mK TM

g hµ

α ρ χ=

+.

1M is the ratio of the magnetic force to gravitational force. Large values of 1M

imply that the magnetic mechanism is very large. When both magnetic and

buoyancy forces cause convection, the Rayleigh number depends on 1M and both

are coupled. When the buoyancy force has a negligible influence (that is, for very

large 1M ), we define another parameter, referred to as the magnetic Rayleigh

67

number, 21 o ( ∆ ) (1 )M mR R M µ K T h χ µ κ= = + . The latter definition is also

applicable to a thin layer of ferromagnetic fluid where the surface tension force is

important. When 1M = 0, we obtain the nonmagnetic classical Rayleigh-Bénard

problem for buoyancy induced convection (Finlayson, 1970).

(iv) Non-buoyancy-magnetization parameter

We define this parameter 3M to be

( )

o

o3

1

1

MH

Mχ

+

=+

.

The parameter 3M measures the departure of linearity in the magnetic equation

of state. 3M = 1 corresponds to linear magnetization. As the equation of state

becomes more nonlinear (i.e. 3M large), the fluid layer is destabilized slightly.

When 3M →∞ , which means very strong nonlinearity of magnetization of the

fluid, the entire problem reduces to the classical Rayleigh-Bénard problem

(Finlayson, 1970).

(v) Taylor number

We define the Taylor number Ta to be

222 R hTa Ω ρµ

=

.

The Taylor number is a dimensionless number measuring the influence of

rotation on a convecting system. It depends on the scale of the convective cell, the

rate of rotation, and kinematic viscosity. The Taylor number Ta characterizes the

importance of centrifugal forces (or the so-called inertial forces) due to rotation of a

fluid about a vertical axis relative to viscous forces. If Ta is greater than unity, then

rotational effects are significant.

68

CHAPTER IV

EFFECT OF TEMPERATURE MODULATION

ON RAYLEIGH-BÈNARD CONVECTION

IN A ROTATING LAYER OF

A FERROMAGNETIC FLUID

4.1 Introduction

Convective instability in ferromagnetic fluids has been the subject of interest

because of their remarkable physical properties and their marketable applications

(Rosensweig, 1986 and Fertman, 1990). Ferromagnetic fluids are formed by

suspending submicron sized particles of magnetite in a carrier medium such as

kerosene, heptane or water. To prevent the particles from agglomerating in the

presence of a magnetic field they are surrounded by a surfactant such as oleic acid.

The combination of the short range repulsion due to the surfactant and the thermal

agitation yields a material which behaves as a continuum and can experience forces

due to magnetic polarization. The fluids are usually good insulators and forces due

to interaction of magnetic fields with currents of free charge, such as found in

magnetohydrodynamics, are negligible. The presence of a ferromagnetic fluid can

distort an external magnetic field if magnetic interaction (dipole-dipole) takes

place, but this is negligible for small particle concentrations. Experience also

suggests that hysteresis is unlikely in these fluids with the exception of rapidly

changing external magnetic fields.

Ferro fluid technology is the basis of a wide variety of products used for high

technology applications in the semiconductor and computer industries. Ferro fluids

are also used in a wide variety of thermoelectric cooling modules which prove

instrumental for the refrigeration of semiconductor process equipment, laser diodes,

medical treatment and optical communication equipment. Ferro fluids have been

found to be an essential element in a nuclear magnetic resonance probe

69

differentiating free and shale oil in oil prospecting. They are used extensively for

the study of magnetic domain structures in magnetic tapes, rigid discs, crystalline

and amorphous alloys, garnets steels and geological rocks. Other commercial uses

are ink jet printing, magneto gravimetric preparations of nonferrous metals,

pumping without moving parts and biotechnology.

Thermo-mechanical interactions in fluids make possible onset of convection

induced by externally applied temperature gradients. The well-known example of

thermo-mechanical interaction is buoyancy-induced convection, in which case the

driving force is the gravity force as long as the density is a function of temperature.

Convective instability analyses are useful for predicting the critical temperature

gradient above which motion occurs.

Finlayson (1970) made a detailed study of convective instability in a

ferromagnetic fluid. He predicted the critical temperature gradient for the onset of

convection when only the magnetic mechanism is important as well as when both

the magnetic and buoyancy mechanisms are operative. The magnetic mechanism is

shown to predominate over the buoyancy mechanism in fluid layers which are

about 1 mm thick.

Qin and Kaloni (1994) developed a nonlinear stability analysis based on energy

method to study the effects of buoyancy and surface tension in a ferromagnetic

fluid layer which is heated from below. The free surface is assumed to be flat and

non-deformable. The possibility of the existence of subcritical instabilities is

pointed out.

Weilepp and Brand (1996) presented a linear stability analysis of a layer of a

magnetic fluid with a deformable free surface, which is heated from below and

exposed to a uniform, vertically applied magnetic field. In this configuration the

temperature dependence of the surface tension, the buoyancy and the focusing of

the magnetic field due to surface fluctuations act as destabilizing effects. It is

70

demonstrated that there is no oscillatory instability in the regions of the parameter

space considered in this problem.

Siddheshwar and Abraham (2003) examined the thermal instability in a layer of

a ferromagnetic fluid when the boundaries of the layer are subjected to

synchronous/asynchronous imposed time-periodic boundary temperatures (ITBT)

and time-periodic body force (TBF). It is shown that the stability or instability of

ferrofluids can be controlled with the help of ITBT and TBF.

Govender (2003b) investigated analytically the Coriolis effect on centrifugally

driven convection in a rotating porous layer using the linear stability theory. The

problem corresponding to a layer placed far away from the axis of rotation was

identified as a distinct case and therefore justifying special attention. The stability

of the basic centrifugally driven convection is analysed. The marginal stability

criterion is established as a characteristic centrifugal Rayleigh number in terms of

the wavenumber and the Taylor number.

Ramanathan and Suresh (2004) examined the effect of magnetic field dependent

viscosity and anisotropy of porous medium on the onset of ferroconvection. The

effect of variable-viscoity is found to have stabilizing influence on the onset of

convection. Ramanathan and Muchikel (2006) investigated the effect of

temperature dependent viscosity on ferroconvection in a sparsely distributed porous

medium using the Brinkman model. It is found that stationary mode of instability is

preferred to oscillatory mode and that variable-viscosity has destabilizing effect on

the onset of ferroconvection.

Sunil et al. (2008) developed generalized energy method, which gives sufficient

condition for the stability, for convection problem in a magnetized ferrofluid with

magnetic field dependent (MFD) viscosity heated from below. Both linear and

nonlinear analyses are carried out and comparison of results shows a marked

difference in the stability boundaries and thus indicates that the sub-critical

instabilities are possible.

71

Saravanan (2009) made a theoretical investigation to study the influence of

magnetic field on the onset of convection induced by centrifugal acceleration in a

magnetic fluid filled porous medium. The layer is assumed to exhibit anisotropy in

mechanical as well as thermal sense. Numerical solutions are obtained using the

Galerkin method. It is found that the magnetic field has a destabilizing effect and

can be suitably adjusted depending on the anisotropy parameters to enhance

convection. The effect of anisotropies of magnetic fluid filled porous media is

shown to be qualitatively different from that of ordinary fluid filled porous media.

Hashim and Siri (2009) applied the linear stability theory to investigate the

effects of rotation and feedback control on the onset of steady and oscillatory

thermocapillary convection in a horizontal fluid layer heated from below with a

free-slip bottom. The thresholds and codimension-2 points for the onset of steady

and oscillatory convection are determined.

Vanishree and Siddheshwar (2010) performed a linear stability analysis for

mono-diffusive convection in an anisotropic rotating porous medium with

temperature-dependent viscosity. Some new results on the parameters’ influence on

convection in the presence of rotation, for both high and low rotation rates, are

presented.

Singh and Bajaj (2009) investigated numerically the effect of frequency of

modulation, applied magnetic field, and Prandtl number on the onset of a periodic

flow in the ferrofluid layer using the Floquet theory. Some theoretical results have

also been obtained to discuss the limiting behavior of the underlying instability

with the temperature modulation. Depending upon the parameters, the flow patterns

at the onset of instability have been found to consist of time-periodically oscillating

vertical magnetoconvective rolls.

Belyaev and Smorodin (2010) used the Langevin law of magnetization to study

the linear stability of a convective flow in a flat vertical layer of ferrofluid subject

to a transverse temperature gradient and a uniform magnetic field. The stability of

72

the flow against planar, spiral and three-dimensional perturbations is examined, and

the stability boundaries and characteristics of critical disturbances are determined.

The competition between the monotonic mode and two types of wave modes is

analyzed taking into account the properties of the fluid (magnetic susceptibility and

Prandtl number) and the magnetic field strength. The domain of parameters where

the oscillatory thermomagnetic wave instability exists is found.

The problem of control of convection is of relevance and interest in innumerable

ferromagnetic fluid applications and is also mathematically quite challenging. The

unmodulated Rayleig-Benard problem of convection in a ferromagnetic fluid with

or without the rotation effect has been extensively studied. However, attention has

not been paid to the effect of thermal modulation on Rayleigh-Benard convection in

a rotating layer of a ferromagnetic fluid. It is with this motivation that we study, in

this dissertation, the problem of Rayleigh-Benard-Marangoni convection in a

sinusoidally heated ferromagnetic fluid layer in the presence of rotation with

emphasis on how the stability criterion for the onset of ferroconvection would be

modified in the presence of both rotation and temperature modulation.

4.2 Mathematical Formulation

We consider a ferromagnetic fluid layer confined between two infinite

horizontal surfaces with height “h”. A vertical downward gravity force acts on the

fluid together with a uniform, vertical magnetic field oH . A Cartesian frame of

reference is chosen with the origin in the lower boundary and the z-axis vertically

upwards. The fluid layer is subjected to rotation with an angular velocity Ω . The

axis of rotation is taken along the z-axis (Fig. 5.1). The Boussinesq approximation

is applied to account for the effect of density variation. With these assumptions the

basic governing equations are

0q•∇ = , (4.1)

73

( ) ( ) 2R 2q q q q p g H B q

tρ Ω ρ µ• •

∂+ ∇ + × = −∇ + + ∇ + ∇ ∂

, (4.2)

2R 1

, ,o oVH

V H V H

M DT M DHC H T K TT Dt T Dt

ρ µ µ• •

∂ ∂ − + = ∇ ∂ ∂ , (4.3)

( )1R RT Tρ ρ α= − − , (4.4)

where ( )wvuq ,,= is the fluid velocity, t is the time, Rρ is a reference density,

p is the pressure, g is the acceleration due to gravity, Ω is the constant angular

velocity, H is the magnetic field, B is the magnetic induction and µ is the

coefficient of viscosity, VHC is the specific heat at constant volume and constant

magnetic field, oµ is the magnetic permeability, T is the temperature, M is the

magnetization, 1K is the thermal conductivity, α is the coefficient of thermal

expansion and RT is a reference temperature.

Maxwell’s equations simplified for a non-conducting fluid with no displacement

current take the form

0B•∇ = , 0H∇× = , (4.5)

and ( )oB M Hµ= + . (4.6)

Since the magnetization M is aligned with the magnetic field and is a function

of temperature and magnetic field, we may write

( ),HM M H TH

= . (4. 7)

The magnetic equation of state is linearized about the magnetic field oH and the

reference temperature RT to become

74

( ) ( )oo m RM M χ H H K T T= + − − − , (4. 8)

where χ is the magnetic susceptibility and mK is the pyromagnetic coefficient. The surface temperatures are

[ ]1 cos ( )2RTT tε ω∆

+ + at 0z = (4.9)

and

[ ]1 cos ( )2RTT tε ω φ∆

− − + at z h= , (4.10)

where T∆ is the temperature difference between the two surfaces in the

unmodulated case, ε is the amplitude of the thermal modulation, ω is the

frequency and φ is the phase angle.

We consider three types of thermal modulation, namely,

Case (a): symmetric (in-phase, 0φ = )

Case (b): asymmetric (out-of-phase, φ π= )

Case (c): only lower wall temperature modulation (φ i= − ∞ ).

4.3 Basic State

In the undisturbed state, the temperature HT , the pressure Hp , the magnetic

field HH , magnetic induction HB and magnetization HM satisfy the following

equations

,H HH H

p Hg Bz z

ρ∂ ∂− = −

∂ ∂ (4.11)

20 1

,

H HR VH H H

V H

M TC H K TT t

ρ µ •

∂ ∂ − = ∇ ∂ ∂

(4.12)

and

( )1 .H R RT Tρ ρ α= − − (4.13)

75

The solution of Eq. (4.12) subject to the boundary conditions (4.9) and (4.10)

consists of both steady and oscillating parts and the same is given by

( ) 2 Re ( ) ( ) ,2

z h z h i tH R

TT T h z a e a e eh

λ λ ωε λ λ − −∆ = + − + + − (4.14)

where ( )2

iT e eae e

φ λ

λ λλ− −

−

∆ −= −

, ( )1 22

21 ,

2hiK

ωλ

= −

12

1

KKC

= ,

1 o,

HR VH H

V H

MC C HT

ρ µ •

∂= − ∂

and Re stands for the real part. Eq. (4.14)

shows that the conduction profile departs from linearity if the frequency of

modulation is fairly large.

4.4 Linear Stability Analysis

Let the basic state be perturbed by an infinitesimal thermal perturbation so that

, , ,

, , ,H H H H

H H H

q q q p p p T T T

H H H B B B M M M

ρ ρ ρ′ ′ ′ ′= + = + = + = +

′ ′ ′= + = + = + (4.15)

where prime indicates that the quantities are infinitesimal perturbations.

Substituting (4.15) into (4.1) to (4.8) and using basic state solution, we obtain the

following equations

,R Tρ α ρ′ ′= − (4.16)

( ) ( )

( )

o 13

2o o o

2

11

,

R

mm

q q p gt

K T T H K Th z

HM H qz

ρ Ω ρ

µ ε χχ

µ µ

′ ∂ ′ ′ ′+ × = −∇ + ∂ ∆ ∂ ′ ′− − + − + ∂

′∂ ′+ + + ∇∂

(4.17)

76

( )

11 1 o

22o 1

1 ,1

m R

m R

T T TC C w K Tt h z t z

K T T T w K Th z

Φε µ

µ εχ

′ ′ ∂ ∆ ∂ ∂ ∂′− − − ∂ ∂ ∂ ∂

∆ ∂ ′ ′+ − = ∇ + ∂

(4.18)

0q• ′∇ = , (4.19)

22 ,R R

wt zζρ ρ Ω µ ζ

′∂ ∂= + ∇

∂ ∂ (4.20)

( )2

201 20

1 1 0 ,mM TKH zz

ΦΦ χ ′ ′∂ ∂′+ ∇ + + − = ∂∂

(4.21)

where 1 Re ( ) ( )z h z h i tT a e a e eλ λ ωλ λ − − = + − , ( ), ,q u v w′ ′ ′ ′= , ζ is the

z-component of the vorticity given by v ux y

ζ′ ′∂ ∂

= −∂ ∂

,

H Φ′ ′= ∇ and Φ ′ is the

magnetic potential.

Upon substituting (4.16) into (4.17), eliminating the pressure term p′

and

rendering the resulting equation and Eqs. (4.18) –(4.21) dimensionless by using the

transformations

( )

* * *

11

* * * * *2

1 121 1

, , ,

1

, , , , , ,

m

w Tw TT K T hK

C h

t x y zt x y zh h hC h K

K C h

ΦΦ

χ

ζζ

′ ′ ′= = =

∆ ∆ +

= = =

(4.22)

we obtain

77

( )

( ) ( )

2 2 1 2 21 1

21 1

1 1 1Pr

1 ,

w Ta R M f Tt z

R M fz

ζ ε

ε Φ

∂ ∂−∇ ∇ + = − − ∇ ∂ ∂

∂+ − ∇

∂

(4.23)

( )( ) 22 21 1 ,T M f w M T

t t zΦε

∂ ∂ ∂+ − − − = ∇ ∂ ∂ ∂

(4.24)

2 1 21 ,

PrwTa

t zζ

∂ ∂−∇ = ∂ ∂

(4.25)

2

23 12 ,TM

zzΦ

∂ ∂+ ∇ = ∂∂

(4.26)

where Re ( ) ( )z z i tf A e A e eλ λ ωλ λ − − = + − , ( )

2

ie eAe e

φ λ

λ λλλ

− −

−

−=

− ,

( )1 2

12

i ωλ = −

, ω is the dimensionless frequency of modulation given by

21

1

C hK

ω ω= , 2 2

21 2 2x y

∂ ∂∇ = +

∂ ∂ and

22 2

1 2z∂

∇ = ∇ +∂

and the asterisks have been

dropped for simplicity. The dimensionless parameters are

1

1Pr (Prandtl number) ,

R

CK

µρ

=

31

1(Rayleigh Number) ,R C g T hR

Kα ρ

µ∆

=

( )

20

1 (buoyancy-magnetization parameter) ,1

m

R

K TM

g hµχ α ρ

∆=

+

( )

20

21

(magnetization parameter) ,1

m RK TM

Cµ

χ=

+

78

( )

o

3

1 o(non-buoyancy magnetization parameter) ,

1

MH

Mχ

+ =

+ and

222 (Taylor number) .R hTa Ω ρµ

=

To this end we note that the typical values of 2M are of the order of 610 −

(Finlayson, 1970). Hence we neglect 2M and proceed further.

We now confine ourselves to stress-free, isothermal boundaries. While this case

is admittedly artificial, it is mathematically important because we can arrive at an

exact solution and the qualitative features of the problem can be disclosed. The

boundary conditions are then

2

2 0ww Tz zzΦ ζ∂ ∂ ∂

= = = = =∂ ∂∂

at 0, 1z = . (4.27)

The magnetic boundary conditions in (4.27) are based on the assumption of

infinite magnetic susceptibility (Finlayson, 1970; Maruthamanikandan, 2005). It is

convenient to express the entire problem in terms of w . Upon combining Eqs.

(4.23) - (4.26), we obtain an equation for the vertical component of the velocity w

in the form

( )

( )

22 2 2 2 2

3 12

2 22 2

3 12 2

22 2 2

3 1 12

2 41 3 1

1 1Pr Pr

1 1Pr

1 1 2 .Pr

M wt t t z

wTa Mt z z

R M f wt z

R M M f wt

ε

ε

∂ ∂ ∂ ∂ −∇ −∇ −∇ + ∇ ∇ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + − ∇ + ∇ ∂ ∂ ∂ ∂ ∂ = − − ∇ + ∇ − ∇ ∂ ∂ ∂ + −∇ − ∇ ∂

(4.28)

79

The boundary conditions (4.27) can also be expressed in terms of w in the form

(Chandrasekhar, 1961)

2 4 6 8

2 4 6 8 0 .w w w wwz z z z

∂ ∂ ∂ ∂= = = = =∂ ∂ ∂ ∂

(4.29)

4.5 Method of Solution

Let us now seek the eigenfunctions w and eigenvalues R of Eq. (4.28) for the

basic temperature profile that departs from the linear profile by quantities of

order ε . It follows that the eigenfunctions and eigenvalues of the problem at hand

differ from those associated with the problem of Rayleigh-Bénard convection in a

rotating layer ferromagnetic fluid by quantities of orderε . We therefore assume the

solution of Eq. (4.28) in the form

2

0 1 2w w w wε ε= + + + ⋅⋅⋅ ⋅ ⋅ ⋅ (4.30a)

20 1 2R R R Rε ε= + + + ⋅⋅ ⋅ ⋅ ⋅ ⋅ (4.30b)

where 0R is the critical Rayleigh number for the unmodulated Rayliegh-Benard

convection in a ferromagnetic fluid in the presence of rotation. Upon substituting

(4.30) into (4.28) and equating the coefficients of like powers of ε we obtain the

following system of equations up to ( )2O ε

0 0 ,Lw = (4.31)

( )

( )

22 2 2

1 1 3 1 1 1 02

22 2 2

0 3 1 1 1 02

1 1Pr

1 1 2 ,Pr

L w R M M wt z

R M M f wt z

∂ ∂= − ∇ + + ∇ ∇ ∂ ∂

∂ ∂− − ∇ + + ∇ ∇ ∂ ∂

(4.32)

80

( )

( )

( )

( )

22 2 2

2 1 3 1 1 1 12

22 2 2

2 3 1 1 1 02

22 2 2

0 3 1 1 1 12

22 2 2

1 3 1 1 1 02

1 1Pr

1 1Pr

1 1 2Pr

1 1 2 ,Pr

Lw R M M wt z

R M M wt z

R M M f wt z

R M M f wt z

∂ ∂= − ∇ + + ∇ ∇ ∂ ∂

∂ ∂+ − ∇ + + ∇ ∇ ∂ ∂

∂ ∂− − ∇ + + ∇ ∇ ∂ ∂

∂ ∂− − ∇ + + ∇ ∇ ∂ ∂

(4.33)

where

( )

22 2 2 2 2

3 12

2 22 2

3 12 2

22 2 2

0 3 1 1 12

1 1Pr Pr

1 1 .Pr

L Mt t t z

Ta Mt z z

R M Mt z

∂ ∂ ∂ ∂= − ∇ − ∇ − ∇ + ∇ ∇ ∂ ∂ ∂ ∂

∂ ∂ ∂+ − ∇ + ∇ ∂ ∂ ∂

∂ ∂− − ∇ + + ∇ ∇ ∂ ∂

(4.34)

The zeroth order problem is equivalent to the problem of rotating Rayleigh-

Benard ferroconvection in the absence of thermal modulation. The stability analysis

of the rotating ferroconvection problem has been investigated by Gupta and Gupta

(1979), who showed that overstability cannot occur if the Prandtl number Pr 1> .

Hence the principle of exchange of stabilities is valid for the unmodulated problem

if Pr 1> . The marginally stable solution for the unmodulated problem is

( )0 exp sin ,w i l x m y zπ= + (4.35) where l and m are wave numbers in the x and y directions. Upon substituting (4.35)

into (4.31), we obtain the expression for the Rayleigh number given by

81

( ) ( )32 2 2 2 2

3

02 2 2

3 1

,(1 )

M TaR

M M

π α π α π

α π α

+ + +

= + +

(4.36)

where 2 2 2α l m= + . We assume that the Prandtl number Pr 1> and proceed to

study the onset of stationary convection in a rotating ferromagnetic fluid layer in

the presence of thermal modulation. Since changing the sign of ε amounts to a

shift in the time origin and such a shift does not affect the stability of the problem,

it follows that all the odd coefficients 1 3, ,.....R R in (4.30b) must vanish.

Following Venezian (1969) and, Siddheshwar and Abraham (2003), we obtain the

following expression for 2R

( ) ( )2

2 2 2 2 2 22 3 3 1

1(1 2 ) ,

n n

n n

b CR K n n M M

Dπ α π α

∞

== + + +∑ (4.37)

where

2 2 2 20 3 1

3 2 23 1

(1 2 )

2 (1 )

R α π M M αK

π M M α

+ + = + +

,

( ) ( )

4 2 224 42 4 2 4

16 ,1 1

nnb

n n

π ω

ω π ω π= + + + −

( ) ( )( ) ( )

( )( )( ) ( )

2 22 2 2 2 2 23

42 2 2 2 2 23

2 2 2 2 2 2 2 23

2 2 2 2 2 2 20 3 1

12Pr Pr

1 ,

nC n n M

n n M

n Ta n n M

R n n M M

ω π α π α

π α π α

π π α π α

α π α π α

= + + +

− + +

− + +

+ + + + 2 21 2nD A A= +

with

82

( ) ( )( ) ( )

( )( )( ) ( )

2 22 2 2 2 2 21 3

42 2 2 2 2 23

2 2 2 2 2 2 2 23

2 2 2 2 2 2 20 3 1

12Pr Pr

1 ,

A n n M

n n M

n Ta n n M

R n n M M

ω π α π α

π α π α

π π α π α

α π α π α

= + + +

− + +

− + +

+ + + +

and

( ) ( )( ) ( )

( ) ( )( )

32 2 2 2 2 2

2 32

32 2 2 2 2 23

2 2 2 2 2 2 2 2 203 3 1

Pr21Pr

1 .Pr

A n n M

n n M

Rn Ta n M n M M

ω π α π α

ω π α π α

ω π π α α π α

= + +

− + + +

− + − + +

The value of R obtained by this procedure is the eigenvalue corresponding to the

eigenfunction w , which though oscillating, remains bounded in time. Since R is a

function of the horizontal wave number α and the amplitude of the modulation ε ,

we may write

20 2( , ) ( ) ( ) .....R R Rα ε α ε α= + + (4.38)

2

0 2 ......α α ε α= + + (4.39)

The critical value of the Rayleigh number R is computed up to ( )2O ε by

evaluating 0R and 2R at 0 cα α= , where cα is the value at which 0R is

minimum. It is only when one wishes to evaluate 4R , 2α must be taken into

account (Venezian, 1969). In view of this, we may write

2

0 2( , ) ( ) ( ) .....c c cR R Rα ε α ε α= + + (4.40)

where 0cR and 2cR are respectively the value of 0R and 2R evaluated at cα α= .

If 2cR is positive, supercritical instability exists and R has the minimum at 0ε = .

83

On the other hand, when 2cR becomes negative, subcritical instability is possible.

We evaluate 2cR in the following three cases:

Case (a): when the oscillating temperature field is symmetric so that the wall

temperatures are modulated in phase ( 0φ = ).

Case (b): when the oscillating temperature field is asymmetric corresponding to an

out-of-phase modulation (φ π= ).

Case (c): when only the temperature of the bottom wall is modulated (φ i= − ∞ ). In Eq. (4.37), the summation extends over even values of n for case (a), odd

values for case (b) and all values for case (c).

84

CHAPTER V

RESULTS, DISCUSSION AND

CONCLUDING REMARKS

5.1 Results and Discussion

The problem considered is that of determining the onset of convection for a

rotating ferromagnetic fluid layer heated from below when, in addition to a fixed

temperature difference between the walls, a perturbation is applied to the wall

temperatures varying sinusoidally in time. The analysis presented in this

dissertation is based on the assumption that the amplitude of the temperature

modulation is small compared with the imposed steady temperature difference. It

should be remarked that the validity of the results obtained depends on the range of

the frequency of modulation ω . When ω is small, the period of modulation

becomes large so that the disturbances may grow to such an extent that the finite

amplitudes become significant. This means that the assumption that nonlinear terms

are small is violated. On the other hand, in the limit as ω →∞ , the effect of

modulation is confined to a narrow boundary layer and outside this boundary layer

the basic temperature field has essentially a linear gradient varying in time. Thus

the effect of temperature modulation is perceptible for moderate values of ω

(Venezian, 1969; Malashetty and Swamy 2007). We present below the results

concerning three different thermal excitations, viz., symmetric temperature

modulation, asymmetric temperature modulation and the bottom wall temperature

modulation.

In Figs. 5.2 through 5.5, the variation of the correction Rayleigh number 2cR

with the frequency of modulation ω is exhibited for the case in which the bounding

wall temperatures are modulated in phase. It is observed that 2cR is negative,

meaning the presence of symmetric modulation paves the way for subcritical

motions; with convection occurring at an earlier point than in the corresponding

85

unmodulated system. Figure 5.2 shows the variation of 2cR with ω for different

values of the buoyancy-magnetization parameter 1M and for fixed values of Pr ,

Ta and 3M . The parameter 1M is the ratio of magnetic force to gravitational force.

It is seen that 2cR increases with an increase in 1M , indicating that the effect of

magnetic mechanism has a stabilizing effect on the system. Further, the presence of

symmetric modulation tends to reduce the stabilizing influence of the magnetic

mechanism for fairly large values of ω . It is also clear from Fig. 5.2 that 2cR first

decreases with an increase in ω , then reaches the peak negative value at 20ω =

and increases with further increase in ω . This means that the system is destabilized

for small values of ω and stabilized for large values of ω . As ω is sufficiently

large, 2cR tends to zero so that the effect of modulation disappears altogether.

It is worth remarking that, for symmetric excitation, the temperature profile

consists of the steady straight line section plus a parabolic profile which oscillates

in time. As the amplitude of the modulation increases, the parabolic part of the

profile becomes more and more significant. It is known that a parabolic profile

results in finite amplitude motions. The influence of a variety of nonlinear basic

temperature profiles on unmodulated ferroconvection is discussed at length in the

work of Maruthamanikandan (2005). The variation of 2cR with ω for different

values of the magnetization parameter 3M and for fixed values of Pr , Ta and 1M

is depicted in Fig. 5.3. The parameter 3M measures the departure of linearity in the

magnetic equation of state. It is observed that an increase in 3M is to increase the

value of 2cR , indicating that 3M has a stabilizing effect on the system.

Figure 5.4 shows the variation of 2cR with ω for different values of the Taylor

number Ta and for fixed values of Pr , 1M and 3M . The Taylor number

characterizes the importance of centrifugal forces due to rotation of a fluid about a

vertical axis relative to viscous forces. The stabilizing effect of rotation is obvious

from Fig. 5.4. We further note that, as with the magnetic mechanism, the effect of

symmetric modulation is to reduce the stabilizing influence of rotation for fairly

86

large values of ω . The variation of 2cR with ω for different values of the Prandtl

number Pr and for fixed values of Ta , 1M and 3M is displayed in Fig. 5.5. It

should be remarked that the expression for 0R does not involve the Prandtl number

Pr and the Prandtl number Pr affects only 2R . This is unsurprising as the

boundary temperatures are time-dependent. We observe from Fig. 5.5 that the

effect of increasing Pr is to destabilize the system. It is also observed that

supercritical motion is more likely for low Prandtl number ferro fluids.

Figures 5.6 through 5.9 show the variation of 2cR with ω for the case in which

the excitation is asymmetric. We see that 2cR is positive rather than negative,

implying that subcritical motions are ruled out. This is due to the fact that, when the

excitation is asymmetric, the temperature field has essentially a linear gradient

varying in time. We observe that both 1M and 3M have opposing influences with

reference to the symmetric and asymmetric excitations, whereas Ta and Pr have

identical influences on the stability of the system. The variation of 2cR with ω for

the case in which only the bottom wall temperature is modulated is exhibited in

Figs. 5.10 through 5.13. We observe from these figures that the effect of 1M , 3M

and Pr on the stability of the system is similar to that corresponding to the case of

asymmetric excitation. However, the effect of rotation is to stabilize the system at

low frequencies and the trend reverses for moderately large values of ω . Further,

we find that, as with the symmetric modulation, the effect of modulation is to

reduce the influences of the magnetic mechanism and rotation for fairly large

values ω .

5.2 Concluding Remarks

The combined effect of thermal modulation and rotation on the onset of

stationary convection in a rotating ferromagnetic fluid layer is investigated and the

following conclusions are drawn:

87

1. The effect of magnetic mechanism is to stabilize the system when the bounding

wall temperatures are modulated in phase and to destabilize the system in the

case of asymmetric excitation and bottom wall temperature modulation.

2. When the temperature modulation is symmetric, supercritical motion is possible

for low Prandtl number ferromagnetic fluids.

3. The effect of rotation is to stabilize the system when the excitation is both

symmetric and asymmetric. However, for the case of bottom wall temperature

modulation, the effect of rotation is to stabilize the system for low frequencies

and to destabilize the system for moderately large values of the frequency of

modulation.

4. The presence of symmetric modulation leads to subcritical motion. This

particular result contrasts sharply with that corresponding to the case in which

the excitation is asymmetric and only the bottom wall temperature is modulated.

5. The effect of Prandtl number is to destabilize the system in all the three cases. 6. The effect of modulation is to reduce the influences of the magnetic mechanism

and rotation for large values of the frequency irrespective of the type of

modulation.

z = 0

Ferromagnetic Fluid

[ ]1 1 cos2RT T tε ω+ ∆ +

( )1 1 cos2RT T tε ω φ+ ∆ − +

z 0H

Ω

y

z = h

x

g

Figure 5.1: Configuration of the problem.

0

50

100

150

200

250

-12 -10 -8 -6 -4 -2 0 2

Figure 5.2: Variation of 2cR with ω for different values of 1M .

Symmetric temperature modulation

Pr = 10, Ta = 100, M3 = 1

R2c

ω

M1 = 1 M1 = 10 M1 = 100

0

50

100

150

200

250

-5 -4 -3 -2 -1 0 1

Figure 5.3: Variation of 2cR with ω for different values of 3M .

Symmetric temperature modulation

ω

R2c

Pr = 10, Ta = 100, M1 = 10

M3 = 1 M3 = 5, 10

0

50

100

150

200

250

-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2

Figure 5.4: Variation of 2cR with ω for different values of Ta .

Symmetric temperature modulation

R2c

ω

Pr = 10, M1 = 10, M3 = 1

Ta = 100 500 1000

0

50

100

150

200

250

-5 -4 -3 -2 -1 0 1

Figure 5.5: Variation of 2cR with ω for different values of Pr .

R2c

ω

Symmetric temperature modulation

Ta = 100, M1 = 10, M3 = 1

Pr = 5, 10, 15

0

20

40

60

80

100

0 20 40 60 80

Figure 5.6: Variation of 2cR with ω for different values of 1M .

R2c

ω

Asymmetric temperature modulation

Pr = 10, Ta = 100, M3 = 1

M1 = 1

M1 = 10 M1 = 100

0

20

40

60

80

100

0 10 20 30 40 50

Figure 5.7: Variation of 2cR with ω for different values of 3M .

R2c

ω

Asymmetric temperature modulation

Pr = 10, Ta = 100, M1 = 10

M3 = 1

M3 = 5 M3 = 10

0

20

40

60

80

100

0 10 20 30 40 50 60

Figure 5.8: Variation of 2cR with ω for different values of Ta .

R2c

ω

Asymmetric temperature modulation

Pr = 10, M1 = 10, M3 = 1

Ta = 100, 500, 1000

0

20

40

60

80

100

0 20 40 60 80

Figure 5.9: Variation of 2cR with ω for different values of Pr .

R2c

ω

Asymmetric temperature modulation

Ta = 100, M1 = 10, M3 = 1

Pr = 5

Pr = 10 Pr = 15

0

20

40

60

80

100

0 20 40 60 80

Figure 5.10: Variation of 2cR with ω for different values of 1M .

R2c

ω

Lower wall temperature modulation

Pr = 10, Ta = 100, M3 = 1

M1 = 1

M1 = 10 M1 = 100

0

20

40

60

80

100

0 10 20 30 40 50

Figure 5.11: Variation of 2cR with ω for different values of 3M .

R2c

ω

Lower wall temperature modulation

Pr = 10, Ta = 100, M1 = 10

M3 = 1

M3 = 5

M3 = 10

0

20

40

60

80

100

0 20 40 60

Figure 5.12: Variation of 2cR with ω for different values of Ta .

R2c

ω

Lower wall temperature modulation

Pr = 10, M1 = 10, M3 = 1

Ta = 100 Ta = 1000

Ta = 500

0

20

40

60

80

100

0 20 40 60 80

Figure 5.13: Variation of 2cR with ω for different values of Pr .

R2c

ω

Lower wall temperature modulation

Ta = 100, M1 = 10, M3 = 1

Pr = 5

Pr = 10 Pr = 15

100

BIBLIOGRAPHY Abdullah, A.A. and Lindsay, K.A. (1991). Bénard convection in a nonlinear

magnetic fluid under the influence of a non-vertical magnetic field. Continuum Mech. Thermodyn., Vol. 3, pp. 13–26.

Abraham, A. (2002a). Rayleigh-Bénard convection in a micropolar ferromagnetic

fluid. Int. J. Engg. Sci., Vol. 40, pp. 449–460. Abraham, A. (2002b). Convective instability in magnetic fluids and polarized

dielectric liquids. Ph.D. Thesis, Bangalore University (India). Ageev, V.A., Balyberdin, V.V., Veprik, I. Yu., Ievlev, I.I., Legeida, V.I. and

Fedonenko, A.I. (1990). Magnetic fluid convection in a non-uniform magnetic field. MHD, Vol. 26, pp. 184–188.

Aggarwal A.K. and Suman Makhija. (2009). Thermal stability of Couple-Stress

Fluid in presence of Magnetic Field and Rotation. Indian Journal of Biomechanics: Special Issue, NCBM 7-8 March 2009.

Aniss S., Belhaq M. and Souhar M. (2001). Effects of a Magnetic Modulation on

the Stability of a Magnetic Liquid Layer Heated From Above. J. Heat Transfer, Vol. 123, pp. 428 – 433.

Aniss S., Belhaq M., Souhar M. and Velarde M.G. (2005). Asymptotic study of

Rayleigh-Benard convection under time periodic heating in Hele-Shaw cell. Phys. Scr., Vol. 71, pp. 395 – 401.

Aniss, S., Brancher, J.P. and Souhar, M. (1993). Thermal convection in a

magnetic fluid in an annular Hele-Shaw cell. JMMM, Vol. 122, pp. 319–322. Anton, I., De Sabata, I. and Vekas, L. (1990). Applications oriented researches on

magnetic fluids. JMMM, Vol. 85, pp. 219–226. Auernhammer, G.K. and Brand, H.R. (2000). Thermal convection in a rotating

layer of a magnetic fluid. Eur. Phys. J. B, Vol. 16, pp. 157–168. Bailey, R.L. (1983). Lesser known applications of ferrofluids. JMMM, Vol. 39,

pp. 178–182.

101

Bajaj, R. (2005). Thermodiffusive magnetoconvection in ferrofluids with two-frequency gravity modulation. JMMM, Vol. 288, pp. 483 – 494.

Bajaj, R. and Malik, S.K. (1996). Pattern selection in magnetic fluids with surface

adsorption. Int. J. Engg. Sci., Vol. 34, pp. 1077–1092. Bajaj, R. and Malik, S.K. (1997). Convective instability and pattern formation in

magnetic fluids. J. Math. Anal. Applns., Vol. 207, pp. 172–191. Banerjee K. and Bhattacharjee J. K. (1984). Onset of convection in 3He-4He

mixtures near superfluid transition temperature: Effect of temperature modulation. Proc. R. Soc. Lond. A, Vol. 395, pp. 353-358.

Barletta A. and Nield D.A. (2009). Effect of pressure work and viscous dissipation

in the analysis of the Rayleigh-Benard problem. Int. J. Heat and Mass Transfer, Vol. 52, pp. 3279 – 3289.

Bashtovoi, V.G. and Pavlinov, M.I. (1978). Convective instability of a vertical

layer of a magnetizable fluid in a uniform magnetic field. Magnitnaya Gidrodinamika, Vol. 14, pp. 27–29.

Belyaev A.V. and Smorodin B.L. (2010). The stability of ferrofluid flow in a

vertical layer subject to lateral heating and horizontal magnetic field. Journal of Magnetism and Magnetic Materials, Vol. 322, pp. 2596-2606.

Bénard, H. (1901). Les tourbillions cellularies dans une nappe liquide transportant

de la chaleut par convection en regime permanent. Ann. d. Chimie et de Physique, Vol. 23, pp. 32–40.

Berkovskii, B.M. and Bashtovoi, V.G. (1971). Gravitational convection in a

ferromagnetic liquid. Magnitnaya Gidrodinamika, Vol. 7, pp. 24–28. Berkovskii, B.M., Medvedev, V.F. and Krakov, M.S. (1993). Magnetic fluids -

Engineering applications. Oxford University Press, Oxford. Berkovsky, B.M., Fertman, V.E., Polevikov, V.E. and Isaev, S.V. (1976). Heat

transfer across vertical ferrofluid layers. Int. J. Heat Mass Trans., Vol. 19, pp. 981–986.

Bhadauria B. S. (2002a). Thermal modulation of Rayleigh-Benard convection.

Z. Naturforsch, Vol. 57, pp. 780 – 786.

102

Bhadauria B. S. (2002b). Effect of modulation on Rayleigh-Benard convection - II. Z. Naturforsch, Vol. 58, pp. 176 – 182.

Bhadauria B. S. and Debnath L. (2004). Effects of modulation on Rayleigh-

Benard convection. Int. J. Mathematics and Mathematical Sciences, Vol. 19, pp. 991 - 1001.

Bhadauria B. S. (2006). Time periodic heating of Rayleigh-Benard convection in a

vertical magnetic field. Phys. Scr. Vol. 73, pp. 296 - 302. Bhadauria B. S. (2007a). Magnetofluidconvection in a rotating porous layer under

modulated temperature on the boundaries. J. Heat Transfer, Vol. 129, pp. 835 – 843.

Bhadauria B.S. (2007b). Double-diffusive convection in a porous medium with

modulated temperature on the boundaries. Trans. Porous Media, Vol. 70, pp. 191 – 211.

Bhadauria B.S. (2007c). Double diffusive convection in a rotating porous layer

with temperature modulation on the boundaries. Jl. Porous Media, Vol. 10, pp. 569-584.

Bhadauria B. S. (2007d). Fluid convection in a rotating porous layer under

modulated temperature on the boundaries. Transport in Porous Media, Vol. 67, pp. 297-315.

Bhadauria B. S. (2008a). Combined effect of temperature modulation and

magnetic field on the onset of convection in an electrically conducting-fluid-saturated porous medium. J. Heat Transfer, Vol. 130, p. 052601.

Bhadauria B. S. (2008b). Effect of temperature modulation on the onset of Darcy

convection in a rotating porous medium. Jl. Porous media, Vol. 11, pp. 361 - 375.

Bhadauria B. S. and Srivastava A. K. (2010). Magneto-double diffusive

convection in an electrically conducting-fluid-saturated porous medium with temperature modulation of the boundaries. Int. Jl. Heat Mass Trans.,Vol. 53, pp. 2530-2538.

Bhadauria B. S., Bhatia P. K. and Debnath L. (2009). Weakly non-linear

analysis of Rayleigh–Benard convection with time periodic heating. Int. Jl. Non-Linear Mechanics, Vol. 44, pp. 58-65.

103

Bhadauria B.S. and Sherani A. (2008). Onset of Darcy convection in a magnetic fluid saturated porous medium subject to temperature modulation of the boundaries. Trans. Porous Media, Vol. 73, pp. 349 – 368.

Bhattacharjee J. K. (1989). Rotating Rayleigh-Benard convection with

modulation. J. Phys. A: Math. Gen., Vol. 22, pp. L1135 - L1139. Bhattacharjee J. K. (1990). Convective instability in a rotating fluid layer under

modulation of the rotating rate. Phys. Rev. A., Vol. 41, pp. 5491 – 5494. Blennerhassett, P.J., Lin, F. and Stiles, P.J. (1991). Heat transfer through

strongly magnetized ferrofluids. Proc. R. Soc. Lond. A, Vol. 433, pp. 165–177. Block, M.J. (1956). Surface tension as the cause of Bénard cells and surface

deformations in a liquid film. Nature, Vol. 178, pp. 650–651. Blums, E. (1987). Free convection in an isothermic magnetic fluid caused by

magnetophoretic transport of particles in the presence of a non-uniform magnetic field. JMMM, Vol. 65, pp. 343–346.

Bodenschatz, E., Pesch, W. and Ahlers, G. (2000). Recent developments in

Rayleigh-Bénard convection. Ann. Rev. Fluid Mech., Vol. 32, pp. 709–778. Booker J.R. (1976). Thermal convection with strongly temperature-dependent

viscosity. J. Fluid Mech., Vol. 76, pp. 741–754. Brenner H. (1970). Rheology of two-phase systems. Ann. Rev. Fluid Mech.,

Vol. 2, pp. 137 - 176. Busse F.H. (1975). Nonlinear interaction of magnetic field and convection.

J. Fluid Mech., Vol. 71, pp. 193–206. Busse, F.H. and Frick, H. (1985). Square-pattern convection in fluids with

strongly temperature-dependent viscosity. J. Fluid Mech., Vol. 150, pp. 451–465. Candel P. S. (2006). Effects of a temperature modulation in phase at the frontier on

the convective instability of a viscoelastic layer. Comptes Rendus Mecanique, Vol. 334, pp. 205 – 211.

Carey V.P. and Mollendorf J.C. (1980). Variable viscosity effects in several

natural convection flows. Int. J. Heat Mass Trans., Vol. 23, pp. 95–108.

104

Chakrabarti A. and Gupta A.S. (1981). Nonlinear thermohaline convection in a rotating porous media. Mech. Research Comm., Vol. 8, pp. 9 - 22.

Chakraborty, S. and Borkakati, A.K. (2002). Effect of variable viscosity on

laminar convection flow of an electrically conducting fluid in uniform magnetic field. Theoret. Appl. Mech., Vol. 27, pp. 49–61.

Chandra, K. (1938). Instability of fluids heated from below. Proc. R. Soc. Lond. A,

Vol. 164, pp. 231–242. Chandrasekhar, S. (1961). Hydrodynamic and hydromagnetic stability. Oxford

University Press, Oxford. Char, M.I., Chiang, K.T. and Jou, J.J. (1997). Oscillatory instability analysis of

Bénard-Marangoni convection in a rotating fluid with internal heat generation. Int. J. Heat Mass Trans., Vol. 40, pp. 857–867.

Chauhan D.S. and Rastogi P. (2010). Radiation Effects on Natural Convection

MHD Flow in a Rotating Vertical Porous Channel Partially Filled with a Porous Medium. Applied Mathematical Sciences, Vol. 4, pp. 643 – 655.

Colinet, P., Legros, J.C. and Velarde, M.G. (2001). Nonlinear dynamics of

surface-tension-driven instabilities. Wiley-VCH, Berlin. Cowley, M.D. and Rosensweig, R.E. (1967). The interfacial stability of a

ferromagnetic fluid. J. Fluid Mech., Vol. 30, pp. 671–688. Cox S.M. (1998a). Rotating Convection in a Shear Flow. Proceedings:

Mathematical, Physical and Engineering Sciences, Vol. 454, pp. 1699-1717. Cox S.M. (1998b). Long-Wavelength Rotating Convection between Poorly

Conducting Boundaries. SIAM Journal on Applied Mathematics, Vol. 58, pp. 1338-1364.

Daniels P.G. (1978). Finite Amplitude Two-Dimensional Convection in a Finite

Rotating System. Proceedings of the Royal Society of London (A): Mathematical and Physical Sciences, Vol. 363, pp. 195-215.

Das A. and Kumar K. (2008). Model for modulated and chaotic waves in zero-

Prandtl-number rotating convection. Pramana – Journal of Physics, Vol. 71, pp. 545 – 557.

105

Davis S.H. (1967). Convection in a box: Linear theory. J. Fluid Mech., Vol. 30, pp. 465 - 478.

Debler W.R. and Wolf L.W. (1970). The effects of gravity and surface tension

gradients on cellular convection in fluid layers with parabolic temperature profiles. Trans. ASME : J. Heat Trans., Vol. 92, pp. 351–358.

Donnelly R. J. (1964). Experiments on the stability of viscous flow between

rotating cylinders. III Enchancement of stability by modulation. Proc. Roy. Soc.Lond. A, Vol. 281, p. 130.

Drazin P. and Reid W. (1981). Hydrodynamic instability. Cambridge University

Press, Cambridge. Einarsson T. (1942). The nature of springs in Iceland. (Ger) Rit. Visindafelug Isl.,

Vol. 26, p. 1. Elder J.W. (1958). An experimental investigation of turbulent spots and breakdown

to turbulence. Ph.D. thesis. Cambridge University. Elder J.W. (1966). Numerical experiments with free convection in a vertical slot.

J. Fluid Mech., Vol. 24, p. 823 - 843. Eltayeb I.A. (1972). Hydromagnetic Convection in a Rapidly Rotating Fluid Layer.

Proceedings of the Royal Society of London (A): Mathematical and Physical Sciences, Vol. 326, pp. 229-254.

Falsaperla P., Mulone G. and Straughan B. (2010). Rotating porous convection

with prescribed heat flux. Int. Jl. Engg. Science, pp. 685-692. Fertman V. (1990). Magnetic fluids guide book: Properties and applications.

Hemisphere Publishing Corporation, New York. Finlayson B.A. (1972). The method of weighted residuals and variational

principles. Academic Press, New York. Finlayson, B.A. (1970). Convective instability of ferromagnetic fluids. J. Fluid

Mech., Vol. 40, pp. 753–767. Frank F.C. (1965). On dilatancy in relation to seismic sources. Rev. Geophys.,

Vol. 3, pp. 484 - 503.

106

Galdi G.P. and Straughan B. (1985). A Nonlinear Analysis of the Stabilizing Effect of Rotation in the Benard Problem. Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, Vol. 402, pp. 257-283.

Gatica J.E., Viljoen H.J. and Hlavacek V. (1987). Thermal instability of

nonlinear stratified fluids. Int. Comm. Heat Mass Transfer, Vol. 14, pp. 673 – 686.

Gebhart, B., Jaluria, Y., Mahajan, R.L. and Sammakia, B. (1988). Buoyancy

induced flows and transport. Hemisphere Publishing Corporation, New York. Gershuni, G.Z. and Zhukhovitsky, E.M. (1976). Convective stability of

incompressible fluids. Keter, Jerusalem. Getling A.V. and Brausch O. (2003). Cellular flow patterns and their evolutionary

scenarios in three dimensional Rayleigh-Benard convection. Phys. Rev. E., Vol. 67, p. 46313 - 46317.

Getling, A.V. (1998). Rayleigh-Bénard convection - Structures and dynamics.

World Scientific, Singapore. Gill A.E. (1966). The boundary layer regime for convection in a rectangular cavity.

J. Fluid Mech., Vol. 26, p. 515 - 536. Gotoh, K. and Yamada, M. (1982). Thermal convection in a horizontal layer of

magnetic fluids. J. Phys. Soc. Japan, Vol. 51, pp. 3042–3048. Gouesbet, G., Maquet, J., Roze, C. and Darrigo, R. (1990). Surface-tension and

coupled buoyancy-driven instability in a horizontal liquid layer: overstability and exchange of stability. Phys. Fluids A, Vol. 2, pp. 903–911.

Govender S. (2004). Stability of convection in a gravity modulated porous layer

heated from below. Transport in Porous Media, Vol. 57, pp. 113-123. Govender S. (2003a). Oscillatory convection induced by gravity and centrifugal

forces in a rotating porous layer distant from the axis of rotation. International Journal of Engineering Science, Vol. 41, pp. 539 – 545.

Govender S. (2003b). Coriolis Effect on the Linear Stability of Convection in a

Porous Layer Placed Far Away from the Axis of Rotation. Transport in Porous Media, Vol. 51, pp. 315 – 326.

107

Govender S. and Vadas P. (2007). The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium. Transport in Porous Media, Vol. 69, pp. 55-66.

Guo W. and Narayanan R. (2007). Onset of Rayleigh-Marangoni convection in a

cylindrical annulus heated from below. J. Coll. Interface Science, Vol. 314, pp. 727 - 732.

Gupta, M.D. and Gupta, A.S. (1979). Convective instability of a layer of a

ferromagnetic fluid rotating about a vertical axis. Int. J. Engg. Sci., Vol. 17, pp. 271–277.

Hashim I. and Siri Z. (2009). Feedback control of thermocapillary convection in a

rotating fluid layer with free-slip bottom. Sains Malaysiana, Vol. 38, pp. 119 – 124.

Hashim, I. and Arifin, N.M. (2003). Oscillatory Marangoni convection in a

conducting fluid layer with a deformable free surface in the presence of a vertical magnetic field. Acta Mech., Vol. 164, pp. 199–215.

Hashim, I. and Wilson, S.K. (1999a). The effect of a uniform vertical magnetic

field on the linear growth rates of steady Marangoni convection in a horizontal layer of conducting fluid. Int. J. Heat Mass Trans., Vol. 42, pp. 525–533.

Hashim, I. and Wilson, S.K. (1999b). The effect of a uniform vertical magnetic

field on the onset of oscillatory Marangoni convection in a horizontal layer of conducting fluid. Acta Mech., Vol. 132, pp. 129–146.

Herron I.H. (2000). On the principle of exchange of stabilities in Rayleigh-Benard

convection. SIAM Jl. Appl. Math., Vol. 61, pp. 1362 – 1368. Hirayama O. and Takaki R. (1993). Thermal convection of a fluid with

temperature-dependent viscosity. Fluid Dyn. Res., Vol. 12, pp. 35–47. Horng, H.E., Hong, C.Y., Yang, S.Y. and Yang, H.C. (2001). Novel properties

and applications in magnetic fluids. J. Phys. Chem. Solids, Vol. 62, pp. 1749–1764.

Hossain, M.A., Kabir, S. and Rees, D.A.S. (2002). Natural convection of fluid

with variable viscosity from a heated vertical wavy surface. ZAMP, Vol. 53, pp. 48–52.

108

Howell, J.R. and Menguc, M.P. (1998). Handbook of heat transfer fundamentals. Tata McGraw Hill, New York.

Hughes, W.F. and Young, F.J. (1966). Electromagnetodynamics of fluids. John

Wiley & Sons, New York. Jeffreys, H. (1926). The stability of a layer of fluid heated below. Phil. Mag.,

Vol. 2, pp. 833–844. Joseph D.D. (1976). Stability of fluid motions I and II. Springer Verlag. Julien K., Legg S., Mcwilliams J. and Werne J. (1996). Rapidly rotating

turbulent Rayleigh-Bénard convection. Journal of Fluid Mechanics, Vol. 322, pp. 243-273.

Kaloni P. N. and Lou J. X. (2004). Convective instability of magnetic fluids.

Physical Review E, Vol. 70, p. 026313. Kaloni, P.N. (1992). Some remarks on the boundary conditions for magnetic fluids.

Int. J. Engg. Sci., Vol. 30, pp. 1451–1457. Kaloni, P.N. and Lou, J.X. (2004). Weakly nonlinear instability of a ferromagnetic

fluid rotating about a vertical axis. JMMM, Vol. 284, pp. 54 – 68. Kamiyama, S. and Satoh, A. (1989). Rheological properties of magnetic fluids

with the formation of clusters: Analysis of simple shear flow in a strong magnetic field. J. Colloid Interface Sci., Vol. 127, pp. 173–188.

Kamiyama, S., Koike, K. and Wang, Z.S. (1987). Rheological characteristics of

magnetic fluids. JSME Int. J., Vol. 30, pp. 761–766. Kamiyama, S., Sekhar, G.N. and Rudraiah, N. (1988). Mixed convection of

magnetic fluid in a vertical slot. Rep. Inst. High Speed Mech. (Tohoku University), Vol. 56, pp. 1–12.

Kays, W.M. and Crawford, M.E. (1980). Convective heat and mass transfer.

McGraw-Hill, New York. Kelly R. E. and Or A.C. (2002). The effects of thermal modulation upon the onset

of Marangoni-Benard convection. J. Fluid Mech., Vol. 456, pp. 161 - 182. Klarsfeld S. (1970) Champs de temperature associe’s aux mouments de convection

naturelle dams un milieu porehx limite. Rev. Gen. Thermique, Vol. 108, p. 1403.

109

Krishnamurti, R. (1968a). Finite amplitude convection with changing mean

temperature. Part I. Theory. J. Fluid Mech., Vol. 33, pp. 445–456. Krishnamurti, R. (1968b). Finite amplitude convection with changing mean

temperature. Part II. Experimental test of the theory. J. Fluid Mech., Vol. 33, pp. 457–475.

Kumar, D. and Chandra, P. (1988). Flow of a magnetic fluid in a channel.

JMMM, Vol. 73, pp. 361–366. Lalas, D.P. and Carmi, S. (1971). Thermoconvective stability of ferrofluids.

Phys. Fluids, Vol. 14, pp. 436–438. Liu I. C. (2004) The effect of modulation on onset of thermal convection of a

second-grade fluid layer. Int. J Non-Linear Mech., Vol. 39, pp. 1647 - 1657. Lord Rayleigh, O.M. (1916). On convection currents in a horizontal layer of fluid,

when the higher temperature is on the under side. Phil. Mag., Vol. 32, pp. 529–546.

Lorenz E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci., Vol. 20,

pp. 130–141. Low, A.R. (1929). On the criterion for stability of a layer of viscous fluid heated

from below. Proc. R. Soc. Lond. A, Vol. 125, pp. 180–192. Luo, W., Du, T. and Huang, J. (1999). Novel convective instabilities in a

magnetic fluid. Phys. Rev. Lett., Vol. 82, pp. 4134–4137. Ma T. and Wang S. (2007). Rayleigh-Benard convection: dynamics and structure

in the physical system. Comm. Math. Sci., Vol. 5, pp. 553 – 574. Ma. T and Wang S. (2004). Dynamic bifurcation and stability in the Rayleigh-

Benard convection. Comm. Math. Sci., Vol. 2, pp. 159 – 183. Maekawa, T. and Tanasawa, I. (1988). Effect of magnetic field on the onset of

Marangoni convection. Int. J. Heat Mass Trans., Vol. 31, pp. 285–293. Magnan J.F. and Reiss E.L. (1988). Rotating Thermal Convection: Neosteady and

Neoperiodic Solutions. SIAM Journal on Applied Mathematics, Vol. 48, pp. 808-827.

110

Mahabaleswar U. S. (2007). Combined effect of temperature and gravity modulations on the onset of magneto-convection in weak electrically conducting micropolar liquids. Int. J. Eng. Sci, Vol. 45, pp. 525 – 540.

Mahmud M.N., Idris R. and Hashim I. (2009). Effect of a magnetic field on the

onset of Marangoni convection in a micropolar fluid. Proc. of World Academy of Science, Engineering and Technology, Vol. 38, pp. 866 – 868.

Mahr, T. and Rehberg, I. (1998). Magnetic Faraday-instability. Europhys. Lett.,

Vol. 43, pp. 23–28. Malashetty M. S. and Basavaraja D. (2002). Rayleigh-Benard convection subject

to time dependent wall temperature/gravity in fluid-saturated anisotropic porous medium. Int. J. Heat Mass Transfer, Vol. 38.

Malashetty M. S. and Basavaraja D. (2004). Effect of time periodic boundary

temperatures on the onset of double diffusive convection in a horizontal anisotropic porous layer. Int. J. Heat Mass Transfer, Vol. 47, pp. 2317 - 2327.

Malashetty M. S. and Swamy M. (2008) Effect of thermal modulation on the

onset of convection in a rotating fluid layer. Int. J. Heat Mass Trans., Vol. 51, pp. 2814 – 2823.

Malashetty M. S. and Wadi S. V. (1999). Rayleigh-Benard convection subject to

time dependent wall temperature in a fluid-saturated porous layer. Fluid Dynamics Research, Vol. 24, pp. 293-308.

Malashetty M.S. and Basavaraja D. (2002). Rayleigh-Benard convection subject

to time dependent wall temperature/gravity in a fluid-saturated anisotropic porous medium. Heat and Mass Trans., Vol. 38, pp. 551 – 563.

Malashetty M.S. and Heera R. (2008). Linear and non-linear double diffusive

convection in a rotating porous layer using a thermal non-equilibrium model. International Journal of Non-Linear Mechanics, Vol. 43, pp. 600-621.

Malashetty M.S. and Swamy M. (2007). Effect of temperature modulation on the

onset of stationary convection in a rotating sparsely packed porous layer. Canadian Journal of Physics, Vol. 85, pp. 927-945.

Malashetty M.S. and Swamy M. (2007). Combined effect of thermal modulation

and rotation on the onset of stationary convection in a porous layer. Transport in Porous Media, Vol. 69, pp. 313 - 330.

111

Malashetty M.S. and Swamy M. (2007a). Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer. Transport in Porous Media, Vol. 69, pp. 313 - 330.

Malkus, W.V.R. and Veronis, G. (1958). Finite amplitude cellular convection.

J. Fluid Mech., 4, pp. 225–260. Martinez-Mardones J., Laroze D. and Bragard J. (2006). Stationary convection

in a rotating binary magnetic fluid. J. of Phys. (conference series), Vol. 134. Maruthamanikandan S. (2003). Effect of radiation on Rayleigh-Bénard

convection in ferromagnetic fluids. Int. Jl. Appl. Mech. Engg., Vol. 8, pp. 449 – 459.

Maruthamanikandan S. (2005) Convective instabilities in Newtonian

ferromagnetic, dielectric and other complex liquids. Ph.D. Thesis. Massaioli F., Benzi R. and Succi S. (1993). Exponential tails in two-dimensional

Rayleigh-Benard convection. Europhys. Lett., Vol. 21, pp. 305 – 310. Morimoto, H., Kobayashi, T. and Maekawa, T. (1998). Microgravity experiment

and linear and nonlinear analysis of the dissipative structure of thermomagnetic convection. Proc. 6th Int. Conf. on Electrorheological Fluids, Magnetorheological Suspensions and their Applications, pp. 511–515.

Mukutmoni D. and Yang K.T. (1994). Flow transitions and pattern selection of

the Rayleigh-Benard problem in rectabgular enclosures. Sadhana, Vol. 19, pp. 649 – 670.

Muller, H.W. (1998). Parametrically driven surface waves on viscous ferrofluids.

Phys. Rev. E, Vol. 58, pp. 6199–6205. Murty Y.N. (2006). Effect of throughflow and magnetic field on Marangoni

convection in micropolar fluids. Appl. Math. Comp., Vol. 173. Nakatsuka, K., Jeyadevan, B., Neveu, S. and Koganezawa, H. (2002). The

magnetic fluid for heat transfer applications. JMMM, Vol. 252, pp. 360–362. Natalia S. (2008). Effect of vertical modulation on the onset of filtration

convection. J. Mathematical Fluid Mech., Vol. 10, p. 567. Neuringer, J.L. and Rosensweig, R.E. (1964). Ferrohydrodynamics, Phys. Fluids,

Vol. 7, pp. 1927–1937.

112

Nield D.A. (1964). Surface tension and buoyancy effects in cellular convection.

J. Fluid Mech., Vol. 19, pp. 341–352. Nield D.A. (1966). Surface tension and buoyancy effects in the cellular convection

of an electrically conducting liquid in a magnetic field. ZAMP, Vol. 17, pp. 131–139.

Nogotov, E.F. and Polevikov, V.K. (1977). Convection in a vertical layer of a

magnetic liquid in the magnetic field of a current carrying sheet. Magnitnaya Gidrodinamika, Vol. 13, pp. 28–34.

Normand, C., Pomeau, Y. and Velarde, M.G. (1977). Convective instability: A

Physicists approach. Rev. Mod. Phys., Vol. 49, pp. 581–624. Odenbach, S. (1993). Drop tower experiments on thermomagnetic convection.

Microgravity Sci. Tech., Vol. 6, pp. 161– 63. Odenbach, S. (1995a). Convection driven by forced diffusion in magnetic fluids

under the influence of strong magnetic field gradients. JMMM, Vol. 149, pp. 116–118.

Odenbach, S. (1995b). Microgravity experiments on thermomagnetic convection in

magnetic fluids. JMMM, Vol. 149, pp. 155–157. Odenbach, S. (1996). Convection in ferrofluids caused by forced diffusion in a

magnetic field gradient. MHD, Vol. 32, pp. 429–434. Odenbach, S. (1999). Microgravity research as a tool for the investigation of

effects in magnetic fluids. JMMM, Vol. 201, pp. 149–154. Odenbach, S., Rylewicz, T. and Heyen, M. (1999). A rheometer dedicated for the

investigation of viscoelastic effects in commercial magnetic fluids. JMMM, Vol. 201, pp. 155–158.

Om, Bhadauria B.S. and Khan A. (2009). Modulated Centrifugal Convection in a

Vertical Rotating Porous Layer Distant from the Axis of Rotation. Transport in Porous Media, Vol. 79, pp. 255-264.

Om, Bhadauria B.S. and Khan A. (2009). Modulated Centrifugal Convection in a

Vertical Rotating Porous Layer Distant from the Axis of Rotation. Transport in Porous Media, Vol. 79, pp. 255-264.

113

Ostrach S. (1972). Natural convection in enclosures. Advances in Heat Transfer, Vol. 8, Academic Press, New York.

Palm E. (1960). On the tendency towards hexagonal cells in steady convection.

J. Fluid Mech., Vol. 8, pp. 183–192. Pardeep Kumar and Mahinder Singh. (2009). Rotatory Thermosolutal

Convection in a Couple-Stress Fluid, Z. Naturforsch. Vol. 64, pp. 448 – 454. Pardeep Kumar, Mohan H. and Lal R. (2006). Effect of magnetic field on

thermal instability of a rotating Rivlin-Ericksen viscoelastic fluid. International Journal of Mathematics and Mathematical Sciences, Vol. 2006, pp. 1–10.

Parekh, K., Upadhyay, R.V. and Mehta, R.V. (2000). Magnetocaloric effect in

temperature-sensitive magnetic fluids. Bull. Matl. Sci., Vol. 23, pp. 91–95. Parmentier, P.M., Regnier, V.C., Lebon, G. and Legros, J.C. (1996). Nonlinear

analysis of coupled gravitational and capillary thermoconvection in thin fluid layers. Phys. Rev. E, Vol. 54, pp. 411–423.

Paul S., Kumar K., Verma M.K., Carati D., De A.K. and Eswaran V. (2010).

Chaotic travelling rolls in Rayleigh-Bénard convection. Pramana, Vol. 74, pp. 75-82.

Pearson, J.R.A. (1958). On convection cells induced by surface tension. J. Fluid

Mech., Vol. 4, pp. 489–500. Pellew, A. and Southwell, R.V. (1940). On maintained convective motion in a

fluid heated from below. Proc. R. Soc. Lond. A, Vol. 176, pp. 312–343. Petrelis, F., Falcon, E. and Fauve, S. (2000). Parametric stabilization of the

Rosensweig instability. Eur. Phys. J. B, 15, pp. 3–6. Plapp B.B., Egolf D.A. and Bodenschatz E. (1998). Dynamics and Selection of

Giant Spirals in Rayleigh-Bénard Convection. Phys. Rev. Lett., Vol. 81, pp. 5334–5337.

Platten, J.K. and Legros, J.C. (1984). Convection in liquids. Springer, Berlin. Popplewell, J. (1984). Technological applications of ferrofluids. Phys. Tech.,

Vol. 15, pp. 150–162.

114

Qin, Y. and Kaloni, P.N. (1994). Nonlinear stability problem of a ferromagnetic fluid with surface tension effect. Eur. J. Mech. B/Fluids, Vol. 13, pp. 305–321.

Racca, R.A. and Annett, C.H. (1985). Simple demonstration of Rayleigh-Taylor

instability. Am. J. Phys., Vol. 53, pp. 484–486. Rajagopal K.R., Ruzica M. and Srinivasa A.R. (1996). On the Oberbeck-

Boussinesq approximation. Math. Model. Methods Appl. Sci., Vol. 6, p. 1157. Raju V. R. K. and Bhattacharyya S. N. (2009). Onset of thermal instability in a

horizontal layer of fluid with modulated boundary temperatures. Journal of Engineering Mathematics, Vol. 66, pp. 343-351.

Ramachandramurthy V. (1987). Marangoni and Rayleigh-Bènard convection.

Ph.D. Thesis, Bangalore University, India. Ramanathan A. and Muchikel N. (2006). Effect of temperature dependent

viscosity on ferroconvection in a porous medium. Int. J. Appl. Mech. and Engg., Vol. 11, pp. 93 – 104.

Ramaswamy B. (1993). Finite element analysis of two dimensional rayleigh-

benard convection with gravity modulation effects. International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 3, pp. 429 – 444.

Resler Jr., E.L. and Rosensweig, R.E. (1964). Magnetocaloric power. AIAA J.,

Vol. 2, pp. 1418–1422. Riahi D.H. (1994). The effect of coriolis force on nonlinear convection in a porous

medium. Int. J. Math. & Math. Sci., Vol. 17, pp. 515-536. Riahi D.N. (1992). Weakly nonlinear oscillatory convection in a rotating fluid.

Proceedings: Mathematical and Physical Sciences, Vol. 436, pp. 33-54. Richter, F.M., Nataf, H.C. and Daley, S.F. (1983). Heat transfer and horizontally

averaged temperature of convection with large viscosity variations. J. Fluid Mech., Vol. 129, pp. 173–192.

Roberts P.H. and Stewartson K. (1974). On finite amplitude convection in a

rotating magnetic system. Philosophical Transactions of the Royal Society of London (A): Mathematical and Physical Sciences, Vol. 277, pp. 287-315.

Rogers J.L and Schatz M.F. (2000). Superlattice patterns in vertically oscillated

Rayleigh-Benard convection. Phys. Rev. Lett., Vol. 85, pp. 4281 – 4284.

115

Rogers J.L., Schatz M.F., Bougie J.L. and Swift J.B. (2000). Rayleigh-Benard

convection in a vertically oscillating fluid layer. Phys. Rev. Lett., Vol. 84, pp. 87 – 90.

Rosenblat S. and Herbert D. M. (1970). Low-frequency modulation of thermal

instability. J. Fluid Mech., Vol. 43, p. 385. Rosenblat S. and Tanaka G. A. (1971). Modulation of thermal convection

instability. Phys. Fluids, Vol. 14, p. 1319. Rosensweig, R.E. (1986). Ferrohydrodynamics. Cambridge University Press.

Cambridge. Rossby H.T. (1969). A study of Bénard convection with and without rotation.

Journal of Fluid Mechanics. Vol. 36, pp. 309-335. Rubio A., Lopez J. M. and Marques F. (2008). Modulated rotating convection:

radially travelling concentric rolls. Journal of Fluid Mechanics, Vol. 608, pp. 357-378.

Rubio A., Lopez J. M. and Marques F. (2008). Modulated rotating convection:

radially travelling concentric rolls. Journal of Fluid Mechanics, Vol. 608, pp. 357-378.

Rudiger S. and Knobloch E. (2003). Mode interaction in rotating Rayleigh-Benard

convection. Fluid Dynamics Research, Vol. 33, pp. 477 – 492. Rudraiah, N. and Sekhar, G.N. (1992). Thermohaline convection in magnetic

fluids. In: Electromagnetic forces and applications (Eds.: Tani, J. and Takagi, T.), Elsevier Sci. Publishers, pp. 119–122.

Rudraiah, N., Shivakumara, I.S. and Nanjundappa, C.E. (1998). Effect of non-

uniform concentration distribution on double diffusive convection in magnetic fluids. Ind. J. Engg. Matl. Sci., Vol. 5, pp. 427–435.

Rudraiah, N., Shivakumara, I.S. and Nanjundappa, C.E. (2002). Effect of basic

temperature gradients on Marangoni convection in ferromagnetic fluids. Ind. J. Pure Appl. Phys., Vol. 40, pp. 95–106.

Russell, C.L., Blennerhassett, P.J. & Stiles, P.J. (1999). Supercritical analysis of

strongly nonlinear vortices in magnetized ferrofluids. Proc. R. Soc. Lond. A, Vol. 455, pp. 23–67.

116

Russell, C.L., Blennerhassett, P.J. and Stiles, P.J. (1995). Large wave number

convection in magnetized ferrofluids. JMMM, Vol. 149, pp. 119–121. Saravanan S. (2009). Centrifugal acceleration induced convection in a magnetic

fluid saturated anisotropic rotating porous medium. Trans. Porous Media, Vol. 77, pp. 79 – 86.

Sarma, G.S.R. (1979). Marangoni convection in a fluid layer under the action of a

transverse magnetic field, Space Res., Vol. 19, pp. 575–578. Sarma, G.S.R. (1981). Marangoni convection in a liquid layer under the

simultaneous action of a transverse magnetic field and rotation. Adv. Space Res., Vol. 1, p. 55.

Sarma, G.S.R. (1985). Effects of interfacial curvature and gravity waves on the

onset of thermocapillary convection in a rotating liquid layer subjected to a transverse magnetic field. Physio-Chem. Hydrodyn., Vol. 6, pp. 283–300.

Schneider K.J. (1963). Investigation of the influence of free thermal convection on

heat transfer through globular materials. 12th Int. Congress of Refrigeration, Munich, Paper II-4.

Scholten, P.C. (1978). Thermodynamics of magnetic fluids. Hemisphere,

Washington. Schwab, I., Hildebrandt, U. and Stierstadt, K. (1983). Magnetic Bénard

convection. JMMM, Vol. 39, pp. 113–114. Schwab, L. (1990). Thermal convection in ferrofluids under a free surface. JMMM,

Vol. 85, pp. 199–202. Schwab, L. and Stierstadt, K. (1987). Field-induced wavevector selection by

magnetic Bénard convection. JMMM, Vol. 65, pp. 315–316. Sekar R., Vaidyanathan G. and Ramanathan A. (2000). Effect of rotation on

ferro-thermohaline convection. JMMM, Vol. 218, pp. 266–272. Sekar R., Vaidyanathan G., Hemalatha R. and Sendhilnathan S. (2006). Effect

of sparse distribution of pores in a Soret-driven ferrothermohaline convection. JMMM, Vol. 302, pp. 20–28.

117

Sekhar G.N. (1990). Convection in magnetic fluids. Ph.D. Thesis, Bangalore University (India).

Sekhar G.N. and Jayalatha G. (2010). Rayleigh-Bénard convection in liquids with

temperature dependent viscosity. Int. Jl. Thermal Sciences, Vol. 49, pp. 67 – 75. Sekhar, G.N. (1991). Convection in magnetic fluids with internal heat generation.

Trans. ASME : J. Heat Trans., Vol. 113, 122–127. Sekhar, G.N. and Rudraiah, N. (1991). Convection in magnetic fluids with

internal heat generation. Trans. ASME : J. Heat Trans., Vol. 113, 122–127. Selak, R. and Lebon, G. (1997). Rayleigh-Marangoni thermoconvective instability

with non-Boussinesq corrections. Int. J. Heat Mass Trans., Vol. 40, pp. 785–798. Severin, J. and Herwig, H. (1999). Onset of convection in the Rayleigh-Bénard

flow with temperature-dependent viscosity: An asymptotic approach. ZAMP, Vol. 50, pp. 375–386.

Sharma R.C. and Mehta C.B. (2005). Thermosolutal convection in compressible,

rotating, couple-stress fluid. Indian J. Phys., Vol. 79, pp. 161-165. Sharma R.C. and Monica Sharma. (2004). Effect of suspended particles on

couple-stress fluid heated from below in the presence of rotation and magnetic field. Indian Jl. Pure. Appl. Math., Vol. 35, pp. 973 – 989.

Sharma V. and Gupta S. (2008). Thermal convection of micropolar fluid in the

presence of suspended particles in rotation. Arch. Mech., Vol. 60, pp. 403–419. Sharma V., Sunil and Gupta U. (2006). Stability of stratified elastico-viscous

Walters’ (Model B′) fluid in the presence of horizontal magnetic field and rotation in porous medium. Arch. Mech., Vol. 58, pp. 187–197.

Shivakumara, I.S., Rudraiah, N. and Nanjundappa, C.E. (2002). Effect of non-

uniform basic temperature gradient on Rayleigh-Bénard-Marangoni convection in ferrofluids. JMMM, Vol. 248, pp. 379–395.

Shliomis, M.I. (1968). Equations of motion of a fluid with hydromagnetic

properties. Sov. Phys. JETP, Vol. 26, pp. 665–669. Shliomis, M.I. (1973). Convective instability of a ferrofluid. Fluid Dynamics - Sov.

Res., Vol. 6, pp. 957–962.

118

Siddheshwar P. G. and Abraham A. (2003). Effect of time-periodic boundary temperatures/body force on Rayleigh-Bénard convection in a ferromagnetic fluid. Acta Mech., Vol. 161, pp. 131–150.

Siddheshwar P. G. and Abraham A. (2009). Rayleigh-Benard Convection in a

Dielectric Liquid: Imposed Time-Periodic Boundary Temperatures. Chamchuri Journal of Mathematics, Vol. 1, pp. 105 – 121.

Siddheshwar P. G. and Pranesh S. (2000). Effect of temperature/gravity

modulation on the onset of magneto-convection in electrically conducting fluids with internal angular momentum. Int. J. Magn. Magn. Mater., Vol. 219, pp. 153 - 162.

Siddheshwar P.G. and Pranesh S. (1999). Effect of temperature/gravity

modulation on the onset of magneto-convection in weak electrically conducting fluids with internal angular momentum. Int. J. Magn. Magn. Mater., Vol. 192, pp. 159 – 176.

Siddheshwar, P.G. (1993). Rayliegh-Bénard convection in a magnetic fluid with

second sound. Proc. Japan Soc. Magnetic Fluids, Sendai (Japan), pp.32–36. Siddheshwar, P.G. (1995). Convective instability of ferromagnetic fluids bounded

by fluid-permeable, magnetic boundaries. JMMM, Vol. 149, pp. 148–150. Siddheshwar, P.G. (2004). Thermorheological effect on magnetoconvection in

weak electrically conducting fluids under 1g or µg. Pramana–J. Phys. (India), Vol. 62, pp. 61–68.

Siddheshwar, P.G. and Abraham, A. (1998). Convection in a ferromagnetic fluid

occupying a vertical enclosure. Ind. J. Engg. Matl. Sci., Vol. 5, pp. 423–426. Siddheshwar, P.G. and Abraham, A. (2003). Effect of time-periodic boundary

temperatures/body force on Rayleigh-Bénard convection in a ferromagnetic fluid. Acta Mech., Vol. 161, pp. 131–150.

Singh J. and Bajaj R. (2008). Temperature modulation in Rayleigh–Bénard

convection. The ANZIAM Journal, Vol. 50, pp. 231-245. Singh J. and Bajaj R. (2009). Temperature modulation in ferrofluid convection.

Phys. Fluids, Vol. 21, p. 064105.

119

Siri Z. and Hashim I. (2008). Control of oscillatory Marangoni convection in a rotating fluid layer. Int. Comm. in Heat and Mass Trans., Vol. 35, pp. 1130 – 1133.

Siri Z. and Hashim I. (2008). Control of oscillatory Marangoni convection in a

rotating fluid layer. Int. Comm. in Heat and Mass Trans., Vol. 35, pp. 1130 – 1133.

Snyder, S.M., Cader, T. and Finlayson, B.A. (2003). Finite element model of

magnetoconvection of a ferrofluid. JMMM, Vol. 262, pp. 269–279. Song H. and Tong P. (2010). Scaling laws in turbulent Rayleigh-Bénard

convection under different geometry, Europhysics Letters, Vol. 90. p. 44001. Sparrow, E.M., Goldstein, R.J. and Jonsson, V.K. (1964). Thermal instability in

a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile. J. Fluid Mech., Vol. 18, pp. 513–528.

Spiegel, E.A. and Veronis, G. (1960). On the Boussinesq approximation for a

compressible fluid. Astrophys. J., Vol. 131, pp. 442–447. Sprague E., Julien K., Serre E., Sanchez-Alvarez J.J. and Crespo del Arco E.

(2005). Pattern formation in Rayleigh-Benard convection in a rapidly rotating cylinder. Proceedings of the Fourth International Symposium on Turbulence and Shear Flow Phenomena.

Sprague E., Julien K., Serre E., Sanchez-Alvarez J.J. and Crespo del Arco E.

(2005). Pattern formation in Rayleigh-Benard convection in a rapidly rotating cylinder. Proceedings of the Fourth International Symposium on Turbulence and Shear Flow Phenomena.

Sri Krishna C.V. (2001). Effect of non-inertial acceleration on the onset of

convection in a second order fluid-saturated porous medium. Int. J. Engg. Sci., Vol. 39, pp. 599 – 609.

Srimani P.K. (1981). Finite amplitude cellular convection in a rotating and non-

rotating fluid saturated porous layer. Ph.D. Thesis, Bangalore University. Srivastava K. M. (1976). On the temperature modulation of thermal convection

instability in hydromagnetics. Astron. and Astrophys., Vol. 46, pp. 361 – 368. Stengel K.C., Oliver D.S. and Booker J.R. (1982). Onset of convection in a

variable viscosity fluid. J. Fluid Mech., Vol. 120, pp. 411–431.

120

Stiles, P.J. and Kagan, M. (1990). Thermoconvective instability of a horizontal

layer of ferrofluid in a strong vertical magnetic field. JMMM, Vol. 85, pp. 196 –198.

Stiles, P.J., Lin, F. and Blennerhassett, P.J. (1992). Heat transfer through weakly

magnetized ferrofluids. J. Colloid Interface Sci., Vol. 151, pp. 95–101. Straughan B. (2002). Sharp global nonlinear stability for temperature-dependent

viscosity convection. Proc. R. Soc. Lond. A, Vol. 458, pp. 1773–1782. Straughan, B. (2004). The energy method, stability and nonlinear convection.

Springer, New York. Sunil and Mahajan A. (2009a). A nonlinear stability analysis for

thermoconvective magnetized ferrofluid saturating a porous medium. Trans. in Porous Media, Vol. 76, pp. 327 - 343.

Sunil and Mahajan A. (2009b). A nonlinear stability analysis for rotating

ferrofluid heated from below. Appl. Math. and Computation, Vol. 204, pp. 299 – 310.

Sunil, Bharti P. K. and Sharma R. C. (2004). Thermosolutal convection in a

ferromagnetic fluid. Arch. Mech., Vol. 56, pp. 117–135. Sunil, Chand P. and Bharti P.K. (2007). Double-diffusive convection in a

micropolar ferromagnetic fluid. Appl. Math. and Computing, Vol. 189, pp. 1648 – 1661.

Sunil, Chand P., Bharti P.K. and Mahajan A. (2008). Thermal convection in a

micropolar ferrofluid in the presence of rotation. JMMM, Vol. 320, pp. 316 – 324.

Sunil, Sharma A. and Shandil R.G. (2008). Effect of magnetic field dependent

viscosity on ferroconvection in the presence of dust particles. J. Appl. Math. and Computing, Vol. 27, pp. 7 - 22.

Sunil, Sharma P. and Mahajan A. (2008). A nonlinear stability analysis for

thermoconvective magnetized ferrofluid with magnetic field dependent viscosity. Int. Comm. Heat and Mass Trans., Vol. 35, pp. 1281 – 1287.

121

Suresh G. and Vasanthakumari B. (2009). Comparison of theoretical and computational ferroconvection induced by magnetic field dependent viscosity in an anisotropic porous medium. Int. Jl. Recent Trends in Engg., Vol. 1, pp. 41 – 45.

Takashima, M. (1970). Nature of the neutral state in convective instability induced

by surface tension and buoyancy. J. Phys. Soc. Japan, Vol. 28, p. 810. Tangthieng, C., Finlayson, B.A., Maulbetsch, J. & Cader, T. (1999). Heat

transfer enhancement in ferrofluids subjected to steady magnetic fields. JMMM, Vol. 201, pp. 252–255.

Taylor G. (1954). Diffusion and mass transport in tubes. Proc. Phys. Soc., Vol. 67,

p. 857. Thess, A. and Nitschke, K. (1995). On Bénard-Marangoni instability in the

presence of a magnetic field. Phys. Fluids, Vol. 7, pp. 1176–1178. Thomson, J. (1882). On a changing tessellated structure in certain liquids. Proc.

Glasgow Phil. Soc., Vol. 13, pp. 469–475. Tong P. and Shen Y. (1992). Relative velocity fluctuations in turbulent Rayleigh-

Benard convection. Phys. Rev. Lett., Vol. 69, pp. 2066 – 2069. Torrance K.E. and Turcotte, D.L. (1971). Thermal convection with large

viscosity variations. J. Fluid Mech., Vol. 47, pp. 113–125. Tritton, D.J. (1979). Physical fluid dynamics. Van Nostrand Reinhold Company,

England. Turner, J.S. (1973). Buoyancy effects in fluids. Cambridge University Press,

Cambridge. Upadhyay, T.J. (2000). Thermomagnetic behaviour of certain magnetic fluids.

Ph.D. Thesis, Bhavnagar University (India). Vanishree R. K. and Siddheshwar P.G. (2010). Effect of rotation on thermal

convection in an anisotropic porous medium with temperature-dependent viscosity. Transp Porous Med., Vol. 81 pp. 73 – 87.

Venezian G. (1969). Effect of modulation on the onset of thermal convection. J.

Fluid Mech., Vol. 35, p. 243.

122

Venkatasubramanian, S. and Kaloni, P.N. (1994). Effects of rotation on the thermoconvective instability of a horizontal layer of ferrofluids. Int. J. Engg. Sci., Vol. 32, pp. 237–256.

Veronis G. (1968). Large-amplitude Bénard convection in a rotating fluid. Journal

of Fluid Mechanics, Vol. 31, pp. 113-139. Veronis, G. (1959). Cellular convection with finite amplitude in a rotating fluid.

J. Fluid Mech., Vol. 5, pp. 401–435. Veronis, G. (1966). Motions at subcritical values of the Rayleigh number in a

rotating fluid. J. Fluid Mech., Vol. 24, pp. 545–554. Veronis, G. (1968). Effect of a stabilizing gradient of solute on thermal convection.

J. Fluid Mech., Vol. 34, pp. 315–336. Weilepp, J. and Brand, H.R. (1996). Competition between the Bénard-Marangoni

and the Rosensweig instability in magnetic fluids. J. Phys. II (France), Vol. 6, pp. 419–441.

White D.B. (1988). The planforms and onset of convection with a temperature-

dependent viscosity. J. Fluid Mech., Vol. 191, pp. 247–286. Wilson, S.K. (1993a). The effect of a uniform magnetic field on the onset of steady

Bénard-Marangoni convection in a layer of conducting fluid. J. Engg. Math., Vol. 27, pp. 161–188.

Wilson, S.K. (1993b). The effect of a uniform magnetic field on the onset of

Marangoni convection in a layer of conducting fluid. Quart. J. Mech. Appl. Math., Vol. 46, pp. 211–248.

Wilson, S.K. (1994). The effect of a uniform magnetic field on the onset of steady

Marangoni convection in a layer of conducting fluid with a prescribed heat flux at its lower boundary. Phys. Fluids, Vol. 6, pp. 3591–3600.

Xi H.W. and Gunton J.D. (1993). Spiral-pattern formation in Rayleigh-Benard

convection. Phys. Rev. E., Vol. 47, pp. 2987 – 2990. Yamaguchi, H., Kobori, I., Uehata, Y. and Shimada, K. (1999). Natural

convection of magnetic fluid in a rectangular box. JMMM, Vol. 201, pp. 264–267.

123

Yamaguchi, H., Zhang, Z., Shuchi, S. and Shimada, K. (2002). Gravity simulation of natural convection in magnetic fluid. JSME Int. J., Vol. 45, pp. 61–65.

Yanagisawa T. and Yamagishi Y. (2005). Rayleigh-Benard convection in

spherical shell with infinite Prandtl number at high Rayleigh number. Journal of Earth Simulator, Vol. 4, pp. 11 – 17.

You, X.Y. (2001). Estimating the onset of natural convection in a horizontal layer

of a fluid with a temperature-dependent viscosity. Chem. Engg. J., Vol. 84, pp. 63–67.

Zebib, A. (1996). Thermal convection in a magnetic fluid. J. Fluid Mech., Vol. 321,

pp. 121–136. Zeng M., Wang Q., Ozoe H., Wang G. and Huang Z. (2009). Natural convection

of diamagnetic fluid in an enclosure filled with porous medium under magnetic field. Prog. Comp. Fluid Dyn., Vol. 9, pp. 77 – 85.

Zhong F., Ecke R.E. and Steinberg V. (1991). Rotating Rayleigh-Bénard

convection: Kuppers-Lortz transition. Physica D: Nonlinear Phenomena, Vol. 51, pp. 596 – 607.

Zhong F., Ecke R.E. and Steinberg V. (1993). Rotating Rayleigh-Bénard

convection: asymmetric modes and vortex states. Journal of Fluid Mechanics, Vol. 249, pp. 135-159.

Zhou Q., Sun C. and Xia K.Q. (2007). Morphological evolution of thermal plumes

in turbulent Rayleigh-Benard convection. Phys. Rev. Lett., Vol. 98, p. 74501. Zierep, J. and Oertel, Jr. (1982). Convective transport and instability phenomena.

G. Braun, Karlsruhe.