prr.hec.gov.pkprr.hec.gov.pk/jspui/bitstream/123456789/7365/1/Amir khan Thesis.pdf · THESIS APPROVAL This is to certify that the thesis submitted by Mr. Amir Khan titled ”Fractional

FRACTIONAL ORDER GENERALIZED FLUID

FLOW MODELS: AN ANALYTICAL APPROACH

By

Amir Khan

Department of Mathematics

University of Malakand, Chakdara

Dir(Lower), Khyber Pakhtunkhwa, Pakistan

(2015)



By

Amir Khan

Supervised by

Dr. Gul Zaman




(2015)



by

Amir Khan

A Dissertation Submitted in Partial Fulfillment

of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in

MATHEMATICS

Supervised by

Dr. Gul Zaman




(2015)

THESIS APPROVAL

This is to certify that the thesis submitted by Mr. Amir Khan titled ”Fractional

Order Generalized Fluid Flow Models: An Analytical Approach” is hereby

recommended as partial fulfillment for the award of Ph.D degree in Mathematics.

——————–

Prof. Dr. Inayat Ali Shah

(External Examiner)

Islamia College University, Peshawar

——————–

Dr. Gul Zaman

(Supervisor)

University of Malakand, Chakdara, Dir

——————–

Dr. Gul Zaman

(Chairman)

University of Malakand, Chakdara, Dir

Dedicated to

my parents

brothers

and

sisters

Contents

Contents i

Acknowledgment iv

Abstract v

1 Introduction 1

1.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Structure of Dissertation . . . . . . . . . . . . . . . . . . . . . . 5

2 Preliminaries 7

2.1 Newtonian and Non-Newtonian Fluids . . . . . . . . . . . . . 7

2.2 Fluids of Differential Type . . . . . . . . . . . . . . . . . . . . . 9

2.3 Fluids of Rate Type . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Fractional Differentiation . . . . . . . . . . . . . . . . . . . . . . 14

2.8 Magnetohydrodynamic (MHD) Fluid . . . . . . . . . . . . . . 14

2.9 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9.1 Laplace Transform . . . . . . . . . . . . . . . . . . . . . 17

2.9.2 Fourier Sine Transform . . . . . . . . . . . . . . . . . . . 18

2.10 Fox H-function . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

CONTENTS

3 Generalized MHD Second Grade Fluid 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Impulsive Flow of GSGF Through PorousMedium . . . . . . . 20

3.2.1 Governing Equation . . . . . . . . . . . . . . . . . . . . 20

3.2.2 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.3 Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.4 Limiting Case . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . 30

3.3 Flow Induced by Constant Pressure Gradient . . . . . . . . . . 34

3.3.1 Problem Formulation and Solution . . . . . . . . . . . . 34

3.3.2 Graphical Results . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Flow Due to Uniform and Non-Uniform Accelerating Plate . . 41

3.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Some Exact Solution of Generalized Jeffrey Fluid 46

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Flow Between Two Side Walls Perpendicular to The Plate . . . 47

4.2.1 Mathematical Modelling . . . . . . . . . . . . . . . . . . 47

4.2.2 Impulsive Motion of The Plate (m = 0) . . . . . . . . . 49

4.2.3 Impulsive Acceleration of The Plate (m = 1) . . . . . . 52

4.2.4 Non-Uniform Acceleration of The Plate (m = 2) . . . . 52

4.2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . 54

4.3 Oscillatory Flow Passing Through a Rectangular Duct . . . . . 58

4.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 58

4.3.2 Solution of the Problem . . . . . . . . . . . . . . . . . . 61

4.3.3 Volume Flux . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . 67

5 Generalized Oldroyd-B Fluid 72

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

ii

CONTENTS

5.2 MHD Oscillatory Flow of GOB Fluid . . . . . . . . . . . . . . . 72

5.2.1 Development of the Flow . . . . . . . . . . . . . . . . . 73

5.2.2 Calculation of Velocity field . . . . . . . . . . . . . . . . 74

5.2.3 Calculation of Shear Stress . . . . . . . . . . . . . . . . . 77

5.2.4 Particular Cases . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.5 Discussion of the Results . . . . . . . . . . . . . . . . . 81

5.3 GOB Fluid Between Two Side Walls . . . . . . . . . . . . . . . 86

5.3.1 Mathematical Formulation . . . . . . . . . . . . . . . . 86

5.3.2 Calculation of the Velocity Field . . . . . . . . . . . . . 87

5.3.3 Calculation of the Shear Stress . . . . . . . . . . . . . . 88

5.3.4 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.5 Numerical Results and Discussion . . . . . . . . . . . . 93

Bibliography 97

iii

Acknowledgments

In the name of Almighty ALLAH, the most Benevolent, the most Merci-

ful and the Creator of the universe, who inculcated His countless blessings

upon me to fulfill the requirements of this dissertation. I offer peace for our

beloved Prophet Hazrat Muhammad (peace be upon him), who is forever a

symbol of complete guidance in every walk of life for humanity.

I especially thank to my supportive supervisor Dr. Gul Zaman for his un-

conditional support, guidance, encouragement and contributions through-

out this research. May Almighty Allah shower upon him abundant bless-

ings and wisdom to maintain his tireless effort in supervision as he contin-

ues helping other students. I will always be grateful for the opportunities

he has provided and the time he spent for the sake of my career. I, hereby,

thank the Dean of Sciences Dr. Rahmat Ali Khan and all teachers who

taught us and those who did not but helped in one way or the other; thank

you a lot for your efforts to make me knowledgeable in modeling ideas.

Words wane in expressing my veneration for my loving parents, I love the

most, who nurtured and guided me to inspired to learn. I owe my heartiest

gratitude for their assistance and never ending prayers for my success. I

would never have been able to stand today without their continuous sup-

port and generous help.

I am also thankful to my family and my friends, Dr. Ghaus ur Rahman, Tahir

Khan, Gulam Hussain, Anwarudin, Abdul Salam, Abdullah and Zakir ullah for

their help throughout the study.

My final appreciation goes to all my humble friends for contributing in di-

verse ways in ensuring the successful completion of this work.

Abstract

In this thesis some new results regarding non-Newtonian fluids with

fractional derivatives under different circumstances have been obtained.

The non-Newtonian fluids under discussion are fractional second grade

fluid, fractional Jeffrey fluid and fractional Oldroyd-B fluid. The similar

solutions for ordinary second grade fluid, ordinary Jeffrey fluid and ordi-

nary Oldroyd-B fluid are obtained as limiting cases of general solutions.

Governing equations are achieved by using approach of fractional calculus.

Laplace and Fourier sine transforms are used to obtain analytic solution for

velocity field and associated shear stress. The obtained solutions are ex-

pressed in series form using Fox H-function. Magnetohydrodynamic flow

of generalized second grade fluid induced by constant pressure gradient in

a porous medium is also discussed. The series solutions satisfy all the initial

and boundary conditions. The effects of different parameters on the flow are

analyzed graphically.

Some exact solutions are established for the magnetohydrodynamics flow

of generalized second grade fluid due to impulsive motion of a flat plate

passing through a porous space. Some new results are established corre-

sponding to generalized Jeffrey fluid produced by a flat plate between two

side walls perpendicular to the flat plate. The flow of generalized Jeffrey

fluid is set into motion by (i) impulsive motion of the plate, (ii) impulsive

accelerating plate, and (iii) non-uniformly accelerated plate. Unsteady mag-

netohydrodynamic flow of generalized Jeffrey fluid in a long porous rectan-

gular duct oscillating parallel to its length is also spotlighted. The volume

flux due to sine and cosine oscillations of the rectangular duct are achieved.

The oscillatory motion of magnetohydrodynamic flow of an incompress-

ible generalized Oldroyd-B fluid is studied. In particular, results regarding

Maxwell fluids are also obtained as limiting case of the general solutions.

Chapter 1

Introduction

1.1 Brief History

One of the most alluring and fascinating subject of applied Mathematics

is fluid mechanics, which has a wide range of applications in our daily

life [1]. The investigation of fluid mechanics serves to know numerous dif-

ferent aspects of applied sciences and engineering for instance, fluid control

frameworks, energy conversion systems, oceanography, climatology, bio-

engineering and food industry. Fluid is a substance having the capability to

flow and containing the shape of a vessel [2]. Fluid mechanics is classified

into two classes i.e fluid dynamics and fluid statics. Fluid dynamics deals

with motion of fluid whereas fluid static deals with fluid at rest. We focus

on fluid in motion and the forces acting on it. Fluid consists of plasmas,

gasses and liquids.

The investigation of fluid motion is scientifically complicated. Fluid me-

chanics is a dynamic region of exploration for the scientists with an expan-

sive number of unsolved problems. Fluids behavior are studied by taking

some properties of the fluid into consideration, for example, velocity, den-

sity, pressure and temperature. These properties of fluids are taken as func-

1

1.1 BRIEF HISTORY

tions of time and space. Fluid dynamics needs both scientific examination

and experimentation. By utilizing systematic methodologies we discover

answers for certain simplified and idealized fluid problems. Using famous

results of classical mechanics such as Newton’s law of conservations and

motion, we can analyze many fluid flow problems. The fluid motion are ex-

plained by a set of differential equations. Commonly, the classical Navier-

Stokes equation are employed to depict the fluid motion.

The behavior of numerous fluids such as paints, foams, polymeric liquids,

slurries, food products, gel, sludge, blood, oils, tooth pastes etc are can’t be

satisfactorily depicted by the traditional Newtonian model [3]. Hence, it is

very important to learn about the flow behavior of non-Newtonian fluids

due to its vast application in industry. Because of this reason it is extremely

important to obtain exact solutions for non-Newtonian fluids. Exact solu-

tions are very important in the sense that they show us accuracy and pre-

ciseness of approximate solutions obtained by numerical methods. For non-

Newtonian fluids, finding exact solutions are not a straight forward job. As

the non-Newtonian fluids deals with a large group of fluids, consequently,

no universal constitutive model has been developed which represent all

properties of non-Newtonian fluids. Corresponding to non-Newtonian flu-

ids, the equations of motion as compare to Navier-Stokes equations have

high order. Moreover, the existence of non-linear terms in the equations of

motion complicated the study of non-Newtonian fluids. Some recent stud-

ies with respect to non-Newtonian fluids are given in [4–6].

Fluids are mixture of various substances such as red cells, oils, water, par-

ticles and other long chain molecules. In general, the shear rate varies non-

linearly with the viscosity function and the elasticity is felt through time-

dependent and elongational impacts. Such type of fluids are known as vis-

coelastic fluids [7–10]. Various constitutive equations have been developed

as there is no universal model which satisfies all properties of viscoelastic

2

1.1 BRIEF HISTORY

fluids.

To investigate the rheological properties of such fluids numerous mathe-

matical models have been formed. In general, these models are divided

into three main classes, namely, fluid of integral type, rate type and dif-

ferential type. In all these models, differential type fluid model received

much concentration [11–16]. These models best explains shear thinning,

shear thickening, normal stress differences and non-linear creep character-

istics exhibited by some non-Newtonian fluids. The drawback of this model

is that it cannot explain the stress relaxation time contained in some fluids.

Among differential type fluids model, second grade fluids have been widely

discussed in various kind of circumstances by several mathematicians and

researchers. Another type of fluid is that of the rate type fluid, whose mod-

els can describe shear thinning, stress relaxation, non-linear creep, normal

stress differences and shear thickening. Integral type fluids are those fluids

in which the materials have considerable memory e.g polymer melts.

The notion of integration and derivation [17, 18] of fractional order can be

followed back to the origin of differential calculus itself. In the late 17th

century G. W. Leibniz, the founder of modern calculus, made a few com-

ments on the significance and prospect of fractional derivative of order 1/2

. Though a comprehensive examination was firstly completed by Liouville

by describing the first outcast of fractional integration operator. Further

examinations and improvements by Riemann prompted the construction

of Riemann-Liouville fractional integral operator, which has been an im-

portant foundation for fractional calculus [19] ever since. Preceding to Li-

ouville and Riemann, Euler stepped in the investigation of fractional inte-

gration [20, 21], when he examined the simple case of fractional integrals

of monomials of arbitrary real order in the heuristic fashion of the time;

which show the way to develop the Gamma function for fractional powers

of the factorial. At present, we have many types of fractional integral oper-

3

1.1 BRIEF HISTORY

ators, from infinite-sum to divided-difference type, however the Riemann-

Liouville operator is used most commonly when fractional integration is

carried out.

Several authors [22–24] proposed that integer-order models for viscoelastic

materials seem to be inappropriate from both quantitative and qualitative

points of view. In the meantime, they developed laws of deformation of

fractional order for the formulation of viscoelastic behavior of real materials.

Caputo [25] amplified Zener′s model [26] from integer-order to fractional-

order.

Recently, numerous authors have made utilization of rheological equations

with fractional derivatives to depict the characteristics of polymers. Several

problems regarding fluid models have been solved using fractional deriva-

tives [27–30]. In general, the constitutive equations are obtained from the

well known non-Newtonian models by replacing the integer order time

derivatives by fractional order derivatives. For example, the fractional order

Maxwell fluid modeled the prediction of the dynamic mechanical proper-

ties of a viscous damper containing a viscoelastic fluid in the form of silicon

gel [31, 32]. Tan et al. [33] studied the flow near a wall which is set by the

sudden motion of the plate with the help of fractional order Maxwell model.

Exact solutions for the unsteady rotational flow of a non-Newtonian fluid

in an annular pipe have been determined by Tong and Liu [34]. Bagley [35]

proved that fractional derivative models of viscoelastic type fluids were in

harmony with the molecular theory and attain the fractional differential

equation of order 1/2. Friedrich [36] developed the fractional derivative

method into rheology to investigate various problems. Li and Jiang [37]

employed the fractional calculus to examine the behavior of Xanthan gum

and Sesbania gum in their experiments and attained adequate results.

4

1.2 STRUCTURE OF DISSERTATION

1.2 Structure of Dissertation

The thesis is classified into 5 chapters. Chapter 2 deals with some basic pre-

liminaries. It contains a brief introduction about rate type and differential

type fluids, constitutive equations, equations of motion, continuity equa-

tion and some integral transforms i.e Laplace and Fourier transforms. These

techniques are used to solve the partial differential equations occurring in

the mathematical modeling of various types of fluid flows. Fractional dif-

ferential equation is also explained here. A special type function known as

Fox H-function is also highlighted which is used to write the lengthy solu-

tion in a compressed form.

In chapter 3, we obtain analytic solutions for the velocity field and adequate

shear stress corresponding to the magnetohydrodynamic flow of general-

ized second grade fluid due to the abrupt motion of a flat plate passing

through a porous space. The generalized second grade fluid is passing

through a porous medium. Laplace transform method is used for the frac-

tional calculus to obtained exact solutions for the profiles of velocity field

and the corresponding shear stress. The solutions obtained here are written

in terms of Fox H-function [38] satisfying all the imposed initial and bound-

ary conditions. Finally, the effects of different parameters on the motion are

analyzed graphically. Velocity and shear stress profiles of magnetohydro-

dynamic flow induced by constant pressure gradient in a porous medium

of generalized second grade fluid are also obtained. Moreover, flow due to

uniform and non-uniform motion of the plate are also investigated.

In chapter 4, we present exact solutions for the unsteady flow of a general-

ized Jeffrey fluid which have been set into motion by (i) impulsive motion of

the plate (ii) impulsive accelerating plate and (iii) non-uniformly accelerated

plate. We establish analytic solutions for the velocity field and the associated

shear stress corresponding to the unsteady flow of an incompressible gener-

5

1.2 STRUCTURE OF DISSERTATION

alized Jeffrey fluid between two side walls perpendicular to the plate. The

obtained solutions, expressed under series form in terms of Fox H-function,

are established by means of Fourier sine and Laplace transforms. The sim-

ilar solution for ordinary Jeffrey fluid can be obtained as limiting case of

general solution. Finally, the influence of the fractional parameters on the

motion of generalized Jeffrey fluids is underlined by graphical illustrations.

Also some new exact solutions corresponding to unsteady magnetohydro-

dynamic flow of generalized Jeffrey fluid in a long porous rectangular duct

oscillating parallel to its length is discussed. The exact solutions are estab-

lished by means of the double finite Fourier sine transform and discrete

Laplace transform.

In chapter 5, we succeeded to provide some exact solutions for the un-

steady oscillatory flow of an incompressible generalized magnetohydrody-

namic Oldroyd-B fluid with constant pressure gradient. The analytic solu-

tions for the profiles of velocity and shear stress are obtained by means of

Fourier sine and Laplace transforms. Fox H-function is used to show the

final solution in a more compact form. Similar solutions for ordinary mag-

netohydrodynamic Oldroyd-B, generalized and ordinary magnetohydrody-

namic Maxwell, generalized and ordinary magnetohydrodynamic second

grade fluids are obtained as particular cases of general solutions. Also some

new exact solutions corresponding to the unsteady flow of a generalized

Oldroyd-B fluid produced by a suddenly moved plate between two side

walls perpendicular to the plate is discussed. Finally, the influence of the

different parameters on the flow is highlighted by graphical illustrations.

6

Chapter 2

Preliminaries

2.1 Newtonian and Non-Newtonian Fluids

Newtonian fluids (named after Sir Isaac Newton) are the fluids whose stress

versus strain rate graph is linear and passes through the origin. Following

are the characteristics of Newtonian fluids, which at constant temperature

and pressure are experimentally conducted [39]:

(i) Shear viscosity is independent of shear rate.

(ii) Shear stress is the only stress formed in simple shear flows, the differ-

ence between two normal stresses is zero.

(iii) Viscosity is constant over a wide range of applied shear. As we stopped

the applying shear the stress in the fluid goes to zero.

(iv) The viscosity of Newtonian fluids are in simple proportion to one an-

other when measured in different types of deformation.

Generally, the following equation describe flow of Newtonian fluid

τ = µdu

dy, (2.1)

where dynamic viscosity of the fluid is represented by µ, shear stress and

the rate of deformation normal to the direction of shear are denoted by τ

and du/dy, respectively.

7

2.1 NEWTONIAN AND NON-NEWTONIAN FLUIDS

Non-Newtonian fluids are those fluids which does not obey the above

characteristics. Most of the fluids which are used in daily life does not fol-

low the above characteristics. Numerous empirical model have been pro-

posed to signify the examined non-linear relationship of τ and du/dy. To

view this behavior the power law model is used

τ = k(du

dy)

n

, (2.2)

where the index of flow consistency is denoted by k and index of flow be-

havior is represented by n. We get Newtons law of viscosity Eq. (2.1) by

taking k = µ and n = 1. Retaining the signs of du/dy and τ, Eq. (2.2) is

modified as

τ = k(du

dy)

n−1

(du

dy) = η

du

dy, (2.3)

where the fluid apparent viscosity is denoted by η = k( dudy )

n−1. Which

shows that, at a given pressure and temperature, the viscosity of non-Newtonian

fluids is a function of velocity gradient.

There are several types of non-Newtonian fluids, the most familiar are shear

thinning, shear thickening and bingham plastic. The apparent viscosity

for shear thinning fluids decreases with increasing shear stress whereas for

shear thickening fluids it increases with increasing shear stress. In shear

thickening, the fluids becomes more viscous when shear is applied. Nu-

merous polymer solutions are shear thinning. Common examples of shear

thinning are water-corn starch and water-sand mixture. Bingham plastic

is another type of non-Newtonian fluids, which is neither a liquid nor a

solid. Mayonnaise and toothpaste are common examples of Bingham fluid.

In general, the non-Newtonian fluids are divided into three major classes

which are given as (i) fluids of differential type (ii) fluids of rate type and

(iii) fluids of integral type.

Fluids of differential and rate type are discussed here, as they will be used

in the upcoming chapters.

8

2.3 FLUIDS OF RATE TYPE

2.2 Fluids of Differential Type

Many substances have the potential to flow but their flow behavior can-

not be characterized by the classical fluid model e.g polymeric fluids, food

products, slurries, geological materials and liquid foams. Numerous ideal-

ized fluid models have been suggested to describe the dissimilarities from

the classical Newtonian fluids. The first fluid model which diverge from

Newtonian fluid is known as differential type fluid or informally Rivlin-

Ericksen fluid.

A subclass of fluid of differential type has gained a particular consideration

known as second grade fluid. Dunn investigated highly specially fluids of

differential type, and obtained a result that the stationary state for all of

these fluids is unstable [40]. The stability flows in infinite domains of sec-

ond grade fluids was discussed by Galdi et al. [41]. They take positive val-

ues for material parameters and obtained that the stationary state is always

conditionally stable i.e, for a little interruption the stationary state is always

stable. This result also holds for the flows in bounded domains.

2.3 Fluids of Rate Type

The flow behavior of numerous types of fluids are inappropriately signified

by Navier-Stokes equations. Rate type models are formed to explain the

response of inhomogeneous fluids whose material properties can depend

upon the shear rate and the mean normal stress. The rate type models are

specially helpful in explaining the behavior of biological fluids, food prod-

ucts and geological fluids. Among the various subclasses of rate type fluids

the Maxwell model is considered to be the simplest model which describes

the stress relaxation effects. The first systematic frame work that describes

the response of rheological behavior of rate type viscoelastic fluids is de-

9

2.4 CONSTITUTIVE EQUATIONS

veloped by Oldroyd [42]. This unique and wonderful work, perceived the

limitations forced by frame invariance, introduced convective derivatives of

the suitable physical quantities to get legitimately frame invariant constitu-

tive relations, included the notion that the present state of stress in a body

can rely on the historical backdrop of distortion of the body, and even gave

unequivocal formulae for processing the development of the material sym-

metry due to distortion all with in the setting of a fully three-dimensional

structure.

2.4 Constitutive Equations

Constitutive equation is a relation between rate of deformation and stress.

Generally, constitutive equation specify the rheological properties of vari-

ous materials. These relations are not universal but provide properties for

some specific class of substances and hold true for a specific class of phys-

ical processes. In other words, constitutive equation explains an ideal ma-

terial which is a mathematical model for illustrating the properties of some

types of real materials. Corresponding to different materials the constitu-

tive equation satisfies some common rules e.g the symmetry principle and

the objectivity principle. The constitutive equation for the non-Newtonian

fluids lead to the flow problems in which the order of the differential equa-

tions exceeds the number of available conditions. In the following constitu-

tive equations of the fluids studied in this thesis are given.

(i). Newtonian Fluid

Newtonian fluid is considered to be the simplest constitutive equation

T = S − pI, S = µA1, (2.4)

where the Cauchy stress tensor is represented by T, extra stress tensor is

denoted by S, hydrostatic pressure is symbolized by p, and unit tensor and

10

2.4 CONSTITUTIVE EQUATIONS

dynamic viscosity are signified by I and µ, respectively.

A1 = L + LT, (2.5)

is the first Rivlin-Ericksen tensor. Here the velocity gradient and the trans-

pose operation are indicated by L and superscript notation T, respectively.

(ii). Second Grade Fluid

Constitutive equation for the differential type second grade fluid is

T = S − pI, S = µA1 + α1A2 + α2A21, (2.6)

where normal stress moduli are represented by α1 and α2 that meet the fol-

lowing conditions

α1 + α2 = 0, 0 ≤ α1 and µ ≥ 0. (2.7)

The kinematic tensors A1 and A2 are

A1 = L + LT, A2 =dA1

dt+ A1L + LTA1, L = ∇v, (2.8)

where the convective time derivative is denoted by ddt and the velocity of

the fluid is represented by v.

(iii). Maxwell Fluid

Constitutive equation for the rate type Maxwell fluid is

T = −pI + S, (1 + λD

Dt)S = µA1, (2.9)

where the relaxation time is denoted by λ and the upper convective deriva-

tive is represented by DDt , defined as

DS

Dt=

dS

dt+ (V.∇)S − LS − SL. (2.10)

(iv). Oldroyd-B Fluid

Constitutive equation for another rate type model known as Oldroyd-B

fluid is given by

T = −pI + S, (1 + λD

Dt)S = µ(1 + θ

D

Dt)A, (2.11)

where θ is retardation time.

11

2.5 CONTINUITY EQUATION

2.5 Continuity Equation

Consider a surface S with control volume V in space. Let us assume that no

fluid can leave or enter the surface S. As it is well known result from Physics

that mass can neither be destroyed nor created, so the entire mass with in

the control volume V is conserved in time. Hence, we can write in this case

d

dt

∫

VρV = 0, (2.12)

where ρ is the density of the fluid at time t.

With the use of Reynolds transport theorem, the above equation can also be

written asd

dt

∫

VρdV =

∫

V(

∂ρ

∂t+∇.(ρv))dV = 0. (2.13)

As the control volume V was taken to be arbitrary, the necessary and suffi-

cient condition for conservation of mass is

∂ρ

∂t+∇.(ρv) = 0. (2.14)

Above equation is called the continuity equation in differential form for a

compressible fluid. For an incompressible fluid (having constant density)

the above equation takes the form

∇.v = trA1 = 0. (2.15)

In cylindrical coordinates (r, θ, z) the above incompressibility condition can

be written as1

r

∂(rvr)

∂r+

1

r

∂(vθ)

∂θ+

∂(vz)

∂z= 0. (2.16)

In the above equation vr, vθ and vz are the physical components of the ve-

locity field v.

12

2.6 EQUATIONS OF MOTION

2.6 Equations of Motion

The general equations of motion are derived by applying Newtons law to a

small but finite fluid particle. The differential form of the equation of motion

is given by

divT + ρb = ρa, (2.17)

where b is the body force and a is the acceleration. In cylindrical compo-

nents

T =

σrr τrθ τrz

τθr σθθ τθz

τzr τzθ σzz

,

b =

br

bθ

bz

and a =

ar

aθ

az

,

where σrr, σθθ and σzz are normal stresses, while τrθ, τθz and τrz are tangential

shear stresses. In cylindrical coordinates Eq. (2.17) can be written as

1

r

∂(rσrr)

∂r+

1

r

∂(τrθ)

∂θ− τrθ

r+

∂(τrz)

∂z+ ρbr = ρ(

∂(vr)

∂t+ vr

∂(vr)

∂r− vθ

r

∂vr

∂θ

− (vθ)2

r+ vz

∂vr

∂z), (2.18)

1

r2

∂(r2τrθ)

∂r+

1

r

∂(σθθ)

∂θ+

∂(τθz)

∂z+ ρbθ = ρ(

∂(vθ)

∂t+ vr

∂(vθ)

∂r+

vθ

r

∂vθ

∂θ

+vr(vθ)

2

r+ vz

∂vθ

∂z), (2.19)

1

r

∂(rσrr)

∂r+

1

r

∂(τrθ)

∂θ− τrθ

r+

∂(τrz)

∂z+ ρbr = ρ(

∂(vr)

∂t+ vr

∂(vr)

∂r− vθ

r

∂vr

∂θ

− (vθ)2

r+ vz

∂vr

∂z). (2.20)

13

2.8 MAGNETOHYDRODYNAMIC (MHD) FLUID

2.7 Fractional Differentiation

The non-Newtonian fluids with fractional derivatives have meet many suc-

cess in the description of complex dynamics. Generally, the governing equa-

tion corresponding to motion of a fluid with fractional derivatives are ob-

tained from the governing equation of the ordinary fluid by replacing the

inner time derivatives by the so called Riemann-Liouville differential oper-

ator

Dαt [ f (t)] =

1

Γ(1 − α)

d

dt

∫ t

0

f (τ)

(t − τ)αdτ, 0 < α < 1, (2.21)

or by the Caputo differential operator

Cαt [ f (t)] =

1

Γ(1 − α)

∫ t

0

f′(τ)

(t − τ)αdτ, 0 < α < 1, (2.22)

where Γ(·) is the Gamma function. The Riemann-Liouville differential op-

erator given by Eq. (2.21) can be written in the equivalent form

Dαt [ f (t)] =

1

Γ(1 − α)

f (0)

tα+

1

Γ(1 − α)

∫ t

0

f′(τ)

(t − τ)αdτ, 0 < α < 1. (2.23)

It is important to note that if f (0) = 0, then the Riemann-Liouville differ-

ential operator is equal with the Caputo differential operator. Therefore, for

f (0) = 0, we have

Dαt [ f (t)] = Cα

t [ f (t)]. (2.24)

2.8 Magnetohydrodynamic (MHD) Fluid

For MHD fluid, the equation which governs the flow is

ρ(dV

dt) = divT + J × B, (2.25)

14

2.9 INTEGRAL TRANSFORMS

where J is the current density and B is the total magnetic field. Neglecting

the displacement current, the Maxwell’s equations are given by

∇ · B = 0, (2.26)

∇ · E = 0, (2.27)

∇× B = µmJ, (2.28)

∇× E = −∂B

∂t0. (2.29)

Here E is the electric field, µm is magnetic permeability, and J is the current

density. Due to modified Ohm’s law we have

J = σ(E + V × B), (2.30)

where σ signifies the finite electrical conductivity. Additionally, we assume

that electrical field is zero and magnetic Reynold number is very small.

Therefore, the induced magnetic field b in B = Bo + b is negligible and

thus

J × B = −σBoV, (2.31)

where Bo is applied magnetic field. Using the above equation, Eq. (2.25)

becomes

ρ(dV

dt) = divT − σBoV. (2.32)

2.9 Integral Transforms

The integral transform of a function g(t), a ≤ t ≤ b, is denoted by Ig(t) =

G(s), and is defined as

Ig(t) = G(s) =∫ b

aK(t, s)g(t)dt, s < 0, (2.33)

where K(t, s) is a given function of two variables t and s, called the kernel of

the transformation. The operator I is generally called an integral transform

15


operator. The transform function G(s) is referred as the image of the given

object function g(t), and s is called the transform variable.

There are many important integral transforms including Hankel, Laplace,

Fourier, Legendre and Hilbert. These integral transforms are defined by

choosing different kernels K(t, s) and different values for a and b used in

(2.33). It can be proved that the integral transform of a function is unique.

This result follows from the argument that an integral transformation sim-

ply means a unique mathematical operator in which a real or complex-

valued function g is transformed into another new function G = Ig, or

into a set of data that can be measured numerically and experimentally.

Integral transformation convert a complex mathematical model to a rela-

tively simple model, which can easily be solved. This makes the integral

transforms very important. For finding solutions of initial boundary value

problems concerning differential equations, the differential operators are

changed by relatively simpler algebraic operations involving G, which can

easily be solved. Then by applying the inverse transformation, solution of

the original problem is obtained in the original variables. So, the next task

is to compute the inverse integral transform approximately or exactly. Gen-

erally, in practice it is difficult to reconstruct g from Ig = G, to make the in-

tegral transformation effective. However, this difficulty has been resolved

by different researchers in different ways. In some applications the trans-

form function G itself has some practical and physical meaning, and needs

to be studied in its own right. Integral transforms have been proved to be

a very systematic, powerful and efficient tool for finding exact solutions of

different types of problems in engineering involving differential equations.

A large number of different integral transforms exist which are used for this

purpose. In the following, we introduce Laplace and Fourier transforms

along with their inverses, as they are used in our present work.

16


2.9.1 Laplace Transform

The Laplace transform of a function g(t), 0 ≤ t < ∞, is defined as

L {g(t)} = g(s) =∫ ∞

oe−stg(t)dt, (2.34)

where s is a complex number. The inverse Laplace transform L −1 is defined

as

L−1{g(s)} = g(t) =

1

2πi

∫ c+i∞

c−i∞est g(s)ds, (2.35)

where c is a real number. In practice, it is quite tedious to apply the above

integral to find inverse Laplace transform, and so we will not be using it.

Instead, we can simply use the other methods to find inverse Laplace trans-

form to obtain the corresponding function g(t) from g(s). For instance, in

order to avoid the lengthy calculations of residues and contour integrals, we

have used the discrete inverse Laplace transform method. Not all functions

are Laplace transformable. For a function g(t) to be Laplace transformable,

it must satisfy the Dirichlet conditions, a set of sufficient but not necessary

conditions. These are:

(i). g(t) must be piecewise continuous; that is, it must be single valued but

can have a finite number of finite isolated discontinuities for t > 0.

(ii). g(t) must be of exponential order; that is, g(t) must remain less than

M exp(a0t) as t approaches ∞, where M is a positive constant and a0 is a

real positive number.

For example, such functions as: tan(t), cot(t) and exp(t2) are not Laplace

transformable. Laplace transform is now used in a great extent in solving

partial differential equations, initial and boundary value problems, integral

equations and difference equations in many fields.

17

2.10 FOX H-FUNCTION

2.9.2 Fourier Sine Transform

Many initial and boundary value problems in applied mathematics, mathe-

matical physics and engineering can be solved by using Fourier sine trans-

form. The Fourier sine transform of a function f (x) is defined as

Fs{ f (x)} = Fs(k) =∫ ∞

osin(kx) f (x)dx, (2.36)

and the inverse Fourier sine transform is

F−1s {Fs(k)} = f (x) =

∫ ∞

osin(kx)Fs(k)dk. (2.37)

2.10 Fox H-function

The Fox function, also referred as the Fox′s H-function, generalizes the Mellin-

Barnes function. The importance of the Fox function lies in the fact that it

includes nearly all special functions occurring in applied mathematics and

statistics as special cases. In 1961, Fox defined the H-function as the Mellin-

Barnes type path integral:

Hm,n

p,q

−σ

∣

∣

∣

∣

∣

∣

(ak, Ak)p1

(bk, Bk)q1

=

1

2πi

∫

l

∏mk=1 (bk − Bks)∏

nj=1 Γ(1 − aj + sAj)

∏qk=m+1 Γ(1 − bk + Bks)∏

pj=n+1 (aj − sAj)

σsds,

where l is a suitable contour, the orders (m, n, p, q) are integers 0 ≤ m ≤q, 0 ≤ n ≤ p and the parameters aj ∈ R, Aj > 0, j = 1, 2, ..p, bk ∈ R, Bk > 0,

k = 1, 2, ..q, are such that Aj(bk + i) 6= Bk(aj − i − 1), i = 0, 1, 2...

H1,s

s,t+1

−σ

∣

∣

∣

∣

∣

∣

(1 − a1, A1), ..., (1 − as, As)

(1, 0), (1 − b1, B1), ..., (1 − bt, Bt)

=

∞

∑r=0

Γ(a1 + A1r)...Γ(as + Asr)

r!Γ(b1 + B1r)...Γ(bt + Btr)σr.

18

Chapter 3

Generalized MHD Second Grade

Fluid

3.1 Introduction

This chapter is divided into two sections. In these sections we solve two

problems related to generalized second grade fluids (GSGF). The second

grade fluids (SGF), which is a subclass of fluids of differential type, have

been introduced by Rivlin and Ericksen [43]. Bandelli [44] investigated ana-

lytic solutions of SGF for start-up flows using integral transform technique.

Tan and Xu not only discussed the abrupt flow of a flat plate but in addi-

tion they also explored unsteady motions of GSGF passing through parallel

plates [45, 46]. Mahmood et al. [47] found the analytic solutions of oscil-

latory flow between two cylinders taking GSGF into consideration. Tripa-

thy [48] examined peristaltic motion in a cylindrical tube of GSGF.

In the most recent couple of decades the investigation of motion of fluids

through porous medium have got much consideration because of its sig-

nificance to the field of industry as well as to the academia. Such move-

ments have numerous applications, for example, movement of blood in the

19

3.2 IMPULSIVE FLOW OF GSGF THROUGH POROUS

MEDIUM

cardiovascular framework [49–51], food industry, oil exploitation, watering

system problems, cellulose solutions and bio-engineering. Also, the human

lungs are considered to be porous layer in biophysical sciences. MHD mo-

tions of viscoelastic fluids saturating the porous medium have considerable

significance [52–61].

3.2 Impulsive Flow of GSGF Through Porous

Medium

In this section, analytic solutions are obtained for incompressible MHD mo-

tion of electrically conducting GSGF. The GSGF bounded by a flat plate at

y = 0 is passing through a porous medium y > 0. Along the y-axis, the

uniform magnetic field βo is stressing the GSGF. The magnetic Reynolds

number is taken to be considerably small, hence the induced magnetic field

is ignored. At time t = 0, the plate as well as the fluid are at rest. Af-

ter time t = 0, the flat plate abruptly starts its motion, hence allowing the

fluid to flow with a constant velocity A. Fractional calculus is used to ob-

tain the governing equation. Laplace transform method is used to obtain

analytic solutions for the profiles of velocity field (VF) and the adequate

shear stress (SS). The results obtained are written in series form using Fox

H-function and satisfies the initial condition (IC) and boundary conditions

(BCs). Finally, the effects of different parameters on the motion are analyzed

graphically.

3.2.1 Governing Equation

The Cauchy stress tensor T for GSGF is

T = S − pI, S = µW1 + α1W2 + α2W21, (3.1)

20


MEDIUM

α1 + α2 = 0, α1, µ ≥ 0. (3.2)

where S is extra stress tensor, pI is the indeterminate spherical stress, α1 and

α2 are the normal stress moduli, µ is the dynamic viscosity, and W1, W2 are

the kinematic tensors which are given by

W1 = L + LT, W2 = Dβt W1 + W1L + LTW1. (3.3)

Here velocity gradient is denoted by L and fractional differentiation of order

β is denoted by Dβt . According to the definition of Riemann-Liouville [20],

the fractional differentiation operator Dβt is written as

Dβt [g(t)] =

1

Γ(1 − β)

d

dt

∫ t

0

g(τ)

(t − τ)βdτ, 0 ≤ β < 1. (3.4)

By taking β = 1 in the above model we get ordinary SGF model. For

the MHD flow saturating the porous space, the continuity and momentum

equations have the following form

∇ · V = 0, ρ(dV

dt) = divT − σβ2

oV + R, (3.5)

where T is Cauchy stress tensor, V is velocity vector, ρ is density of the

fluid, d/dt is the convective time derivative, σ is electrical conductivity and

for porous medium the Darcy’s resistance is denoted by R.

For the GSGF the Darcy’s law is

R = −φ

κ(µ + αD

βt )V, (3.6)

where α = µθβ, θ is the retardation time, κ > 0 is the permeability of the

porous medium and φ (0 < φ < 1) is the porosity. We take the VF and the

associated SS in the following form

V = (u(y, t), 0, 0), S = S(y, t). (3.7)

The velocity of the fluid along x-axis is denoted by u. Putting Eq. (3.7) into

Eq. (3.1) and using the IC

S(y, 0) = 0, y > 0, (3.8)

21


MEDIUM

we get Syy = Szz = Sxz = Syz = 0 and the relevant fractional differential

equation

Sxy(y, t) = (µ + αDβt )∂yu(y, t). (3.9)

Assuming that there are no body forces, the equation of linear momentum

takes the form

∂ySxy(y, t)− ∂x p − σβ2ou − φ

κ(µ + αD

βt )u(y, t) = ρ∂tu(y, t),

∂y p = ∂z p = 0, (3.10)

where the pressure gradient is taken along x-axis and is denoted by ∂x p. We

obtain the governing equation by putting Sxy from Eq. (3.9) into (3.10)

ρ∂tu(y, t) = (µ + αDβt )∂

2yu(y, t)− ∂x p − σβ2

ou(y, t)− φ

κ(µ + αD

βt )u(y, t).

(3.11)

For the VF, following are the IC and BCs

u(y, 0) = 0, u(0, t) = A, y > 0, t > 0, (3.12)

u(y, t), ∂yu(y, t) → 0, t > 0, y → ∞. (3.13)

3.2.2 Velocity Field

We use the following non-dimensional quantities

u∗ = uU , y∗ = yU

ν , t∗ = tU2

ν , θβ∗ = θβU2β

νβ , A∗ = AU ,

τ =Sxy

ρU2 , K = κU2

φν2 , M2 = σνβ2o

ρU2 , P = − ∂p∂x

νρU2 .

(3.14)

After excluding asterisks, the governing equation and Eq. (3.9) are rewritten

in their dimensionless form as

∂tu(y, t) = P + (1 + θβDβt )∂

2yu(y, t)− 1

K(1 + θβD

βt )u(y, t)− M2u(y, t),

(3.15)

22


MEDIUM

τ(y, t) = (1 + θβDβt )∂yu(y, t), (3.16)

along with the IC and BCs

u(y, 0) = 0; u(0, t) = A; y > 0, t > 0, (3.17)

u(y, t), ∂yu(y, t) → 0, t > 0, y → ∞.

To solve the above fractional differential equation, we apply the LT for se-

quential fractional derivatives [62] to Eq. (3.15)

u(y, q) =∫ ∞

ou(y, t)e−qtdt, q > 0. (3.18)

Using the IC (3.17), we get the following fractional differential equation

∂2yu(y, q)−

(

1

K+

q + M2

1 + θβqβ

)

u(y, q) +P

q(1 + θβqβ)= 0, q > 0, (3.19)

u(0, q) =A

q; q > 0,

∂yu(y, q) → 0 as y → ∞ and q > 0. (3.20)

Using the BCs (3.20), we obtain the following solution of Eq. (3.19)

u(y, q) =A

qe−

√By − C

Be−

√By +

C

B, (3.21)

where

B =1

K+

q + M2

1 + θβqβand C =

P

q(1 + θβqβ). (3.22)

23


MEDIUM

In series form Eq. (3.21) is expressed as

u(y, q) = P∞

∑a1=0

∗∑

∞

∑h1=0

(−1)ζ+a1+h1+1θβ(a1+d1+h1+r+s)M2c1+2g1ye1

h1!g1! f1!e1!d1!c1!b1!a1!r!s!K−b1+e1/2− f1−1

× Γ(b1 + a1)Γ(c1 − b1)Γ(d1 + b1)Γ( f1 − e1/2)Γ(g1 − f1)Γ(h1 + f1)

q−a1−b1+c1− f1−h1−d1−βr−s+1Γ(−1)Γ(−a1)Γ(b1)Γ(−b1)Γ(e1/2)

× Γ(r + e1/2)Γ(s − e1/2)Γ(a1 + 1)

Γ(−e1/2)Γ(e1/2)Γ( f1)Γ(− f1)+ P

◦∑

∞

∑c1=0

(−1)ξ+c1θβ(a1+d1)

a1!b1!c1!d1!

× Γ(a1 + 1)Γ(b1 + 1)Γ(c1 − b1)Γ(d1 + 1)qa1+b1+d1−c1−1Kb1+1M2c1

Γ(−1)Γ(−1)Γ(b1)Γ(−b1)

+∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑h1=0

∞

∑r=0

∞

∑s=0

A(−1)e1+ f1+g1+h1+r+sθβ(h1+r+s)M2g1ye1

e1! f1!g1!h1!r!s!Ke1/2− f1q− f1−h1−βr−s+1

× Γ( f1 − e1/2)Γ(g1 − f1)Γ(h1 + f1)Γ(r + e1/2)Γ(s − e1/2)

Γ(e1/2)Γ(−e1/2)Γ(e1/2)Γ( f1)Γ(− f1), (3.23)

where

∗∑ =

∞

∑b1=0

∞

∑c1=0

∞

∑d1=0

∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑r=0

∞

∑s=0

,◦∑ =

∞

∑a1=0

∞

∑b1=0

∞

∑d1=0

,

ξ = a1 + b1 + d1 and ζ = b1 + c1 + d1 + e1 + f1 + g1 + r + s.

By taking inverse LT of Eq. (3.23), we attain the VF as

u(y, t) = P∞

∑a1=0

∗∑

∞

∑h1=0

(−1)ζ+a1+h1+1Γ(a1 + 1)Γ(b1 + a1)

a1!b1!c1!d1!e1! f1!g1!h1!r!s!Γ(−1)Γ(−a1)

× Γ(g1 − f1)Γ(h1 + f1)Γ(r + e1/2)Γ(s − e1/2)θβ(a1+d1+h1+r+s)

Γ(e1/2)Γ( f1)Γ(− f1)Γ(−e1/2)Γ(e1/2)K−b1+e1/2− f1−1M−2c1−2g1

× Γ(c1 − b1)t−a1−b1+c1− f1−h1−d1−βr−sye1Γ(d1 + b1)Γ( f1 − e1/2)

Γ(b1)Γ(−b1)Γ(−a1 − b1 + c1 − f1 − h1 − d1 − βr − s + 1)

+ P◦∑

∞

∑c1=0

(−1)ξ+c1 θβ(a1+d1)t−a1−b1−d1+c1Kb1+1M2c1Γ(a1 + 1)

a1!b1!c1!d1!Γ(−1)Γ(−1)Γ(−a1 − b1 − d1 + c1 + 1)

× Γ(b1 + 1)Γ(c1 − b1)Γ(d1 + 1)

Γ(b1)Γ(−b1)+

∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑h1=0

∞

∑r=0

∞

∑s=0

A

e1! f1!g1!

× Γ( f1 − e1/2)Γ(g1 − f1)Γ(h1 + f1)Γ(r + e1/2)Γ(s − e1/2)

h1!r!s!Γ(e1/2)Γ(−e1/2)Γ(e1/2)Γ( f1)Γ(− f1)Ke1/2− f1

× (−1)e1+ f1+g1+h1+r+s M2g1ye1t− f1−h1−βr−sθβ(h1+r+s)

Γ(− f1 − h1 − βr − s + 1). (3.24)

24


MEDIUM

We use Fox H-function to write the above solution in a compact form as

u(y, t) = P∞

∑a1=0

∗∑

(−1)ζ+a1+1θβ(a1+d1+r+s)t−a1−b1+c1− f1−d1−βr−sye1

a1!b1!c1!d1!e1! f1!g1!r!s!K−b1+e1/2− f1−1M−2c1−2g1

× H1,9

9,11

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (1 − b1 − c1, 0), (−b1, 0), (1 − d1 − b1, 0),

(1 − g1 + f1, 0), (1 − f1, 1), (1 − s + e1/2, 0),

(1 − f1 + e1/2, 0), (1 − r − e1/2, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(0, 1), (1 + e1/2, 0), (1 − e1/2, 0),

(a1 + b1 − c1 + f1 + d1 + βr + s,−1).

+ P◦∑

(−1)ξθβ(a1+d1)t−a1−b1−d1Kb1+1

a1!b1!d1!

× H1,4

4,6

M2t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (−b1, 0), (1 + b1, 1), (−d1, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(0, 1), (a1 + b1 + d1, 1).

+ A∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑r=0

∞

∑s=0

(−1)e1+ f1+g1+r+s M2g1ye1t− f1−βr−s+1

θ−β(r+s)e1! f1!g1!r!s!Ke1/2− f1

× H1,5

5,7

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0).

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0), (1 + e1/2, 0),

(0, 1), (1 − e1/2, 0), ( f1 + βr + s,−1).

. (3.25)

3.2.3 Shear Stress

To find the analytic solution of SS, we apply LT to Eq. (3.16)

τ(y, q) = (1 + θβqβ)∂yu(y, q). (3.26)

Using Eq. (3.21) to eliminate u(y, q) from Eq. (3.26)

τ(y, t) = −(1 + θβqβ)Ae−

√By√

B

q+

P

qB−1/2e−

√By, (3.27)

25


MEDIUM

where

B =1

K(1 + θβqβ)((1 + θβq) + K(q + M2)).

In series form Eq. (3.27) is expressed as

τ(y, q) = P∗∑

∞

∑h1=0

∞

∑t=0

∞

∑u=0

(−1)ζ+h1+t+uΓ(h1 + f1)M+2c1+2g1

b1!c1!d1!e1! f1!g1!h1!r!s!t!u!Γ(1/2)

× Γ(b1 + 1/2)Γ(c1 − b1)Γ(d1 + b1)Γ( f1 − e1/2)Γ(g1 − f1)

Γ(b1)Γ(−1/2)Γ(−b1)Γ(e1/2)Γ( f1)Γ(− f1)Γ(−e1/2)Γ(e1/2)

× Γ(s − e1/2)Γ(t − 1/2)Γ(u + 1/2)Γ(r + e1/2)Kb1−e1/2+ f1+1ye1

Γ(−1/2)θβ(−d1−h1−r−s−t−u)q−b1+c1− f1−h1−d1−βr−βt−s−u+1

+∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑h1=0

∗∗∑

∞

∑r=0

∞

∑s=0

A(−1)e1+ f1+g1+h1+ζ1+r+s+1

e1! f1!g1!h1!i1!j1!k1!l1!m1!r!s!

× θβ(h1+k1+l1+m1+r+s)Γ( f1 − e1/2)Γ(g1 − f1)Γ(h1 + f1)

Γ(e1/2)Γ(−e1/2)Γ(e1/2)Γ( f1)Γ(− f1)Ke1/2− f1−i1+1/2

× M2g1+2j1ye1Γ(i1 − 1/2)Γ(j1 − i1)Γ(k1 + i1)Γ(l1 − 1/2)

Γ(1/2)Γ(i1)Γ(1/2)Γ(−i1)Γ(1/2)q− f1−h1−i1−k1−βl1−m1−βr−s+1

× Γ(s − e1/2)Γ(r + e1/2)Γ(m1 − 1/2), (3.28)

where

∗∗∑ =

∞

∑i1=0

∞

∑j1=0

∞

∑k1=0

∞

∑l1=0

∞

∑m1=0

and ζ1 = i1 + j1 + k1 + l1 + m1.

26


MEDIUM

We get the analytic solution of SS by taking the inverse LT of Eq. (3.28)

τ(y, t) = P∗∑

∞

∑h1=0

∞

∑t=0

∞

∑u=0

(−1)ζ+h1+t+uΓ(t − 1/2)Γ(u + 1/2)

b1!c1!d1!e1! f1!g1!h1!r!s!t!u!Γ(1/2)

× Γ(b1 + 1/2)Γ(c1 − b1)Γ(d1 + b1)Γ( f1 − e1/2)Γ(g1 − f1)

Γ(b1)Γ(−1/2)Γ(−b1)Γ(e1/2)Γ( f1)Γ(− f1)Γ(−e1/2)Γ(e1/2)

× Γ(s − e1/2)Kb1−e1/2+ f1+1ye1t−b1+c1− f1−h1−d1−βr−βt−s−u+1

Γ(−b1 + c1 − f1 − h1 − d1 − βr − βt − s − u + 1)

× Γ(r + e1/2)Γ(h1 + f1)

M−2c1−2g1Γ(−1/2)θβ(−d1−h1−r−s−t−u)

+∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑h1=0

∗∗∑

∞

∑r=0

∞

∑s=0

A(−1)e1+ f1+g1+h1+ζ1+r+s+1

e1! f1!g1!h1!i1!j1!k1!l1!m1!r!s!

× Γ( f1 − e1/2)Γ(g1 − f1)Γ(h1 + f1)Γ(r + e1/2)

Γ(e1/2)Γ(−e1/2)Γ(e1/2)Γ( f1)Γ(− f1)Ke1/2− f1−i1+1/2

× M2g1+2j1ye1Γ(i1 − 1/2)Γ(j1 − i1)Γ(k1 + i1)Γ(l1 − 1/2)

θ−β(h1+k1+l1+m1+r+s)Γ(1/2)Γ(i1)Γ(1/2)Γ(−i1)Γ(1/2)

× Γ(s − e1/2)Γ(m1 − 1/2)t− f1−h1−i1−k1−βl1−m1−βr−s

Γ(− f1 − h1 − i1 − k1 − βl1 − m1 − βr − s + 1). (3.29)

27


MEDIUM

The SS is written in a compact form by using Fox H-function as

τ(y, t) = P∗∑

∞

∑t=0

∞

∑u=0

(−1)ζ+t+uKb1−e1/2+ f1+1ye1

r!s!t!u!b1!c1!d1!e1! f1!g1!

× t−b1+c1− f1−d1−βr−βt−s−u+1

M−2c1−2g1θβ(−d1−r−s−t−u)

× H1,10

10,12

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1 + 1/2, 0), (1 − c1 + b1, 0), (1 − d1 − b1, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − r − e1/2, 0), (−t + 3/2, 0), (1/2 − u, 0),

(1 − s + e1/2, 0).

(3/2, 0), (1/2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0), (3/2, 0),

(b1 − c1 + f1 + d1 + βr + βt + s + u,−1).

+∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∗∗∑

∞

∑r=0

∞

∑s=0

A(−1)e1+ f1+g1+ζ1+r+s+1ye1

e1! f1!g1!i1!j1!k1!l1!m1!r!s!M−2g1−2j1

× t− f1−i1−k1−βl1−m1−βr−s

θβ(−k1−l1−m1−r−s)Ke1/2− f1−i1+1/2

× H1,10

10,12

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−i1 + 3/2, 0), (1 − j1 + i1, 0), (1 − k1 − i1, 0),

(1 − l1 + 1/2, 0), (−m1 + 3/2, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0).

(1/2, 0), (1 − i1, 0), (1/2, 0), (1 + i1, 0), (1/2, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0),

( f1 + i1 + k1 + βl1 + m1 + βr + s,−1).

. (3.30)

28


MEDIUM

3.2.4 Limiting Case

By putting β → 1 in Eqs. (3.25) and (3.30), we obtain VF and associated SS

for an ordinary SGF.

u(y, t) = P∞

∑a1=0

∗∑

(−1)ζ+a1+1θa1+d1+r+st−a1−b1+c1− f1−d1−r−sye1


× H1,9

9,11

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (1 − b1 − c1, 0), (−b1, 0), (1 − d1 − b1, 0),

(1 − g1 + f1, 0), (1 − f1, 1), (1 − s + e1/2, 0),

(1 − f1 + e1/2, 0), (1 − r − e1/2, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(0, 1), (1 + e1/2, 0), (1 − e1/2, 0),

(a1 + b1 − c1 + f1 + d1 + r + s,−1).

+ P◦∑

(−1)ξθa1+d1t−a1−b1−d1Kb1+1

a1!b1!d1!

× H1,4

4,6

M2t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (−b1, 0), (1 + b1, 1), (−d1, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(0, 1), (a1 + b1 + d1, 1).

+ A∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑r=0

∞

∑s=0

(−1)e1+ f1+g1+r+s M2g1ye1t− f1−r−s+1

θ−(r+s)e1! f1!g1!r!s!Ke1/2− f1

× H1,5

5,7

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0).

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0), (1 + e1/2, 0),

(0, 1), (1 − e1/2, 0), ( f1 + r + s,−1).

, (3.31)

29


MEDIUM

τ(y, t) = P∗∑

∞

∑t=0

∞

∑u=0

(−1)ζ+t+uKb1−e1/2+ f1+1ye1

r!s!t!u!b1!c1!d1!e1! f1!g1!

× t−b1+c1− f1−d1−r−t−s−u+1

M−2c1−2g1θ−d1−r−s−t−u

× H1,10

10,12

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1 + 1/2, 0), (1 − d1 − b1, 0), (1 − c1 + b1, 0),

(1 − g1 + f1, 0), (1 − f1 + e1/2, 0), (1 − f1, 1),

(1 − r − e1/2, 0), (−t + 3/2, 0), (1/2 − u, 0),

(1 − s + e1/2, 0).

(3/2, 0), (1/2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0), (3/2, 0),

(b1 − c1 + f1 + d1 + r + t + s + u,−1).

+∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∗∗∑

∞

∑r=0

∞

∑s=0

A(−1)e1+ f1+g1+ζ1+r+s+1ye1

e1! f1!g1!i1!j1!k1!l1!m1!r!s!M−2g1−2j1

× t− f1−i1−k1−l1−m1−r−s

θ−k1−l1−m1−r−sKe1/2− f1−i1+1/2

× H1,10

10,12

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−i1 + 3/2, 0), (1 − j1 + i1, 0), (1 − k1 − i1, 0),

(1 − l1 + 1/2, 0), (−m1 + 3/2, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0).

(1/2, 0), (1 − i1, 0), (1/2, 0), (1 + i1, 0), (1/2, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0),

( f1 + i1 + k1 + l1 + m1 + r + s,−1).

. (3.32)

3.2.5 Numerical Results

Several graphs are presented here for the analysis of some important phys-

ical aspects of the obtained solutions. The numerical results shows the pro-

files of VF and the adequate SS for the MHD flow. We analyze these results

30


MEDIUM

through different parameters of interest.

In Fig. (3.1) the effect of viscoelastic parameter θ on VF and the adequate

SS profiles has been shown. The graphs shows VF and the corresponding

SS for three different values of θ. It is observed that the profiles of VF and

SS are magnifying by increasing the values of θ. Fig. (3.2) shows the vari-

ation of the fractional parameter β. The VF along with the SS are changing

their monotonicity by increasing β. Fig. (3.3) shows the effect of the perme-

ability K of the porous medium. As expected, the velocity profiles increases

with the increase of the permeability K of the porous medium which is the

consequence that K reduces the drag force. Similarly, the profile of SS also

increases with the increase of K. Fig. (3.4) shows the variation of magnetic

parameter M. It is observed that by increasing the magnetic parameter M

the velocity decreases. The higher this value, the more prominent is the

reduction in velocity. This is due to the transverse magnetic field which

build up a drag force that opposes the flow. Also, it has been noticed that

by increasing the transverse magnetic field results in thinning the boundary

layer thickness. Thus the parameters M and K have opposite effects on the

velocity profile.

31


MEDIUM

0 1 2 3 40

0.2

0.4

0.6

q=1

q=2

q=3

y

u(y,

t)

0 1 2 3 40

0.02

0.04

0.06

0.08

q=1

q=2

q=3

y

t

Figure 3.1: VF u(y, 6) and SS τ(y, 6) profiles given by Eqs. (3.25) and (3.30)

for K = 3, β = 0.5, M = 5, P = 1.5, A = 2 and different values of θ.

0 1 2 3 40

0.2

0.4

0.6

b=0.1

b=0.3

b=0.5

y

u(y,

t)

0 1 2 3 40.02-

0

0.02

0.04

0.06

0.08

0.1

b=0.1

b=0.3

b=0.5

y

t


for K = 3, θ = 2, M = 5, P = 2, A = 2 and different values of β.

32


MEDIUM

0 1 2 3 40

0.2

0.4

0.6

K=2

K=2.5

K=3

y

u(y,

t)

0 1 2 3 40

0.02

0.04

0.06

0.08

K=2

K=2.5

K=3

y

t


for θ = 2, β = 0.5, M = 5, P = 2, A = 2 and different values of K.

0 1 2 3 40

0.2

0.4

0.6

M=3

M=5

M=7

y

u(y,

t)

0 1 2 3 40

0.02

0.04

0.06

0.08

M=3

M=5

M=7

y

t


for K = 3, θ = 2, β = 0.5, P = 2, A = 2 and different values of M.

33

3.3 FLOW INDUCED BY CONSTANT PRESSURE GRADIENT

3.3 Flow Induced by Constant Pressure Gradient

3.3.1 Problem Formulation and Solution

We take the same flow as discussed in the last section. The governing equa-

tion remains the same as given in Eq (3.11). We consider that the fluid as

well as the plate are at rest, and after time t = 0, the fluid suddenly starts

motion in the x-direction due to a constant pressure gradient. Following are

the IC and BCs of the given problem

u(y, 0) = 0; u(0, t) = 0; y > 0, t > 0,

u(y, t), ∂yu(y, t) → 0 as y → ∞ and t > 0.

(3.33)

Using the BCs (3.33), the solution of Eq. (3.19) is

u(y, q) = −C

Be−

√By +

C

B. (3.34)

We express Eq. (3.34) in series form as

u(y, q) = P∞

∑a1=0

∗∑

∞

∑h1=0

(−1)ζ+a1+h1+1θ(a1+d1+h1+r+s)β M2c1+2g1ye1

h1!g1! f1!e1!d1!c1!b1!a1!r!s!K−b1+e1/2− f1−1

× Γ(b1 + a1)Γ(c1 − b1)Γ(d1 + b1)Γ( f1 − e1/2)Γ(g1 − f1)Γ(h1 + f1)

q−a1−b1+c1− f1−h1−d1−βr−s+1Γ(−1)Γ(−a1)Γ(b1)Γ(−b1)Γ(e1/2)

× Γ(r + e1/2)Γ(s − e1/2)Γ(a1 + 1)

Γ(−e1/2)Γ(e1/2)Γ( f1)Γ(− f1)+ P

◦∑

∞

∑c1=0

(−1)ξ+c1θ(a1+d1)β

a1!b1!c1!d1!

× Γ(a1 + 1)Γ(b1 + 1)Γ(c1 − b1)Γ(d1 + 1)qa1+b1+d1−c1−1Kb1+1M2c1

Γ(−1)Γ(−1)Γ(b1)Γ(−b1). (3.35)

34


To get the analytic solution of VF, we apply discrete inverse LT to Eq. (3.35)

u(y, t) = P∞

∑a1=0

∗∑

∞

∑h1=0

(−1)ζ+a1+h1+1Γ(a1 + 1)Γ(b1 + a1)

a1!b1!c1!d1!e1! f1!g1!h1!r!s!Γ(−1)Γ(−a1)

× Γ(g1 − f1)Γ(h1 + f1)Γ(r + e1/2)Γ(s − e1/2)θ(a1+d1+h1+r+s)β

Γ(e1/2)Γ( f1)Γ(− f1)Γ(−e1/2)Γ(e1/2)K−b1+e1/2− f1−1M−2c1−2g1

× Γ(c1 − b1)t−a1−b1+c1− f1−h1−d1−βr−sye1Γ(d1 + b1)Γ( f1 − e1/2)

Γ(b1)Γ(−b1)Γ(−a1 − b1 + c1 − f1 − h1 − d1 − βr − s + 1)

+ P◦∑

∞

∑c1=0

(−1)ξ+c1 θ(a1+d1)βt−a1−b1−d1+c1Kb1+1M2c1Γ(a1 + 1)

a1!b1!c1!d1!Γ(−1)Γ(−1)Γ(b1)Γ(−b1)

× Γ(b1 + 1)Γ(c1 − b1)Γ(d1 + 1)

Γ(−a1 − b1 − d1 + c1 + 1). (3.36)

Using Fox H-function Eq. (3.36) becomes

u(y, t) = P∞

∑a1=0

∗∑

(−1)ζ+a1+1θ(a1+d1+r+s)βye1

a1!b1!c1!d1!e1! f1!g1!r!s!

× t−a1−b1+c1− f1−d1−βr−s

K−b1+e1/2− f1−1M−2c1−2g1

× H1,9

9,11

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1, 0), (1 − b1 − c1, 0), (1 − d1 − b1, 0),

(1 − g1 + f1, 0), (1 − f1, 1), (1 − s + e1/2, 0),

(−a1, 0), (1 − f1 + e1/2, 0), (1 − r − e1/2, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0), (1 − e1/2, 0),

(1 + f1, 0), (0, 1), (1 + e1/2, 0), (1 − e1/2, 0),

(1 − f1, 0), (a1 + b1 − c1 + f1 + d1 + βr + s,−1).

+ P◦∑

(−1)ξθ(a1+d1)βt−a1−b1−d1Kb1+1

a1!b1!d1!

× H1,4

4,6

M2t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (−b1, 0), (1 + b1, 1), (−d1, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(0, 1), (a1 + b1 + d1, 1).

. (3.37)

Shear stress

Substituting u(y, q) from Eq. (3.34) in Eq. (3.26), we get

τ(y, t) =P

qB−1/2e−

√By. (3.38)

35


τ(y, q) is rewritten in series form as

τ(y, q) = P∗∑

∞

∑h1=0

∞

∑t=0

∞

∑u=0

(−1)ζ+h1+t+uΓ(h1 + f1)M+2c1+2g1

b1!c1!d1!e1! f1!g1!h1!r!s!t!u!

× Γ(b1 + 1/2)Γ(c1 − b1)Γ(d1 + b1)Γ( f1 − e1/2)Γ(g1 − f1)

Γ(−1/2)Γ(b1)Γ(−b1)Γ(e1/2)Γ( f1)Γ(− f1)Γ(−e1/2)Γ(e1/2)

× Γ(s − e1/2)Γ(t − 1/2)Γ(u + 1/2)Γ(r + e1/2)Kb1−e1/2+ f1+1ye1

Γ(1/2)Γ(−1/2)θ(−d1−h1−r−s−t−u)βq−b1+c1− f1−h1−d1−βr−βt−s−u+1. (3.39)

We get the SS by applying the inverse LT to Eq. (3.39)

τ(y, t) = P∗∑

∞

∑h1=0

∞

∑t=0

∞

∑u=0

(−1)ζ+h1+t+uΓ(t − 1/2)Γ(u + 1/2)

b1!c1!d1!e1! f1!g1!h1!r!s!t!u!Γ(1/2)

× Γ(b1 + 1/2)Γ(c1 − b1)Γ(d1 + b1)Γ( f1 − e1/2)Γ(g1 − f1)

Γ(−1/2)Γ(b1)Γ(−b1)Γ(e1/2)Γ( f1)Γ(− f1)Γ(−e1/2)Γ(e1/2)

× Γ(s − e1/2)Kb1−e1/2+ f1+1ye1t−b1+c1− f1−h1−d1−βr−βt−s−u+1

Γ(−b1 + c1 − f1 − h1 − d1 − βr − βt − s − u + 1)M−2c1−2g1

× Γ(r + e1/2)Γ(h1 + f1)

Γ(−1/2)θ(−d1−h1−r−s−t−u)β. (3.40)

And by employing Fox H-function, we can also write the above equation as

τ(y, t) = P∗∑

∞

∑t=0

∞

∑u=0

(−1)ζ+t+uKb1−e1/2+ f1+1ye1

b1!c1!d1!e1! f1!g1!r!s!t!u!

× t−b1+c1− f1−d1−βr−βt−s−u+1


× H1,10

10,12

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1 + 1/2, 0), (1 − c1 + b1, 0), (1 − d1 − b1, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0), (−t + 3/2, 0),

(1/2 − u, 0).

(3/2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0), (1/2, 0),

(3/2, 0), (b1 − c1 + f1 + d1 + βr + βt + s + u,−1).

. (3.41)

Particular Case

By putting β → 1 in Eqs. (3.37) and (3.41), we get the VF and associated SS

36


of an ordinary SGF.

u(y, t) = P∞

∑a1=0

∗∑

(−1)ζ+a1+1θa1+d1+r+sye1

a1!b1!c1!d1!e1! f1!g1!r!s!

× t−a1−b1+c1− f1−d1−r−s

K−b1+e1/2− f1−1M−2c1−2g1

× H1,9

9,11

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1, 0), (1 − b1 − c1, 0), (1 − d1 − b1, 0),

(1 − g1 + f1, 0), (1 − f1, 1), (1 − s + e1/2, 0),

(−a1, 0), (1 − f1 + e1/2, 0), (1 − r − e1/2, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0), (1 − e1/2, 0),

(1 + f1, 0), (0, 1), (1 + e1/2, 0), (1 − e1/2, 0),

(1 − f1, 0), (a1 + b1 − c1 + f1 + d1 + r + s,−1).

+ P◦∑

(−1)ξθa1+d1t−a1−b1−d1Kb1+1

a1!b1!d1!

× H1,4

4,6

M2t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (−b1, 0), (1 + b1, 1), (−d1, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(0, 1), (a1 + b1 + d1, 1).

, (3.42)

τ(y, t) = P∗∑

∞

∑t=0

∞

∑u=0

(−1)ζ+t+uKb1−e1/2+ f1+1ye1

b1!c1!d1!e1! f1!g1!r!s!t!u!

× t−b1+c1− f1−d1−r−t−s−u+1


× H1,10

10,12

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1 + 1/2, 0), (1 − c1 + b1, 0), (1 − d1 − b1, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0), (−t + 3/2, 0),

(1/2 − u, 0).

(3/2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0), (1/2, 0),

(3/2, 0), (b1 − c1 + f1 + d1 + r + t + s + u,−1).

. (3.43)

37


3.3.2 Graphical Results


ical aspects of the obtained solutions. The numerical results shows the pro-

files of VF and the adequate SS for the MHD flow. We analyze these results

through different parameters of interest.

In Fig. (3.5) the impact of viscoelastic parameter θ on profiles of VF and

SS has been shown. The graphs are depicted for three dissimilar values of

θ. From these figures it is observed that the profile of velocity is reducing

while the SS profile is extending by increasing θ. Fig. (3.6) shows the vari-

ation of the fractional parameter β. The velocity along with the SS profiles

changed its monotonicity by increasing β. Fig. (3.7) depicts the impact of

permeability K of the porous medium. As expected, the velocity profile is

amplifying by enlarging the permeability K, which is the consequence that

K reduces the drag force. Similarly, the profile of SS also increases with the

increase of K. Fig. (3.8) shows the variation of magnetic parameter M. It is

observed that by magnifying M the velocity is diminishing. This is due to

the transverse magnetic field which build up a drag force that opposes the

flow. Also, it has been noticed that by increasing the transverse magnetic

field results in thinning the boundary layer thickness. Thus the impact of M

and K have opposite effects on the velocity profile.

38


0 1 2 30

0.5

1

1.5

q=1.3

q=1.5

q=1.7

y

u(y

)

0 1 2 30

0.1

0.2

0.3

0.4

0.5

q=1.3

q=1.5

q=1.7

y

t

Figure 3.5: VF and SS profiles given by Eqs. (3.37) and (3.41) for K = 2, β =

0.6, t = 4, M = 0.3, P = 1.2 and different values of θ.

0 1 2 30

0.5

1

1.5

b=0.4

b=0.6

b=0.8

y

u(y

)

0 1 2 30

0.1

0.2

0.3

0.4

b=0.4

b=0.6

b=0.8

y

t

Figure 3.6: VF and SS profiles given by Eqs. (3.37) and (3.41) for K = 2, θ =

1.5, t = 4, M = 0.3, P = 1.2 and different values of β.

39


0 1 2 30

0.5

1

1.5

K=1.5

K=2

K=2.5

y

u(y

)

0 1 2 30

0.1

0.2

0.3

0.4

0.5

K=1.5

K=2

K=2.5

y

t

Figure 3.7: VF and SS profiles given by Eqs. (3.37) and (3.41) for θ = 1.5, β =

0.6, t = 4, M = 0.3, P = 1.2 and different values of K.

0 1 2 30

0.5

1

1.5

M=0.3

M=1.3

M=2.3

y

u(y

)

0 1 2 30

0.1

0.2

0.3

0.4

M=0.3

M=1.3

M=2.3

y

t

Figure 3.8: VF and SS profiles given by Eqs. (3.37) and (3.41) for K = 2, β =

0.6, t = 4, θ = 1.5, P = 1.2 and different values of M.

40

3.4 FLOW DUE TO UNIFORM AND NON-UNIFORM ACCELERATING

PLATE

3.4 Flow Due to Uniform and Non-Uniform Ac-

celerating Plate

We take an unsteady incompressible flow of homogenous and electrically

conducting second-grade fluid bounded by a rigid plate at y = 0. The plate

is taken normal to y−axis and the fluid saturates the porous medium y >

0. The electrically conducting fluid is stressed by a uniform magnetic field

βo parallel to the y−axis, while the induced magnetic field is neglected by

choosing a small magnetic Reynolds number. Initially, both the plate and

the fluid are at rest, and after time t = 0, it is suddenly set into motion by

translating the flat plate in its plane, with a constant velocity A. The IC and

BCs corresponding to uniform accelerating plate are

u(y, 0) = 0, u(0, t) = At, t > 0, y > 0,

u(y, t), ∂yu(y, t) → 0 t > 0, y → ∞.

(3.44)

The IC and BCs corresponding to non-uniform accelerating plate are

u(y, 0) = 0; u(0, t) = At2; t > 0, y > 0,

u(y, t), ∂yu(y, t) → 0 t > 0, y → ∞.

(3.45)

41


PLATE

3.4.1 Results

After solving the governing equation (3.11) by using the conditions (3.44),

we get the VF and SS profiles corresponding to uniform accelerating plate

u(y, t) = P∞

∑a1=0

∗∑

(−1)ζ+a1+1θβ(a1+d1+r+s)t−a1−b1+c1− f1−d1−βr−sye1


× H1,9

9,11

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (1 − b1 − c1, 0), (−b1, 0), (1 − d1 − b1, 0),

(1 − g1 + f1, 0), (1 − f1, 1), (1 − s + e1/2, 0),

(1 − f1 + e1/2, 0), (1 − r − e1/2, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(0, 1), (1 + e1/2, 0), (1 − e1/2, 0),

(a1 + b1 − c1 + f1 + d1 + βr + s,−1).

+ P◦∑

(−1)ξθβ(a1+d1)t−a1−b1−d1Kb1+1

a1!b1!d1!

× H1,4

4,6

M2t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (−b1, 0), (1 + b1, 1), (−d1, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(0, 1), (a1 + b1 + d1, 1).

+ A∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑r=0

∞

∑s=0

(−1)e1+ f1+g1+r+s M2g1ye1θ(r+s)β

e1! f1!g1!r!s!t−(− f1−βr−s+1)Ke1/2− f1

× H1,5

5,7

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0).

(1 − e1/2, 0), (1 − f1, 0),

(1 + f1, 0), (0, 1), (1 + e1/2, 0),

(1 − e1/2, 0), ( f1 + βr + s − 1,−1).

, (3.46)

42


PLATE

τ(y, t) = P∗∑

∞

∑t=0

∞

∑u=0

(−1)ζ+t+uKb1−e1/2+ f1+1ye1

r!s!t!u!b1!c1!d1!e1! f1!g1!

× t−b1+c1− f1−d1−βr−βt−s−u+1

M−2c1−2g1θβ(−d1−r−s−t−u)

× H1,10

10,12

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1 + 1/2, 0), (1 − c1 + b1, 0), (1 − d1 − b1, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − r − e1/2, 0), (−t + 3/2, 0), (1/2 − u, 0),

(1 − s + e1/2, 0).

(3/2, 0), (1/2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0), (3/2, 0),

(b1 − c1 + f1 + d1 + βr + βt + s + u,−1).

+∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∗∗∑

∞

∑r=0

∞

∑s=0

A(−1)e1+ f1+g1+ζ1+r+s+1ye1

e1! f1!g1!i1!j1!k1!l1!m1!r!s!

× M2g1+2j1t− f1−i1−k1−βl1−m1−βr−s+1

θ(−k1−l1−m1−r−s)βKe1/2− f1−i1+1/2

× H1,10

10,12

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−i1 + 3/2, 0), (1 − j1 + i1, 0), (1 − k1 − i1, 0),

(1 − l1 + 1/2, 0), (1 − f1, 1), (1 − s + e1/2, 0),

(1 − r − e1/2, 0), (1 − f1 + e1/2, 0), (−m1 + 3/2, 0),

(1 − g1 + f1, 0).

(1/2, 0), (1 − i1, 0), (1 + i1, 0), (1 − e1/2, 0),

(1 + f1, 0), (1 − e1/2, 0), (1/2, 0), (1/2, 0),

(1 + e1/2, 0), (1 − f1, 0), (0, 1),

( f1 + i1 + k1 + βl1 + m1 + βr + s − 1,−1).

.

(3.47)

After solving the governing equation (3.11) by using conditions (3.45), we

obtain the VF and associated SS corresponding to non-uniform accelerating

43


PLATE

plate

u(y, t) = P∞

∑a1=0

∗∑

(−1)ζ+a1+1θ(a1+d1+r+s)βt−a1−b1+c1− f1−d1−βr−sye1


× H1,9

9,11

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − b1 − c1, 0), (−b1, 0), (1 − d1 − b1, 0),

(−a1, 0), (1 − f1 + e1/2, 0), (1 − g1 + f1, 0),

(1 − f1, 1), (1 − s + e1/2, 0), (1 − r − e1/2, 0).

(2, 0), (2, 0), (1 − b1, 0), (1 + b1, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(0, 1), (1 + e1/2, 0), (1 − e1/2, 0),

(a1 + b1 − c1 + f1 + d1 + βr + s,−1).

+ P◦∑

(−1)ξθ(a1+d1)βt−a1−b1−d1Kb1+1

a1!b1!d1!

× H1,4

4,6

M2t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−a1, 0), (−b1, 0), (1 + b1, 1), (−d1, 0).

(2, 0), (1 − b1, 0), (2, 0), (1 + b1, 0),

(0, 1), (a1 + b1 + d1, 1).

+ 2A∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∞

∑r=0

∞

∑s=0

(−1)e1+ f1+g1+r+s M2g1ye1θ(r+s)β

t−(− f1−βr−s+2)e1! f1!g1!r!s!Ke1/2− f1

× H1,5

5,7

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0).

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (1 − e1/2, 0),

(0, 1), ( f1 + βr + s − 2,−1).

, (3.48)

44


PLATE

τ(y, t) = P∗∑

∞

∑t=0

∞

∑u=0

(−1)ζ+t+uKb1−e1/2+ f1+1ye1

b1!c1!d1!e1! f1!g1!r!s!t!u!

× t−b1+c1− f1−d1−βr−βt−s−u+1

M−2c1−2g1θ(−d1−r−s−t−u)β

× H1,10

10,12

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−b1 + 1/2, 0), (1 − c1 + b1, 0), (1 − d1 − b1, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0),

(−t + 3/2, 0), (1/2 − u, 0).

(3/2, 0), (1 − b1, 0), (1 + b1, 0), (0, 1),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (1 − e1/2, 0), (1/2, 0), (3/2, 0)

, (b1 − c1 + f1 + d1 + βr + βt + s + u,−1).

+∞

∑e1=0

∞

∑f1=0

∞

∑g1=0

∗∗∑

∞

∑r=0

∞

∑s=0

2A(−1)e1+ f1+g1+ζ1+r+s+1ye1

e1! f1!g1!i1!j1!k1!l1!m1!r!s!

× t− f1−i1−k1−βl1−m1−βr−s+2

M−2g1−2j1θ(−k1−l1−m1−r−s)βKe1/2− f1−i1+1/2

× H1,10

10,12

θβ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(−i1 + 3/2, 0), (1 − j1 + i1, 0), (1 − k1 − i1, 0),

(1 − l1 + 1/2, 0), (−m1 + 3/2, 0), (1 − f1, 1),

(1 − s + e1/2, 0), (1 − r − e1/2, 0),

(1 − f1 + e1/2, 0), (1 − g1 + f1, 0).

(1/2, 0), (1 − i1, 0), (1 + i1, 0), (1/2, 0),

(1 − e1/2, 0), (1 − f1, 0), (1 + f1, 0),

(1 + e1/2, 0), (0, 1), (1 − e1/2, 0), (1/2, 0),

( f1 + i1 + k1 + βl1 + m1 + βr + s − 2,−1).

. (3.49)

45

Chapter 4

Some Exact Solution of

Generalized Jeffrey Fluid

4.1 Introduction

This chapter is classified into two sections. In these sections, we solve two

problems related to generalized Jeffrey fluid (GJF) [63, 64]. Amongst the

non-Newtonian fluids, the Jeffrey fluid (JF) is considered to be one of the

simplest type of model which best explain the rheological effects of vis-

coelastic fluids. The JF is relatively simple linear model using the time

derivatives instead of convective derivatives. Khadrawi et al. [65] solved

some basic viscoelastic fluid problems using JF. Qi and Xu [66] solved Stoke’s

first problem for a viscoelastic fluid with the GJF. Nadeem et al. [67] investi-

gated stagnation flow of a JF over a shrinking sheet. Khan et al. [68] studied

some unsteady flows of JF which lies between two side walls.

46

4.2 FLOW BETWEEN TWO SIDE WALLS PERPENDICULAR TO THE PLATE

4.2 Flow Between Two Side Walls Perpendicular

to The Plate

In this section we presents some new exact solutions corresponding to three

unsteady flow problems of GJF produced by a flat plate which lies between

two side walls. The flow of GJF is set into motion by (i) impulsive mo-

tion of the plate, (ii) impulsive accelerating plate, and (iii) non-uniformly

accelerated plate. Governing equation is achieved by using the approach

of fractional calculus. The analytic solutions are established through finite

Fourier sine transform (FFST) and discrete Laplace transform (LT). The se-

ries solution, satisfying all IC and BCs, are stated in Fox H-function. The

similar solutions for ordinary JF, performing the same motion, are obtained

as limiting case of the general solutions. Also, the obtained results are ana-

lyzed graphically through various pertinent parameter.

We take an unsteady GJF saturating the space above a plane wall which is

perpendicular to the y−axis and between two side walls perpendicular to

the plane. At first the fluid as well as the plane wall is at rest and at time

t = 0+ the fluid is set into flow by translating the bottom wall in its own

plane, with a time dependent velocity Vtm.

4.2.1 Mathematical Modelling

The Cauchy stress tensor for unsteady and incompressible GJF is [54]

T = S − pI, S =µ

1 + λ

[

A + θβ

(

∂βA

∂tβ+ (V.∇)A

)]

, (4.1)

where θ and λ are retardation and relaxation times, respectively. For the

following problem, we consider the VF and an extra stress of the form

V = u(y, z, t)i, S = S(y, z, t), (4.2)

47


where u is the velocity and i is the unit vector along the x-direction. The

continuity equation for such flows is automatically satisfied. We take the

extra stress S independent of x as the VF is independent of x. Also, at t = 0

the fluid being at rest is given by

S(y, z, 0) = 0, (4.3)

therefore from Eqs. (4.1) and (4.2) it results that Sxx = Syz = Syy = Szz = 0

and the relevant fractional order differential equations

τ1 =µ

(1 + λ)(1 + θβ ∂β

∂tβ)∂yu(y, z, t), (4.4)

τ2 =µ

(1 + λ)(1 + θβ ∂β

∂tβ)∂zu(y, z, t), (4.5)

where τ1 = Sxy and τ2 = Sxz are the tangential stresses. In the absence of

body forces, the balance of linear momentum becomes

∂yτ1 + ∂zτ2 = ∂x p + ρ∂tu, ∂y p = 0, ∂z p = 0. (4.6)

Here the constant density of the fluid is denoted by ρ. Putting τ1 and τ2

from Eq. (4.4) and Eq. (4.5) into Eq. (4.6) and neglecting pressure gradient

in the flow direction, we obtain the fractional differential equation

∂tu(y, z, t) =ν

(1 + λ)(1 + θβ ∂β

∂tβ)(∂2

y + ∂2z)u(y, z, t), (4.7)

where ν = µ/ρ is the kinematic viscosity of the fluid. The associated IC and

BCs are

(y, z, 0) = ∂tu(y, z, 0) = 0; y ≥ 0 and 0 ≤ z ≤ h,

u(0, z, t) = Vtm; f or t ≥ 0 and 0 ≤ z ≤ h,

u(y, 0, t) = u(y, h, t) = 0; y, t ≥ 0.

(4.8)

Here h is the distance between the two side walls. Furthermore, we employ

the natural conditions as well

u(y, z, t), ∂yu(y, z, t) → 0 as y → ∞, z ∈ (0, h) and t > 0. (4.9)

48


4.2.2 Impulsive Motion of The Plate (m = 0)

To get the analytic solution of VF, we multiply both sides of the governing

equation (4.7) by sin(nπzh ), and then integrate the obtained result from 0 to

h with respect to z, we get the following fractional differential equation

(1 + λ)∂

∂tun(y, n, t) = ν(1 + θβ ∂β

∂tβ)

∂2un(y, n, t)

∂y2− ν(

nπ

h)(1 + θβ ∂β

∂tβ)

× un(y, n, t). (4.10)

We take the LT of Eq. (4.10) to get the image function un(y, n, s) of un(y, n, t),

along with the boundary and natural conditions

∂2

∂y2un(y, n, s)−

[

ξ2 +s(1 + λ)

(1 + θβsβ)

]

un(y, n, s) = 0, (4.11)

u(0, n, s) =V

s, u(y, n, s) → 0 as y → ∞, z ∈ (0, h) and t > 0, (4.12)

where ξ = nπh . The solution of above differential equation is in the following

form

un(y, n, s) =V

sexp

[

− y

√

ξ2 +s(1 + λ)

ν(1 + θβsβ)

]

. (4.13)

To avoid difficult calculations of contour integrals and residues, we take the

discrete inverse LT technique to get analytic solution for the VF, but first we

express Eq. (4.13) in series form as

un(y, n, s) = V∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

(−1)j+n+p+qyjξ j−2nν−nθ−n−p+β

j!n!q!p!Γ(n)Γ(−n)Γ( j2)

× λn−qΓ(q − n)Γ(n − j2)Γ(p + n)

s−n+β(p+n)+1. (4.14)

Now employing inverse LT to Eq. (4.14), to obtain

un(y, n, t) = V∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

(−1)j+n+p+qyjξ j−2nν−nθ−n−p+β


× λn−qΓ(p + n)t−n+β(p+n)Γ(q − n)Γ(n − j2)

Γ(−n + β(p + n) + 1). (4.15)

49


Taking the inverse FFST to get the analytic solution of the VF

u(y, z, t) =2

h

∞

∑m=1

sin(mπz

h)un

=2V

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

(−1)j+n+p+qyjξ j−2n

j!n!q!p!

×λn−qθ−n−p+βt−n+β(p+n)Γ(p + n)Γ(q − n)Γ(n − j2)

νnΓ(n)Γ(−n)Γ( j2)Γ(−n + β(p + n) + 1)

.

(4.16)

Fox H-function is used to write Eq. (4.16) in a solid form as

u(y, z, t) =2V

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

(−1)j+n+qξ j−2nν−nλn−q

j!n!q!y−jθn−βtn−βn

×H1,3

3,5

[

tβ

θ

∣

∣

∣

∣

∣

∣

(1 − n, 1), (1 − q + n, 0), (1 − n + j2 , 0)

(0, 1)(1 − n, 0)(1 + n, 0)(1 − j/2, 0)(n − βn, β)

]

.

(4.17)

To get the SS, we apply LT to Eqs. (4.4) and (4.5), to obtain

τ1 =µ(1 + θβsβ)

1 + λ

∂u(y, z, s)

∂y, (4.18)

τ2 =µ(1 + θβsβ)

1 + λ

∂u(y, z, s)

∂z. (4.19)

Taking inverse FFST to Eq. (4.13) to get u(y, z, s) and then putting it into Eq.

(4.18), we obtain

τ1 =2Vξ

hs

µ(1 + θβsβ)

1 + λ

∞

∑n=1

sin(nπz

h)exp

[

− y

√

ξ2 +s(1 + λ)

ν(1 + θβsβ)

]

×[

√

1 +s(1 + λ)

ξ2ν(1 + θβsβ)

]

. (4.20)

50



τ1 =∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

∞

∑m=0

(−1)j+n+p+q+w+y+z+x+m

j!n!q!p!w!y!z!x!m!

× Γ(p + n)Γ(q − n)Γ(n − j2)Γ(w − 1

2)Γ(x + 12)

Γ(n)Γ(−n)Γ( j2)Γ(

12)Γ(

−12 )Γ(1

2)Γ(m)Γ(−m)

× 2Vρνyjξ j−2n+2mνm−nλn−qΓ(m − 12)Γ(y − m)Γ(z + m)

θ−(−n−p+w+x+y+z+β)hs−n+β(p+n)+m−βw−βy+ 12

. (4.21)

Taking the inverse LT of (4.21), we obtain

τ1 =2ρV

h

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

∞

∑m=0

Γ(y − m)

× Γ(q − n)Γ(n − j2)Γ(w − 1

2)Γ(x + 12)Γ(m − 1

2)


12)Γ(

−12 )Γ(1

2)Γ(m)Γ(−m)

× Γ(p + n)Γ(z + m)t−n+β(p+n)+m−βw−βy− 12

Γ(−n + β(p + n) + m − βw − βy + 12)

× λn−qθ−n−p+w+x+y+z+βyjξ j−2n+2mνm−n+1

(−1)−(j+n+p+q+w+y+z+x+m) j!n!q!p!w!y!z!x!m!. (4.22)

In order to obtain a more suitable form of τ1, we use Fox H-function as

τ1 =2ρV

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

× t−n+β(p+n)−βw−βy− 12 (−1)j+n+p+q+w+y+z+xyjξ j−2n

ν−n+1λn−qθ−n−p+w+x+y+z+β j!n!q!p!w!y!z!x!

× H1,8

8,10

−ξ2t

ν

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − p + n, 0), (1 − q + n, 0), (1 − n + j2 , 0),

(1 + 12 , 1), (1 − y,−1), (1 − z, 1),

(1 − w + 12 , 0), (1 − x − 1

2 , 0).

(1 + 12 , 0), (1 − 1

2 , 0),

(1, 1), (1,−1), (0, 1), (1, 1),

(1 − n, 0), (1 + n, 0), (1 − j2 , 0),

(1 + n − β(p + n − w − y)− 12 , 1).

. (4.23)

In the similar fashion we can find τ2(y, z, t) from Eqs. (4.17) and (4.19).

51


4.2.3 Impulsive Acceleration of The Plate (m = 1)

Following the same practice as we have done in the last section, the analytic

solution of VF and the associated SS for the flow of JF due to impulsive

accelerating (uniform motion) of plate are given by

u(y, z, t) =2V

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0


j!n!q!y−jθn−β−1tn−βn−1

×H1,3

3,5

tβ

θ

∣

∣

∣

∣

∣

∣

(−n + 1, 1), (−q + 1 + n, 0), (−n + 1 + j2 , 0).

(0, 1), (1 − n, 0), (n + 1, 0), (1 − j/2, 0), (n − βn + 1, β).

.

(4.24)

τ1 =2ρV

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0



× H1,8

8,10

−ξ2t

ν

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − p + n, 0), (1 − q + n, 0), (1 − n + j2 , 0),

(1 + 12 , 1), (1 − y,−1), (1 − z, 1),

(1 − w + 12 , 0), (1 − x − 1

2 , 0).

(1 + 12 , 0), (1 − 1

2 , 0),

(1, 1), (1,−1), (0, 1), (1, 1),

(1 − n, 0), (1 + n, 0), (1 − j2 , 0),

(1 + n − β(p + n − w − y)− 32 , 1).

. (4.25)

4.2.4 Non-Uniform Acceleration of The Plate (m = 2)

Adopting the same methodology of the last section, the resultant expression

for the VF and corresponding SS for the flow of JF due to non-uniformly

52


accelerated plate are given by

u(y, z, t) =2V

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0


j!n!q!y−jθn−βtn−βn−2

×H1,3

3,5

tβ

θ

∣

∣

∣

∣

∣

∣

(−n + 1, 1), (−q + n + 1, 0), (−n + 1 + j2 , 0).

(0, 1), (−n + 1, 0), (n + 1, 0), (−j/2 + 1, 0), (−βn + 2 + n, β).

.

(4.26)

τ1 =2ρV

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0



× H1,8

8,10

−ξ2t

ν

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − p + n, 0), (1 − q + n, 0), (1 − n + j2 , 0),

(1 + 12 , 1), (1 − y,−1), (1 − z, 1),

(1 − w + 12 , 0), (1 − x − 1

2 , 0).

(1 + 12 , 0), (1 − 1

2 , 0),

(1, 1), (1,−1), (0, 1), (1, 1),

(−n + 1, 0), (n + 1, 0), (− j2 + 1, 0),

(n + 1 − β(p + n − w − y)− 52 , 1).

. (4.27)

Special case

By putting β → 1 in Eqs. (4.17) and (4.23), we get the VF and the adequate

SS profiles for an ordinary JF produced by the abrupt motion of the flat plate

u(y, z, t) =2V

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0


j!n!q!y−jθn−1

×H1,3

3,5

[

tθ

∣

∣

∣

∣

∣

∣

(1 − n, 1), (1 − q + n, 0), (1 − n + j2 , 0).

(0, 1), (1 − n, 0), (1 + n, 0), (1 − j/2, 0), (0, 1).

]

.

(4.28)

53


τ1 =2ρV

h

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

× tp−w−y− 12 (−1)j+n+p+q+w+y+z+xyjξ j−2n

ν−n+1λn−qθ−n−p+w+x+y+z+ j!n!q!p!w!y!z!x!

× H1,8

8,10

−ξ2t

ν

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − p + n, 0), (1 − q + n, 0), (1 − n + j2 , 0),

(1 + 12 , 1), (1 − y,−1), (1 − z, 1),

(1 − w + 12 , 0), (1 − x − 1

2 , 0).

(1 + 12 , 0), (1 − 1

2 , 0),

(1, 1), (1,−1), (0, 1), (1, 1),

(1 − n, 0), (1 + n, 0), (1 − j2 , 0),

(1 + n − (p + n − w − y)− 12 , 1).

. (4.29)

Similarly, we can get VF and SS profiles for an ordinary JF due to impulsive

accelerating plate and non-uniformly accelerating plate by putting β → 1 in

Eqs. (4.26), (4.27), (4.28) and (4.29).

4.2.5 Results and Discussion

Unsteady flows of GJF produced by abrupt motion of the flat plate which

lies between two side walls, are examined. Exact analytical solutions are es-

tablished for such flow problem using FFST and LT technique. The obtained

solutions are expressed in series form using Fox H-function. Several graphs

are presented here for the analysis of some important physical aspects of

the obtained solutions. Solutions regarding ordinary JF are also obtained as

limiting case of general solutions. The numerical results shows the profiles

of VF and the adequate SS for the flow given by Eqs. (4.17) and (4.23). We

analyze these results through different parameters of interest.

The effects of fractional parameters β of the model are important for us to

be discussed. In Fig. (4.1), we depict the profiles of VF and SS for three

54


different values of β. It is observed from these figures that the flow VF as

well as the SS increases with increasing β, which corresponds to the shear

thinning phenomenon. Fig. (4.2) are sketched to show the VF and the SS

profiles at different values of λ. It is noticeable that VF as well as the SS de-

creases by increasing λ. In order to study the effects of material parameter

θ, we have plotted Fig. (4.3), where it appears that the VF is also a strong

function of the material parameter θ of JF. It can be observed that by increas-

ing the material parameter θ acts as an increase of the magnitude of velocity

components near the plate, and this again corresponds to the shear-thinning

behavior of the examined non-Newtonian fluid. Fig. (4.4) presents, the VF

and the SS profiles at different values of y. It is noticeable that velocity and

SS decreases by increasing y. Also, by increasing y the velocity becomes

steady, which shows that the BC (4.9) is satisfied.

0 0.2 0.4 0.6 0.8 10.01-

0

0.01

0.02

0.03

0.04

b=0.3

b=0.6

b=0.9

y

u(y

)

0 0.2 0.4 0.6 0.8 11- 10

3-´

0

1 103-

´

2 103-

´

3 103-

´

4 103-

´

5 103-

´

b=0.3

b=0.6

b=0.9

y

t

Figure 4.1: VF and SS profiles when K = 2, t = 4, M = 0.3, P = 1.2, A = 1, λ =

6 and for different values of β.

55


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

l=4

l=6

l=8

y

u(y

)

0 0.2 0.4 0.6 0.8 10

1 103-

´

2 103-

´

3 103-

´

4 103-

´

l=4

l=6

l=8

y

t

Figure 4.2: VF and SS profiles when K = 2, β = 0.6, t = 4, M = 0.3, P = 1.2, A

= 1 and for different values of λ.

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

q=8

q=10

q=12

y

u(y

)

0 0.2 0.4 0.6 0.8 10

1 103-

´

2 103-

´

3 103-

´

q=8

q=10

q=12

y

t

Figure 4.3: VF and SS profiles when β = 0.6, t = 4, M = 0.3, P = 1.2, A = 1, λ

= 6 and for different values of θ.

56


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

y=0.2

y=0.4

y=0.6

t

u(y

)

0 0.2 0.4 0.6 0.8 10

1 103-

´

2 103-

´

3 103-

´

4 103-

´

y=0.2

y=0.4

y=0.6

t

t

Figure 4.4: VF and SS profiles when K = 2, β = 0.6, t = 4, P = 1.2, A = 1, λ = 6

and for different values of y.

57

4.3 OSCILLATORY FLOW PASSING THROUGH A RECTANGULAR DUCT

4.3 Oscillatory Flow Passing Through a Rectan-

gular Duct

In this section, we presents some new exact solutions corresponding to un-

steady MHD [69]− [72] flow of GJF in a long porous rectangular duct oscil-

lating parallel to its length. The exact solutions are established by means of

the double FFST and discrete LT. The series solution of VF, associated SS and

volume flow rate in terms of Fox H-function, satisfying all imposed IC and

BCs, have been obtained. Also, the obtained results are analyzed graphi-

cally through various pertinent parameters.

Much attention has been given to the flows of rectangular duct because of

its wide range applications in industries. Gardner et al. [73] discussed MHD

duct flow of bi-cubic B-spline finite element in two-dimensions. Fetecau [74]

investigated the motions of Oldroyd-B fluid in a channel of rectangular

cross-section. Nazar et al. [75] examined oscillating flow passing through

rectangular duct for Maxwell fluid using integral transforms. Unsteady

MHD flow of Maxwell fluid passing through porous rectangular duct was

studied by Sultan et al. [76]. Tsangaris and Vlachak [77] discussed analytic

solution of oscillating flow in a duct of Navier-Stokes equations.

We take an incompressible flow of GJF in a porous rectangular duct under

an imposed transverse magnetic field whose sides are at x = 0, x = d, y = 0

and y = h. At time t = 0+ the duct begins to oscillate along z−axis.

4.3.1 Problem Formulation

The continuity and momentum equations for the MHD flow passing through

a porous medium are

∇ · V = 0, ρ(dV

dt) = divT + J × B + R, (4.30)

58


where velocity is represented by V, density by ρ, Cauchy stress tensor by

T, Darcy’s friction in the porous medium by R and magnetic body force by

J × B, which is defined as

J × B = −σβ2oV, (4.31)

where the current density is denoted by J, magnetic field by B, electrical

conductivity by σ and strength of magnetic field by βo. For the GJF the

Darcy’s friction is defined as follow

R = − µφ

κ(1 + λ)(1 + θβ ∂β

∂tβ)V, (4.32)

where φ (0 < φ < 1) and κ(> 0) are the porosity and the permeability of the

porous medium. We take the VF and extra stress in the following problem

as

V = (0, 0, w(x, y, t)), S = S(x, y, t), (4.33)

where w is the velocity in the z-direction. The continuity equation for such

flows is automatically satisfied. Also, at t = 0 the fluid being at rest is given

by

S(x, y, 0) = 0, (4.34)

therefore from Eqs. (4.1) and (4.33) it results that Sxx = Syy = Syz = Szz = 0,

and the relevant fractional differential equations are

τ1 =µ

(1 + λ)(1 + θβ ∂β

∂tβ)∂xw(x, y, t), (4.35)

τ2 =µ

(1 + λ)(1 + θβ ∂β

∂tβ)∂yw(x, y, t). (4.36)

We denote the tangential stresses Sxy and Sxz by τ1 and τ2, respectively. Af-

ter solving Eqs. (4.30), (4.35) and (4.36), we get the governing Eq. (in the

absence of pressure gradient in the flow direction) as

(1 + λ)∂tw(x, y, t) = ν(1 + θβ ∂β

∂tβ)(∂2

x + ∂2y)w(x, y, t)− νK(1 + θβ ∂β

∂tβ)

59


× w(x, y, t)− H(1 + λ)w(x, y, t), (4.37)

where H =σB2

0ρ is the magnetic parameter, K = φ

κ is the porosity parameter

and ν = µ/ρ is the kinematic viscosity. The associated IC and BCs are

w(x, y, 0) = ∂tw(x, y, 0) = 0, (4.38)

w(0, y, t) = w(x, 0, t) = w(d, y, t) = w(x, h, t) = Ucos(ωt), (4.39)

or w(0, y, t) = w(x, 0, t) = w(d, y, t) = w(x, h, t) = Usin(ωt), (4.40)

t > 0, 0 < x < d and 0 < y < h.

The solutions of problems (4.37), (4.38), (4.39) and (4.37), (4.38), (4.40) are

denoted by u(x, y, t) and v(x, y, t), respectively. We define the complex VF

as

F(x, y, t) = u(x, y, t) + iv(x, y, t), (4.41)

which is the solution of the problem

(1 + λ)∂tF(x, y, t) = ν(1 + θβ ∂β

∂tβ)(∂2

x + ∂2y)F(x, y, t)− νK(1 + θβ ∂β

∂tβ)

× F(x, y, t)− H(1 + λ)F(x, y, t), (4.42)

F(x, y, 0) = ∂tF(x, y, 0) = 0, (4.43)

F(0, y, t) = F(x, 0, t) = F(d, y, t) = F(x, h, t) = Ueiωt, (4.44)

t > 0, 0 < x < d and 0 < y < h.

The solution of the problem (4.42)-(4.44) will be obtained by means of the

double FFST and LT.

The double FFST of function F(x, y, t) is denoted by

Fmn(t) =∫ d

0

∫ h

0sin(

mπx

d)sin(

nπy

h)F(x, y, t)dxdy, m, n = 1, 2, 3, .. (4.45)

.

60


4.3.2 Solution of the Problem

To solve the problem, first we multiply by sin(mπxd ) and sin(nπy

h ) to both

sides of Eq. (4.42), then taking the double integration w.r.t x and y over

[0, d]× [0, h], and using Eq. (4.45), we get

(1 + λ)∂Fmn(t)

∂t+ νλmn(1 + θβ ∂β

∂tβ)Fmn(t) + H(1 + λ)Fmn(t) + Fmn(t)

× νK(1 + θβ ∂β

∂tβ) = νλmnU

[1 − (−1)m][1 − (−1)n]

ζmλn(1 + iωθβ)eiωt, (4.46)

where

ζm =mπ

d, λn =

nπ

hand λmn = ζ2

m + λ2n.

The FFST Fmn(t) have to satisfy the following IC

Fmn(0) = ∂tFmn(0) = 0. (4.47)

We apply LT to Eq. (4.46) and using IC (4.47) to get

Fmn(s) =νλmnU[1 − (−1)m](1 + iωθβ)[1 − (−1)n]

ζmλn(s − iω)[(1 + λ)(s + H) + ν(1 + θβsβ)(λmn + K)]. (4.48)

To keep away from complicated calculations of contour integrals and residues,

we express Eq. (4.48) in series form as

Fmn(s) =νλmnU[1 − (−1)m](1 + iωθβ)[1 − (−1)n]

ζmλn(s − iω)

∞

∑p=0

∞

∑q=0

∞

∑r=0

×∞

∑s=0

∞

∑l=0

νp+1λsθqβKp−r Hl(λmn)r+1Γ(q − p)Γ(r − p)

(−1)−(p+q+r+s+l)q!r!s!l!Γ(p)Γ(p)Γ(1 + p)Γ(1 + p)

× Γ(s + p + 1)Γ(l + p + 1)

sl−qβ+p+1. (4.49)

We apply the discrete inverse LT to Eq. (4.49), to obtain

Fmn(t) =eiωtU[1 − (−1)m][1 − (−1)n]νλmn(1 + iωθβ)

ζmλn

∞

∑p=0

∞

∑q=0

×∞

∑r=0

∞

∑s=0

∞

∑l=0

νp+1λsθqβKp−r Hl(λmn)r+1Γ(q − p)Γ(r − p)

(−1)−(p+q+r+s+l)q!r!s!l!Γ(p)Γ(p)Γ(1 + p)Γ(1 + p)

× Γ(s + p + 1)Γ(l + p + 1)tl−qβ+p

Γl − qβ + p + 1. (4.50)

61


Taking the inverse FFST, we get the analytic solution of the VF

F(x, y, t) =4

dh

∞

∑m=1

∞

∑n=1

sin(ζmx)sin(λny)Fmn(x, y, t)

=4eiωtU(1 + iωθβ)

dh

∞

∑m=1

∞

∑n=1

sin(ζx)sin(λny)

[1 − (−1)m]−1[1 − (−1)n]−1

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

∞

∑l=0

νp+1λsθqβKp−r Hl(λmn)r+1tl−qβ+p

ζmλn(−1)−(p+q+r+s+l)q!r!s!l!

× Γ(q − p)Γ(r − p)Γ(s + p + 1)Γ(l + p + 1)

Γ(p)Γ(p)Γ(1 + p)Γ(1 + p)Γl − qβ + p + 1. (4.51)

We write Eq. (4.51) in terms of Fox H-function as

F(x, y, t) =4eiωtU(1 + iωθβ)

dh

∞

∑m=1

∞

∑n=1

sin(ζmx)sin(λny)

[1 − (−1)m]−1[1 − (−1)n]−1

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

(−1)p+q+r+sνp+1λsθqβKp−r(λmn)r+1t−qβ+p

ζmλnq!r!s!

× H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − q + p, 0), (1 − r + p, 0),

(−s − p, 0), (−p, 1).

(0, 1), (1 − p, 0), (1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

. (4.52)

or

F(x, y, t) =16eiωtU(1 + iωθβ)

dh

∞

∑c=0

∞

∑e=0

sin(ζcx)sin(λey)

ζcλe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

(−1)p+q+r+sνp+1λsθqβKp−r(λce)r+1t−qβ+p

q!r!s!

× H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − q + p, 0), (1 − r + p, 0),

(1 − s − p, 0), (−p, 1).

(0, 1), (1 − p, 0), (1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

, (4.53)

where

ζc = (2m + 1)π

d, λe = (2n + 1)

π

h, c = 2m + 1, e = 2n + 1.

62


From Eq. (4.53), we obtain the VF due to cosine oscillations of the duct

u(x, y, t) =16U(cos(ωt)− ωθβsin(ωt))

dh

∞

∑c=0

∞

∑e=0

sin(ζcx)sin(λey)

ζcλe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0


q!r!s!

× H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − q + p, 0), (1 − r + p, 0),

(1 − s − p, 0), (−p, 1).

(0, 1), (1 − p, 0), (1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

, (4.54)

and the VF due to sine oscillations of the duct

v(x, y, t) =16U(sin(ωt)− ωθβcos(ωt))

dh

∞

∑c=0

∞

∑e=0

sin(ζcx)sin(λey)

ζcλe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0


q!r!s!

× H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − q + p, 0), (1 − r + p, 0),

(1 − s − p, 0), (−p, 1).

(0, 1), (1 − p, 0), (1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

. (4.55)

The tangential tensions are denoted by τ1c(x, y, t), τ2c(x, y, t) (for cosine os-

cillations of the duct) and τ1s(x, y, t), τ2s(x, y, t) (for sine oscillations of the

duct).

If we introduce

τ1(x, y, t) = τ1c(x, y, t) + iτ1s(x, y, t), (4.56)

τ2(x, y, t) = τ2c(x, y, t) + iτ2s(x, y, t), (4.57)

in Eqs. (4.35) and (4.36), we get

τ1(x, y, t) =µ

(1 + λ)(1 + θβ ∂β

∂βt)∂xF(x, y, t), (4.58)

63


τ2(x, y, t) =µ

(1 + λ)(1 + θβ ∂β

∂βt)∂yF(x, y, t). (4.59)

We apply LT to Eqs. (4.58) and (4.59), to obtain

τ1(x, y, s) =µ(1 + θβsβ)

1 + λ∂x F(x, y, s), (4.60)

τ2(x, y, s) =µ(1 + θβsβ)

1 + λ∂y F(x, y, s). (4.61)

Taking the inverse FFST of Eq. (4.48) to get F(x, y, s) and then by putting it

into Eq. (4.60), we get

τ1(x, y, s) =4µ(1 + θβsβ)

dh(1 + λ)

∞

∑m=1

∞

∑n=1

cos(ζmx)sin(λny)

λn(s − iω)

× [1 − (−1)m][1 − (−1)n]Uνλmn(1 + iωθβ)

[(1 + λ)(s + H) + ν(1 + θβsβ)(λmn + K)], (4.62)

or

τ1(x, y, s) =16µ(1 + θβsβ)

dh(1 + λ)

∞

∑c=0

∞

∑e=0

cos(ζcx)sin(λey)Uνλce

λe(s − iω)

× (1 + iωθβ)

[(1 + λ)(s + H) + ν(1 + θβsβ)(λce + K)], (4.63)

where

ζc = (2m + 1)π

d, λe = (2n + 1)

π

h, c = 2m + 1, e = 2n + 1.


τ1(x, y, s) =16µU(1 + iωθβ)

dh(s − iω)

∞

∑c=0

∞

∑e=0

cos(ζcx)sin(λey)

λe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

∞

∑l=0

νp+1λsθqβKp−r Hl(λce)r+1

(−1)−(p+q+r+s+l)q!r!s!l!

× Γ(q − p − 1)Γ(r − p)Γ(s + p + 2)Γ(l + p + 1)

Γ(p)Γ(p + 1)Γ(2 + p)Γ(1 + p)sl−qβ+p+1, (4.64)

64


using the inverse LT of the last equation, we obtain

τ1(x, y, t) =16µeiωtU(1 + iωθβ)

dh

∞

∑c=0

∞

∑e=0

cos(ζcx)sin(λey)

λe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

∞

∑l=0

νp+1λsθqβKp−r Hl(λce)r+1tl−qβ+p

(−1)−(p+q+r+s+l)q!r!s!l!

× Γ(q − p − 1)Γ(s + p + 2)Γ(l + p + 1)

Γ(l − qβ + p + 1)Γ(p)Γ(p + 1)Γ(2 + p)Γ(1 + p). (4.65)

Lastly, we write the stress field in Fox H-function as

τ1(x, y, t) =16µeiωtU(1 + iωθβ)

dh

∞

∑c=0

∞

∑e=0

cos(ζcx)sin(λey)

λe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

νp+1λsθqβKp−r(λmn)r+1t−qβ+p

(−1)−(p+q+r+s)q!r!s!

×H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(2 − q + p, 0), (1 − r + p, 0),

(−1 − s − p, 0), (−p, 1).

(0, 1), (−p, 0), (−1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

. (4.66)

From Eq. (4.66), we obtain the tangential tension due to cosine oscillations

τ1c(x, y, t) =16Uµ(cos(ωt)− ωθβsin(ωt))

dh

∞

∑c=0

∞

∑e=0

cos(ζcx)sin(λey)

λe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

(−1)p+q+r+sνp+1λsθqβKp−r

(λmn)−r−1tqβ−pq!r!s!

×H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(2 − q + p, 0), (1 − r + p, 0),

(−1 − s − p, 0), (−p, 1).

(0, 1), (−p, 0), (−1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

, (4.67)

65


and the tangential tension corresponding to sine oscillations

τ1s(x, y, t) =16Uµ(sin(ωt)− ωθβcos(ωt))

dh

∞

∑c=0

∞

∑e=0

cos(ζcx)sin(λey)

λe

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

(−1)p+q+r+sνp+1λsθqβKp−r

(λmn)−r−1tqβ−pq!r!s!

×H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(2 − q + p, 0), (1 − r + p, 0),

(−1 − s − p, 0), (−p, 1).

(0, 1), (−p, 0), (−1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

. (4.68)

In the similar fashion, we can find τ2c(x, y, t) and τ2s(x, y, t) from Eqs. (4.48)

and (4.61).

4.3.3 Volume Flux

The volume flux due to cosine oscillations is given by

Qc(x, y, t) =∫ d

0

∫ h

0u(x, y, t)dxdy, (4.69)

putting u(x, y, t) from Eq. (4.54) into the above equation, we obtain the vol-

ume flux of the rectangular duct due to cosine oscillations

u(x, y, t) =64U(cos(ωt)− ωθβsin(ωt))

dh

∞

∑c=0

∞

∑e=0

1

(ζcλe)2

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

νp+1λsθqβKp−r(λce)r+1t−qβ+p

(−1)−(p+q+r+s)q!r!s!

× H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − q + p, 0), (1 − r + p, 0),

(1 − s − p, 0), (−p, 1).

(0, 1), (1 − p, 0), (1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

. (4.70)

66


Similarly, we obtain the volume flux of the rectangular duct due to the sine

oscillations

v(x, y, t) =64U(sin(ωt)− ωθβcos(ωt))

dh

∞

∑c=0

∞

∑e=0

1

(ζcλe)2

×∞

∑p=0

∞

∑q=0

∞

∑r=0

∞

∑s=0

νp+1λsθqβKp−r(λce)r+1t−qβ+p

(−1)−(p+q+r+s)q!r!s!

× H1,4

4,6

Ht

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − q + p, 0), (1 − r + p, 0),

(1 − s − p, 0), (−p, 1).

(0, 1), (1 − p, 0), (1 − p, 0),

(−p, 0), (−p, 0), (qβ − p, 1).

. (4.71)

4.3.4 Numerical Simulations

The numerical results shows the profiles of VF and the adequate SS for the

flow corresponding to the cosine oscillations of the duct. We analyze these

results through different parameters of interest. The effects of relaxation

time λ of the model are important for us to be discussed. In Fig. (4.5) we

illustrate the profiles of VF and SS for three dissimilar values of λ. From the

graphs it is seen that the VF and SS of the flow decreases with increasing

λ, which shows the shear thickening phenomenon. Fig. (4.6) are drawn to

demonstrate the flow VF and the adequate SS at different values of retarda-

tion time θ. It is noticeable that VF as well as the SS increases by increasing

θ. In order to study the effect of frequency of oscillation ω, we have plotted

Fig. (4.7), where it come out that the velocity is also a strong function of ω of

the GJF. The outcome of ω on the VF profile for cosine oscillation is similar

to that of the retardation time θ. In Fig. (4.8) we demonstrate the VF and

SS profiles for various values of magnetic parameter H. It is examined from

these figures that the VF and SS both decreases with increasing H, which

relates to the shear thickening phenomenon. In Fig. (4.9) we concentrate

on the impact of K. It is noticeable that the flow velocity along with the SS

67


increases by increasing K. In order to study the effects of t, we have plotted

Fig. (4.10), in which the out come is that the velocity is a strong function of

t of the GJF. It can be observed that the increase of t acts as an increase of

the magnitude of velocity components near the plate, and this corresponds

to the shear-thinning behavior of the examined non-Newtonian fluid. Fig.

(4.11) presents the VF and the associated SS at various values of y. It is

noticeable that velocity and SS both decreases by increasing y.

0 1 2 30

5 104-

´

1 103-

´

1.5 103-

´

l=1.0

l=1.4

l=1.8

t

u

0 1 2 35- 10

3-´

0

5 103-

´

0.01

0.015

l=1.0

l=1.4

l=1.8

t

t

Figure 4.5: VF and SS profiles for λ when x = 0.5, y = 0.3, U = 0.2, H = 0.5, K

= 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

68


0 1 2 30

5 104-

´

1 103-

´

1.5 103-

´

q=0.4

q=0.6

q=0.8

t

u

0 1 2 35- 10

4-´

0

5 104-

´

1 103-

´

1.5 103-

´

q=0.4

q=0.6

q=0.8

t

t

Figure 4.6: VF and SS profiles for θ when x = 0.5, y = 0.3, U = 0.2, H = 0.5, K

= 0.6, d = 1, h = 2, λ = 1.4, ω = 0.5 and ν = 0.1.

0 1 2 30

1 104-

´

2 104-

´

3 104-

´

w=0.5

w=0.7

w=0.9

t

u

0 1 2 35- 10

4-´

0

5 104-

´

1 103-

´

1.5 103-

´

w=0.5

w=0.7

w=0.9

t

t

Figure 4.7: VF and SS profiles for ω when x = 0.5, y = 0.3, U = 0.2, H = 0.5, K

= 0.6, d = 1, h = 2, θ = 0.6, λ = 1.4 and ν = 0.1.

69


0 1 2 30

5 105-

´

1 104-

´

1.5 104-

´

2 104-

´

H=0.1

H=0.5

H=0.9

t

u

0 1 2 35- 10

4-´

0

5 104-

´

1 103-

´

1.5 103-

´

H=0.1

H=0.5

H=0.9

t

t

Figure 4.8: VF and SS profiles for H when x = 0.5, y = 0.3, U = 0.2, λ = 1.4, K

= 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

0 1 2 30

5 105-

´

1 104-

´

1.5 104-

´

2 104-

´

K=2

K=4

K=6

t

u

0 1 2 31- 10

4-´

0

1 104-

´

2 104-

´

3 104-

´

K=2

K=4

K=6

t

t

Figure 4.9: VF and SS profiles for K when x = 0.5, y = 0.3, U = 0.2, H = 0.5, λ

= 1.4, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

70


0 1 2 30

5 105-

´

1 104-

´

1.5 104-

´

t=6

t=7

t=8

y

u

0 1 2 32- 10

4-´

0

2 104-

´

4 104-

´

6 104-

´

t=6

t=7

t=8

y

t

Figure 4.10: VF and SS profiles for t when x = 0.5, λ = 1.4, U = 0.2, H = 0.5, K

= 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

0 1 2 30

1 104-

´

2 104-

´

3 104-

´

y=0.1

y=0.3

y=0.6

t

u

0 1 2 34- 10

7-´

2- 107-

´

0

2 107-

´

4 107-

´

6 107-

´

8 107-

´

y=0.1

y=0.3

y=0.6

t

t

Figure 4.11: VF and SS profiles for y when x = 0.5, λ = 1.4, U = 0.2, H = 0.5,

K = 0.6, d = 1, h = 2, θ = 0.6, ω = 0.5 and ν = 0.1.

71

Chapter 5

Generalized Oldroyd-B Fluid

5.1 Introduction

The foremost rate type model, which is so far utilized broadly, is because

of Maxwell [78]. Lately, in view of the fundamental work of Maxwell, Ra-

jagopal and Srinivasa [79] build up an efficient thermodynamic structure

for an assortment of rate type of viscoelastic fluid. In the midst of them,

the Oldroyd-B fluid is by all accounts agreeable to investigation and more

critical to analyze. It has some accomplishment in portraying the reaction

of some polymeric fluids being seen as a standout amongst the best mod-

els for depicting the reaction of a sub-class of such fluids. The generalized

Oldroyd-B (GOB) fluid is observed to be very adaptable for portraying the

motions of viscoelastic fluids. Amid the most recent years, a great deal of

work with respect to GOB have been carried out [80–87].

5.2 MHD Oscillatory Flow of GOB Fluid

In this section, we obtained analytic solution regarding MHD incompress-

ible flow of GOB fluid. The GOB fluid, saturating above a flat plate, has a

72

5.2 MHD OSCILLATORY FLOW OF GOB FLUID

constant pressure gradient in the direction of flow. At first, the GOB fluid

and the flat plate are at rest, and after some time t, the flat plate starts oscil-

latory motion with a velocity Vcos(wt) or Vsin(wt), where V is taken to be

constant. The governing equation is developed with the help of fractional

calculus. FFST and discrete LT are used to solve the problem. The results of

the problem are written in series form using Fox H-function. The obtained

results are obeying all the given IC and BCs. In particular, results regarding

Maxwell and second grade fluids are obtained as limiting cases. At the end,

graphs are shown to analyze the impact of various important parameters

which are used in the model.

5.2.1 Development of the Flow

For an incompressible and unsteady GOB fluid the constitutive equation

is [88]

T = −pI + S; (1 + λα Dα

Dtα)S = µ(1 + θβ Dβ

Dtβ)A. (5.1)

Here relaxation time is denoted by λ and retardation time by θ, fractional

parameters are represented by α and β satisfying 0 ≤ α ≤ β ≤ 1, and

DαS

Dtα= Dα

t S + (V.∇)S − LS − SLT, (5.2)

DβA

Dtβ= D

βt A + (V.∇)A − LA − ALT. (5.3)

In the above relations, A = L + LT and ∇ is the gradient operator. We

can obtain the ordinary Oldroyd-B model by putting α = β = 1 in the above

model.

We consider the VF and SS in the following form

V = u(y, t)i, S = S(y, t). (5.4)

Substituting Eq. (5.4) into Eq. (5.1) and considering

S(y, 0) = 0, y > 0, (5.5)

73


we get Syy = Szz = Sxz = Syz = 0 where Sxy = Syx, and

(1 + λαDαt )Sxy = µ(1 + θβD

βt )∂yu(y, t). (5.6)

The equation of motion, with the magnetic body force σB20u, is given by

∂ySxy − ∂x p − σB20u = ρ∂tu, ∂y p = ∂z p = 0, (5.7)

After solving Eqs. (5.6) and (5.7), we obtain the governing equation

(1 + λαDαt )∂tu(y, t) = ν(1 + θβD

βt )∂

2yu(y, t)− M(1 + λαDα

t )u(y, t)

+1

ρ(1 + λαDα

t )∂xP, (5.8)

where M = σB20u. Following are the IC and BCs along with the natural

conditions

(y, 0) = 0 = ∂tu(y, 0); y > 0,

u(0, t) = Vsin(wt) or u(0, t) = Vcos(wt), t > 0

u(y, t), ∂yu(y, t) → 0 as y → ∞, and t > 0.

(5.9)

5.2.2 Calculation of Velocity field

We use non-dimensional parameters as

u∗ = uV , y∗ = yV

ν , t∗ = tVν ,

λ∗ = (λVν )α, θ∗ = ( θV2

ν )β, M∗ = MνV2 .

(5.10)

After excluding asterisks, the governing equation and Eq. (5.6) are rewritten

in their dimensionless form as

(1 + λDαt )∂tu(y, t) = ν(1 + θD

βt )∂

2yu(y, t)− M(1 + λDα

t )u(y, t)

+1

ρ(1 + λDα

t )∂xP, (5.11)

(1 + λDαt )Sxy = µ(1 + θD

βt )∂yu(y, t), (5.12)

74


with the given IC and BCs as

(y, 0) = 0 = ∂tu(y, 0); y > 0,

u(0, t) = Vsin(wt) or u(0, t) = Vcos(wt), t > 0,

u(y, t), ∂yu(y, t) → 0 as y → ∞, and t > 0.

(5.13)

To find the analytic solution, we multiply Eq. (5.11) by sin(ξy) and then

take the anti-derivative w.r.t y from 0 to ∞, to get

(1 + λDαt )∂tus(ξ, t) = ν(1 + θD

βt )(ξysin(wt)− (ξy)2us(ξ, t))

−M(1 + λDαt )us(ξ, t)− A

1

ξ(1 + λ

t−α

Γ(1 − α))(1 − (−1)n), (5.14)

where us(ξ, t) is the FFST of u(y, t) satisfying the IC

us(ξ, 0) = 0 = ∂tus(ξ, 0), ξ > 0. (5.15)

Taking the LT of Eq. (5.14) and using the IC (5.15), we get

us(ξ, s) =1

(S + M)((1 + λsα)) + νξ2(1 + θsβ)[ν(1 + θsβ)ξ

w

s2 + w2

− A(1 + λDαt )

ξS(1 − (−1)n)]. (5.16)

The last equation is expressed in series form as

us(ξ, s) =w

(s2 + w2)

∞

∑i=0

∞

∑o=0

∞

∑k=0

∞

∑l=0

(−1)i+o+k+lξ−(2i+1)λkθl Mo

o!k!l!s−i+o−αk−βlνi

Γ(o − i)Γ(k − i)Γ(l + i)

Γ(−1)Γ(i)Γ(i)Γ(−i)− A(1 − (−1)n)

∞

∑j=0

∞

∑m=0

∞

∑n=0

∞

∑p=0

× (−1)j+m+n+pξ2j−1νjλnθmMpΓ(m − j)Γ(n + j)Γ(p + j + 1)

m!n!p!Γ(−1)Γ(−j)Γ(j)Γ(−j − 1)s1+p−αn−βm. (5.17)

75


Taking the discrete inverse LT of Eq. (5.17)

us(ξ, t) = sin(wt)∞

∑i=0

∞

∑o=0

∞

∑k=0

∞

∑l=0

(−1)i+o+k+lξ−(2i+1)ν−iλkθl Mo

o!k!l!Γ(−1)Γ(i)Γ(i)Γ(−i)

× Γ(o − i)Γ(k − i)Γ(l + i)t−i+o−αk−βl−1

Γ(−i + o − αk − βl)− A(1 − (−1)n)

×∞

∑j=0

∞

∑m=0

∞

∑n=0

∞

∑p=0

(−1)j+m+n+pξ2j−1νjλnθmMp

m!n!p!Γ(1 + p − αn − βm)

× Γ(m − j)Γ(n + j)Γ(p + j + 1)tp−αn−βm

Γ(−1)Γ(−j)Γ(j)Γ(−j − 1). (5.18)

Finally, the analytical solution of VF is obtained here by applying inverse

FFST as

us(y, t) =2

π

∞

∑ξ=1

sin(ξy) sin(wt)∞

∑i=0

∞

∑o=0

∞

∑k=0

∞

∑l=0

(−1)i+o+k+lν−i

o!k!l!Γ(−1)Γ(i)

× ξ−(2i+1)λkθl MoΓ(k − i)Γ(l + i)Γ(o − i)t−i+o−αk−βl−1

Γ(−i)Γ(i)Γ(−i + o − αk − βl)

− 2

π

∞

∑ξ=1

sin(ξy)A(1 − (−1)n)∞

∑j=0

∞

∑m=0

∞

∑n=0

∞

∑p=0

(−1)j+m+n+p

m!n!p!

× ξ2j−1νjλnθmMpΓ(n + j)Γ(p + j + 1)Γ(m − j)tp−αn−βm

Γ(−j)Γ(−1)Γ(−j − 1)Γ(j)Γ(1 + p − αn − βm). (5.19)

Eq. (5.19) is stated in a simple form by using Fox H-function

us(y, t) =2

π

∞

∑ξ=1

sin(ξy)sin(wt)∞

∑i=0

∞

∑o=0

∞

∑k=0

(−1)i+o+kξ−(2i+1)λk

o!k!νiθl Moti−o+α(k)

× H1,3

3,6

θ

tβ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − o + i, 0), (1 − i, 1), (1 − k + i, 0).

(2, 0), (1 − i, 0), (1 − i, 0),

(0, 1), (1 + i, 0), (1 + i − o + αk,−β).

− A(1 − (−1)n)2

π

∞

∑ξ=1

sin(ξy)∞

∑j=0

∞

∑m=0

∞

∑n=0

(−1)j+m+nξ2j−1νjλnθm

m!n!M−ptαn+βm

× H1,3

3,6

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − m + j, 0), (−j, 1), (1 − n − j, 0).

(1 + j, 0), (2, 0), (1 − j, 0),

(2 + j, 0), (0, 1), (αn + βm, 1).

. (5.20)

76


5.2.3 Calculation of Shear Stress

Taking LT of Eq. (5.12)

Sxy =∂u(y, s)

∂y

µ(1 + θsβ)

(1 + λsα). (5.21)

We can obtain u(y, s) from Eq. (5.16) and then substituting it into Eq. (5.21),

we get

Sxy =2µ

π

(1 + θsβ)

1 + λsα

∞

∑ξ=1

ξcos(ξy)

(S + M)((1 + λsα)) + νξ2(1 + θsβ)

[ν(1 + θsβ)ξw

s2 + w2− A(1 + λsα)

ξS(1 − (−1)n)]. (5.22)

Eq. (5.22) can be rewritten in series form as

Sxy =2µ

π

∞

∑ξ=1

cos(ξy)w

s2 + w2

∞

∑i=0

∞

∑o=0

∞

∑k=0

∞

∑l=0

(−1)i+o+k+lξ−2iν−i

o!k!l!s−i+o−αk−βl

× λkθl MoΓ(o − i)Γ(k − i + 1)Γ(l − 1 + i)

Γ(−1)Γ(i)Γ(i − 1)Γ(1 − i)−

2µ

π

∞

∑ξ=1

cos(ξy)A(1 − (−1)n)∞

∑j=0

∞

∑m=0

∞

∑n=0

∞

∑p=0

(−1)j+m+n+pξ2jνj

m!n!p!

× λnθmMpΓ(m − j − 1)Γ(n + j + 1)Γ(p + j + 1)

Γ(−1)Γ(−1 − j)Γ(1 + j)Γ(−j − 1)s1+p−αn−βm. (5.23)

Applying the inverse LT to Eq. (5.23), to obtain the SS

Sxy =2µ

π

∞

∑ξ=1

cos(ξy)sin(wt)∞

∑i=0

∞

∑o=0

∞

∑k=0

∞

∑l=0

(−1)i+o+k+lξ−2iν−i

o!k!l!Γ−i+o−αk−βl

× λkθl Mot−1−i+o−αk−βlΓ(o − i)Γ(k − i + 1)Γ(l − 1 + i)

Γ(−1)Γ(i)Γ(i − 1)Γ(1 − i)−

2µ

π

∞

∑ξ=1

cos(ξy)A(1 − (−1)n)∞

∑j=0

∞

∑m=0

∞

∑n=0

∞

∑p=0

(−1)j+m+n+pξ2jνj

m!n!p!

λnθmMpΓ(m − j − 1)Γ(n + j + 1)Γ(p + j + 1)tp−αn−βm

Γ(−1)Γ(−1 − j)Γ(1 + j)Γ(−j − 1)Γ(1 + p − αn − βm). (5.24)

77


The SS is expressed in Fox H-function as

Sxy =2µ

π

∞

∑ξ=1

cos(ξy)sin(wt)∞

∑i=0

∞

∑o=0

∞

∑k=0

(−1)i+o+kξ−2iν−i

λ−k M−oti−o+αko!k!

× H1,3

3,6

θ

tβ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − k + i, 0), (1 − o + i, 0), (1 − i, 1).

(1 − i, 0), (2, 0), (i + 1, 0), (−i + 1, 0),

(0, 1), (1 + i − o + αk,−β).

− A(1 − (−1)n)2µ

π

∞

∑ξ=1

cos(ξy)∞

∑j=0

∞

∑m=0

∞

∑n=0

(−1)j+m+nξ2j−1νj

λ−nθ−mtαn+βmm!n!

× H1,3

3,6

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − n − j, 0), (1 − m + j, 0), (−j, 1).

(1 + j, 0), (2, 0), (1 − j, 0),

(0, 1), (2 + j, 0), (αn + βm, 1).

. (5.25)

5.2.4 Particular Cases

We obtain profiles for VF and SS for a generalized second grade fluid by

putting λ → 0 and α 6= 0 in Eqs. (5.20) and (5.25) as

u = sin(wt)2

π

∞

∑ξ=1

sin(ξy)sin(wt)∞

∑i=0

∞

∑o=0

(−1)i+oξ−(2i+1)

νi M−oti−oo!

× H1,2

2,5

θ

tβ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − o + i, 0), (1 − i, 1).

(2, 0), (1 − i, 0), (1 − i, 0), (1 − i, 0),

(0, 1), (1 + i − o,−β).

− A(1 − (−1)n)2

π

∞

∑ξ=1

sin(ξy)∞

∑j=0

∞

∑m=0

(−1)j+mξ2j−1

ν−jθ−mtβmm!

× H1,2

2,5

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − m + j, 0), (−j, 1).

(2, 0), (1 + j, 0),

(2 + j, 0), (0, 1), (βm, 1).

, (5.26)

78


Sxy =2µ

π

∞

∑ξ=1

cos(ξy)sin(wt)∞

∑i=0

∞

∑o=0

(−1)i+oξ−2i

νi M−oti−oo!

× H1,2

2,5

θ

tβ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − o + i, 0), (1 − i, 1).

(2, 0), (1 − i, 0), (1 − i, 0),

(0, 1), (1 + i − o,−β).

− A(1 − (−1)n)2µ

π

∞

∑ξ=1

cos(ξy)∞

∑j=0

∞

∑m=0

(−1)j+m+nξ2j−1

ν−jθ−mtβmm!

× H1,2

2,5

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − m + j, 0), (−j, 1).

(2, 0), (1 + j, 0), (0, 1),

(2 + j, 0), (βm, 1).

. (5.27)

Similarly, we obtain VF and SS profiles for the flow of generalized Maxwell

fluid by taking θ → 0 and β 6= 0 in Eqs. (5.20) and (5.25)

u = sin(wt)2

π

∞

∑ξ=1

sin(ξy)sin(wt)∞

∑i=0

∞

∑o=0

(−1)i+oξ−(2i+1)

νi M−oti−oo!

× H1,2

2,6

−λ

tα

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − o + i, 0), (1 + i, 1).

(2, 0), (1 − i, 0), (1 + i, 0),

(0, 1), (1 + i − o + βl,−α).

− A(1 − (−1)n)2

π

∞

∑ξ=1

sin(ξy)∞

∑j=0

∞

∑m=0

(−1)j+mξ2j−1

ν−jtβmm!

× H1,2

2,6

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − m + j, 0), (1 − j, 1).

(2, 0), (1 + j, 0), (0, 1),

(2 + j, 0), (βm,−α).

, (5.28)

79


Sxy =2µ

π

∞

∑ξ=1

cos(ξy)sin(wt)∞

∑i=0

∞

∑o=0

(−1)i+oξ−2iν−i

M−oti−o+1o!

× H1,2

2,5

−λ

tα

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − o + i, 0), (i, 1).

(2, 0), (1 − i, 0), (i, 0),

(0, 1), (1 + i − o,−α).

− A(1 − (−1)n)2µ

π

∞

∑ξ=1

cos(ξy)∞

∑j=0

∞

∑m=0

(−1)j+mξ2j−1νj

λ−nθ−mtαnm!

× H1,2

2,5

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(2 − m + j, 0), (−j, 1).

(2, 0), (2 + j, 0), (0, 1),

(2 + j, 0), (0,−α).

. (5.29)

By putting α, β → 1 in Eqs. (5.20) and (5.25), we get the VF and associated

SS of an ordinary Oldroyd-B fluid

u = sin(wt)2

π

∞

∑ξ=1

sin(ξy)sin(wt)∞

∑i=0

∞

∑o=0

∞

∑k=0

(−1)i+o+kλkθl Mo

ξ2i+1νiti−o+ko!k!

× H1,3

3,6

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − o + i, 0), (1 − k + i, 0), (1 − i, 1).

(2, 0), (1 − i, 0), (1 − i, 0),

(1 + i, 0), (0, 1), (1 + i − o + k,−1).

− A(1 − (−1)n)2

π

∞

∑ξ=1

sin(ξy)∞

∑j=0

∞

∑m=0

∞

∑n=0

(−1)j+m+nξ2j−1νjλn

θ−mM−ptn+mm!n!

× H1,3

3,6

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − m + j, 0), (1 − n − j, 0), (−j, 1).

(2, 0), (1 + j, 0), (1 − j, 0),

(2 + j, 0), (0, 1), (n + m, 1).

, (5.30)

80


Sxy =2µ

π

∞

∑ξ=1

cos(ξy)sin(wt)∞

∑i=0

∞

∑o=0

∞

∑k=0

(−1)i+o+kλk Mo

ti−o+kξ2iνio!k!

× H1,3

3,6

θ

t

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − o + i, 0), (1 − k + i, 0), (1 − i, 1).

(2, 0), (1 − i, 0), (1 − i, 0),

(1 + i, 0), (0, 1), (1 + i − o + k,−1).

− A(1 − (−1)n)2µ

π

∞

∑ξ=1

cos(ξy)∞

∑j=0

∞

∑m=0

∞

∑n=0

(−1)j+m+nνjλnθm

ξ−2j+1tn+mm!n!

× H1,3

3,6

Mt

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − m + j, 0), (1 − n − j, 0), (−j, 1).

(2, 0), (1 + j, 0), (1 − j, 0),

(2 + j, 0), (0, 1), (n + m, 1).

. (5.31)

Similarly, taking α, β → 1 in Eqs. (5.26), (5.27), (5.28) and (5.29), we get the

VFs and the adequate SSs for ordinary Maxwell and ordinary second grade

fluids.

5.2.5 Discussion of the Results


ical aspects of the obtained solutions. The comparison between the models

are also analyzed. The numerical results shows the profiles of VF and the

adequate SS for the oscillatory flow. We analyze these results through dif-

ferent parameters of interest.

More important for us is to show the effects of fractional parameters α and

β of the model. In Fig. (5.1) we depict the profiles of VF and SS for three

different values of α. It is observed from these figures that the profiles of

VF and SS both increases with increasing α. Fig. (5.2) show the variation

of the fractional parameter β. The velocity profile of the fluid is decreased

by increasing β whereas the SS is increased by increasing β. The influence

of relaxation and retardation times λ and θ are depicted in Fig. (5.3) and

81


(5.4). They show opposite effects on fluid oscillation. The effect of increas-

ing λ decreases the profiles of VF and SS while by increasing θ the VF and

SS profiles increases for the GOB fluid. The effect of magnetic parameter M

is shown in Fig. (5.5). The amplitude of fluid oscillation for VF as well as for

the SS decreases by increasing M. Finally, comparison of VF for the three

models i.e generalized and ordinary Oldroyd-B, generalized and ordinary

Maxwell, generalized and ordinary second grade with magnetic effect are

together shown in Fig. (5.6). It is obvious from these graphs that the gener-

alized and ordinary second grade fluids have the smallest amplitude while

the generalized and ordinary Oldroyd-B fluids have largest amplitude of

fluid oscillations for VF.

0 2 4 60

1 104-

´

2 104-

´

3 104-

´

a=0.3

a=0.6

a=0.9

y

u(y

)

0 2 4 60

5 105-

´

1 104-

´

1.5 104-

´

a=0.3

a=0.6

a=0.9

y

t

Figure 5.1: VF and SS profiles given by Eqs. (5.20) and (5.25) when A = 1.6,

β = 0.4, λ = 6, θ = 8, t = 6, ν = 0.186, M = 5 and for different values of α.

82


0 2 4 60

5 105-

´

1 104-

´

1.5 104-

´

2 104-

´

b=0.4

b=0.6

b=0.8

y

u(y

)

0 2 4 60

5 105-

´

1 104-

´

1.5 104-

´

2 104-

´

b=0.4

b=0.6

b=0.8

y

t


α = 0.2, λ = 6, θ = 8, t = 6, ν = 0.186, M = 5 and for different values of β.

0 2 4 60

5 105-

´

1 104-

´

1.5 104-

´

l=6

l=8

l=10

y

u(y

)

0 1 2 3 4 50

5 105-

´

1 104-

´

1.5 104-

´

l=6

l=8

l=10

y

t


α = 0.2, β = 0.4, θ = 8, t = 6, ν = 0.186, M = 5 and for different values of λ.

83


0 2 4 60

5 105-

´

1 104-

´

1.5 104-

´

2 104-

´

q=8

q=10

q=12

y

u(y

)

0 1 2 3 4 50

1 104-

´

2 104-

´

3 104-

´

4 104-

´

q=8

q=10

q=12

y

t


β = 0.4, λ = 6, α = 0.2, t = 6, ν = 0.186, M = 5 and different values of θ.

0 2 4 60

1 104-

´

2 104-

´

3 104-

´

M=10

M=14

M=18

y

u(y

)

0 2 4 60

5 105-

´

1 104-

´

1.5 104-

´

M=10

M=14

M=18

y

t


α=0.2, β = 0.4, λ = 6, θ = 8, t = 6, ν = 0.186 and for different values of M.

84


0 2 4 60

1 104-

´

2 104-

´

3 104-

´

Oldroyd-b

Maxwell

second grade

y

u

0 2 4 60

2 105-

´

4 105-

´

6 105-

´

G. Oldroyd-b

G. Maxwell

G. second

y

u

Figure 5.6: VF profiles of different fluid models when A = 1.6, α = 0.2, β =

0.4, λ = 6, θ = 8, t = 6, M = 5 and ν = 0.186.

85

5.3 GOB FLUID BETWEEN TWO SIDE WALLS

5.3 GOB Fluid Between Two Side Walls

In this section, we present new results regarding GOB fluid which occupy

a space above a flat plate. The flat plate lies vertically amid two side walls.

At first, the fluid as well as the flat plate are at rest and after time t = 0+,

the flow of the fluid is generated by the abrupt motion of the flat plate,

with a constant velocity V. To solve the flow problem, we use the technique

of fractional calculus. The governing equation is formed with the help of

fractional differential equation. FFST and discrete LT are used to achieve

the exact solutions of the flow problem. The obtained results of VF and SS

are written in series form with the help of Fox H-function. We get particular

results for generalized Maxwell and generalized second grade fluids from

the obtained solutions of GOB fluid. At the end, we investigated our results

graphically with the help of different parameters of interest.

5.3.1 Mathematical Formulation

We consider the VF and an extra stress in the following form

V = u(y, z, t)i, S = S(y, z, t). (5.32)

At time t = 0 the fluid being at rest is given by

S(y, z, 0) = 0. (5.33)

Substituting Eq. (5.32) in (5.1) it results Syz = Syy = Szz = 0 and the associ-

ated fractional differential equations

(1 + λαDαt )Sxy = µ(1 + θβD

βt )∂yu(y, z, s), (5.34)

(1 + λαDαt )Sxz = µ(1 + θβD

βt )∂zu(y, z, s). (5.35)

Since, we are neglecting the body forces therefore the balance of linear mo-

mentum becomes

ρ∂tu + ∂x p − ∂ySxy − ∂zSxz = 0, ∂z p = 0 = ∂y p. (5.36)

86


Solving Eqs. (5.34), (5.35) and (5.36), we obtain the governing equation, with

the supposition that in the direction of flow there is no pressure gradient

(1 + λαDαt )∂tu(y, z, t) = ν(1 + θβD

βt )(∂

2y + ∂2

z)u(y, z, t). (5.37)

Following are the associated IC and BCs of the flow problem

(y, z, 0) = 0 = ∂tu(y, z, 0); y > 0 and 0 ≤ z ≤ h,

u(0, z, t) = V; f or t > 0 and 0 < z < h,

u(y, 0, t) = 0 = u(y, h, t); y > 0, t > 0,

(5.38)

along with the natural conditions

u(y, z, t), ∂yu(y, z, t) → 0 as y → ∞, t > 0 and z ∈ (0, h).

The two side walls are h distance apart from each other.

5.3.2 Calculation of the Velocity Field

To get the analytic solution of VF, first we multiply Eq. (5.37) by sin(nπzh ),

and then take the integration w.r.t z from 0 to h

(1 + λαDαt )

∂un(y, n, t)

∂t= ν(1 + θβD

βt )

∂2

∂y2un(y, n, t)− ν(

nπ

h)(1 + θβD

βt )

× un(y, n, t). (5.39)

Taking LT of Eq. (5.39), we get the following fractional differential equation

∂2

∂y2un(y, n, s)−

[

ξ2 +s(1 + λαsα)

(1 + θβsβ)

]

un(y, n, s) = 0, (5.40)

where ξ = nπh . Using the BCs (5.38), we obtain the solution of the above

fractional differential equation in the following form

usn = Vexp

[

− y

√

ξ2 +s(1 + λαsα)

ν(1 + θβsβ)

]

. (5.41)

87


We expressed the last equation in series form as

usn = V∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

(−1)j+n+p+qyjξ j−2nν−nλn−qθ−n−p


× Γ(p + n)Γ(q − n)Γ(n − j2)

s−n+α(q−n)+β(p+n). (5.42)

Taking the inverse LT of Eq. (5.42), we attain

un = V∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

(−1)j+n+p+qyjξ j−2nν−nλn−qθ−n−p


× Γ(p + n)Γ(q − n)Γ(n − j2)t

−n+α(q−n)+β(p+n)

Γ(−n + α(q − n) + β(p + n)). (5.43)

Using the inverse FFST to obtain the analytic solution for the VF

u =2

h

∞

∑m=1

sin(mπz

h)un

=2

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

(−1)j+n+p+qyjξ j−2n

tn−α(q−n)−β(p+n) j!n!q!p!

× θ−n−pν−nλn−qΓ(p + n)Γ(q − n)Γ(n − j2)

Γ(n)Γ(−n)Γ( j2)Γ(−n + α(q − n) + β(p + n))

. (5.44)

To express the obtained result in a simple form we use Fox H-function as

u =2

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

(−1)j+n+qyjξ j−2nν−nλn−q

θntn−α(q−n)−βn j!n!q!

×H1,3

3,5

tβ

θ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − n, 1), (1 − q + n, 0), (1 − n + j2 , 0).

(0, 1), (1 − n, 0), (1 + n, 0), (1 − j/2, 0),

(1 + n − α(q − n)− βn, β).

.

(5.45)

5.3.3 Calculation of the Shear Stress

Taking the LT of Eqs. (5.34) and (5.35), to attain

τ1 =µ(1 + θsβ)

(1 + λsα)

∂u(y, z, s)

∂y, (5.46)

τ2 =µ(1 + θsβ)

(1 + λsα)

∂u(y, z, s)

∂z. (5.47)

88


We can get u(y, z, s) from Eq. (5.41) by taking the inverse FFST. And after

substituting it into (5.46), we obtain

τ1 =2Vξµ

h

µ(1 + θsβ)

1 + λsα

∞

∑n=1

sin(nπz

h)exp

[

− y

√

ξ2 +s(1 + λsα)

ν(1 + θsβ)

]

×[

√

1 +s(1 + λsα)

ξ2ν(1 + θsβ)

]

. (5.48)

In series form Eq. (5.48) can be written as

τ1 =∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

∞

∑m=0


j!n!q!p!w!y!z!x!m!

× 2Vyjξ j−2n+2mνm−nλn−qθ−n−p+w+x+y+zΓ(p + n)Γ(q − n)

hΓ(n)Γ(−n)Γ( j2)Γ(

12)Γ(

−12 )Γ(1

2)Γ(m)Γ(−m)

× Γ(n − j2)Γ(w − 1

2)Γ(x + 12)Γ(m − 1

2)Γ(y − m)Γ(z + m)

s−n+α(q−n)+β(p+n)+m−βw−αx−βy−αz− 12

. (5.49)

Applying the inverse LT to the last equation to get the analytic solution for

SS

τ1 =∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

∞

∑m=0


j!n!q!p!w!y!z!x!m!

× Γ(p + n)Γ(q − n)Γ(n − j2)Γ(w − 1

2)Γ(x + 12)


12)Γ(

−12 )Γ(1

2)Γ(m)Γ(−m)

× Γ(z + m)t−n+α(q−n)+β(p+n)+m−βw−αx−βy−αz−−32

Γ(−n + α(q − n) + β(p + n) + m − βw − αx

× 2Vyjξ j−2n+2mνm−nλn−qΓ(m − 12)Γ(y − m)

−βy − αz − 12)hθn+p−w−x−y−z

. (5.50)

89


And in Fox H-function, the stress field is rewritten as

τ1 =2ρ

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

× (−1)j+n+p+q+w+y+z+xyjξ j−2nν−nλn−q

θn+ptn−α(q−n)−β(p+n) j!n!q!p!w!y!z!x!

× H1,8

8,10

−ξ2t

ν

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − p + n, 0), (1 − q + n, 0), (1 + 12 , 1),

(1 − w + 12 , 0), (1 − x − 1

2 , 0),

(1 − n + j2 , 0), (1 − y,−1), (1 − z, 1).

(1 − n, 0), (1 + n, 0), (1 − j2 , 0), (1,−1),

(1 + 12 , 0), (1 − 1

2 , 0), (1, 1), (0, 1), (1, 1),

(1 + n − α(q − n − x − z)−β(p + n − w − y) + 1

2 , 1).

. (5.51)

In the similar fashion we can obtain τ2(y, z, t), from Eqs. (5.41) and (5.47).

5.3.4 Special Cases

By putting λ → 0 and α 6= 0 in Eqs. (5.45) and (5.51), we obtain VF and SS

corresponding to GSGF

u =2

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

(−1)j+nyjξ j−2nν−n

j!n!θntn−βn−1

×H1,2

2,4

tβ

θ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − n, 1), (1 − n + j/2, 0).

(0, 1), (1 − n, 0), (1 − j/2, 0),

(1 + n − βn, β).

,

(5.52)

90


τ1 =2ρ

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

∞

∑m=0

× (−1)j+n+w+y+z+x+myjξ j−2n+2mνn−m+1θ−n+w+x+y+z

j!n!w!y!z!x!m!tn−β(p+n−w−y)+3/2−m

× H1,7

7,9

tβ

θ

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − n, 1), (1 − n + j2 , 0), (1 − w + 1

2 , 0),

(1 − x − 12 , 0), (1 − m + 1

2 , 0),

(1 − y + m, 0), (1 − z − m, 0).

(0, 1), (1 − n, 0), (1 − j2 , 0),

(1 + 12 , 0), (1 − 1

2 , 0),

(1 − m, 0), (1 + m, 0), (−m, 0),

(1 + n − β(n − w − y)− m + 12 , β).

. (5.53)

We obtain VF and SS for generalized Maxwell fluid by taking θ → 0 and β

6= 0 in Eqs. (5.45) and (5.51)

u =2

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

(−1)j+nyjξ j−2nν−n

j!n!λ−ntn−αn−1

×H1,2

2,4

tα

λ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 + n, 1), (1 − n + j/2, 0).

(0, 1), (1 + n, 0), (1 − j/2, 0),

(1 + n − αn, α).

,

(5.54)

91


τ1 =2ρ

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

∞

∑m=0

× (−1)j+n+w+y+z+x+myjξ j−2n+2mνn−m+1

j!n!w!y!z!x!m!λnt−n+α(−n−x−z)−3/2+m

× H1,7

7,9

tα

λ

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − n + j2 , 0), (1 − w + 1

2 , 0),

(1 − x − 12 , 0), (1 − m + 1

2 , 0),

(1 + n, 1), (1 − y + m, 0), (1 − z − m, 0).

(0, 1), (1 + n, 0), (1 − j2 , 0),

(1 + 12 , 0), (1 − m, 0),

(1 − 12 , 0), (1 + m, 0), (−m, 0),

(1 + n − α(−n − x − z)− m + 12 , α).

. (5.55)

To get VF and SS for an ordinary Oldroyd-B fluid we let α, β → 1 in Eqs.

(5.45) and (5.51)

u =2

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

(−1)j+n+qyjξ j−2nν−nλn−q

θntn−q−p j!n!q!

×H1,3

3,5

tβ

θ

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − n, 1), (n − q + 1, 0), (1 − n + j2 , 0).

(0, 1), (1 − n, 0), (n + 1, 0), (1 − j/2, 0),

(n + 1 − q, 1).

,

(5.56)

92


τ1 =2ρ

hV

∞

∑m=1

sin(mπz

h)

∞

∑j=0

∞

∑n=0

∞

∑q=0

∞

∑p=0

∞

∑w=0

∞

∑y=0

∞

∑z=0

∞

∑x=0

× (−1)j+n+p+q+w+y+z+xyjξ j−2nν−nλn−q

θn+ptn−q−p j!n!q!p!w!y!z!x!

× H1,8

8,10

−ξ2t

ν

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(1 − p + n, 0), (n − q + 1, 0), (1 + 12 , 1),

(1 − w + 12 , 0), (1 − x − 1

2 , 0),

(1 − n + j2 , 0), (−y + 1,−1), (1 − z, 1).

(−n + 1, 0), (1 + n, 0), (1 − j2 , 0), (1,−1),

(1 + 12 , 0), (1 − 1

2 , 0), (1, 1), (0, 1), (1, 1),

(n − q + x + z + w + y + 32 , 1).

.

(5.57)

5.3.5 Numerical Results and Discussion

Various graphs are presented here for the investigation of some important

physical aspects of the obtained solutions. More vital for us is to investigate

about the impacts of fractional parameters α and β. VF and SS profiles are

shown in Fig. (5.7) corresponding to three dissimilar values of α. From the

graphs it is concluded that the velocity of flow increases with increasing α,

though, the SS show an effect opposite to that of velocity by increasing α. In

Fig. (5.8) we show the consequences of β by changing its values. It is noted

that by enlarging the values of β the flow velocity is diminishing while the

associated SS is raising. The effect of material parameter λ is depicted in

Fig. (5.9). It is observed that the VF profile of GOB fluid is reducing and

the adequate SS profile is enhancing by increasing λ. The impact of another

material parameter θ is illustrated in Fig. (5.10). Clearly, it is seen that the

effects of θ is similar to that of λ. To analyze the importance of y we sketched

Fig. (5.11). From the graphs it is examined that velocity is declining and the

SS is magnifying by giving different values to y. The comparison of the

93


models i.e GSGF, GOB fluid and generalized Maxwell fluid are discussed in

Fig. (5.12). It is derived from these graphs that the velocity of generalized

Maxwell fluid as well as ordinary Maxwell fluid has larger magnitude as

compare to the rest of fluid models.

0 0.2 0.4 0.65-

0

5

10

15

a=0.3

a=0.6

a=0.9

y

u

0 0.2 0.4 0.66- 10

3-´

4- 103-

´

2- 103-

´

0

2 103-

´

a=0.3

a=0.6

a=0.9

y

t

Figure 5.7: VF and SS profiles given by Eqs. (5.45) and (5.51) when K = 3, β

= 0.5, t = 6, M = 5, P = 1.5, A = 2 and for different values of α.

0 0.2 0.4 0.60

2

4

6

8

b=0.2

b=0.5

b=0.8

y

u

0 0.2 0.4 0.64- 10

3-´

3- 103-

´

2- 103-

´

1- 103-

´

0

1 103-

´

b=0.2

b=0.5

b=0.8

y

t

Figure 5.8: VF and SS profiles given by Eqs. (5.45) and (5.51) when K = 3, α

= 2, t = 6, M= 5, P = 2, A = 2 and for different values of β.

94


0 0.2 0.4 0.65-

0

5

10

15

l=6

l=9

l=12

y

u

0 0.2 0.4 0.66- 10

3-´

4- 103-

´

2- 103-

´

0

2 103-

´

l=6

l=9

l=12

y

t

Figure 5.9: VF and SS profiles given by Eqs. (5.45) and (5.51) when α = 2, β

= 0.5, t = 6, M = 5, P = 2, A = 2 and for different values of λ.

0 0.2 0.4 0.60

5

10

15

20

q=0.5

q=0.7

q=0.9

y

u

0 0.2 0.4 0.64- 10

3-´

3- 103-

´

2- 103-

´

1- 103-

´

0

1 103-

´

q=0.5

q=0.7

q=0.9

y

t

Figure 5.10: VF and SS τ(y, 6) profiles given by Eqs. (5.45) and (5.51) when

K = 3, β = 0.5, t = 6, M = 5, P = 1.5, A = 2 and for different values of θ.

95


0 0.2 0.4 0.6100-

50-

0

50

100

150

y=1.3

y=1.5

y=1.7

t

u

0 0.2 0.4 0.64- 10

3-´

3- 103-

´

2- 103-

´

1- 103-

´

0

1 103-

´

y=1.5

y=1.7

y=1.9

t

t

Figure 5.11: VF and SS profiles given by Eqs. (5.45) and (5.51) when K = 3, α

= 2, t = 6, M = 5, P = 2, A = 2 and for different values of y.

0 0.2 0.4 0.610-

0

10

20

30

Oldroyd-B

Maxwell

second grade

y

u

0 0.2 0.4 0.60

5

10

15

Gen. Oldroyd-B

Gen. Maxwell

Gen. second grade

y

u

Figure 5.12: VF profiles of different fluid models when α = 2, β = 0.5, t = 6,

M = 5, K = 3, P = 2 and A = 2.

96

Bibliography

[1] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Uni-

versity Press, Cambridge, (1967).

[2] F. M. White, Fluid Mechanics, McGraw Hill, NewYork, (2003).

[3] I. Siddique, Exact solution of some helical flows of Newtonian fluids, Pro-

ceedings of World Academy of Science, Engineering and Technol-

ogy, 33, 664-667 (2008).

[4] G. Bohme, Non-Newtonian Fluid Mechanics, North Holland, Amster-

dam, (1987).

[5] R. B. Bird, R. C. Armstrong, O. Hassager, Dynamics of Polymeric Liq-

uids, Willey, New York, (1977).

[6] J. N. Reddy, An Introduction to Continuum Mechanics and its Applica-

tions, Cambridge University Press, Cambridge, (2013).

[7] N. Makris, Theoretical and experimental investigation of viscous dampers

in applications of seismic and vibration isolation, Ph.D. Thesis, State

Univ. of New York at Buffalo, Buffalo, NY, (1991).

[8] P. N. Srivastava, Non-steady helical flow of a visco-elastic liquid, Arch.

Mech. Stos., 18, 145-150 (1966).

[9] K. Walters, Relation between Coleman-Noll, Rivlin-Ericksen, Green-

Rivlin and Oldroyd fluids, ZAMP, 21, 592-600 (1970).

97

BIBLIOGRAPHY

[10] N. D. Waters, M. J. King, The unsteady flow of an elasto-viscous liquid

in a straight pipe of circular cross section, J. Phys. D: Appl. Phys., 4,

204-211 (1971).

[11] S. Abelman, E. Momoniat, T. Hayat, Steady MHD flow of a third grade

fluid in a rotating frame and porous space, Non linear Anal.: Real World

Appl., 10, 3322-3328 (2009).

[12] M. E. Erdogan, Plane surface suddenly set in motion in a non-Newtonian

fluid, Acta Mech., 108, 179-187 (1995).

[13] M. E. Erdogan, On unsteady motions of fluids of a second-order fluid, Int.

J. Non-Linear Mech., 38, 1045-1051 (2003).

[14] M. E. Erdogan, C. E. Imrak, Effects of the side walls on starting flows in

ducts, Int. J. of Non-Linear Mech., 40, 107-111 (2005).

[15] K. R. Rajagopal, Longitudinal and torsional oscillations of a rod in a non-

Newtonian fluid, Acta Mech., 49, 281-285 (1983).

[16] I. Siddique, D. Vieru, Exact solution for the rotational flow of a general-

ized second grade fluid in a circular cylinder, Acta Mech. Sin., 25, 777-85

(2009).

[17] J. T. Machado, V. S. Kiryakova, F. Mainardi, Recent history of frac-

tional calculus, Commun. Nonlin. Sci. Numer. Simul., 16(3), 1140-

1153 (2011).

[18] V. S. Kiryakova, Generalized fractional calculus and applications, Har-

low, Longman (1994). [Pitman Research Notes in Mathematics, Vol.

301] (1993).

[19] R. Hilfer, Applications of Fractional Calculus in Physics, World Scien-

tific Press, Singapore, (2000).

98

BIBLIOGRAPHY

[20] I. Podlubny, Fractional differential equations [M], Academic Press,

New York, (1999).

[21] L. I. Palade, P. Attane, R. R. Huilgol, B. Mena, Anomalous stability

behavior of a properly invariant constitutive equation which generalizes

fractional derivative models, Int. J. Eng. Sci., 37, 315-329 (1999).

[22] G. L. Slonimsky, On the law of deformation of highly elastic polymeric

bodies, Dokl. Akad. Nauk, BSSR, 140, 343-346 (1961).

[23] M. Stiassnie, On the application of fractional calculus for the formulation

of viscoelastic models, Appl. Math. Modelling, 3, 300-302 (1979).

[24] F. Mainardi, Applications of Fractional Calculus in Mechanics, Transform

Methods and Special Functions, Bulgarian Academy of Sciences, Sofa,

(1998).

[25] M. Caputo, F. Mainardi, A new dissipation model based on memory

mechanism, Pure Appl. Geo phys., 91, 134-147 (1971).

[26] C. M. Zener, Elasticity and Unelasticity of Metals, University of

Chicago Press, Illinois, (1948).

[27] R. L. Bagley, A theoretical basis for the application of fractional calculus

to viscoelasticity, J. Rheology, 27, 201-210 (1983).

[28] H. Qi, H. Jin, Unsteady rotating flows of a viscoelastic fluid with the frac-

tional Maxwell model between coaxial cylinders, Acta Mech. Sin., 22,

301-305 (2006).

[29] W. Shaowei, X. Mingyu, Exact solution on unsteady Couette flow of gen-

eralized Maxwell fluid with fractional derivative, Acta Mech., 187, 103-

112 (2006).

99

BIBLIOGRAPHY

[30] D. Tong, W. Ruihe, Y. Heshan, Exact solutions for the flow of non-

Newtonian fluid with fractional derivative in an annular pipe, Sci. Chi.

Ser. G: Phy. Mech. Ast., 48, 485-495 (2005).

[31] N. Makris, D. F. Dargush, M. C. Constantinou, Dynamical analysis of

generalized viscoelastic fluids, J. Engg. Mech., 119, 1663-1679 (1993).

[32] T. Hayat, S. Nadeem, S. Asghar, Periodic unidirectional flows of a vis-

coelastic fluid with the fractional Maxwell model, Appl. Math. Comput.,

151, 153-161 (2004).

[33] W. Tan, W. Pan, M. Xu, A note on unsteady flows of a viscoelastic fluid

with the fractional Maxwell model between two parallel plate, Int. J. Non-

Linear Mech., 38, 645-650 (2003).

[34] D. Tong, Y. Liu, Exact solutions for the unsteady rotational flow of non-

Newtonian fluid in an annular pipe, Int. J. Eng. Sci., 43, 281-289 (2005).

[35] R. L. Bagley, P. J. Torvik, On the fractional calculus model of viscoelastic

behavior, J. Rheol., 30, 133-155 (1986).

[36] C. Friederich, Relaxation and retardation functions of the Maxwell model

with fractional derivatives, Rheol. Acta, 30, 151-158 (1991).

[37] J. Li, T. Q. Jiang, The research on viscoelastic constitutive relationship

model with fractional derivative operator, South China Technology Uni-

versity Press, Guangzhou, (1993).

[38] A. M. Mathai, R. K. Saxena, H. J. Haubold, The H-functions: Theory

and Applications, Springer, New York, (2010).

[39] L. Debnath, D. Bhatta, Integral Transforms and Their Applications, (Sec-

ond Edition), Chapman and Hall/CRC, Florida, (2007).

100

BIBLIOGRAPHY

[40] J. E. Dunn, R. L. Fosdick, Thermodynamics, stability and boundedness of

fluids of complexity 2 and fluids of second grade, Arch. Rational Mech.

Anal., 56, 191-252 (1974).

[41] G. P. Galdi, M. Padula, K. R. Rajagopal, On the stability of the rest state

of a fluid of second grade in unbounded domains, Arch. Rational Mech.

Anal., 109, 173-182 (1990).

[42] J. G. Oldroyd, On the formulation of rheological equations of state, Proc.

Roy. Soc. London A200, 4, 523-591 (1950).

[43] S. Rivlin, J. L. Ericksen, Stress deformation relation for isotropic material,

J. Rational Mech. Anal., 4, 323-425 (1955).

[44] R. Bandelli, K. R. Rajagopal, Start-up flows of second-grade fluid in do-

main with one finite dimension, Int. J. Non-Linear Mech., 30, 817-839

(1995).

[45] W. C. Tan, M. Y. Xu, The impulsive motion of flat plate in a generalized

second-grade fluid, Mech. Res. Comm., 29, 3-9 (2002).

[46] W. C. Tan, T. Masuoka, Unsteady flows for the generalized second grade

fluid with the fractional derivative model between two parallel plates, Acta

Mech. Sin., 20, 471-476 (2004).

[47] A. Mahmood, C. Fetecau, N. A. Khan, M. Jamil, Some exact solutions

of the oscillatory motion of a generalized second grade fluid in an annular

region of two cylinders, Acta Mech. Sin., 26, 541-550 (2010).

[48] D. Tripathi, Peristaltic flow of a fractional second grade fluid through a

cylindrical tube, Therm. Sci., 15, S167-S173 (2011).

101

BIBLIOGRAPHY

[49] G. Zaman, S. Islam, Y. H. Kang, I. H. Jung, Blood flow of an Oldroyd-

B fluid in a blood vessel incorporating a Brownian stress, Sci. Chi. Phy.

Mech Ast., 55, 125-131 (2012).

[50] G. Zaman, Y. H. Kang, I. H. Jung, A nonlinear mathematical model of

blood flow, Proc. 6th Int. Conf. Scientific Computing and Application,

147-150 (2008).

[51] G. Zaman, Y. H. Kang, I. H. Jung, Orientational stress tensor of polymer

solution with application to blood flow, Sci. Mod. Phy. Letters B, 25,

1157-1166 (2011).

[52] K. Ramakrishnan, K. Shailendhra, Hydromagnetic blood flow through

a uniform channel with permeable walls covered by porous media of finite

thickness, J. Appl. Fluid Mech., 6, 39-47 (2013).

[53] K. Ramesh, M. D. Singh, The influence of heat transfer on peristaltic

transport of MHD second grade fluid through porous medium in a vertical

asymmetric channel, J. Appl. Fluid Mech., 8, 351-365 (2015).

[54] W. C. Tan, T. Masuoka, Stoke’s first problem for second grade fluid in a

porous half space, Int. J. Non-Linear Mech., 40, 515-522 (2005).

[55] W. C. Tan, T. Masuoka, Stoke’s first problem for an Oldroyd-B fluid in a

porous half space, Phys. Fluid, 17, 023-101 (2005).

[56] K. D. Singh, P. Reena, An analysis of an oscillatory rotating MHD

Poiseuille flow with injection/suction and Hall currents, Proc. Indian

Natn. Sci. Acad., 76, 201-207 (2010).

[57] Y. Liu, L. Zheng, Unsteady MHD Couette flow of a generalized Oldroyd-

B fluid with fractional derivative, Comput. Math. Appl., 61, 43-450

(2011).

102

BIBLIOGRAPHY

[58] K. D. Singh, B. P. Garg, Exact solution of an oscillatory free convective

MHD flow in a rotating porous channel with radiative heat, Proc. Nat.

Acad. Sci. India, 80, 81-89 (2010).

[59] A. Khan, G. Zaman, G. Rahman, Hydromagnetic flow near a non-

uniform accelerating plate in the presence of magnetic field through porous

medium, J. Porous Media, 18, 801-809 (2015).

[60] A. Khan, G. Zaman, O. Algahtani, Unsteady flow of viscoelastic fluid

due to impulsive motion of plate, Asian J. Math. Appl., 2014, 191-199

(2014).

[61] A. Khan, G. Zaman, Unsteady magnetohydrodynamic flow of second

grade fluid due to impulsive motion of plate, EJMAA, 3, 215-227 (2015).

[62] I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, (1951).

[63] M. Khan, Partial slip effects on the oscillatory flows of a fractional Jeffrey

fluid in a porous medium, J. Porous Media, 10, 473-487 (2007).

[64] T. Hayat, M. Khan, K. Fakhar, N. Amin, Oscillatory rotating flows of

a fractional Jeffrey fluid filling a porous medium. J. Porous Media, 13,

29-38 (2010).

[65] A. F. Khadrawi, M. A. Al-Nimr, A. Othman, Basic viscoelastic fluid

problems using the Jeffreys model, Chem. Eng. Sci. 60, 7131-7136 (2005).

[66] H. Qi, M. Xu, Stoke’s first problem for a viscoelastic fluid with the gener-

alized Jeffrey model, Acta Mech. Sin. 23, 463-469 (2007).

[67] S. Nadeem, A. Hussain, M. Khan, Stagnation flow of a Jeffrey fluid over

a shrinking sheet, Z. Naturforsch., 65(a), 540-548 (2010).

103

BIBLIOGRAPHY

[68] M. Khan, F. Iftikhar, A. Anjum, Some unsteady flows of a Jeffrey fluid

between two side walls over a plane wall, Z. Naturforsch., 66(a), 745-752

(2011).

[69] M. M. Rashidi, M. Keimanesh, Using differential transform method and

Pade approximant for solving MHD flow in a laminar liquid film from a

horizontal stretching surface, Math. Probl. Eng., 179-193 (2010).

[70] M. M. Rashidi, N. Kaviani, S. Abelman, Investigation of entropy gen-

eration in MHD and slip flow over a rotating porous disk with variable

properties, Int. J. Heat Mass Transfer, 70, 892-917 (2014).

[71] M. M. Rashidi, E. Erfani, Analytical method for solving steady MHD

convective and slip flow due to a rotating disk with viscous dissipation and

ohmic heating, Eng. Computation, 29, 562-579 (2012).

[72] O. A. Beg, M. M. Rashidi, N. Kaviani, M. N. Islam, Entropy genera-

tion in magnetohydrodynamic convective Von Karman swirling flow: Ho-

motopy analysis, Int. J. Appl. Math. Mech., 9, 37-65 (2013).

[73] L. R. T. Gardner, G. A. Gardner, A two-dimensional bi-cubic B-spline

finite element used in a study of MHD duct flow, Comput. Math. Appl.

Mech. Eng., 124, 365-375 (1995).

[74] C. Fetecau, Corina Fetecau, Unsteady flows of Oldroyd-B fluids in a

channel of rectangular cross-section, Int. J. Non-linear Mech., 40, 1214-

1219 (2005).

[75] M. Nazar, F. Shahid, S. Akram, Q. Sultan, Flow on oscillating rectan-

gular duct for Maxwell fluid, Appl. Math. Mech. Eng. Ed, 33, 717-730

(2012).

[76] Q. Sultan, M. Nazar, W. Akhtar, U. Ali, Unsteady flow of a Maxwell

fluid in a porous rectangular duct, Sci. Int. J., 25, 181-194 (2013).

104

BIBLIOGRAPHY

[77] S. Tsangaris, N. W. Vlachakis, Exact solution of the Navier-Stokes equa-

tions for the oscillating flow in a duct of across-section of right-angled

isosceles triangle, Z. Angew. Math. Phys, 54, 1094-1100 (2003).

[78] J. C. Maxwell, On the dynamical theory of gases. Phil. Trans. R. Soc.

Lond. A, 157, 26-78 (1866).

[79] K. R. Rajagopal, A. Srinivasa, Exact Solutions for Some Simple Flows of

an Oldroyd-B Fluid., Acta Mech., 113, 233-239 (1995).

[80] J. Kangand, X. Mingyu, An exact solution for flow past an accelerated

horizontal plate in a rotating fluid with the generalized Oldroyd-B model,

Acta Mech. Sin., 25, 463-469 (2009).

[81] M. Khan, A. Anjum, Q. Haitao, C. Fetecau, On exact solutions for some

oscillating motions of a generalized Oldroyd-B fuid, ZAMP, 61, 133-145

(2010).

[82] H. T. Qi, M. Y. Xu, Stoke’s first problem for a viscoelastic fluid with the

generalized Oldroyd-B model, Acta Mech. Sin., 23, 463-469 (2007).

[83] S. H. A. M. Shah, Some accelerated flows of generalized Oldroyd-B fluid

between two side walls perpendicular to the plate, Non linear Anal.: Real

World Appl., 10, 2146-2150 (2009).

[84] C. Fetecau, Corina Fetecau, M. Kamran, D. Vieru, Exact solutions for

the flow of a generalized Oldroyd-B fluid induced by a constantly accel-

erating plate between two side walls perpendicular to the plate, J. Non-

Newtonian Fluid Mech., 156, 189-201 (2009).

[85] D. Vieru, Corina Fetecau, C. Fetecau, Flow of a generalized Oldroyd-B

fluid due to a constantly accelerating plate, Appl. Math. Comput., 201,

834-842 (2008).

105

BIBLIOGRAPHY

[86] A. Khan, G. Zaman, The motion of a generalized Oldroyd-B fluid between

two side walls of a plate, South Asian J. Math., 5, 42-52 (2015).

[87] A. Khan, G. Zaman, Exact analytic solutions of oscillatory motion of

a generalized MHD Oldroyd-B fluid, Int. J. App. Math., 27, 605-612

(2014).

[88] Corina Fetecau, C. Fetecau, M. Khan, D. Vieru, Decay of a potential

vortex in a generalized Oldroyd-B fluid, Appl. Math. Comput., 205, 497-

506 (2008).

106

BIBLIOGRAPHY

Publications

1. A. Khan, G. Zaman, G. Rahman, Hydromagnetic flow near a non-uniform

accelerating plate in the presence of magnetic field through porous medium,

J. Porous Media, 18, 801-809 (2015).

2. A. Khan, G. Zaman, Unsteady magneto-hydrodynamic flow of second grade

fluid due to uniform accelerating plate, Journal of Applied Fluid Mechan-

ics (JAFM), Accepted.

3. A. Khan, G. Zaman, The oscillating motion of a generalized Oldroyd-B fluid

in magnetic field with constant pressure gradient, Special Topics Reviews

in Porous Media - An International Journal (begel house), 6, 251-260

(2016).

4. A. Khan, G. Zaman, O. Algahtani, Unsteady flow of viscoelastic fluid due

to impulsive motion of plate, Asian J. Math. Appl., 2014, 191-199 (2014).

5. A. Khan, G. Zaman, Unsteady magnetohydrodynamic flow of second grade

fluid due to impulsive motion of plate, EJMAA, 3, 215-227 (2015).

6. A. Khan, G. Zaman, The motion of a generalized Oldroyd-B fluid between

two side walls of a plate, South Asian J. Math., 5, 42-52 (2015).

7. A. Khan, G. Zaman, Exact analytic solutions of oscillatory motion of a

generalized MHD Oldroyd-B fluid, Int. J. App. Math., 27, 605-612 (2014).

8. A. Khan, G. Zaman, MHD oscillating flow of generalized Jeffrey fluid pass-

ing through a porous rectangular duct, J. Porous Media, Accepted.

9. A. Khan, M. Shah, A. Ali, Construction of middle nuclear square loop, J.

Pri. Res. Math., 9, 72-78 (2013).

10. A. Khan, M. Shah, A. Ali, On right alternative loop, International Jour-

nal of Algebra and Statistics, 2, 29-32 (2013).

107

BIBLIOGRAPHY

11. A. Khan, M. Shah, A. Ali, F. Muhammad, On commutative quasigroup,

International Journal of Algebra and Statistics, 3, 42-45 (2014).

12. A. Khan, M. Shah, A. Ali, Construction of right nuclear square loop, Ital-

ian Journal of Pure and Applied Mathematics, (Accepted).

13. A. Khan, M. Shah, A. Ali, On left alternative loops, International Journal

of Pure and Applied Mathematics, (Accepted).

108

prr.hec.gov.pkprr.hec.gov.pk/jspui/bitstream/123456789/7365/1/Amir khan Thesis.pdf · THESIS APPROVAL This is to certify that the thesis submitted by Mr. Amir Khan titled ”Fractional

Documents