ANALYSIS OF NON-NEWTONIAN UNSTEADY NANO FLUIDS USING SERIES SOLUTION SCHEME Mathematics By Syed Asif Hussain Reg No. 2016/ICP/0250
ANALYSIS OF NON-NEWTONIAN
UNSTEADY NANO FLUIDS USING SERIES
SOLUTION SCHEME
Mathematics
By
Syed Asif Hussain
Reg No. 2016/ICP/0250
ABSTRACT
The research work predominantly focuses on the theoretical study of analysis of non-
newtonian unsteady nano fluids using series solution scheme. This study elaborates
the Bioconvection Model for Magneto Hydrodynamics Squeezing Nanofluid Flow
with Heat and Mass Transfer Between Two Parallel Plates Containing Gyrotactic
Microorganisms Under the Influence of Thermal Radiations, Dusty Casson
Nanofluid Flow with Thermal Radiation Over a Permeable Exponentially Stretching
Surface, Unsteady Squeezing Flow of Water Based Carbon Nanotubes in Permeable
Parallel Channels.
The elementary constitutive unsteady equations of velocity, temperature and
concentration are modeled in the form of partial differential equations and converted
to a system of ordinary differential equations by using appropriate similarity
transformation with unsteady dimensionless parameters.
An optimal approach (HAM) have been used to get appropriate results from the
modeled problem. The convergence of HAM method has been identified
numerically.
The first chapter of this thesis contains the introduction, fluid, nano fluids, Non-
Newtonian Fluids, Squeezing Flow, Stretching Sheet, Magneto-hydrodynamic,
Dusty fluid, Carbon nanotubes, Basic idea of ham. In Second chapter we literature
and methodology. In third chapter we investigate the bioconvection magneto
hydrodynamics (MHD) of nano fluid flow squeezed between two parallel plates.
One of the plates is extended and the other is kept fixed. the water is taken as a base
liquid since it is consider as a best liquid for microorganisms life. Thermophoresis
and Brownian motion and their significance also influences Nano fluid model. the
method of combination has also been numerically shown. Nusselt number,
3
Sherwood Number and skin friction variation and effects of these Numbers on
Velocity, temperature and flocky mass and the density of motile microorganism
profiles are checked. It is raising radiation due to temperature also increase the
temperature of fluid layer specially at the in the layers of the boundaries thus the rate
of cooling for nanofluid flow decrease due to this temperature rise. Variation in
bioconvection parameters like in the case of injection and suction, bring variation in
the density of mobile microorganism and was investigated. Along with this some
parameters were graphically plotted and discussed e.g. Thermal radiation parameter
(Rd), Squeezing parameter , Thermophoresis parameter(Nt), Brownian motion
parameter(Nb), Peclet parameter (Pe) , Prandtl number (Pr), Schmidt number(Sc),
Levis number (Le), to comprehend the physical presentation of these parameters in
order to get some conclusion through this research paper. Punjab University
Journal of Mathematics (ISSN 1016-2526) Vol. 51(3) (2019) pp. 113-136.
In fourth chapter of the flow of Casson fluid on permeable exponential stretching
along with dust and liquid phase in the presence of thermal radiation. With the help
of elementary coupled and non-linear variables, the equation mostly used for fluid
velocity and thermal energy has been altered to some suitable differential equation.
The effect of temperature, velocity and concentration on the different types of
parameters like parameter of radiation (RD), parameter of dust particles mass
concentration (𝛾), parameter of unsteadiness (S) and porosity (k), Volume fraction
of dust particle (ɸ𝑑) and Nano particles (ɸ),were studied. The impact of Sherwood
number, skin friction and Nusslet number has been outlined. Correlation is
accommodated the approval of our outcomes with previous successful works and is
found in proper settlement. It has been seen that liquid molecule connection is
directly proportional to the temperature exchange rate and is in reversely
proportional to wall friction. Moreover, with radiation parameter, the profile of dusty
nano fluid is raised productively. Journal of Nanofluids (J. Nanofluids )8,(4)
(2019) pp. 714–724.
In fifth chapter we investigation of 3-D squeezing flows of carbon-nanotube CNTs
in light of water in a pivoting disk with the base wall making permeable has been
introduced. Walls are also keep permeable. Graphs are present to discuss the effect
prominent physical factors on the speed and temperature.
Moreover, for different relevant variables the values for coefficient of Nusselt
number & skin-frictions are arranged in tabular form. The most significant after
effect of this research work is to think about the diverse practices of developing
variables on carbon nanotube (CNTs) in light of the water with change in squeezing
variable and to realize the effects of these rising parameter. Journal of Nanofluids
(J. Nanofluids) 8,(6) (2019) pp. 1319–1328.
FLUIDS FLOW
Presently, the technologies are advancing with a rapid pace and for sure such drift is
moving to the next generation. A comprehensive understanding of the principles of
fluid mechanics and its practical applications in the real world problems are the basic
necessities of the present era. To confront the phenomenon of large number of
complex flows, engineers from all over the world as well as meteorologists,
geophysicists, space researchers, astrophysicists, mathematicians, physical
oceanographers and physicists are being utilizing such understandings. Most often,
the researchers engaged in studying the complex flows involve two or more phases
i.e. reaction kinetics and heat and mass exchange in which the interaction of theses
controlling transport processes play a central role. A quantitative examination via an
appropriate theory is the basic requirement for the comprehension of the physics
involving in the behavior of complex flows as well as to acquire important scale-up
data for its implementation in industry.
NON NEWTONIAN FLUIDS
The studies of non-Newtonian nanofluids have various utilities in the industrial and
technical procedure. The numerous remarkable uses of non-Newtonian fluid
mechanics having one of the most dynamic and significant of all engineering,
medicine and the studies in applied sciences. In metrology, oceanography, and
hydrology the study of fluids is elementary meanwhile the atmosphere and the ocean
is fluids. There are many other fluids in biology and accepting their motion is vital
to effective medicine. The heart pumps a fluid, blood through hosieries of tubes in
the body is an important application of fluid rheology. The most important subclass
of non-Newtonian fluid is the Casson fluid. For the very first time, Casson suggested
the models of Casson fluid that demonstrates a thin shear type of fluid which is
3
supposed to have an unlimited viscosity at a rate of zero trim. The human’s blood is
considered as a Casson fluid as the structural chain of blood cells comprised of
protein, fibrinogen, and rouleaux, etc. Casson fluid has a lot of importance in
manufacturing and scientific point of view.
NANO FLUIDS FLOW
The nanoparticle of size less than 100 nm(Nanometer) adjourned into a base fluid is
known as nanofluid. Nanofluids are castoff in microelectronics, hybrid powered
machines, pharmaceutical procedures, fuel cells, and nanotechnologies’ field. Found
to have a remarkable enhancement in thermal conductivity then assumed from the
traditional heat transfer liquids such as oil, ethylene glycol, water, etc. Due to the
lower thermal conductivity of traditional heat transfer fluids, engineered dispersion
of small fraction of solid nanoparticles. In conventional fluids to change and
enhanced their thermal conductivity and boosting the enactment and compression of
distinct types of engineering instruments such as heat exchangers devices, electronic
devices, aerospace engineering, cooling of reactors and in medicine too. Initially,
the work on nanofluids was mainly concentrated on the measurements with the
function of temperature, nanoparticles concentration and size of nanoparticles.
SQUEEZING FLOW
From the industrial and practical point of view, the squeezing flow amongst various
geometries is worth important. Such kind of flows has many applications in the field
of biomechanics, chemical engineering, mechanical engineering, and food
processing. Squeezing mechanism is also investigated in the composition of
lubrication, machine devices, polymer processing, injection, bearing and automotive
engines. Squeezing flows are in practice since previous decades in different hydro
dynamical machines in discrete geometries in such a way that normal stresses of
upward velocities are forced because of the movement of the geometrical
boundaries.
Bioconvection model for MHD squeezing flow with mass and heat transfer between
two parallel plates containing gyrotactic microorganisms under the influence of
thermal radiation has been studied. It plays an important role in the real world
phenomena and has vast applications which attract the researchers. Squeezing flow
in the parallel plates gained the attention of researchers because of its widespread
significance in different fields, particularly in mechanical engineering, chemical
engineering, and bioengineering, etc.
STRETCHING SHEET
The most commonly utilized sensors in the discipline of chemical and biomedicine
industry comprised of stretching surfaces which acts as sensing elements. For
example, the micro-cantilever is dealt as an appropriate element for sensing to
perceive several bio-warfare means and harmful diseases in the discipline of
chemical and biological industry. During bending over a targeted material, coating
of micro-cantilever with a receptor is accomplished to one of its surfaces.
Nevertheless, the micro-cantilever is normally positioned within a film having cells
of thin fluidity under the effect of influential disturbing squeezing. Such condition
of flow of fluids concerning micro-cantilever can be appropriately formulated since
the flow past the surface of horizontal sensor.
MAGNETO HYDRODYNAMIC
Owing to rapid development in the industry and biomedical applications, the flow
of fluids regarding magnetic field acquired a high engineering concerns in the
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present technology. One of the reasons of using the magneto-hydrodynamic (MHD)
principle is that the field of flow may be distributed in a chosen direction while
switching the structure of boundary layer in the existence of MHD. Besides, the field
of flow can be controlled while modifying the behavior of kinematic flow using the
principle of MHD. Nonetheless, for therapy of several pathogenic conditions, the
idea of using MHD can be intensively applied in the discipline of biomedical
engineering. Accordingly, owing to its countless applications, the MHD appealed
the consideration of various researchers, medical practitioners, fluid dynamists and
physiologists. Besides, a lot of current metal art and metallurgical operations are
controlled using the principles of MHD. Thence, the succeeding reported research
work is focused to confirm the above remarked ideas and applications of MHD.
DUSTY FLUIDS
Most often, dust combines with the particles of fluid and this combination is called
dusty fluid. Examples of dusty fluids are cosmic dusty fluid and motion of dusty air.
In the ionization process, dust particles are resulted along with the emission of
various rays i.e. comet 238. Fluid flows are affected by the elements of dust which
came out with a huge significance i.e. in the petroleum industry and purification of
crude oil. Dusty fluids may be observed in various applications, for instance, the
explosion of nuclear fluids, rocket propulsions, natural wind, spray of paints, the
flow of blood and combustion, etc. Besides, it is applicable in the collection of dust,
cooling of a nuclear reactor, atmosphere fallout, sedimentation, the guided missile,
erosion due to rain, solid fuel rock nozzle performances, and acoustics, etc. Various
researchers have tried to accomplish the approximate numerical solutions. Heat
relocation of radiation is tremendously significant in various industrial operations
and equipment’s at elevated temperatures. The significance of radiative flux is
increasing day by day with the computational and analytical development which is
dependent upon the absolute temperature between two points. The radiations get
increased with the rise in temperature. Particularly, the flows of thermal boundary
layer have substantial concerns with robust thermal radiations.
There is a lot of applications of incompressible sticky fluid flow across the stretching
sheets i.e. in the processes of engineering and technology. These applications
comprise of fiber glass and paper production, plastic sheets, drawing of wires,
blowing of glass, hot rolling, processing of polymers, artificial fibers, etc.
CARBON NANOTUBES
During 1991, nanoparticles were investigated by means of carbon nanotubes
(CNTs). CNTs are made up of prominent molecules of pure carbon atoms which are
thin, long and cylindrical in shape having a diameter of 0.7-50 mm. CNTs have a
discrete significance in nanotechnology, structural composite materials, optics,
conductive plastics, and several others. CNTs have also many applications in
electrical appliances like heating sources, resisters, electrical contacts high-
temperature refractories and in medical devices like biosensors. In everyday life,
CNTs may be used in electromagnetic devices and radios as an antenna. MWCNTs
and SWCNTs are the two main categories of CNTs.
In the area of engineering, science, and industries, porous media possesses many
applications. In the recent past, many of the researchers aimed at working in the field
of fluids transport using saturated porous media. Performing such researches, has
numerous applications, for instance, movement of water in geothermal reservoirs,
geothermal engineering, dissemination of chemical wastes, radioactive transfer of
heat using porous media, heating pipes, etc.
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The foremost objective of the present study is to validate the variations in the warmth
transfer rate of three normally employed base fluids in the existence of MWCNT
and SWCNT. The main purpose was to investigate squeezing flows of carbon
nanotubes in 3-dimensions employing stretching sheets of the porous medium.
Furthermore, magnetic field and viscous dissipation effects were also considered
wherein the fluid motion was perpendicular to the magnetic field. Besides, the
convective boundary conditions were considered below the wall’s surface. The
mathematical modeling was slot in to “t” second section followed by the reduction
of the partial differential system to ordinary differential system using similarity
transformed variables. Subsequently, the ordinary differential equations were
handled numerically and the physical performance of every parameter of MSWCNT
and SWCNT were assigned graphically for temperature, velocity, local Nusselt
number, and skin friction.
BASIC IDEA OF HOMOTOPY
In 1992 Shijun Liao suggested HAM (homotopy analysis method) depend upon
concept of homotopy, a basic idea in differential geometry and topology. The idea
of homotopy can be traced back to a famous mathematician Jules Henri Poincare. In
mathematics, homotopy refers to a non-stop/ Continuous deformation or variations.
For example, a shape of the circle can be deformed into rectangular or elliptic shape,
the coffee cup can deforms continuously into a doughnut shape. Furthermore, the
coffee cup shape cannot be distorted continuously into football form. In
mathematics, a homotopy describes a connection among various things, which
incorporate the same capabilities in a few aspects.
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Literature Review
The bioconvection model for MHD squeezing flow with heat and mass transfer
between two parallel plates containing gyrotactic microorganisms with the influence
of thermal radiation. It has significant applications in the real world phenomena
which attract the researchers. Squeezing flow among the parallel plates gained the
attention of researchers due to general significance in different fields, particularly in
mechanical engineering, chemical engineering, and food industry, etc. (see [1-3]).
Effect of magnetic field on fluid flow is also an essential aspect of research in the
last few decades. In this regard, the following phenomena investigated by many
researchers [4-7]. (i) The velocity parameter of fluid mainly effected due to the
presence of a magnetic field. (ii)The flow of the fluid has remarkable influence when
an electrically conducting fluids passes across the magnetic field. It has numerous
applications in engineering and applied sciences, e.g. in nuclear reactors, boundary
layer control in aerodynamic, plasma and MHD accelerator. Hayat et al studied third
order MHD fluid and analyzed the influence of slip boundaries on velocity. Some
researchers worked on the inclination of the field [8]. The idea of Gyrotactic
microorganisms improves the stability of nanoparticles in the suspension [9].
Among many ways of improving, the stability of nanoparticles and heat transfer in
fluid gyrotactic microorganisms study an important one. A water-based fluid
consisting of motile gyrotactic microorganisms studied numerically in [10-12]. The
micro-organism which is self-impellent can improve the fluid density in a specific
direction, which can cause bio-convection flow. Depend on impellent the micro-
organism is divided into diverse types; including chemotaxis, negative gravitaxis
and gyrotactic. Stimuli for these motile-microorganisms are displacement,
concentration gradient, oxygen and negative gravity between mass and center of
buoyancy respectively. The motion of Nano-particles is not like a motile
microorganism. The motion of Nano-particles is due to thermophoresis and
Brownian motion. Kuznetsov [13] has explained these phenomena. The study of
Kuznetsov is beginning of bio-convection in a horizontal layer having both
gyrotactic micro-organism and nanoparticles. Khan et al [14] studied magnetic and
Navier slip effect in heat and mass transfer in gyrotactic micro-organism in a vertical
surface. Similarly, Khan with Makinde [15] has studied boundary layer flow of
magnetohydrodynamic (MHD) in Nano-fluid consisting gyrotactic organism in the
linearly stretching sheet.
Casson fluid has a lot of importance in manufacturing and scientific point of view.
Attia [16] studied the flows of transient Couette for Casson fluids in the being there
of warmth transfer and MHD within parallel plates. The flow of liquid film of
transfer of heat and Casson fluid in the presence of viscous dissipation and heat flux
was studied by Megahe [17]. The same fluids was investigated by Abol bashariet
[18] in which they have considered the entropy generation of nanoparticles. The
influence of transfer of heat of a Casson fluids in the existence of viscous dissipated
heat source and dependency of temperature was examined by Vijaya et al. [19]
where, in a Casson thin fluid film, the influence of 1st order thermal radiation and
chemical reaction on temperature and mass transfer properties of flow are analyzed.
Various types of other studies [20-25] regarding Casson fluid were conducted as
well.
Because of the nonconventional structure of the nanofluids, it has a lot of
applications i.e. fuel cells, medicinal operations, micro-electronics, hybrid type of
engines and predominately utilized in the area of nanotechnology [26]. One of the
researchers [27] presented a review regarding nanofluids which is based upon the
experiments. Particularly, the flows of nanofluid between the parallel plates is
considered to be one of the typical problem having significant publications.
11
Primarily, it is employed in the industry of petroleum, various sprays for
automobiles, aerodynamic heating, design of the cooling system, MHD pumps and
power generators, accelerators, and refining of crude oil, etc. Some studies [28-31]
examined the fluid flows of nano a fluids’ having a three-dimensional system of
rotating parallel plates under the influence of MHD affects where they have utilized
numerical methods for solving modeled problems in which the effect of attaining
parameters was concisely assigned. The hydromagnetic effect of kerosene alumina
flow of nanofluid for temperature checkup relocation employing a method of
differential transformation was studied by Mahmoodi and Kandelousi [32]. The
MHD and the effect of heat on the flow of nanofluids having a rotating system of
parallel plates was comprehended by Tauseef [33] and Rokni et al. [34]. Lately, the
non-Newtonian 3-dimensional nanofluid and Brownian motion in the existence of a
rotating frame and Hall effect respectively was studied by Shah et al. [35].
The radiations get increased with the rise in temperature. Particularly, the flows of
thermal boundary layer have substantial concerns with robust thermal radiations.
Various researchers [36-40] have investigated thermal radiations of fluids having
electrical conductance and gray for emissions.
There is a lot of applications of incompressible sticky fluid flow across the stretching
sheets i.e. in the processes of engineering and technology. These applications
comprise of fiberglass and paper production, plastic sheets, drawing of wires,
blowing of glass, hot rolling, processing of polymers, artificial fibers, etc. One of
the researchers [41] studied the flow of Jeffery fluid in which the influence of
thermal radiation is examined across the boundary layer in the presence of exterior
exponential stretching. The Carreau fluid flow was examined by Akbar et al. [42] in
which the boundary layer of stagnancy point flow to a reducing sheet having MHD
was studied. Bhatti and Rashidi [43] mathematically investigated entropy’s
generation having non-linear thermal radiations in the existence of MHD boundary
layer of liquid flow using enlarging porous stretching surface. Some of the studies
[44-48] are showing the flow of the boundary layer across the stretching surface for
various non-Newtonian models.
In everyday life, CNTs may be used in electromagnetic devices and radios as an
antenna. MWCNTs and SWCNTs are the two main categories of CNTs. One of the
researchers [49] established an irregular improvement of thermal conductivity in the
suspension of nanotubes. It was demonstrated that nanotubes furnish the most
prominent improvement of thermal conductivity. Currently, lots of researchers are
aiming at the examination of nanofluids which was introduced by Choi [50] in 1995.
According to his findings, the thermal conductive of nano-fluid is much higher than
that of base fluid i.e. water, glycol, etc. Nanofluids are the heat transfer fluids which
are fundamental elements of nanotechnology. CNTs nanofluids possess numerous
applications in electromagnetic, structural materials, chemical, electro-acoustic,
mechanical, cardiac autonomic regulation, transistors cancer treatment, tissues
regeneration, platelet activation, etc. Owing to the mentioned applications, it has
drawn the attention of the researchers to CNTs. One of the researchers [51]
performed experimentations to clearly conclude and verify the results of Choi. Yazid
et al. [52] examined the stability impacts of CNTs nanofluids. One of the researchers
[53] concluded that SWCNTs posses more prominent skin friction and Nusselt
number than MWCNTs when taking into account water as a base fluid. Liu et al.
[54] studied synthetic engine oil and ethylene glycol in the existence of MWCNTs
and reported that the ethylene glycol by CNTs renders much greater thermal
conductive than those of having no CNTs. Wang et al.[55] demonstrated heat
transfer and pressure drop of nanofluids in the existence of carbon nanotubes. The
elementary concept of the present study was demonstrated by Stefan [56] while the
unsteady squeezing flow is investigated by Rashidi et al., [57]. Such researchers
studied the flow models of squeezing flow between the parallel plates. Analytical
13
models were followed for such flows. In the recent past, ol eslami et al., [58] adopted
the analytical technique for the nanofluids having squeezing flow and such technique
is known as Adomian’s decomposition technique. Khan et al., [59] investigated 2-
dimensional squeezing flows for nanofluids in the existence of viscous dissipation
and slip velocity. Kerosine oil and water was taken by them as a base fluid while
copper was taken as a nanoparticle and the solutions for the flow model was acquired
using parameter technique. They also studied the effect of various non-dimensional
parameters over the non-dimensional temperature field and velocity. The influence
of structural over the mechanical glass transition temperature in a porous medium
was investigated by Stefan et al. [60] while the precise flow of 3rd-grade flow
employing porous medium was examined by Ayub et al. [61]. Hayat et al. [62]
examined the Williamson fluid’s flow in a steady state using a porous plate. Using
a porous stretching sheet, Eyring Powell’s fluids was observed by Dawar [63]. Liu
[64] studied the mechanical behavior of the porous concrete i.e. distribution of pores
and permeability. Some of the studies [65-69] observed the fluids employing the
parameter of porosity.
In science and engineering, solving various numerical problems the solution might
be complex, for which computational techniques are utilized. For instance, HAM is
one of the best techniques to solve complex problems. This method may be
employed for solving non-linear differential equations without using linearization
and discretization. This technique was introduced for the first time by Liao [70-76]
and demonstrated that it is the quickest convergent technique for solving specific
problems and it has the ability to provide a single variable solution. The parameters
delivered by this technique may assist in discussing the behavior of the parameters
effortlessly. Owing to the quickest convergent attribute, the majority of the
researchers are utilizing this technique. A tremendous amount of literature is
available regarding the viscous fluids whereas, the relatively lesser extent of
attempts has been performed regarding non-Newtonian fluids.
General Procedure of Homotopy Analysis Method
In 1992, Liao introduced it in his Ph.D. thesis. It is a very effective technique to
control non-linear and linear problems. To comprehend the basic concept of HAM,
assume a differential equation as:
( ) ( ) 0, ,A f s s (2.1)
with boundary conditions:
( , ) 0, .s
(2.2)
In Eq. (2.2) A represent a differential operator, Analytic function is ( )f s , Boundary
operator is and boundary of the domain is .
Generally, the A is a sum of non-linear “ N ” and linear “ L ” operator. Equation (2.1)
is as under:
( ) ( ) ( ) 0L N f s (2.3)
By this technique a homotopy is created by ( , ) : [0,1]H v , It fulfills the Eq.
(2.4)
01 ( ( , )) ( ) ( ) ( ( , )) ( )L v s L s A v s f s (2.4)
In this technique, Eq.(2.4) is known as the oth -order deformation equations. where
is the convergence control parameter, 0,1 is embedding parameter, 0 and
( )s an initial approximation and smoothing function for the solution of equation
(2.3) respectively. When the parameter 0 and 1 , eq.(2.4) be written as:
0( ,0) ( ) 0L v L ( ,1) ( ) ( ) 0,L v A f s separately. Where varies from 0 to 1, the
function ( , )v s starts tracing progressively from 0 ( )s to the curve of solution ( ).u r
Further, we apply Taylor Series to express , v s with respect to , in a way as
below
15
0
1
( , ) ( ) ( ) ,n
n
n
v s s s
(2.5)
and
0
1 ( , )( )
!
n
n n
v ss
n
(2.6)
The Eq. (2.5) convergence is dependent upon the parameter . Liao [70–76], found
that for convergence of series solution, there often occurs such an effective-region
hE that any hE . An approximate solution region is constructed, applying the
problem’s initial conditions. For instance, one may approximately find hE by
illustrating the curves (0) , (0) and so on. The obtained curves are known
as “ -curves”. Suppose that for few values of , the Eq. (2.5), converges at 1 ,
then
0
1
( ,1) ( ) ( ) ( )n
n
v s s s s
(2.7)
Now at this phase, the vector define as
0 1{ ( ), ( ), ..., ( )}n ns s s , (2.8)
Now the thn -order derivative w.r.t to of the deformation of zeroth-order Eq. (2.3)
is taken and the attained result is then divided by n!. It gives us the thn -order
deformation equation for = 0 is
1 1[ ( ) ( )] ( ),n n n n nL s s (2.9)
where
1
1 1
0
1 ( ( , ))( ) ,
( 1)!
n
n n n
A v s
n
(2.10)
and
0, 1
.1, 1
n
n
n
(2.11)
Chapter 3 Bioconvection Model for Magneto Hydrodynamics
Squeezing Nanofluid Flow with Heat and Mass
Transfer between two Parallel Plates containing
Gyrotactic Microorganisms under the Influence of
Thermal Radiations
17
3.1 Introduction
This chapter deals we investigate the bioconvection magneto hydrodynamics
(MHD) of nano fluid flow squeezed between two parallel plates. One of the plates
is extended and the other is kept fixed. The water is taken as a base liquid since it
is considering as a best liquid for microorganism’s life. Some factors lead to a solid
nonlinear standard differential framework. The acquired nonlinear framework has
been unraveled through homotopy examination technique (HAM). Thermophoresis
and Brownian motion and their significance also influences Nano fluid model. The
method of combination has also been numerically shown. Nusselt number,
Sherwood Number and skin friction variation and effects of these Numbers on
Velocity, temperature and flocky mass and the density of motile microorganism
profiles are checked. It is raising radiation due to temperature also increase the
temperature of fluid layer specially at the in the layers of the boundaries thus the rate
of cooling for nanofluid flow decrease due to this temperature rise. Variation in
bioconvection parameters like in the case of injection and suction, bring variation in
the density of mobile microorganism and was investigated. Along with this some
parameters were graphically plotted and discussed e.g. squeezing parameters,
Thermophoresis parameter, Brownian motion parameters, Schmidt number, Levis
number, to comprehend the physical presentation of these parameters.
3.2 Formulation
Modeling and calculation used in this section described as under:
Consider 2-D, unsteady and symmetric natures incompressible flow in two parallel
channels with MHD and thermos radiation impact. The disks are place in Cartesian
coordinate systems. The lower disk place on x-axis while y-axixs normal to the
bellow disk. Distance among the disks is ,y h also note that the lower plate movable
and the lower plate is plae at 0y . Lower plate moves with velocity dh
v tdt
and
the magnetic field 0B in y-axis direction. The 2 and 1 , be the temperature on upper
and lower plates Furthermore upper plates have some auxiliary conditions and the
nanomaterials are separated uniformly. It is presumed that both plates are sustained
at constant temperature. Uniform microorganisms distribution on the upper plate
represented by 2 and lower plate by 1 . The nanomaterials are dispersed uniformly
on the lower disk. The geometrical representation of the nanofluid model displayed
in the fig. 3.1.
FIGURE 3.1: Geometry of the Nanofluid flow.
On the basis of in above analysis, the rudimentary Eqs. are continuity, velocities,
temperature, concentrations and motile-microorganism densities are enunciated [3]
as under,
0,x yu v (3.1)
2
0 ( ),nf t x y x xx yyu uu vv p u v B u t (3.2)
2 2
2 2,nf
pu
t x y y x y
(3.3)
22
0
1,
B x x y y
rdt x y xx yy
Tp f
D T C T C
qT uT vT T T D yc
x y
(3.4)
19
0
,Tt y x B yy xx yy xx
DC C uC D C C T T
(3.5)
* 2
2.t y x n
vN vN uN D
z y
(3.6)
In above mentioned Equations. (3.1-3.6) u and v denotes velocities component,
& C signifies temperature on plate and the volumetric fraction of the nanoparticles,
be density of the motile-microorganism,
p
f
c
c
, while ( )pc
and ( ) fc
embodies heat capacity of nanomaterials and fluids. Furthermore, indicates
viscosities, BD signifies Brownian-diffusions and TD symbolizes coefficient of
thermophores coefficient. Eqs. (3.1-3.6) represents the flow model for nanofluid.
Further * ( )cbw Cv
C y
. In Eq (3.4), rdq is the radiative heat fluctuations is express in
terms of Roseland approximations as:
4*
*
4,
3rdq
y
(3.7)
Where in eq. (3.7) *k and * denoted the mean absorptions-coefficient & Stefan’s-
Boltzman constant. Assuming that the differences in heat insides the flow is such
that 4 can be expresse as a linear-combinations of heat, and expand 4 by Taylor’s
theorem about 0 as beneath:
4 4 3
0 0 04 ..., (3.8)
Neglecting greater order term we get:
4 4 3
0 03 4 , (3.9)
By Putting Eq. (3.8) in Eq. (3.7) we get
3 * 2
0
* 2
16,
3
rdq
y K y
(3.10)
B.Cs for upper and lower disks:
1 0 10, 0, T , , ,u C C N N (3.11)
2 2
0
, 0, , ( ) ( ) 0 and .TB
Ddh Cu T D N N
dt y y
(3.12)
The dimensionless similarity variables for microorganisms flow model as under:
0.5 1 0.5 0.5
0 0
2 0 0 2 0
1 1 1 (1 )( , ) ( ), ( ), ( ), ,
( ) , ( ) 1 ( ) .
t t t v tx y xf u f v f y
bv bx bv b
N NCand
C N N
(3.13)
Puting Eq. (3.13) into the governing Eqs. (3.1-3.6), we develops the successive
transmuted ODE’s as under:
- - -3 - 0,ivf ff f f f f f (3.14)
24
1 Rd 0,3
Nt r f Nb
(3.15)
0,Nt
Le f
Nb
(3.16)
( ) 0.Sc f Pe Pe (3.17)
After simplifications the dimensionless variables as beneath:
2 3
0 2 0
0
0
1 0 1 0 1 0
1
2 0 0 2 02
( ) ( )4, (1 ), , ,
2 ( )3
( ), Pr , , , ,
( )
( ) ( ), , , .
( ) ( )2( )
p T
fp f
p b c c
f n n B
cB DTM t Rd t
b b cc k
c D C b Wv v vNb Sc Pe Le
c D D D
N N C CH
N N Cvb
(3.18)
In the above model, equations (3.14-3.17) unlike parameters are used like Thermal-
radiations ,Rd Peclet number ,Pe Levis number ( ),Le , Brownian-motions ,Nb
Prandtl Pr & Schmidt numbers Sc and thermophoresis parameter ,Nt unsteady
21
squeezing parameter. Where , , and are fixed constants. Besides,
altered B.Cs for lower & upper disk demarcated in Eqs. (3.11) and (3.12) as under:
(1) , (0) = (0) = (1) = 0, f w f f f
Nb Nt
(3.19)
The Skin-frictions, Sherwoods and Nusselt numbers, motile-microorganisms density
Number are define as:
2
0 0 0
0 0 0 0
q2, u = , , ,
K T - T
, , , .
w m nf x x x
w B w n w
m B n n
y y y y
x xq xqC N Sh = Nn
U D C - C D n n
uq q D q D
y y y y
(3.20)
Applying equation. (3.13) non-dimensional form, Nusselt-Number, Skin-frictions,
Sherwoods-Number and motile-microorganisms local densities are as:
0.5 0.5 0.5Re0 , Re 0 , Re 0 & Re 0
2
x
f x x x x x xC f Nu Sh Nn
(3.21)
Where Rex
xU
v
is signifies a local Reynolds number.?
3.3 HAM Solution
In organize to technique equation. (3.14-3.17) with boundary-conditions (3.19), we
apply HAM [70-76]. The initial suppositions and linear-operators ( 0f , 0 , 0 , 0 ) and
(fL , L , L
, L) These are chosen as follows:
2 3
0 0
0
0
( ) 3 2 , ( ) 1 ,
1( ) ( ),
φ ( ) 1
f
Nt Nt Nt Nt NbNb
(3.22)
The linear operators taken as:
(f) , ( )= , ( )= , ( )= .fL f L L L
(3.23)
The above mentioned differential operators’ contents are shown below:
2 3
9 10 1 2 3 4 5 6 7 8( ) ( ) ( ) ( ) 0.f L L L L
(3.24)
Here 12
1i
i
where 1, 2,3...i denotes arbitrary constants.
The resultant nonlinear operators are given by: , , , and fN N N N
4 2
4 2
3
3
( ; ) ( ; ) ( ; )( ; ) ( ; ) ( ; )
( ; )3 ( ; ) ( ; ) ,
f
f f fN f f f
ff M f
(3.25)
2
2
2
4 ( ; ) ( ; ) ( ; ), ( ; ), ( ; ) 1 Pr ( ; )
3
( ; ) ( ; ) ( ; ),
N f Rd f
Nb Nt
(3.26)
2
2
2
2
( ; ) ( ; )( ; ), ( ; ), ( ; ) ( ; )
( ; ),
N f Le f
Nt
Nb
(3.27)
2
2
2
2
( ; ) ( ; )( ; ), ( ; ), ( ; ) ( ; )
( ; ) ( ; ) ( ; )( ; ) .
N f Sc f
Pe Pe
(3.28)
3.1.1 0th Order Deformation Problems
0( 1) ( ) ( ; ) ( ; ) ,f f ff f h N f L (3.29)
0(1 ) ( ; ) ( ) ( ; ), ( ; ), ( ; ) ,h N f L (3.30)
0( 1) ( ) ( ; ) ( ; ), ( ; ), ( ; ) ,h N f L (3.31)
0( 1) ( ) ( ; ) ( ; ), ( ; ), ( ; ) .h N f L
(3.32)
The subjected B.C’s are derived as:
23
0
0 1 0
1 0
1
0 1
0
1
( ; ) ( ; ) ( ; ) , ( ; ) , ,
( ; ) ( ; ) 1, ( ; ) , ( ; ) ,
( ; ) , ( ; ) 1, ( ; ) .
f ff o f w o
Nt Nb
(3.33)
Where [0,1] interval is the imbedding parameter ,f ,
and were apply to
regulate convergence of the solution. Where 0 and 1 we have:
( ;1) ( ) 0, ( ;1) ( ) 0, ( ;1) ( ) 0 and ( ;1) ( ).f f
For expanding the upper terms of by Taylor’s series expansion we take:
1 10 0
1 10 0
( , ) ( ) ( ), ( , ) ( ) ( ),
( , ) ( ) ( ), ( , ) ( ) ( ).
i ii i
i ii i
f f f
(3.34)
Where
1 ( ; ) 1 ( ; )( ) , ( ) ,
i! i!
1 ( ; ) 1 ( ; )( ) , ( ) .
i! i!
i i
i i
ff
=0 =0
=0 =0
(3.35)
3.1.2 Ith Order Deformation Problem
1 1
1 1
( ) ( ) ( ), ( ) ( ) ( ),
( ) ( ) ( ), ( ) ( ) ( ).
f
f i i i f i ii i i i
i i i i i i i i
f f h h
h h
L L
L L
(3.36)
The subsequent B.C’s as:
(0) = (0) = (1) = 0, (1) = (0) (1) 0,
(0) (0) 0, (1) (0) (1) 0.
i i i i ii ii
i i i ii ii
f f f f
Nb Nt
(3.37)
1 10 01 1 1 1 1 1( ) 3 ,f iv i i
k ki i i k k i k k i i if f f f f f f Mf
(3.38)
1 1 10 0 01 1 1 1 1
4( ) 1 Pr ,
3
i i ik k ki i k k i i k k i k k ib Rd t f
(3.39)
101 1 1 1 ( ) , i
ki i i k k i i
NtLe f
Nb
(3.40)
1 1 1
0 0 01 1 1 1 1( ) .i i ik k ki i i k k i i k k i k kSc f Pe Pe
(3.41)
Where
1, if 1
0, if 1. i
(3.42)
3.4 Convergence of Solution
Once we compute the perturbation solution for velocities, density of motile
microorganism, heat and concentrations-function by utilizing HAM technique,
the supporting parameter are , , f ,
. For the solution convergences’ these
parameter are very important. To get likely region of the -Curves graph of
(0), (0), (0)f & (0) for 13th order-approximations are graphed in the figs.
(3.1-3.2). The -curve successively displays valid regions. In HAM scheme the
convergences regions is essential to determines the meaningful perturbation
solution of prevailing problem of (0), (0), (0)f & (0). The , , f , &
parameter are engaged to controls the solution. Moreover, the -curve are
strategized at 13th order-approximations. As from -curve we observe the
suitable ranges , , f ,&
are 2.3 0.2, 2.2 0.1,f and 2.1 0.4.
25
2.5 0.5 .
FIGURE 3.2: The combined curve of functions ( ), ( )f at 13th order approximation.
FIGURE 3.3: The combined curve of functions ( ) and ( ) at 13th order approximation.
Table. 3.1.
HAM convergences up to 13th Order-approximation while, 0.5,M Nt
Pr 0.7, 1, 0.6, 0.8, 0.1, and 0.7.Nb Le Sc Rd Pe
Approximation
Order.
(0)f
(0)
(0)
(0)
1 3.98886 -0.0207921 0.953750 1.00025
3 3.97650 -0.0387064 0.913169 1.07375
5 3.97584 -0.0396453 0.910490 1.08915
7 3.97583 -0.0396677 0.910383 1.09009
9 3.97583 -0.0396682 0.910379 1.09014
13 3.97583 -0.0396682 0.910379 1.09014
3.5 Results and Discussion
This subsection of the thesis, we have disclosed physical interpretation of sundry
parameters involving into the problem and to comprehend the effects of numerous -
dimensionless physicals-quantity on velocity ( )f , energy/heat , Concentrations
& Density of motile microorganism profiles. The subsequent outcomes
with full details are attained. The model of the fluid flow problem shown by in fig.
3.1 for readers understanding. The h-curves are elaborated in Figs. (3.2-3.3). Figs.
(3.4-3.7). represent the squeezing fluid parameter impacts over , , f
& . When disks/plates are moving aside, then take positive values in that
case and when disks are coming nearer, the negative values are considered.
27
Obviously, fig. 3.4 indicates the impact of the flow when disks are moving away and
that is contrary case of whilst disks coming closer. With the rise of value fluid
velocities also growing. Visibly velocity rises in the channel while fluid sucked
inside. Covertly when fluid injected out, then the plates come closer to each other.
This way brings about a drop within the fluid and consequently declines the velocity.
With varying, values of the impact of f revealed in fig. 3.4. The figures. 3.5.
& 3.6. display the effect of on the heat and concentration profiles separately. Due
to squeezing of the liquid, the speed will increase and finally falls the temperature
of the liquid because warm nanomaterials are escaping hastily which leads to
decrease temperature and the concentration of the liquid reduces. The Fig. 3.7.
Indicates variation in densities of the motile-microorganisms for various value of .
The density of microorganisms illustrates variations. With changing values,
the is a decreasing factor, when parameter changes negatively and it shows
increasing function for positive value of . The figure. 3.8 exhibits the effect of
velocity profile for numerous values of magnetic M parameter. It illustrates that with
an growth in vale of M , velocity field falls, because Lorentzs-force work towards
against the flows and those areas wherein its effects dominate, it decreases velocity.
After some distance it rises. Demonstrates The features of magnetic M parameter
on heat profile, is illustrated in figure 3.9 which is growing for greater value and
falls for the smaller value of M . Due to Lorentzs-force reducing which depends
upon magnetic M , so reducing M reduce Lorentz forces and therefore drops ( ).
profile. The influence of Pr over heat and concentration profiles are
displyed in Figures. 3.10. & 3.11 respectively. Obviously it is observed that heat and
concentrations profiles vary conversely with Pr that is heat distributions descent
with huge numbers of Pr and increase for smaller value of Pr . The physical, point
view the fluid have a small numbers of Pr has larger thermal diffusivities and this
effects is reverse for greater values of Pr because of this facts massive value of Pr
leads the thermal boundaries layers to reduce. The impact is even more diverse for
less quantity of Pr as thermos boundaries layers thicknes is comparatively larger.
However, growing behavior of concentrations profiles is illustrated figure. 3.11 for
greater value of Pr . The effects of thermophoretic Nt parameter over ( ). is shown
by figure. 3.12 and it is investigated that heat profiles is raised by varying .Nt By
Kinetic-Molecular theories, growing variety of nanoparticles and growing active
actives nanoparticles both can causes to rise in the warmness factor. The figure 3.13
illustrates the varying in Nt changes the concentration distribution . In case of
injection, the decrements in concentration profile is slower as compared to the
suction case. The consequence of Nb on and profiles shown in figures.
3.14. & 3.15. The Temperature distribution is increases by varying value of
Nb as illustrated through figure. 3.14. According to Kinetics-molecular theories the
temperature of the liquid rises due to the rise of Brownian-motions. Likewise, figure.
3.15 highlight the impression of Nb with respect to profile in closed interval
[0,1] .The rising effect of is seen for both suctions and injections case in the
figure. 3.15. The rapid increase was detected in for fluid suctions as compare
to fluids injections cases. The figure. 3.16 represents effect of Peclet number Pe
over . The values of density field of motile microorganism increase with
increase value of Pe . Fig. 3.17 shows the impacts of on density field of motile
microorganism . The values of density field of the motile microorganisms
decrease with rise in the value of .Sc Actually, Schmidt numbers is the ratio of
kinematicviscosities to mass flux. So when kinematic viscosity increases, then
spontaneously the Sc increases and decreases. The inspiration of Le on
is illustrated by figure. 3.18, where it falls when Le is rises. Because it is ratio of
29
thermal-diffusivities to the mass diffusivities. When thermal diffusivities reduces
automatically, it reduce Le and reduce concentrations profile. The impact of
radiations Rd parameter over is shown by figure 3.19. It is simply found that
heat distribution falls with rising value of Rd . Commonly it is observe, radiating a
fluids or some others things can causes to reduce the heat of that specific objects.
3.5.1 Table Discussions
Table.1 displays numerical values of HAM solutions at different approximation
using various values of different parameters. It is clear from the table.1 that
homotopy analysis technique is a quickly convergent technique. Physical quantities
such as skin friction co-efficient, heat flux, mass flux and Local-density number of
motile microorganism for motivation of science and technology are calculate by
Tab’s: (3.2-3.5). Table: 2 displays the impact of inserting parameters and M on
Skin friction .fC It is seen that increasing value of and M decreases the skin friction
.fC Table.3 examines the influences of embedding parameters , , PrNb Nt and Rd on
heat flux .Nu
It is seen that increasing values of Pr increase the heat flux ,Nu where
, and Rd Nt Nb decrease the heat flux when it increased. Table.4 inspects the
influences of , and Le Nb Nt on mass flux .Sh The increasing values of and Le Nb
increase the mass flux where and SrNt reduces the mass flux .The influences of
, and PeSc on 0 are shown in Table.5. The increasing values of Pe increases
0 , while the higher value of and Sc reduce 0
3.5.2 GRAPHS
FIGURE 3.4: Effect of on .f When
0.8 and 1.9.M
FIGURE 3.5: Effect of on . When
0.8, 0.4, 0.1, 0.4,Pr 0.6.Le Nt Rd Nb
31
FIGURE 3.6: Outcome of our . When
0.8, 0.4, 0.1, 0.6,Pr 0.6, 0.5.Le Nt Nb M
FIGURE 3.7: Effect of on . When
0.8, 0.4, 0.1, 0.3, 1.Le Sc Nt Pe Nb M
FIGURE 3.8: Effect of M on .f When
0.8 and 0.9.
FIGURE 3.9: Effect of M on . When
0.8, 0.3, 1, 0.6, 0.1,Pr 0.5.Le Rd Nt Nb
33
FIGURE 3.10: Effect of Pr on . When
0.8, 0.3, 0.4, 0.6, 0.1, 1.Le Rd Nt Nb M
FIGURE 3.11: Effect of Pr on . When
0.8, 0.3, 0.4, 0.6, 0.1,Pr 0.2, 1.Le Nt Nb M
FIGURE 3.12: Outcome our Nt on . When
0.8, 0.3, 0.4, 0.1,Pr 0.6, 2.Le Rd Nb M
FIGURE 3.13: Effect of Nt on . When
0.8, 0.3, 0.4, 0.1,Pr 0.6, 2.Le Nb M
35
FIGURE 3.14:Outcome our Nb on .
0.8, 0.3, 0.4, 0.1,Pr 0.6, 2.Le Rd Nt M
FIGURE 3.15: Effect of Nb on . When
0.8, 0.3, 0.4, 0.1, Pr 0.6, 1.Le Nt M
FIGURE 3.16: Effect of Pe on . When
0.8, 0.3, 0.4, 0.1, 0.6, 2.Le Sc Nb Nt M
FIGURE 3.17: Effect of Sc on . When
0.8, 0.3, 0.4, 0.1, 0.6,Pr 0.5, 1.Le Nb Nt M
37
FIGURE 3.18: Effect of Le on . When
0.8, 0.4, 0.4, 0.3, 0.1,Pr 0.6, 1.Le Sr Nb Nt M
FIGURE 3.19: Outcome our Rd on .
0.8, 4, 0.1, 0.6,Pr 0.5, 1.9.Le Nb Nt M
3.5.3 TABLES
Table. 3.2.
Numeric value of the skin-Frictions Coefficients for numerous parameter
When 1, 0.6, 0.8, 0.7Nb Nt Le Sc Pe and 0.1 .
M
1
2Ref xC
Hayat et al. [77] result
1
2Ref xC
Present results
0.1
1.5
-2.40160
1.1051
0.5
-2.41735
0.9999
1.0
-2.42522
0.9957
0.1
1.5
-2.40788
1.2057
2.0
-2.40828
1.1947
2.5
-2.40948
1.1747
3.0
-2.41426
0.9957
39
Table. 3.3.
Numeric value of Local-Nusselts numbers for dissimilar kind parameter,
when Pr 0.7, 1, 0.6, 0.8, 0.7, 0.1, and 0.5.Le Sc Pe M
Rd
Nt
Nb
Pr
0
Alsaedi et al
[9]. Result
0
Present result
0.5
0.5
0.5
1.0
------
2.0003
0.1
------
1.6202
1.5
------
1.2133
0.5
0.5
0.8167
1.9501
1.0
0.6971
1.5546
1.5
0.5735
1.2013
0.5
0.8943
2.5923
0.5
0.8011
1.7456
1.0
1.0
0.7472
1.0072
0.5
1.5
0.8943
1.1196
2.0
1.0270
1.7456
Table. 3.4.
Numeric type value of Local-Sherwoods number for dissimilar Parameter
when Le=0.6, Sc=0.8, Pr=Pe=0.7, λ=M=0.6, and ω=0.1.
Le Nb Nt 0
Alsaedi et al. [9]
result
0
Present results
0.1 0.2 0.5 0.4471 0.8518
0.5 0.5878 0.9618
0.1 0.2 0.5878 0.6624
0.6 0.9582 0.7910
1.0 1.0320 0.8518
0.2 0.5 0.5878 0.8615
1.0 0.8588 0.9020
1.5 -0.3914 0.8901
Table. 3.5.
Numeric value of Local-density-number of motilemicroorganism densities number for
numerous kinds of parameters when 0.5, Pr 1, 0.8, and 0.1,M Sc
and 0.7.Pe
Sc
Pe
0
Alsaedi et al. [9]
result
0
Present results
1.0 0.5 0.5 ------ 1.6434
1.5 ------ 1.2526
2.0 ------ 0.9542
1.0 0.5 ------ 2.1053
1.0 ------ 1.8599
1.5 ------ 0.9552
0.5 0.5 1.3811 1.6864
1.5 1.4764 1.7801
2.0 1.5731 2.1053
41
3.4 Main Observations
The problem of steady bioconvection flow between two parallel plates is studied by
assuming It is the plates capable to expand and contract. For the purpose a simple form
of similarity variables is used by reducing the supporting boundary condition in the
form. As mentioned in the previous lines that embedding parameters are graphically
presented and discussed and the different types of parameters and numbers are studied
and investigate. The base points are:
1. The concentration field of suction and injection reversely effected by
Thermophoretic and Brownian motion parameters.
2. Variation in various physical parameters and convergence of the homotopy were
numerically analysed.
3. For various bioconvectoion parameters the are precisely analysed.
4. Changes in the motile microorganisms density were analyzed for various
bioconvection parameter .
5. Skin friction .fC decrease continuesly by increasing the and M values.
6. heat flux Nu increase by increasing Pr and decrease by increasing .
7. Mass flux is in direct propotion to and Le Nbas when and Le Nb increase mass flux
also increases while mass flux reduce with Nt .
8. A very prominent variation in the density of motile microorganism observed for the
Brownian (Nb), thermophoretic (Nt) and suction/ injection parameters.
Chapter 4
Dusty Casson Nanofluid Flow with Thermal Radiation Over a
Permeable Exponentially Stretching Surface
43
4.1 Introduction
In this section, of the flows of Casson fluid on permeable exponential stretching
along with dust and liquid phase in the presence of thermal radiation. With the help
of elementary coupled and non-linear variables, the equation mostly used for fluid
velocity and thermal energy has been altered to some suitable differential equation.
Thus for the solution of model equations logical methodology has been applied. The
effect of temperature, velocity and concentration on the different types of parameters
like parameter of radiation, parameter of dust particles mass concentration,
parameter of unsteadiness and porosity, Volume fraction of dust particle and Nano
particles were studied. The impact of skin friction, Nusslet number and Sherwood
number has been outlined. Correlation is accommodated the approval of our
outcomes with previous successful works and is found in proper settlement. It has
been seen that liquid molecule connection is directly proportional to the temperature
exchange rate and is in reversely proportional to wall friction. Moreover, with
radiation parameter, the profile of dusty nano fluid is raised productively.
4.2 Formulation
Assume the unsteady 2-D thermally and electrically conducting boundary layer flow
of the dusty Casson nanofluid through porous stretching plate surface. The speed of
fluids flow over stretching surface is assumed as
0 /, exp(1 )
w
xUU x t
ct
, (4.1)
where is a exponential parameter. The flow is unsteady which start at 0t and it
became steady for 0.t The Coordinates system is selected in such a way that the
surface is along x -axis and y -axis is perpendicular on it.
The temperature source is considered non uniform where the dust particles are taken
uniform in the size. The shape of the nano and dust particles has been considered
spherical. The volumetric fraction nano particles, dust particles and number of
density of the dust particles have been considered. The heat maintained near the
surface is
0 /2, exp1
.x
w x tct
(4.2)
The necessary constitutive equations of Casson fluid as
..
.
..
.
.
; ,2 2
; .2 2
c fc fmn
d c f
mn d
c fc fmn
d c f
mn c f
e
e
y
y
p
p (4.3)
where .c f
mn yield the shear stress tensor of Casson fluid, .c f
denoted Casson fluid
plastic viscousity, d deformations rate to itselfs,.c f the critical amount of current
product depend on Casson fluid. Assume the situation in which certain liquid
necessitate gradual growing the shear stress to sustain the constant rate of strain. In
the case .d c f the Casson fluid posses the kienematic viscousity .c f is the function
of . .,c f c f
and Casson parameter such that
. .. . .
.
.
. 2.
. 2
2, , ,
2
11
1, 1 .
c f c fdc f c f c f
c f
d
c f
c fc f
c f
u
y y
yy
pp
(4.4)
Here .c f present dynamic viscosity of Casson fluid, . 2c f
d
yp and yp show yield
stress of liquid. From the denotation of described the product of .
c f and 2 d to
the yield stress yp show the resistivity in liquid greater if o and the liquid present
the inviscid behavior if o which vanishes the viscosity of Casson fluid.
For dusty nanofluid, boundary layer equations are
45
0,x yu v (4.5)
2
2
11 1 ,1n d d nf pf
u u u uu КN u u
t x y y
(4.6)
,0p pu
x y
(4.7)
,p p p
p p p
u u uNm u КN u u
t x y
(4.8)
2
.p t x y nf yy pnf
qN rc T uT T k T u uy
(4.9)
The corresponding B.Cs as
, , , , , , y 0,
0, as, 0, .
w w w
p p
u U x t V x t T T x t where
y u T T y
(4.10)
Here the velocity & particle phase of fluid along & x y are represented by ,u
and . ,p pu The nf is dynamic viscosity and
nf is nanofluid density,
represents stokes resistance, also N and m are number density and mass
concentration of the dust particle’s respectively, where is electrical
conductivity. Also f nfc and
nfk are specific heat capacitance and effective
thermal conductivity where t denotes time.
Here 0 /, exp(1 )
w
xUU x t
ct
shows sheet velocity, c is the positive constant which
measure un-steadiness,
0 /, exp2 1
f x
w
Ux t q
tV
c
is a suction velocity, 0q for
0q injection, 0U is the suction velocity, characteristic length is ,
/20, exp1
w
xx tct
be heat distribution near the surface, 0 represent
reference heat. qr can be described under Rosseland approximation radiative
heat-flux as [36-48]:
* 4
*
4,
3qr
y
(4.11)
where * and * identifies mean absorption coefficients and Stefan-Boltzmann
constant individually. Heat differences in the flow are supposed appropriately
minor provided that 4 may be conveyed as a linear function of heat. Moreover
expanding 4 applying Taylor’s series by ignoring higher order terms yield:
4 3 44 3 , (4.12)
Furthermore, nano fluid constants are specified by [26-35]:
2.51 , 1 ,
1 1.
1
p p p nf fnp p s
s f f snf
f s f f s
c c c
n n
n
(4.13)
In order to measure nfk for various shapes of the nano particles, we assumed the
flowing formula, given by Crosser & Hamilton [78]. Here n is used for nano
particle shapes. Furthermore, 3/ 2n is for cylindrical and 3n for spherical
shapes. Volume fraction is denoted by in the nano particles. Subscript f
indicates fluid whereas subscript s indicates the solid properties.
Introducing the similarity transformations into the governing equations which
transform into the below respective coupled non-linear ordinary differential
equations:
47
12
00
12
00
12
0 0
exp ( ), exp ( ) ,1 2 1
exp ( ), exp ( ) ,1 2 1
, exp , exp .2 1 1
fx x
fx x
p p
x x
w f
UUu f E f f
ct ct
UUu g E g g
ct ct
Uy
ct ct
(4.14)
By inserting equations (4.11) to (4.14) into equations (4.6), (4.8 4.10) and
comparing the coefficients of 0
x L on both sides, we get
2111 1 1 2 2
1
2 0
d sd
f
f f ff S f f
g f f
(4.15)
2
2 2 2 0gg g S g g f g (4.16)
1 21 42Pr
3Pr 1
4 0
nf f
p pf s
Rdk k Ec g f
c c
f f S
(4.17)
The B.C’s:
, 1, 1 0,
0, g 0, g ,
0 .
f q f at
f f nf g
as
(4.18)
Here exponential parameter is , unsteadiness parameter is A where 0A Uc ,
represent mass concentration of a dust particle where
fNm , is a fluid
particle interaction parameter,
0
1Ex
f
ctM e
u
where
01 vct U , and
v m denote relaxation time of the dust relaxation time for the prandtl number
and Eckert number is Ec where 2
0 0 .p fEc U c T
For the sack of interest of the
engineers fC (friction factor) and xNu (Nusselt number) are defined as:
1 21 2 1
Re 1 1 0f xC f
(4.19)
1 2Re 0x x nf fNu k k (4.20)
Here local Reynolds number is 0Re .x fU
4.3 HAM Solution
Liao [70-76] was the pioneer who anticipated the HAM. He applied one of the
fundamental idea of the topology called Homotopy to derive the current method.
Two Homotopic functions has been used to derive present method. These functions
are known as Homotopic function when one of them is continuously distorted into
another. Assume that 1 2, are two function which are nonstop and ,X Y are
mapped from to Y then 1 is called homotopy to 2 while it gives a nonstop
functions
: 0, 1 .Y
(4.21)
x
1, 0
and 2, 1 (4.22)
Then this mapping are known Homotopic. This scheme give us auxiliary
technique and it’s mainly applied to the non-linear differential equations without
linearization and discretization.
HAM which is one of the faster and stronger convergence method. HAM is use both
strong and weak solution of nonlinear problems. Solution results found by HAM
49
hold assisting parameter which control and adjust convergence of solution as well
as bases function. The initial guess is
0 0 0( ) exp 1 , g ( ) 1 , ( ) exp .x xf q q η η η (4.23)
The linear-operators can be
3 2 2
3 2 2( ) , ( )= , ( )= .f g
f f gf g
η η η η
(4.24)
The above-mentioned differential operators satisfy the
3 1 2
4 5
6 7
( exp exp ) 0,
( exp exp ) 0,
( exp exp ) 0,
f
g
η η
η η
η η
(4.25)
Where 7
1i
i
are arbitrary constants. The resultant nonlinear operator through
, & f gN N N for the selected problem in the below given as:
23 2
3 2
( ; ), ( ; )
111 1 1 2
1
2 2 ,
f
d sd
f
f g
f f ff
S f f g f f
η ηη η η η
η η
η η η
η
(4.26)
22 2
2 2( ; ), ( ; ) 2 2 2 ,g
g g g g f gf g g S
η η η
η η η η η η (4.27)
2
2
2
1 4( ; ); ( ; ); ( ; )
3Pr 1
2Pr 4 ,
nf
fp ps f
k RdN f g
kpc pc
Ec g f f f S
η η η η η
η η ηη
(4.28)
4.3.1 0th Deformation Problem
0( 1) ( ) ( , ) ( , ), ( , )f f ff f N g f η η η η (4.29)
0( 1) ( ) ( , ) ( , ), ( , )g g gg g N f g η η η η (4.30)
0( 1) ( ) ( , ) ( , ), ( , ), ( , )N f g η η η η η (4.31)
The subjected B.Cs obtained as
( ; ) ( ; )( ; ) 0, 1, 0,
( ; ) ( ; ) ( ; )0, ( ; ) ( ; ) ,
( ; ) 1, ( ; ) 0
f ff
g f gg f n
η=0
η=0 η
η η
η η η
η=0 η
η ηη
η η
η η ηη η
η η η
η η
(4.32)
where , , 0f . The 0th order deformation problem taken as results. For 0
and 1 we get:
( ;1) ( ); ( ;1) ( ); ( ;1) ( ).f f g g η η η η η η (4.33)
Expand the non-dimensional temperature, concentration and velocity fields
( ; ); ( ; ); ( ; )f g η η η as
10
0
1 ( , )( , ) ( ) ( ) ( ) ,
!
ff f f f
ηη η η
η
MM M M
M
(4.34)
10
0
1 ( , )( , ) ( ) ( ) ( ) ,
!
gg g g g
ηη η η η
η
MM M M
M (4.35)
10
0
1 ( , )( , ) ( ) ( ) , ( ) .
!
ηη η η η
η
MM M M
M (4.36)
The axillary parameters , f and are equation series (4.29-4.31) converges at
1 , we achieve:
1
1
1
( ) ( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ).
o
o
o
f f f
g g g
η η η
η η η
η η η
M M
M M
M M
(4.37)
4.3.2 thM Order
The thM order problem gratifies the succeeding equations are:
1
1
1
( ) ( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ),
f
f f
g
g g
f f R
g g R
R
η η η
η η η
η η η
M M M M
M M M M
M M M M
(4.38)
51
The resultant B.Cs are
, , ,
, ,
f o f o o at o
f o g g o o as
η
η
M M M
M M M M
(4.39)
Here
1
1 1
1 10 0
1 1 1 1 1
11( ) 1 ( )
1
1 1 2
2 2 ,
df
sd n n n n
n nf
R f
f f f f
S f f g f f
η ηM M
M M
M M
M M M M M
(4.40)
1
1 1 10
1
1 1 10
( ) 2 2
( ) 2 ,
g
n nn
n nn
R S g g g g
g g f g
η
η
M
M M M M
M
M M M
(4.41)
1
1 1 1
1 1 10 0 0
1 1 1 1
1 4( ) ( )
3Pr 1
+2Pr 2
+ 4 .
nf
fp ps f
n n n n nn n n
k RdR
kpc pc
Ec g g f f g f
f f S
η ηM M
M M M
M M M M
M M M M
(4.42)
Where
1, if 1
0, if 1
M (4.43)
4.4 Numerical and Graphical Discussions
The aim of this segment of the article is to understand the effects of different
parameters which have been discussed and displayed from Figs. (4.1-4.21), Fig. 4.1
and 4.2 presented volume fraction’s impact on velocity of fluid and on dust particles.
The rise in volume fraction of the dust particles ,d decreases dust particles and
nano-fluid velocity. The density of fluid increases due to which velocity of fluid
decreases and in other words the volume to mixture ratio of dust elements is greater
than mass to dust particles. It also decreases thermal boundary layer thickniss. Fig.
4.3 and 4.4 represents the influence of volume fraction of nano particles on
temperature, velocity of Nano-particles. Graphs clearly indicate that increasing
volume fraction cause to increase the velocities of nanofluid as well as the
temperature profiles. The effect of unsteady parameter is shown in the graphs (4.5-
4.7) to rise the value of unsteady parameters it decreases the speed and warmth of
dust phase because stretching and steadiness can increase the value of velocity of
fluid. This increases the fluid velocity at the edge layer. Fig. (4.8-4.10) depicts the
influence of the interaction parameter with temperature, velocities of Nano-fluid
phase and dust particle phase. Increasing value of interaction parameter increases
velocity of dust phase and decreases velocity of nanofluid. Temperature of dust
phase rises due to growing the value of interaction parameter. The influence of
that is mass concentration on temperature and velocity of the dust phase and
nanofluid is shown in the fig. (4.11-4.13) Rise in the value of mass concentration
parameter drops the value of velocity of phase of dust and nano-fluid. Particles’
weight increase with increasing mass concentration value and due to it velocity
decreases. Fig. 4.14 describe the characteristics of magneticfiled parameter M on
velocity fields of fluids, which have a vital character in speed of the flow. It have
been perceived that growth in ,M rise the porous space which produces resistance in
path of flow and decreases motion of flow. The liquid film particles observe
obstacles in flowing above these dumps. All the way through this motion path is
indistinct and the fluid feels retardation at every single point. The effect of Rd on
the heat distribution ( ) is obtain in the Fig. 4.15 The Rd an key role in broad
external heat conduction while the coefficient of convections temperature broadcast
is minor. If thermal radiations variable Rd is increased then it enhances the
temperature on the edge region in fluid layer. Present intensification clues to fall in
the rate of coling for the flow of nanofluids. Fig. (4.16-4.18) displayed the effect of
53
exponential parameter. By growing rate of exponential parameter rises speed of
Nano-fluid and dust phase where temperature increases for positive . Fig. (4.16-
4.18) reflect the impact of suction/injection on speed and heat of dust phase and
nanofluid. In case of dust phase the speed and heat increases by growing cost of
suction and injection parameter. In nanofluid case the velocity and temperature have
the reverse effect. The speed of the fluid decreases because its concentration
decreases. Adding fluid to stream demand to flow faster while less fluid that is
removal of the fluid from the stream cause to decrease the velocity.
Tables. 4.1 & 4.2 depicts the effect of skin frictions coefficients & Nusselt number
for dusty nanofluids. Tables clearly indicate that increasing magnetic field parameter
with increasing volume fraction of dust elements decreases both the friction
parameter with temperature transfer rate. Rising values of interaction and time
dependent variables decrease the value of skin friction coefficient parameter while
increases the Nusselt number. Radiation parameter has no influence on skin frictions
while reduces temperature transfer rate. By increase in the suction and injection
factor and volume fraction of the nanofluid particles increase heat transfer rate and
friction parameter.
4.4.1 Graphs and Tables
Fig. 4.1: Change in velocity shape f for dissimilar values of
volume fraction of dust particles d when
1, 0.6, 0.5, 0.5, 0.7, 0.2, 0.4q S
Fig. 4.2: Change in dust velocity shape F for dissimilar values of
volume fraction of dust particles d when
1, 0.6, 0.5, 0.4q S
55
Fig. 4.3: Change in velocity shape f for dissimilar values of
volume fraction of nanoparticles when
1, 0.5, 0.3, 0.6, 0.5, 0.7dq S
Fig. 4.4: Change in temperature shape for dissimilar values of
volume fraction of nanoparticles when
1, 0.5, 0.3, 0.7, 0.2, 0.1q Rd S
Fig. 4.5: Outcome our velocity shape f for dissimilar value of
unsteadiness factor S when
0.6, 0.5, 0.3, 0.7, 0.5d
Fig. 4.6: Outcome our dust velocity shape F dissimilar value of
unsteadiness factor S when 3, 0.6, 1.5, 0.3, 0.5dq
57
Fig. 4.7: Change in temperature shape for dissimilar values of
unsteadiness parameter S when 1, 0.6, 0.5, 0.7, 0.2, 0.1q Rd
Fig. 4.8: Change in velocity shape f for dissimilar values of
fluid particle interaction parameter when
1, 0.6, 0.3, 0.5, 0.4, 0.2dq S
Fig. 4.9: Change in dust velocity shape F for dissimilar values of
fluid particle interaction parameter when 1, 0.6, 0.5, 0.3, 0.6, 0.7dq S
Figure. 4.10: Change in temperature shape dissimilar value our
fluid particle interaction parameter when 1, 0.6, 0.5, 0.3, 0.1q Rd S
59
Fig. 4.11: Change in velocity shape f for dissimilar values of
concentration of mass of the dust particles when
1, 0.6, 0.5, 0.3, 0.7, 0.2, 0.4dq S
Fig. 4.12: Change in dust velocity shape F for dissimilar values of
concentration of mass of the dust particles when 1, 0.6, 0.5, 0.3, 0.4dq S
Fig. 4.13: Change in temperature shape for dissimilar values of
concentration of mass of the dust particles when
1, 0.6, 0.5, 0.3, 0.2, 0.1q Rd S
Fig. 4.14: Outcome our velocity shape f for dissimilar value of
Magnetic-filed factor M when
1, 0.6, 0.5, 0.3, 0.7, 0.2d
q S
61
Fig. 4.15: Change in temperature shape for dissimilar values of
radiation parameter Rd when
1, 0.6, 0.5, 0.7, 0.2, 0.1q S
Fig. 4.16: Outcome our velocity shape f for dissimilar
values of exponential Factor when
1, 0.6, 0.3, 0.4, 0.5, 0.7, 0.2dq S
Fig. 4.17: Outcome our dust velocity shape F dissimilar
value of exponential factor when
0.1, 0.6, 0.3, 0.4, 0.5dq S
Fig. 4.18: Change in temperature shape for dissimilar
values of exponential parameter when
1, 0.1, 0.3,, 0.7, 0.2dq S R
63
Fig. 4.19: Change in velocity shape f for dissimilar values of
suction and injection parameter q when
0.6, 0.5, 0.3, 0.7, 2d S
Fig. 4.20: Change in dust velocity shape F for dissimilar values of
suction and injection parameter q when
0.6, 0.3, 0.6, 0.5, 0.4d S
Fig. 4.21: Change in temperature shape for dissimilar values of
suction and injection parameter q when
0.6, 0.5, 0.3, 0.7, 0.2, 0.1Rd S
65
Table. 4.1.
Variation in (0)fC f and (0)Nu Casson nanofluid when ( 0.2 )
Rd d S q
[79]fC
result
fC
Present
results
[79]Nu
result
Nu
Present
results
1 -1.561840 -1.77152 2.254144 4.63400
2 -1.839995 -1.77443 1.808287 4.63336
3 -2.081123 -1.77734 1.539637 4.63318
1 -1.561840 -1.77152 2.254144 4.63400
2 -1.561840 -1.77152 1.542794 3.58159
3 -1.561840 -1.77152 1.157965 3.26566
0.1 -1.561720 -1.90582 2.352369 4.40880
0.2 -1.365966 -1.83707 2.567185 4.40989
0.3 -1.265964 -1.73473 2.766619 4.41112
0.1 -1.561840 -1.76871 2.254144 4.63039
0.2 -1.649159 -1.76973 2.092565 4.62708
0.3 -1.754082 -1.77070 1.926384 4.62422
0.1 -1.123516 -1.77070 0.120926 4.63039
0.5 -1.561840 -1.78109 2.254144 4.63059
0.9 -1.918901 -1.79146 4.204056 4.63078
0.5 -1.561840 -1.77070 2.254144 2.87312
0.7 -1.747527 -1.81456 3.247056 3.57616
0.9 -1.918901 -1.85819 4.204056 4.27843
-1 -1.698309 -1.90224 1.675188 0.40880
0 -1.622546 -1.83236 1.893080 1.46096
1 -1.561840 -1.70639 2.254144 2.87612
Table. 4.2.
Variation in (0)fC g and Nu for dusty nanofluid. when ( 0.2 )
M R d S Sp
[79]fC
result
fC
Present
results
[79]Nu
result
Nu
Present
results
1 -1.541720 -1.06242 2.302369 4.63400
2 -1.817069 -1.06419 1.843278 4.63336
3 -2.056565 -1.06597 1.567336 4.63318
1 -1.541720 -1.06242 2.302369 4.63400
2 -1.541720 -1.06242 1.577610 3.58159
3 -1.541720 -1.06242 1.185285 3.26566
0.1 -1.541720 -1.06242 2.302369 4.40880
0.2 -1.345966 -1.04349 2.547185 4.40989
0.3 -1.245964 -1.03224 2.736619 4.41112
0.1 -1.541720 -1.06242 2.302369 4.63039
0.2 -1.628161 -1.06384 2.135593 4.62708
0.3 -1.732163 -1.06422 1.964310 4.62422
0.1 -1.107813 -1.06242 0.127414 4.63039
0.5 -1.541720 -1.06865 2.302369 4.63059
0.9 -1.897238 -1.07488 4.288405 4.63078
0.5 -1.541720 -1.06242 2.302369 2.87312
0.7 -1.726478 -1.08874 3.314185 3.57616
0.9 -1.897238 -1.11491 4.288405 4.27843
-1 -1.648902 -1.30986 1.729464 0.40880
0 -1.588282 -1.18329 1.944349 1.46096
1 -1.541720 -1.06242 2.302369 2.87612
67
Concluding Remarks
The present study shows transmission of temperature and flow of radiation for dusty
nano fluid flow of Casson flow in the presence of volumetric friction of nano dusty
particles on a rapid absorbing porous enlarge exterior. For the solution of model
qualities analytical techniques were also applied and the convergent were showed in
mathematical form. The impacts of implanted factors were graphically and skin
friction were numerically and graphically showed and discussed. The main points of
conclusion are:
Greater is the value of Nb greater will be the kinetic energy of nanoparticles
inside the fluid which leads to raise the heat field.
Radiation parameter Rd increment causes shrink in Thermal boundary layers
and expand the Nusselt number u .
Increment in Volume friction of Nano particles improve the temperature
transfer rate and friction factors.
Coefficient of skin friction and Nusslet number is decreased by Porosity factor
decrement.
Interaction parameters of fluid particles causes increase in temperature
transfer rate and decrease in friction factors.
Injection / suction parameter decrease heat flow profile rate.
69
5.1 Introduction
This subsection of thesis, we study of 3-D squeezing flow of carbon nanotube
(CNT’s) in light of water in a pivoting channel with the base wall making permeable
has been introduced. Walls are also keep permeable. Proper changes relate to solid
nonlinear Homotopy Analysis Method (HAM) illuminates normal differential
framework with the assistance of similarity factors. Graphs are present to discuss the
effect prominent physical factors on the speed and temperature.
Moreover, for different relevant variables the values for coefficient of skin friction
and Nusselt number are arranged in tabular form. The most significant after effect
of this research work is to think about the diverse practices of developing variables
on carbon nanotube (CNT’s).
5.2 Formulation of the Problem
These examination considerations on 3-measurement squeezing flows of CNTs
nanofluid (SWCNT, MWCNT nano-particles) in view of water in a pivoting disk
channel with constent & penetrable base divider. The stretching wall of the channel
is kept porous. The bottommost wall of the waterway is stretching along 0y and
beside the x axis with the velocity 1 1
w
xU
p t
. Along y-axis, the velocity of this
wall is monitored from 0
1
Vv
t
. The channel tallness is fluctuating with extending
and withering through and through of the dividers. “See Fig.1” This height change
follow fromh tv h . For the expressed issue, all the presumption and condition are
communicated in condition (5.9). The displaying of this issue is as per the following
[80];
Fig. 5.1. Geometrical portrayal of the issue.
0,x yu v (5.1)
1*
1*
2 10,
1
nf
t x y nf x xx yy
nf nf
u uu vu w up u u ut k
(5.2)
*10,
nf
t x y x xx yy
nf nf
v uv vv up v u
(5.3)
1*
1*
2 10,
1
nf
t x y nf x xx yy
nf nf
w uw vw w up w w wt k
(5.4)
0.
nf
t x y xx yy
p nf
T uT vw T TC
(5.5)
If , and w u v the modules of the velocity. For *, , , , ,p nf nf nfnfC p T
demonstrate for reformed weight, kinematic-consistency, dynamic thickness,
thickness, temperature capacitance, and heat.
The limit conditions for the overhead expressed issue as pursues:
0
0.51
, , 0, at 0,1 1
, 0, 0, at .1
w
VdU px dTu v T y
dw t t dw
dV p dTv u T y h t
dh t dh
(5.6)
71
The thickness, dynamic-consistency, and warmth limit our nano-fluids over and
done with numerical conditions are:
1 2
3 42.5
1 , 1 ,
1 2 ln ln 2
, .1
1 2 ln ln 2
nf CNT f p p CNT pnf f
CNTCNT f f
f CNT f
nf nf ff
CNT f f
CNT f
J C J C C
kk k k
k kJ J k k
kk k k
k k
(5.7)
Where ; ; ; & CNT CNT f fk k for thickness of carboon nano-tubes, the liquid
thickness, volumes segment of nanoparticle, the warm conductivitys of the base
liquid and carbon nanotube one-to-one.
So as to improve the conditions (5.1-5.6) the behind definition as:
0.5 1
1, , , , .w h
w w
w
p t T T yu U f w U g v f
p T T h t
(5.8)
Where , , and g f f is the velocity functions laterally the x and y-axis
dimensionless temperature functions and rotational velocity functions in that order,
and is the local parallel parameter.
By replacing equations (5.8) keen on (5.1-5.6), the lastly gained equations as:
12 2 3 ,ivf g J ff f f S f f f (5.9)
12 2 ,g f g J fg gf S g g
(5.10)
42 .
Pr 2
J SJ f
(5.11)
Having boundary conditions;
0 0 1, 0 0, 0 ,
1 1 1 0 1 2.
f g f A
f g f S
(5.12)
The following numbers and parameters in equation (5.9-5.11) are under as [81];
0
*
squeeze variable ,Pr Prandtl number ,
suction variable , rotation variable ,
porosity parameter .
f p f
f
CS
p k
V
ph p
pk
(5.13)
The xNu and xCf are under as follow:
12
0
0
0,nf
lowernf y y
UCf
u
(5.14)
12
00.
nf
uppernf y y h t
UCf
u
(5.15)
0
,lower
nf h w y
X T yNu Qr
k T T
(5.16)
.upper
nf h w y h t
X T yNu Qr
k T T
(5.17)
The simplifying equation (5.14-5.17), xNu and xCf are under as:
1
10.5
2.5Re 0 0,
1x lower
JCf f
(5.18)
0.5
2.5
1
1Re 1 0.
1x upper
Cf fJ
(5.19)
0.5
4Re 0 0,x lowerNu J (5.20)
0.5
4Re 1 0.x upperNu J (5.21)
Following is our required local Reynolds number Rex
dUX
dwdv
df
.
73
5.3 HAM Solution
To achieved the explanation of equations (5.9-5.11) with the B. Cs of Eq. (5.12), we
applying HAM [70-76] with the consequent route. The primary assumptions are
chosen as:
2 33( ) 2 3 1 2 , g ( ) 0, ( ) 1 .
2o o of S A A S A
(5.22)
The values L ,L and L g fare given below:
( ) ,L ( )= ,L (g)=g''.iv
f gL f f (5.23)
Having the following as:
2 3( ) L ( ) L ( ) 0,f a b c d g e h i jL e e e e e e e e (5.24)
There are ( 1 9)ie i constants general solution of the problem:
Where our model resultant nonlinear variables , g fN N and N are mentioned below:
2
12
( ; )( ; ), ( ; ) ( ; ) ( ; ) ( ; )
( ; )( ; ) 2 ( ; ) 3 ( ; ) 0,
2
f
fN g f J f f f
g Sf f f
(5.25)
2
2
1
( ; ) ( ; )( ; ), ( ; ) ( ; ) ( ; )
2
( ; ) ( ; ) ( ; )( ; ) ( ; ) 2 0,
g
g gN f g S g g
g f fJ f g
(5.26)
42( ; ), ( ; ) ( ; ) ( ; ) ( ; ) ( ; ) .
Pr 2
J SN f J f
(5.27)
The model 0th -order problems achieved from Equations. (5.9-5.11) as:
0( ; ), ( ; ) ( 1) ( ) ( ; ) 0,f f fN g f L f f
(5.28)
0( ; ), ( ; ) ( 1) ( ) ( ; ) 0,g g gN g f L g g (5.29)
0( ; ), ( ; ) ( 1) ( ) ( ; ) 0.N f L (5.30)
With (equivalent) sufficient B.Cs are:
1 0
0 1
0 1 1 0
( ; ) ( ; ) 1, ( ; ) , ( ; ) , 0,
2
( ; ) 0, ( ; ) ( ; ) 0, ( ; ) 1.
f S ff f A
g g
(5.31)
Where having two values 0 and 1so
( ;1) ( ), g( ;1) ( ) and ( ;1) ( ) .g f f (5.32)
By Taylor’s series the expanding of ( ; ), ( ; ) and ( ; ) f g are described as
1
1
1
( ) ( ; ) ( ) ,
( ) ( ; ) ( ) ,
( ) ( ; ) ( ) .
q
o qq
q
o qq
q
o qq
f f f
g g g
(5.33)
Here
0 0 0
1 ( ; ) 1 ( ; ) 1 ( ; )( ) ,g ( ) and ( ) .
! ! !q q q
g ff
q q q
(5.34)
The secondary constraint show that 1 , with , and f g are a selection of such that
the above series (5.33) converges we obtained the following:
1
1
1
( ) ( ) ( ),
( ) ( ) ( ),
( ) ( ) ( ).
o qq
o qq
o qq
f f f
g g g
(5.35)
The problem thq -order fulfills below:
1
1
1
( ) ( ) ( ) ,
( ) ( ) ( ) ,
( ) ( ) ( ) .
f
f q q q f q
g
g q q q g q
q q q q
L f f U
L g g U
L U
(5.36)
The equivalent B.Cs are:
(0) 0, (0), (1) 0, (0) 0,
(1) 0, (1) 0, (0) (1) 0.
q q q q
q q q q
f f
f f g g
(5.37)
Here
75
1 1
1 1 1 1 1 1 1 1
0 0
( ) 2 3 ,2
q qiv f
q q k q k k q k q q q q
k k
Sf U J f f f f g f f f
(5.38)
1 1
1 1 1 1 1 1 1 10 0
( ) 2 ,2
q qg
q q q q q k q k k q k qk k
g g U S g g J g f f g f
(5.39)
14
2 1 1 2 10
( ) .2 Pr
q
q q q q k kk
JSU J J f
(5.40)
Where
0, if 1
1, if 1q
Main Observations
The present chapter investigation for the squeezing in 3D flow of CNTs dependent
on water based of radiation due to Temperature and MHD by utilizing homotopy
investigation strategy to be completed. MWCNTs and SWCNTs are utilized in ideal
model. Effects of implanted of variables are sketchily described.
1. With respect to MWCNTs, SWCNTs shows little resistance to the fluid.
That’s why MWCNTs have large skin resistance w.r.t SWCNTs.
2. Temperature profile is same for SWCNTs and MWCNTs with a difference
that SWCNTs has high Nusselt number against the nanoparticle volumetric
fraction.
3. Velocity profiles f in x-direction decrease with increase in A and
shows the oscillation according to increase in , and .
4. When decrement in 0S seen velocity profiles get increase and vice versa.
5. The temperature function increase with the icreasing value of S and
get reduction with the increase in and Pr .
5.4 Result and Discussions
In this segment of the dissertation, we have concisely discussed the effect of non-
dimensional numbers and emergent parameters on the functions of angular and
linear velocity , ,f f g , heat , skin-coefficient 1
2RexCf & Nusselt
numbers 1
2RexNu regarding water based MWCNTs and SWCNTs. These emerging
parameters and non-dimensional numbers are nanoparticle volume fraction , a
rotation parameter , squeeze parameter S , the suction parameter A , Prandtl
number Pr and porosity parameter respectively. Figures 5.2-5.7 are plotted to
observe the influences of these parameters. The influence of while enhancing
quantity ore , ,f f g & are depicted in Figs 5.2(a-d). Fig 5.2(a)
shows the influence on f . It can be clearly understood that in the presence of
both SWCNTs and MWCNTs, increase in the values of indicates improvement
in velocity function f in the y-direction. Consequently, the behavior of f is
dominant in cases when 0S and 0S . Due to the facts that SWCNT is more
denser than MWCNT. Fig 5.2(b) is showing the influence of our f . It can
be seen that in the interval 0 4 in the x-direction, f is showing a decreasing
trend although there is an increasing trend of f in the interval 4 1 . It must
also be noted that for 0S , the trend of f is more prominent. One can observe
the increase in rotational velocity g for 0S as depicted in figure 5.2(c). In case
MWCTs, g have higher values than SWCNTs. Also observed g have higher
values for 0S than for 0S . Figure 5.2(d) shows the relationship between and
. From here we see that the variation in is reducing with the escalating
values of . The case where the channel is contracted or the case where the channel
is stretched out, this result is identical for both cases of SWCNTs and MWCNTs.
The performance of on , ,f f g are depicted in figures 5.3(a-c). Figure
77
5.3(a) is also showing that the performance of on f is dual in nature.
Besides, during SWCNTs and MWCNTs, the increasing trend of is showing the
behavior of reduction in the interval 0 0.6 while in the interval 0.6 1 shows
escalating behavior with the escalating values of . It is significant to mention that
the values of f for MWCNTs is greater than that of SWCNTs. Besides, for
velocity at 0S is less than at 0S . Figure 5.3(b) shows the influence of on f
. It can be observed that in comparison with the variation in , f expresses the
behavior of an oscillatory. At the upper and lower surface of the channel, the
enhancement in in interval 0 0.3 and 0.8 1.0 has lowered f during
both the cases of SWCNTs and MWCNTs. However, the enhancement is observed
in mean location of the channel i.e. 0.3 0.8 . Figure 5.3(c) displays the influence
of on g . It can be seen that the variation in is showing a relation of
reciprocal with g . It can also be observed that in case of MWCNTs, the velocity
function g is greater than SWCNTs at all the time while the velocity increases
more rapidly for 0S in comparison with 0S . We have presented Figure 5.4(a-d)
to observe the influence of variation S on , ,f f g and . The
influence of S on f is graphically shown in Figure 5.4(a). It can be observed
that the enhancement in squeezing parameter S enhanced the velocity function
along y-direction. Besides, by contracting the channel 0S increased the velocity
and vice versa. Figure 5.4(b) and 5.4(c) is showing the same trend wherein the most
prominent locations of then channel, a reverse flow is observed which is responsible
for the reduction in the velocity. However, at the upper and lower surfaces of the
channel, this behavior was disappeared. From Figure 5.4(d), it can be observed that
raising S for both cases of 0S and 0S enhance the temperature of nano-
fluid in SWCNTs and MWCNTs. The influence of A on , ,f f g is
depicted in Figure 5.5(a-c). From figure 5.5(a), it can be seen that the enhancement
in A increases the velocity in y-direction. The location adjacent to the lower
portion of the channel, a prominent increase is observed because of the fluid’s
suction. It is notable that the increase in velocity due to MWCNTs is more prominent
than SWCNTs. The correlation between rate of distribution of velocity of nano-
fluid f and A is depicted in figure 5.5(b). It can be observed that there is a
reverse correlation between f and A wherein there is a decrease in f by
increasing A . It should be also observed that the reverse flow is higher for 0S
in comparison when 0S . Furthermore the f , in case of MWCNT is bigger
than SWCNT. The similar outcome is strategized for rotating velocities-distribution
g in fig. 5.5(c), by differences in g on situation of SWCNT was decreased
greater than in situation MWCNT. Figures 6(a-c) is strategized tore resents impacts
of pores on velocities distributions , ,f f g . The fig. 5.6(a) & 5.6(c)
represents decreasing behaviors in velocities function ,f g , Where in sketch
5.6(b), velocities field f displays a decline in a closed interval .4,1. . This
conduct is lads enlarging of bellow disk. In fig. 5.6(a), it remarkable to indicates the
converse flows is larger when S as matched to S . Beside, this it is seen
from the sketch that for situation of MWCNT, the velocities function
, ,f f g are constantly greater than SWCNT. the fig. 5.7 illustrates the
Pr over distribution. Heat distribution vary conversely by Pr parameter.
Boundaries layers thickness reduces with higher value of Pr variable, this
consequence is extra dispersed for smaller Pr parameter. Clearly from fig. that in
circumstance of MWCNT, heat profile constantly greater than SWCNT.
79
Fig. 5.2. (a): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.2. (b): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.2. (c): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.2. (d): Change in temperature profile for dissimilar values of
volume fraction of nanoparticles
81
Fig. 5.3. (a): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.3. (b): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.3. (c): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.4. (a): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles S
83
Fig. 5.4. (b): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles S
Fig. 5.4. (c): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles S
Fig. 5.4. (d): Change in temperature profile for dissimilar values of
volume fraction of nanoparticles S
Fig. 5.5. (a): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles A
85
Fig. 5.5. (b): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles A
Fig. 5.5. (c): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles A
Fig. 5.6. (a): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.6. (b): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
87
Fig. 5.6. (c): Change in velocity profile f for dissimilar values of
volume fraction of nanoparticles
Fig. 5.7: Change in temperature profile for dissimilar values of
volume fraction of nanoparticles Pr
5.4.1 Discussion
To visualize the impact of the involved parameters on the skin-friction coefficient at
the bottom and top wall of the channel for both cases i.e. (S>0) and 0S , the data
is presented in the Table 5.1. Further Table 5.2 and Table 5.3-5.5 are devoted to
show the impacts of emerging parameter on the local Nusselt number, thermo-
physical properties of SWCNTs , MWCNTs and Nano-fluid of some Thermal
conductivity (nfk ) of CNTs with different volume fractions ( ) respectively.
Table. 5.1.
The skin friction coefficient for numerical values.
A
f loC at
1 2S
f loC at
1 2S
f uppC at
1 2S
f uppC at
1 2S
1.0
-2.39109
-1.82073
0.765365
-0.264043
1.2
-2.67770
-2.09395
0.669857
-0.306960
1.4
2.0
-3.01713
-2.41285
0.564992
-0.349857
2.1
-3.01747
-2.41303
0.565168
-0.349804
2.2
1.0
-3.01780
-2.41320
0.565345
-0.349749
1.1
-3.01781
-2.41320
0.565347
-0.349748
1.2
-3.01783
-2.41320
0.565350
-0.349748
89
Table. 5.2.
The local Nusselt number for the numerical values.
A
lower
Nu at
0.5S
lower
Nu at
0.5S
upp
Nu at
1 2S
upp
Nu at
1 2S
1.0 -0.159557 -0.286224 -0.097682 -0.079128
1.2 -0.158737 -0.261897 -0.073141 -0.100385
1.4 2.0 -0.147609 -0.233884 -0.051804 -0.116670
2.1 -0.147642 -0.233936 -0.051826 -0.116746
2.2 1.0 -0.147675 -0.233987 -0.051844 -0.116823
1.1 -0.162442 -0.257386 -0.057033 -0.128505
1.2 -0.177209 -0.280784 -0.062218 -0.140187
Table. 5.3.
CNTs for Physical properties.
Material
SWCNTs
MWCNTs
Thermal Conductivity
nfk
/W mK
3000
3000
Electrical Conductivity
nf
/S m
7 610 10
7 610 10
Strength Tensile GPa 100+50 =150 100+50 =150
Modulus Young’s GPa 1000+54 =1054 1000+200 =1200
Table. 5.4.
The thermophysical properties of CNTs and nano-fluid of some base fluid.
Physical Properties
Specific
Heat
1 1( / )pc kg k
Density 3
( / )kg m
Thermal Conductivity
( / )k W mk
fluid Base
Water 34.197 10 29.97 10 16.13 10
Kerosene (lamp)
oil
32.090 10
27.83 10
11.45 10
Engine oil 31.910 10 28.84 10 11.44 10
Nanofluids
SWCNT 24.25 10 32.6 10 36.6 10
MWCNT 27.96 10 31.6 10 33 10
Table. 5.5.
volume fraction ( ) with Thermal conductivities (nfk ) of CNTs different.
SWCNT fornfk
MWCNT fornfk
0.000 20.145 10 20.145 10
0.010 20.174 10 20.172 10
0.020 20.204 10 12 10
0.0300 20.235 10 20.228 10
0.040 20.266 10 20.257 10
The investigated three problems in the thesis:
1. Bioconvection Model for Magneto Hydrodynamics Squeezing Nanofluid
Flow with Heat and Mass Transfer between two Parallel Plates containing
Gyrotactic Microorganisms under the Influence of Thermal Radiations
2. Dusty Casson Nanofluid Flow with Thermal Radiation Over a Permeable
Exponentially Stretching Surface
3. Unsteady squeezing flow of Water Based Carbon Nanotubes In permeable
parallel Channels
Chapter 3
In the chapter the bioconvection flow between two parallel plates is studied by
assuming It is the plates capable to expand and contract. The model of dimensional
flow have nanofluid and microorganisms and have been changed into the non-
dimensional, nonlinear system of simple differential equations. For the purpose a
simple form of similarity variables is used by reducingthe supporting boundary
condition in the form.As mentioned in the previous lines that embedding
parameteers a re graphically presented and discussed and the different types of
parameters and numbers are studied and investigate. The base points are:
An increment seen in heat profile is due to Kinetic energy of the Nanoparticles
which is because of greater value of Nb.
An increment in thermal radiation(Rd) raise the boundary layer area
temperature in the fluid which cause drop in the cooling rate of Nano fluid
flow.
Density of motile microorganism increased due to variation in
thermophoretic parameter(Nt).
The concentration field of suction and injection reversely effected by
Thermophoretic and Brownian motion parameters.
93
Variation in various phesical parameters and convergence of the homotopy
were numerically analysed.
For various bioconvectoion parameters the are precisely analysed.
Changes in the motile microorganisms density were analyzed for
various bioconvection parameter .
Skin friction .fC decrease continuesly by increasing the and M values.
heat flux Nu increase by increasing Pr and decrease by increasing .
Mass flux is in direct propotion to and Le Nbas when and Le Nb increase mass
flux also increases while mass flux reduce with Nt .
A very prominent variation in the density of motile microorganism observed for
the Brownian (Nb), thermophoretic (Nt) and suction/ injection parameters.
Chapter 4
The present study shows transmission of temperature and flow of radiation for
dusty nano fluid flow of Casson flow in the presence of volumetric friction of
nano dusty particles on a rapid absorbing porous enlarge exterior. For the solution
of model qualities analytical techniques were also applied and the convergent
were showed in mathematical form. The impacts of implanted factors were
graphically and skin friction were numerically and graphically showed and
discussed. The main points of conclusion are:
Greater is the value of Nb greater will be the kinetic energy of nanoparticles
inside the fluid which leads to raise the heat field.
Radiation parameter Rd increment causes shrink in Thermal boundary layers
and expand the Nusselt number u .
Increment in Volume friction of Nano particles improve the temperature
transfer rate and friction factors.
Coefficient of skin friction and Nusslet number is decreased by Porosity factor
decrement.
Interaction parameters of fluid particles causes increase in temperature
transfer rate and decrease in friction factors.
Injection / suction parameter decrease heat flow profile rate.
Chapter 5
The present chapter investigation for the squeezing in 3D flow of CNTs dependent
on water based of radiation due to Temperature and MHD by utilizing homotopy
investigation strategy to be completed. MWCNTs and SWCNTs are utilized in ideal
model. Effects of implanted of variables are graphically described. The main subject
theme of this research work is shown as:
With respect to MWCNTs, SWCNTs shows little resistance to the fluid.
That’s why MWCNTs have large skin resistance w.r.t SWCNTs.
Temperature profile is same for SWCNTs and MWCNTs with a difference
that SWCNTs has high Nusselt number against the nanoparticle volumetric
fraction.
Velocity profile f in y-direction increase by the increasing in value of
and A , moreover shows dual behaviour with increment and
function of Velocity decrease while with the increase in .
Velocity profiles f in x-direction decrease with increase in A and
shows the oscillation according to increase in , and .
Velocity profiles g increase by the increment, and get decrease with
the increase in , A and parameters.
When decrement in 0S seen velocity profiles get increase and vice
versa.
The temperature function increase with the icreasing value of S and
get reduction with the increase in and Pr .