NUMERICAL STUDY OF HEAT AND MASS

NUMERICAL STUDY OF HEAT AND MASS

TRANSFER IN SOME INTERNAL/EXTERNAL

FLOWS

BY

MUHAMMAD FAROOQ IQBAL

CENTRE FOR ADVANCED STUDIES

IN PURE AND APPLIED MATHEMATICS

BAHAUDDIN ZAKARIYA UNIVERSITY

MULTAN, PAKISTAN.

FEBUARY 2018

NUMERICAL STUDY OF HEAT AND MASS

TRANSFER IN SOME INTERNAL/EXTERNAL

FLOWS

BY


SUPERVISED BY

PROF. DR. MUHAMMAD ASHRAF




MULTAN, PAKISTAN.

FEBUARY 2018

NUMERICAL STUDY OF HEAT AND MASS TRANSFER

IN SOME INTERNAL/EXTERNAL FLOWS

A DISSERTATION SUBMITTED IN PARTIAL

FULFILLMENT OF THE REQUIREMENT

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN MATHEMATICS

BY





MULTAN, PAKISTAN.

FEBUARY 2018

PProophj

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MYMYMYMY PARENTSPARENTSPARENTSPARENTS

Certificate

It is certified that the work contained in this dissertation is original and has been

accomplished completely by MUHAMMAD FAROOQ IQBAL, under my

supervision and that, in my opinion; it is fully adequate, in scope and quality for

the degree of "DOCTOR OF PHILOSOPHY" in Mathematics.

(Supervisor)

PROF. DR. MUHAMMAD ASHRAF

Centre for Advanced Studies

in Pure and Applied Mathematics,

Bahauddin Zakariya University,

Multan.

Declaration

I hereby declare that work contained in this dissertation has not been

previously published in any form and shall not, in future, be submitted for

obtaining Ph.D. degree of any other university. All sources of information have

been acknowledged in this dissertation.


ACKNOWLEDGEMENTS

Praise and glory is to Allah, The Cherisher and The Sustainer of the worlds.

It is mere His will and grace which enabled me to accomplish this uphill task. I bow

to Him in reverence and gratitude. Peace and blessing to Hazrat Muhammad (صلى الله عليه وسلم)

who is the cause of the birth of cosmos.

Afterward, I would like to express my deep gratitude and sincere sentiments to my research supervisor and Director CASPAM PROF. Dr. Muhammad Ashraf. With great patience, he is always willing to extend valuable advice.

Thanks are also due to Dr. Kashif Ali and Dr. Shahzad Ahmad for their kind cooperation and continuous encouragement. Acknowledgement is also gratefully extended to all my honorable teachers who taught me during my studies.

I am grateful to all my friends especially Mufti Hussain Ahmad, Mr. Muhammad Ibrahim, Mr. Ahmad Hassan, Dr. Ali Ahmad, Mr. Muhammad Zubair Qureshi, Mr. Naeem Shafqat, and Dr. Muhammad Salman, whose support was essential to the realization of this work.

In the end, I am extremely grateful to my wife and children, whose continuous support was unfathomably needed for the completion of this task.


Numerical Study of Heat and Mass Transfer in Some

Internal/External Flows

Ph.D. Thesis

By

Muhammad Farooq Iqbal

Centre for Advanced Studies in Pure and Applied Mathematics

(CASPAM)

Bahauddin Zakariya University, Multan

TABLE OF CONTENTS

1 Abstract and Literature Review 1

2 Heat and Mass Transfer in Unsteady Titanium Dioxide Nano-Fluid Flow

Between Two Orthogonally Moving Porous Disks 9

2.1 Introduction 10

2.2 Mathematical Formulation 10

2.3 Numerical Solution 14

2.4 Results and Discussion 15

2.5 Conclusions 17

3 On the Combined Effect of Heat and Mass Transfer in MHD Flow of a Nano-

fluid Between Expanding / Contracting Walls of a Porous Channel

34

3.1 Introduction 35




3.5 Conclusions 42

4 On the Non –Newtonian Flow Driven by a Curved Stretching Sheet

60

4.1 Introduction 61




4.5 Conclusions 67

5 On the Suitability of Rosseland Approximation for Thermal Radiation in Flow over a Rotating Cone Embedded in Porous Media

78

5.1 Introduction 79




5.5 Conclusions 84

Possible Future Work 122

List of Publications 126

References 128

1

CHAPTER 1

ABSTRACT AND LITERATURE REVIEW

2

ABSTRACT

The purpose of this thesis is to present the numerical investigation of the heat and the mass

transfer rate in some internal/external flow problems related to the orthogonally moving disks,

expending contracting channel with porous walls, curved stretching sheet, and rotating cone.

Using the similarity variables, the governing coupled partial differential equations are converted

into non-linear ordinary differential equations which are solved numerically by using the quasi-

linearization and finite difference discretization. We have studied the momentum as well as the

heat and the mass transfer properties of the Nano-fluids and the Casson fluid. Impacts of the

relevant parameters on the behaviours of velocity, temperature, and concentration profiles are

demonstrated graphically. The skin-friction coefficient and heat and mass transfer rates are also

tabularized for different governing parameters. Further, two different models for the thermal

radiation in the flow of fluid over the rotating cone have also been compared, and some

interesting numerical results have been obtained.

3

LITERATURE REVIEW :

Flows completely bounded by the solid surfaces are called the internal flows.

Therefore internal flows include the flows through pipes, ducts, nozzles, diffusers, sudden

expansion and contractions, valves, and fittings. The external flows are the flows over the bodies

immersed in an unbounded fluid. The flows over the semi-infinite flat plate, flow around

buildings and flows over the cylinder are examples of the external flows. Depending on the

geometry, external flows can be very simple or quite complicated. Both internal and external

flows may be laminar or turbulent, compressible or incompressible.

One of the most important flows is the flow due to rotating disk. Karman [1] presented

initial work on the rotating disk flows. He applied a special substitution (known as the Von

Karman transformation) for deriving ordinary differential equations from the partial differential

equations (Navier-Stokes equations). Numerical solution for the problem of flow over a rotating

disk was presented by Cochran [2] which was further extended by Stewartson k. [3] to the flow

of fluid between both disks rotating with some angular velocities. Chapples and Stokes V. K. [4]

and Mellor G. J. et al. [5] investigated the flow of fluid when a disk was rotating and the second

one was kept static. Thermal analysis for the fluid flow through two rotating disks was presented

by Arora and Stokes [6]. Soong and Yan [7] worked on porous rotating disks. Soong C. Y. et al.

[8] studied the fluid flow among two disks rotating coaxially. Domairry and Aziz [9] studied

electrical conduction in the fluid between two plates (one permeable and the other impermeable).

Many extensions of this work may be found in the literature. For example, Joneidi et al. [10]

incorporated the magnetic field effects in the problem, whereas Hayat et al. [11] considered the

case of squeezing walls. Hussain et al. [12] examined the MHD flow of fluid and the heat

transfer in the fluid squeezed between the two disks. Time dependent squeezing MHD flow was

investigated by Shaban [13]. This work was further extended by Turkyilmazoglu [14] by

studying the shrinking of the rotating disks. In a series of papers, Gao et al. [15-18] presented

different methods for the measurement and the characterizing of low-velocity oil-water two-

phase fluid flows. Nano-fluid flows through different geometries under magnetic field or slip

effects have been studied by many researchers (for example, Makinde and Aziz [19], Bachok et

al. [20], Safaei et al.[21,22], Goodarzi et al. [23], Togun et al. [24], Hashmi et al. [25], Das et al.

[26] etc.) Majority of papers were based on constant thermo-physical properties. However, some

4

works (Lai & Kulacki [27], Prasad et al. [28] etc.) may be found in the literature where the

variation of these properties with temperature has been considered. Vajjha and Das [29] showed

experimentally that very low fraction of volume of the nanoparticles can notably change thermo-

physical properties of the nano-fluid. An efficient analytical method known as the optimum

homotopy analysis method (Liao [30], and Fan and You [31]) was employed to solve the coupled

non-linear equations. Magnetic field effects on fluid flows have wide range of the applications in

the field of engineering, with intensity and the orientation of the magnetic field being the most

important factors. Hayat T. et al. [32] investigated MHD effects on the nano-fluid flow with the

convective boundary conditions. Sheikholeslami et al. [33] presented flow of Cu-water nano-

fluid. Hsiao [34, 35] analyzed MHD fluid flow of nano-fluid with slip boundary conditions along

with the mixed convection and Ohmic dissipation effects. Zhang et al. [36] analyzed the flow

and the heat transfer under MHD and thermal radiation effects. Gangi and Malvandi [37] studied

the convection in nano-fluid in a vertical enclosure. Hayat T. et al. [38] studied the MHD 3-

dimensional flow of the nano-fluid over the porous & shrinking surface. Yang L. and Shen H.

[39] analyzed the effects of the distribution of the porous media on the improvement of the

performance of an isothermal chamber. The unsteady MHD flow (squeezing) of the fluid

between both parallel disks was considered by Azimi and Riazi [40], by using the Galerkin

optimal homotopy asymptotic method (GOHAM). Ganji D. D. et al. [41] studied the MHD

squeezing fluid flow among the permeable disks. Radiative effect in the time-dependent

axisymmetric flow (squeezing) of the Jeffery fluid through two parallel disks was presented by

Hayat T. et al. [42]. The solution of the MHD squeezing fluid flow among two permeable disks

was given by the Domairry G. & Aziz A. [43]. Xin-hui SI et al. [44] analyzed the laminar,

unsteady, two-dimensional, and the incompressible flow of the micropolar fluid between the two

porous & coaxial disks moving orthogonally. This work has further been extended by Sun Yina

et al. [45] by considering the case of counter rotating disks.

In this thesis, we have studied the heat transfer and the mass transfer in an unsteady

hydromagnetic (MHD) hydro based nano-fluid ( 2TiO nano-particles) between the two permeable

coaxial disks moving orthogonally with suction and viscous dissipation effects. Combination of

the iterative and the direct method is used to solve the sparse systems of linear algebraic

5

equations arising from the finite difference (FD) discretization of the quasi-linearized self similar

ODEs. The numerical results have been physically interpreted in chapter no. 2 of the thesis.

Study of the combined forced convection and the free convection fluid flow through the

two heated parallel vertical walls in the porous medium along with the viscous dissipation effect,

the walls are heated asymmetrically and the symmetrically, has been presented by Ingham D. B.

et al. [46]. Makinde and Mhone [47] examined the magnetohydrodynamic oscillatory flow

through the channel filled with the porous medium. Mehmood and Ali [48] studied the effect of

slip condition on the free convective oscillatory flow through the vertical channel with the

temperature (periodic) distribution. Makinde and Beg [49] devoted their study to investigate the

inherent irreversibility and thermal stability in a chemically reactive flow through the channel

with isothermal walls. Ajibade and Jha [50] investigated the effects of the suction and the

injection on the oscillatory flow through the parallel plates. In [51, 52], the problem was

extended to study the heat generation/absorption and time dependent boundary condition

(respectively). Umavathi [53] employed the Darcy-Brinkman–Forchheimer model to discuss the

natural convection fluid flow in the channel (vertical) filled with the porous medium.

Srinivasacharya and Kaladhar [54] studied the chemically reacting couple stress fluid in the

channel along with the Soret and the Dufour effects. Garg B. P. et al. [55] analyzed the influence

of Hall current in magneto-hydrodynamic viscoelastic fluid in the porous channel. Further,

several investigators made significant contributions in peristalsis under different assumptions

(for example, Kothandapani and Srinivas [56], Gnaneswara et al. [57], Ali et al. [58], Sarkar et

al. [59], Mekheimer and Abd Elmaboud [60]). Laminar and incompressible natural convective

transport inside vertical channel with porous medium has been considered by Kaladhar et al.

[61]. Falade et al. [62] investigated the effect of the suction and injection on the unsteady

oscillatory fluid flow through the channel (vertical) with the non-uniform wall temperature.

Manipulation of the heat convection of the copper particles in the blood has been considered

peristaltically, two phase model of the fluid flow is used in the channel with insulating walls

[63].

Motivated from above studies, the flow in the parallel-plate channel has been considered

in chapter no. 3 of the thesis where we have investigated the laminar flow of nano-fluid in the

channel (porous) with expanding & contracting walls via a similarity transformation. We have

6

observed that the permeability Reynolds number enhances both of the heat transfer rate and the

shear stress when the porous walls of the channel are expanding, on the other hand the viscous

dissipation always boosts the heat transfer rate at the walls, irrespective of movement of the

walls.

Fluid flow and the heat transfer analysis over the stretching surface has gained a special

focus of recent researchers and engineers on account of its considerable applications in the

engineering and industrial processes (particularly, in the production of polymer films or thin

films, in the glass fiber and production of the paper, food manufacturing, drawing of the plastic

thin films and the wires, films of the liquid in the process of condensation, the growing of

crystal, the manufacturing and extraction of the polymer and the rubber sheets etc). Flow due to

stretching of the surface was first examined by Crane L. J. [64]. Subsequently, the fluid flow

problems due to stretching surfaces under the diverse configurations have been studied by

numerous researchers. Magnetohydrodynamic boundary layer flow of the Casson fluid over the

exponentially porous shrinking sheet has been considered by Nadeem S. et al. [65]. Ali et al. [66]

analyzed the heat transfer of MHD boundary layer flow of Casson fluid. Influence of the

chemical reaction on MHD flow of the Casson nano-fluid caused by nonlinearly stretching sheet,

immersed in the porous medium, under the radiative effect and the convective boundary

condition, were studied numerically by Imran and Shafie [67]. Analysis of the rate of transfer

heat was carried out, by Abbas et al. [68], for stretching flow over the curved surface under two

thermal conditions, namely, the prescribed surface temperature (PST) and the prescribed heat

flux (PHF). Further, two-dimensional flow of non-Newtonian MHD flow of Casson fluid has

been considered by Veena et al. [69], for both PST & PHF. Some references may be found in the

scientific literature where study of the flow of viscous fluid and the heat transfer rate is done,

under the effect of applied H-field over the sheet which is curved and stretching. The Bi-

dimensional .boundary layer fluid flow of the electrically .conducting micropolar fluid, subject to

the transversely applied magnetic field, over the stretching curved sheet has been studied by

Naveed et al. [70]. The study has been further extended by the same authors (please see [71]) to

study the effect of the nano nature of the fluid. The exothermic-endothermic reaction impacts on

the MHD flow of the viscous fluid over the curved (stretching) surface were presented by Imtiaz

et al. [72].

7

Above mentioned studies motivated us to numerically study the problem of casson fluid

flow due to the curved stretching sheet, in chapter 4 of the thesis, under the action of transverse

magnetic field. Physical features of the problem in terms of the local Nusselt number, coefficient

of the skin-friction, flow velocity and thermal profiles are discussed through tables and graphs.

Thermal radiation has turn out to be a main branch of the engineering sciences and a very

important part of various applications in chemical, environmental, mechanical, solar power and

aerospace engineering. Thermal transport through radiations is much important in modern

industries for the design of efficient equipments, missiles, satellites, aircrafts, nuclear power

plants, gas turbines and the space vehicles or the various propulsion devices. Due to its useful

applications, the thermal radiation problem has fascinated several researchers during the last

three decades (some references are [73-86]). Radiation effects on boundary layer flow alongside

a symmetric wedge were studied by Mukhopadhyay [87], for the fluid viscosity varying linearly

with temperature. It was concluded that, with the increase of temperature-dependent fluid

viscosity parameter, the velocity of fluid was raised up to the cross-over point, and after that, the

velocity was found to decrease but the temperature kept on increasing. Further, flow separation

could be controlled due to variable fluid viscosity. Radiative effects on the thermal boundary

layer fluid flow induced by the stretching sheet immersed in an incompressible and the

micropolar fluid with the constant temperature on the surface, has been studied by Ishak [88].

It was observed that the rate of heat transfer at the surface reduces due to the radiative

effects. Flow and the heat transfer of the viscous nano-fluid over the nonlinearly stretching sheet,

with the radiation and variable temperature of the wall, were studied by Fekry et al. [89]. It was

noted that rise in the parameter of the thermal radiation and the nonlinear stretching sheet yielded

a reduction in the fluid temperature, which led to an increase in the heat transfer rate at the sheet.

Thermal radiation effects on heat and mass transfer past a moving vertical cylinder have been

discussed by Gnaneswara [90]. At minute values of the radiation parameter, the flow velocity

and temperature increased sharply near the surface of the cylinder, as the time passed. The mixed

convection, unsteady flow through the vertical porous plate (impulsively started) with the

thermal radiation, heat generation, chemical reaction, time dependent suction velocity, induced

magnetic field and diffusion (thermal) under the stable heat and fluxes of mass were numerically

8

analyzed by Shakhaoath et al. [91]. It was noted that the temperature decreased as the suction

and heat source were strengthened.

Effects due to the radiation parameter and the heat source/sink on the steady, two

dimensional MHD boundary layer flow of the heat and the mass transfer past a shrinking sheet

with the wall mass suction was investigated by Babu et al. [92]. Two-phase flow model of the

dusty fluid flow due to the linearly stretching of the cylinder immersed in the porous medium

under the radiation effects was studied by Manjunatha et al. [93]. Flow was described in the

terms of ‘dusty gas’ model suggested by Saffman, which treated the discrete phase and the

continuous phase as the two continua occupying the same space. Similarly, many more

references (for example, [94, 95] and the references therein) may be found in the scientific

literature where the linearized form of the radiation has been adopted. However, there are still

some references dealing with the flows in various geometries where nonlinear modeling of the

thermal radiation is used. For instance, in [96], radiation effects in the two-dimensional flow of

the second-grade fluid which is electrically conducting were examined with non-linear radiative

heat flux. Entropy generation in MHD Williamson nano-fluid over the porous shrinking sheet

has been analyzed by Bhatti et al. [97], with nonlinear thermal radiation and the chemical

reaction. The solution of the highly nonlinear coupled ordinary differential equations was

obtained by employing a combination of the Successive linearization method (SLM) and the

Chebyshev spectral collocation method. Similar nonlinear thermal radiation has been utilized in

the work [98].

After a comprehensive literature review, we have noticed that no comparison of results

has been made by using the linearized and the nonlinear forms of thermal radiation. Therefore,

the chapter 5 of the present thesis is devoted to giving a (quantitative as well as qualitative)

comparison of the results for the flow due to a rotating cone. Effect of the governing parameters

has been obtained through tables and figures.

9

CHAPTER 2

HEAT AND MASS TRANSFER IN UNSTEADY TITANIUM

DIOXIDE NANO-FLUID FLOW BETWEEN TWO

ORTHOGONALLY MOVING POROUS DISKS

10

2.1. INTRODUCTION

In this chapter, the numerical investigations of flow and mass and heat transfer in MHD

unsteady viscous flow of hydro based nano-fluid (containing nano-particles of Titanium dioxide)

between two coaxial permeable disks are presented using quasi-linearization method. The disks

are moving orthogonally. The impact of suction and viscous dissipation are taken into account.

The linear algebraic system of equations is then solved iteratively using the finite difference

discretization of self-similar ODEs.

2.2. MATHEMATICALFORMULATION

Consider the flow, heat and concentration of an incompressible and electrically conducting nano-

fluid between the two permeable coaxial disks which move orthogonally, where )(2 ta is the distance

between them, as shown in the Fig. 2.1. The assumptions regarding the flow, heat and concentration

may be summarised as

o The nano-particles 2TiO are inserted in the base fluid

o The flow is laminar and unsteady

o The fluid is of constant density and viscosity

o The disks are uniformly moving up or down at time dependent rate )(ta′

o The walls are located at a distance )(2 ta which is function of time

o cylindrical coordinate system may be chosen for this problem with origin at the centre of

both disks

o The symbols 1u and 2u represent the velocity components in the directions of 1x and 2x

respectively, where 1x represents the radius and 2x represents the z-axis

o The Joule heating, viscous dissipation and external magnetic field are taken into account

o The external applied magnetic field is acting in the perpendicular direction of the flow

o The suction through the surfaces of the disks is also taken into account

All the thermo physical properties are assumed to be constant. Water is taken as base fluid,

which is thermally stable with nano-particles and has no slip taking place among them. The

magnetic field (induced) supposed to be minor upon comparing with applied field. Magnetic

11

Reynolds number means a ratio of the product of the fluid velocity & length (characteristic) to

magnetic diffusivity. The magnetic Reynolds number, here, is utilized as a part of correlation of

magnetic force lines transportation in a fluid and seepage of those lines from the fluid. For

smaller values of the magnetic Reynolds number, the magnetic field has a tendency of relaxing

towards a purely diffusive state. No applied polarization and electric field are additionally

supposed here. Both permeable disks possess same value of the permeability. The governing

partial differential equations for the conservation of mass, momentum, heat and concentration

may be written, mathematically, as

Continuity Equation

1 1 2

1 1 2

0,u u u

x x x

∂ ∂+ + =

∂ ∂ (2.1)

r- Components of Flow Equation

22 20 11 1 1 1 1 1 1

1 2 2 2 21 2 1 1 1 2 1

1 1( ) ,e

nfnf nf

B uu u u u u u u pu u

t x x x r x x x x

συ

ρ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ + = + − + − −∂ ∂ ∂ ∂ ∂ ∂ ∂

(2.2)

z- Components of Flow Equation

2 22 2 2 2 2 2

1 2 2 21 2 1 1 1 2 2

1 1( ) ,nf

nf

u u u u u u pu u

t x x x x x x xυ

ρ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ + = + + −∂ ∂ ∂ ∂ ∂ ∂ ∂

(2.3)

Heat Equation

( ) ( )

222 21

1 2 0 121 2 2 2

1nfnf e

p pnf nf

uT T T Tu u B u

t x x x xc c

µα σ

ρ ρ

∂∂ ∂ ∂ ∂+ + = + + ∂ ∂ ∂ ∂ ∂

, (2.4)

Concentration Equation

21 2

1 2

n n nn

C C Cu u D C

t x x

∂ ∂ ∂+ + = ∇

∂ ∂ ∂, (2.5)

Where the pressure is denoted byp , the density is denoted bynfρ and nano-fluid’s kinematics

viscosity is denoted by nfυ . Moreover, conductivity is denoted by eσ , magnetic field force is

12

denoted by , T is the temperature, nfα is the thermal diffusivity and the nC is the

concentration which may be expressed as

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

1

2.5

1

, , 1 ,

1 , 1 , ,

2 2 2 .

nf nf snf nf nf nf

f fp nf

p p p nf fs f nf

nfs f f s s f f s

f

k

c

c c c

kk k k k k k k k

k

ρ ρυ µ ρ α ϕ ϕ

ρ ρρ

ϕ ρ ρ ϕ ρ µ µ ϕ

ϕ ϕ

−

−

−

= = = − +

+ − = = −

+ − − + + − =

(2.6)

where the densities of the solid and of the fluid, respectively, are denoted byfρ and sρ , the heat

capacitance is denoted by ( )nfpcρ and nano-fluid’s thermal conductivity is denoted by nfk [99].

The conditions on the surface of the boundary for the flow, temperature and concentration are

2 1 2 1 1( ); 0, ( ), , n nx a t u u Aa t T T C C′= − = = − = =

2 1 2 2 2( ); 0, ( ), , .n nx a t u u Aa t T T C C′= = = = = (2.7)

Where wall permeability is denoted by A. 1T and Cn1 are the temperature and concentration at the

lower disk respectively. These quantities at the upper disk are 2.T and Cn2. The temperature at the

lower disk is greater than the temperature at the upper one. Similar relation between the

concentrations at the surfaces of the two disks holds. The temperatures and concentrations are

fixed at the boundaries.

After eliminating the pressure field term from the flow equations (2.2) and (2.3) and introducing

the following similarity transformations

121 22

2 2

1 2 1 2

2, ( , ) , ( , ),

( , ) , ( , ) ,

f f

n n

n n

xxu F t u F t

a a aC C T T

t tC C T T

ξ

ν νξ ξ ξ

χ ξ θ ξ

= =− =

− −= =

− −

(2.8)

we obtain the dimensionless ODEs

2

(3 ) 2 0,nf ft

f f nf

aF F F F FF M Fξξξξ ξξ ξξξ ξξ ξξξ ξξ

υ ρα ξ

υ υ ρ+ + − − − = (2.9)

( ) ( )( )2

2.52 22 1 0,f fr c t

nf nf nf

k aF MF F P E

kξξ ξ ξ ξξ

υθ ξα θ ϕ θ

α α−

− − + + + − − =

(2.10)

0B

13

2 (2 )t fa F Dξ ξξχ ν ξα χ χ+ − = . (2.11)

With transformed B. Cs. of the form

1 Re, 0, 1, 1;

1 Re, 0, 0, 0.

at F F

and

at F F

ξ

ξ

ξ θ χ

ξ θ χ

= − = − = = =

= = = = = (2.12)

Wall expansion ratio is expressed as f

taa

υα

)(′= , the Reynolds number is expressed as

f

aAa

υ2Re

′= and the magnetic parameter is expressed as

f

e aBM

µσ 22

0= . Further, the Prandtl

number is denoted by ( )

f

fp

r k

cP

µ= , Schmidt number is denoted by DSc fν

= , the Eckert

number is denoted by ( )( )fp

c cTT

UE

21

2

−= , and the reference velocity is denoted by 1

2

fxU

a

υ= .

It is important to take a note that Equation (2.1) is identically fulfilled. This gives evidence that

the velocity and continuity equation are compatible and, therefore, signifies the motion of the

fluid.

Further, if we use the transformationRe

Ff = and keeping in view the work of [100] with α

being constant, we obtain ( )f f ξ= , ( )χ χ ξ= , ( )θ θ ξ= , 0tfξξ = . Moreover, tθ as well as tχ

reduce to zero. Accordingly, we obtain the flow, heat and concentration equations given below

2Re ( 3 ) 0,nf f

f nf

f Mf ff f fξξξξ ξξ ξξξ ξξξ ξξ

υ ρα ξ

υ ρ− − + + = (2.13)

( ) ( )( )2.52 2 22Re Re 1 0,f fr c

nf nf

kf P Mf f E

kξξ ξ ξ ξξ

υθ ξα θ ϕ

α−

− − + + − =

(2.14)

(2Re )Sc f ξ ξξξα χ χ− = , (2.15)

With the boundary conditions of flow heat and concentration

1 1, 0, 1,at f fξξ θ= − = − = = . 1;χ =

1at ξ = 1, 0, 0,f fξ θ= = =

. 0.χ = (2.16)

14

2.3. NUMERICAL SOLUTION

Quasi-linearization is applied to make the vector sequences as ( ) mf , ( ) mθ & ( ) mχ , which

converge to the numerical solutions of the corresponding differential equations (2.13), (2.14) &

(2.15). To make the sequence ( ) mf , we linearize equation (2.13) as

( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( 1) ( ) ( 1) ( )( ) ( )

( 1) ( ) ( 1)( ) ( )

: ( , , , , ) ( 3 ) 2Re ,

( , , , , )

nf f

f nf

m m m m m m m m mm m

m m mm m

We put H f f f f f f f f ff Mf

H HH f f f f f f f f f

f f

H Hf f f f

f f

ξ ξξ ξξξξ ξξξ ξξξ ξξ ξξξξ ξξξ ξξ

ξ ξξ ξξξ ξξξξ ξ ξξ

ξξ ξξ ξξξ ξξξξ ξξξ

υ ρα η

υ ρ

+ +

+ +

≡ + + − −

∂ ∂+ − + − +∂ ∂

∂ ∂− + −

∂ ∂( ) ( )( ) ( 1) ( )

( )0,m m m

m

Hf f

fξ ξξξξ ξξξξξξξξ

+∂+ − =∂

( )

( )

( 1) ( 1) ( ) ( 1)

( 1) ( ) ( ) ( )

2Re 3

2Re 2Re

nf fm m m m

f nf

m m m m

f f f f M

f f f f

ξξξξ ξξξ ξξ

ξξξ ξξξ

υ ρηα α

υ ρ+ + +

+

+ − + + −

− = −

(2.17)

Now the system of the linear differential equations is obtained by the equation (2.17), where mf

represents the solution of the m th equation. In order to find solution of the system of these

ODEs, the first, second, third and fourth order derivative terms are approximated by their central

differences. In this way, the sequence ( )mf is constructed by the linear system given below:

( 1)mDf E+ = with ( )( )mn nD D f×≡ and ( )( )

1 ,mnE E f×≡ (2.18)

We have n grid points where, equations (2.14) and (2.15) are linear in θ and χ correspondingly,

to make the sequence ( )mθ and ( ) mχ which may be written a

( ) ( )( ) ( )

( ) ( ) ( )( )

11 1 2

2.51 12 2

2Re Re

1 0,

mm f m

nf

f m mc r

nf

f

kE P f Mf

k

ξξ ξ

ξξ ξ

υθ ξα θ

α

ϕ

++ +

−+ +

− − +

− + =

(2.19)

( ) ( )( ) ( )11 12Re 0,mm mSc fξξ ξχ ξα χ++ +− − = (2.20)

15

Remarkably ( 1)mf +

, in the above frameworks of conditions, should be known. Moreover, the

central difference approximations are then substituted for derivatives. We summarize

computational method as follows:

o Make accessible the initial supposition( ) ( ) ( )000 ,, χθf fulfilling the BCs predefined in

Eq.(2.16)

o Differential equations given by Eq.(2.18) of linear system has been solved to get( )1f

o Utilize ( )1f for linear system solution emerging from the finite difference discretization

of Eqs. (2.19) and (2.20), to get ( )1θ and ( )1χ .

o The values of ( ) ( ) ( )1 1 1, &f g θ are taken to be the new initial guesses and after repeating the

method to obtain sequences ( ) ( ) ( ) , &m m mf g θ which, individually, converge to

, &f g θ ( Eqs. (2.13), (2.14) and (2.15)).

o Production of all sequences until

( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 6, , 10m m m m m m

L L Lmax f f g g θ θ

∞ ∞ ∞

+ + + −− − − <

Scheme of polynomial extrapolation is also utilized to enhance the requirement of exactness of

the solution, finally.

2.4. RESULTS AND DISCUSSION

Values of the shear stress, the rate of mass and heat transfer at the surfaces of the disks

characterise the physical quantities of our concern. These quantities are related to( )1−′′f , ( )1−′χ

and ( )1−′θ . It would be sufficient to present the results only at surface of the lower disk because

of the symmetry of the problem under consideration. For present investigations the parameters

characterising the flow, heat and mass transfer are Re (Reynolds number), M (magnetic

parameter), φ (volume fraction parameter of the nano-particle), α (ratio of wall expansion), Ec

(Eckert number) and Sc (Schmidt number). Importantly, 0<α means disks are approaching

towards each other and 0>α means the disks are receding, thoughReis taken to be negative for

the suction.

16

The effects of above mentioned parameters on( )1−′′f , ( )1−′θ and ( )1−′χ , flow velocity ( )f ξ′ ,

temperature field ( )θ ξ & the concentration ( )χ ξ are studied. Since, the nano-fluid (water-based)

is considered here which contains 2TiO (nano-particles), the values given below are utilized for

the solid particles and the base-fluid:

( )4250, 4179, 8.9538, 997.1, 0.613,s p s f ffc k kρ ρ= = = = = ( ) 686.2,p s

c = & 2.6Pr= .

Necessarily, the case 0=φ relates to the water without the nano-particles.

Table 2.1 explains the convergence of the numerical outcomes due to decreasing step-size. The

values 0.02, 0.01& 0.005h h h= = = are taken for the grid sizes. This convergence test has provided

us affirmation of our computational technique. From Table 2.2, it is obvious that the mass and

heat transfer rates, and the shear stress increase due to the imposed magnetic field at the disks,

whether disks are drawing closer or receding.

Nano-particles addition to the base fluid increases the rate of transfer of heat whereas

reducing the shear stress while not notably the transfer rate of mass, for both cases of moving

away and closing disks as predicted here in Table 2.3. It is obvious from Table 2.4 that the

suction increases both heat transfer rate and shear stress due to the movement of disks away from

each other, but lowers the shear stress whereas boosting the transfer rate of heat for the other

case. In both the cases the rate of mass transfer decreases, remarkably. Table 2.5 shows that the

( )1−′θ ,& ( )1' −χ as well as the values of the shear stress at the surface of the disks show

increasing trend with expanding parameter as the disks move to get closer while different pattern

is observed when disks are receding. Table 2.6 demonstrates that for the two cases ofα rate of

transfer of heat due to the viscous dissipation always increases at the disks. Impact of the

Schmidt number is to reduce the transfer rate of mass at disks in either case ofα as demonstrated

in Table 2.7.

Fig. 2.2 represents the streamlines for the present flow, heat and concentration problem.

Figures 2.3-2.6 are that, for all cases ofα , the external field of magnet tends to decrease the flow

velocity in the center of both the disks. Therefore, for present area, field of magnet behaves like a

drag force which is called the Lorentz force and hence it lowers the velocity of the fluid. This

provides us results of production of thermal energy which is remarkably built the temperature of

fluid as shown in Fig. 2.4 and Fig. 2.6.

17

Compared to the velocity field, the impact of the nano structure of the fluid is more

prominent on the temperature distribution, regardless of the disks are approaching or moving

away shown in Figs. 2.7-2.10.

Then again, the impact of the permeability Reynolds number Re is same as that ofM for

0>α , though a contrary trend might be observed here for 0<α as shown in Figs. 2.11-2.16.

Fig. 2.17 illustrates that, as the values of α are changed from negative to positive,

velocity increases just a midst region between the disks. Fig. 2.18 uncovers that, for the disks

moving towards each other, as ratio of wall expansion α increases the temperature distribution

over the entire domain also increases. In addition, the concentration distribution increases in

lower half of the plane 0η = , while opposite trend is noted close to the upper disk as appeared in

the Fig. 2.19. In other case, the profiles of the temperature increase only in a midst of the disks.

Figures 2.20 and 2.21 demonstrate that viscous dissipation significantly build temperature

dispersion over disks, apart from the disks are getting closer or receding. Finally, Fig. 2.22

demonstrates that the impact of the Schmidt number Sc on the concentration is same for both the

cases of α .

2.5. CONCLUSIONS

The numerical investigations of the problem of flow, heat and concentration of unsteady flow of

a nano-fluid with constant density and viscosity between two coaxial porous disks are presented

here taking into account the suction through the boundaries. The disks are moving orthogonally.

We want to explore that how the governing parameters, ,M Re, φ , α , ,Sc andEc affect the

flow velocity, the temperature distribution and the concentration along with the heat and mass

transfer rate.

For ( 0>α ):

1) At the surface of the disks, shear stress increases with M andRe, while contrary

impact is found for φ&α .

2) Rate of heat transfer at the disks increases with M , Re and φ , while decreases with

α .

18

3) Rate of mass transfer decreases withRe,φ , Sc and α while a contrary trend is noted

for the external magnetic field.

For ( 0<α ):

4) Shear stress at the disks increases with M and α while contrary impact is observed

for φandRe.

5) Rate of transfer of heat at the disks increases with M ,Re,α and φ .

6) Rate of transfer of mass at the disks decreases withSc ,Reand φ though an opposite

pattern is noted for α and M .

19

Table 2.1. ( )f ξ for 4,1.0,8Re −==−= αφ , 3=M , 2.0=Ec , and . 0.1Sc = .

( )f ξ

ξ 1st Grid

size

2nd Grid size

3rd Grid size

Extrapolated values

0 0.2 0.4 0.6 0.8

0 0.3018240 0.5755708 0.7971402 0.9455484

0 0.3018321 0.5755883 0.7971641 0.9455688

0 0.3018342 0.5755927 0.7971701 0.9455739

0 0.3018348 0.5755941 0.7971721 0.9455756

Table 2.2. Effect of M on ( )1−′θ , ( )1−′′f ,.and ( )1' −χ .for Re 8, 0.1ϕ=− = , 1.0=Sc and

2.0=Ec .

3=α 3−=α

M ( )1−′θ ( )1' −χ ( )1−′′f ( )1−′θ ( )1' −χ

1

2

3

4

5

1.7397

1.7695

1.7994

1.8294

1.8595

2.7556

2.8499

2.9460

3.0438

3.1435

-0.2161

-0.2163

-0.2166

-0.2169

-0.2171

2.6130

2.6530

2.6930

2.7329

2.7727

8.8637

9.1308

9.4012

9.6749

9.9518

-0.2790

-0.2793

-0.2796

-0.2799

-0.2808

Table 2.3. Effect of φ on. ( )1−′θ , ( )1−′′f and ( )1' −χ .for Re 8, 3M=− = , 1.0=Sc , 2.0=Ec

( )1−′′f

20

3=α 3−=α

( )1−′θ ( )1' −χ ( )1−′′f ( )1−′θ ( )1' −χ

0

0.02

0.04

0.06

0.08

1.8304

1.8220

1.8149

1.8088

1.8037

2.2662

2.3771

2.4996

2.6345

2.7829

-0.2169

-0.2168

-0.2167

-0.2167

-0.2166

2.7325

2.7225

2.7137

2.7059

2.6991

7.2501

7.6087

8.0003

8.4274

8.8932

-0.2799

-0.2798

-0.2797

-0.2797

-0.2796

Table 2.4. Effect of Re on ( )1−′′f , ( )1−′θ , and ( )1' −χ for ,1.0=φ ,3=M 2.0,2.6Pr == Ec ,

and 1.0=Sc .

3=α 3−=α

Re ( )1−′θ ( )1' −χ ( )1−′′f ( )1−′θ ( )1' −χ

-3 -6 -9 -12 -15

1.6163 1.7507 1.8174 1.8561 1.8813

0.7315 1.9976 3.4358 4.9428 6.4839

-0.3467 -0.2625 -0.1964 -0.1453 -0.1064

3.7274 2.9363 2.6108 2.4476 2.3517

8.7523 8.8845 9.7417 10.9532 12.3242

-0.4356 -0.3357 -0.2545 -0.1904 -0.1408

φ ( )1−′′f

( )1−′′f

21

Table 2.5.Effect of α on ( )1−′′f , ( )1−′θ , and ( )1' −χ for 3,8Re,1.0 =−== Mφ , 2.0=Ec , and

1.0=Sc .

α ( )1−′θ ( )1' −χ

-4 -2 0 2 4

2.8985 2.5064 2.1829 1.9155 1.6935

11.6856 7.6177 5.1074 3.5168 2.4826

-0.2916 -0.2680 -0.2462 -0.2261 -0.2075

Table 2.6. Influence of . Ec on ( )1−′θ for 1.0,8 =−= φR , 3=M , and 1.0=Sc .

( )1−′θ

Ec 3=α 3−=α 0

0.2

0.4

0.6

0.8

0

2.9460

5.8919

8.8379

11.7839

0

9.4012

18.8023

28.2035

37.6047

( )1−′′f

22

Table 2.7. Effect of Sc on ( )1' −χ for 1.0,8Re =−= φ , 3=M , and 2.0=Ec .

( )1' −χ

Sc 3=α 3−=α

0

0.1

0.2

0.3

0.4

-0.5000

-0.2166

-0.0841

-0.0302

-0.0103

-0.5000

-0.2796

-0.1472

-0.0738

-0.0357

23

Figure 2.1. Physical model.

Figure 2.2. The Streamlines.For ,4,1.0,8Re −==−= αφ and 3=M .

r

z

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

24

Figure 2.3. Velocity profiles for Re 2, 0.05, 2, 0.1,Ecϕ α= − = = = and 1=Sc .

Figure 2.4. Thermal profiles for Re 2, 0.05, 2, 0.1,Ecϕ α= − = = = and 1=Sc .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

η

f ' (

η )

M = 0

M = 2

M = 4M = 6

M = 8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

η

θ (

η )

M = 0

M = 2

M = 4M = 6

M = 8

25

Figure 2.5. Velocity profiles forRe 2, 0.05, 2ϕ α= − = = − , ,1.0=Ec and 1=Sc .

Figure 2.6. Thermal profiles forRe 2, 0.05, 2ϕ α= − = = − , ,1.0=Ec and 1=Sc .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

η

f ' (

η )

M = 0

M = 2

M = 4M = 6

M = 8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

η

θ (

η )

M = 0

M = 2

M = 4M = 6

M = 8

26

Figure 2.7. Velocity profiles for ,1,1.0,2,2Re ===−= ScEcα and 1=M .

Figure 2.8. Thermal profiles for ,1,1.0,2,2Re ===−= ScEcα and 1=M .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

η

f ' (

η )

φ = 0

φ = 0.02

φ = 0.04

φ = 0.06

φ = 0.08

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

η

θ (

η )

φ = 0

φ = 0.02

φ = 0.04

φ = 0.06

φ = 0.08

27

Figure 2.9. Velocity profiles for ,1,1.0,2,2Re ==−=−= ScEcα and 1=M .

Figure 2.10. Thermal profiles for ,1,1.0,2,2Re ==−=−= ScEcα and 1=M .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

η

f ' (

η )

φ = 0

φ = 0.02

φ = 0.04

φ = 0.06

φ = 0.08

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

η

θ (

η )

φ = 0

φ = 0.02

φ = 0.04

φ = 0.06

φ = 0.08

28

Figure 2.11. Velocity profiles for 5,1.0,1 === αφM , ,1.0=Ec and 1=Sc .

Figure 2.12. Thermal profiles for the 1, 0.05, 5M ϕ α= = = , ,1.0=Ec and 1=Sc .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

2.5

η

f ' (

η )

R = -3

R = -5

R = -7R = -10

R = -12

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

η

θ (

η )

R = -3

R = -5

R = -7R = -10

R = -12

29

Figure 2.13. Concentration profiles for 5,1.0,1 === αφM , ,1.0=Ec and 1=Sc .

Figure 2.14. Velocity profiles for 5,1.0,1 −=== αφM , ,1.0=Ec and 1=Sc .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

χ

( η

)

R = -3

R = -5

R = -7R = -10

R = -12

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

η

f ' (

η )

R = -3

R = -5

R = -7R = -10

R = -12

30

Figure 2.15. Thermal profiles for 5,1.0,1 −=== αφM , ,1.0=Ec and 1=Sc .

Figure 2.16. Concentration profiles for 5,1.0,1 −=== αφM , ,1.0=Ec and 1=Sc .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

η

θ (

η )

R = -3

R = -5

R = -7R = -10

R = -12

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

χ

( η

)

R = -3

R = -5

R = -7R = -10

R = -12

31

Figure 2.17. Velocity profiles for 1, 0.05,Re 3M ϕ= = = − , ,1.0=Ec and 1=Sc .

Figure 2.18. Thermal profiles for 1, 0.05,Re 3M ϕ= = = − , ,1.0=Ec and 1=Sc .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

2

2.5

η

f ' (

η )

α = -5

α = -2

α = 1

α = 2

α = 5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

η

θ (

η )

α = -5

α = -2

α = 1

α = 2

α = 5

32

Figure 2.19. Concentration profiles for 1, 0.05,Re 3M ϕ= = = − , ,1.0=Ec and 1=Sc .

Figure 2.20. Thermal profiles for 1, 5,Re 3M α= = = − , 0.05,ϕ = and 1=Sc .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

χ

( η

)

α = -5

α = -2

α = 1

α = 2

α = 5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

η

θ (

η )

Ec = 0

Ec = 0.1

Ec = 0.2Ec = 0.3

Ec = 0.4

33

Figure 2.21. Thermal profiles for 1, 5,Re 3M α= = − = − , ,2.0=φ and 1=Sc .

Figure 2.22. Concentration profiles for 1, 5,Re 3M α= = = − , 0.05,ϕ = and 1.0=Ec .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

η

θ (

η )

Ec = 0

Ec = 0.1

Ec = 0.2Ec = 0.3

Ec = 0.4

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

χ

( η

)

Ec = 0

Ec = 0.1

Ec = 0.2Ec = 0.3

Ec = 0.4

34

CHAPTER 3

ON THE COMBINED EFFECT OF HEAT AND MASS

TRANSFER IN MHD FLOW OF A NANO-FLUID

BETWEEN EXPANDING / CONTRACTING WALLS OF

A POROUS CHANNEL

35

3.1. INTRODUCTION

Flow, heat and mass transfer rate in a porous channel with infinite parallel walls are

described numerically for incompressible laminar viscous nano-fluid. The channel

walls may be contracting or expanding. The influence of the external magnetic field

as well as the viscous dissipation is also taken into account. A set of nonlinear

ordinary differential equations (ODEs) is obtained by introducing similarity variables

in the governing partial differential equations (PDEs) of the flow, heat and

concentration. The iterative (Successive over Relaxation) and a direct method are

employed to solve sparse system of linear algebraic equations. The shear stress and

the rate of heat and mass transfer are analyzed keeping in view the role of the

permeability Reynolds, expanding/contracting walls, the magnetic field and the

viscous dissipation.

3.2. MATHEMATICAL FORMULATION

Consider the flow, heat and concentration of an electrically conduction nano-fluid

through a semi-infinite open ended porous channel with width 2a as shown in the Fig.

3.1. The assumptions regarding the flow, heat and concentration may be summarised as

o The nano-particles 2 3Al O are inserted in the base fluid

o The flow is laminar and unsteady

o The walls are uniformly moving up or down at time dependent rate )(ta′

o The walls are located at a distance )(2 ta which is function of time

o One end of the channel is unobstructed whereas a solid membrane closes the other

end

o The x-axis may be chosen as a line of symmetry in a rectangular coordinate

system for this problem

o The symbols u and v represent the velocity components in the directions of

increasingx andy respectively

o The Joule heating, viscous dissipation and external magnetic field are taken into

account

o The external applied magnetic field is acting in the perpendicular direction of the

flow

36

The equations of the continuity, momentum (in both the directions), heat and

concentration with constant density and viscosity for the problem under consideration

take the following mathematical form

0u u

x y

∂ ∂+ =

∂ ∂, (3.1)

( )2

021 nfnf

nf nf

B uu u u pu v u

t x y x

συ

ρ ρ∂ ∂ ∂ ∂+ + = − + ∇ −

∂ ∂ ∂ ∂, (3.2)

( )21nf

nf

v v v pu v v

t x y yυ

ρ∂ ∂ ∂ ∂+ + = − + ∇

∂ ∂ ∂ ∂, (3.3)

2 202 2( ) ( )

( ) ( )nf nf

nfp nf p nf

B uT T T uu v T

t x y c y c

µ σα

ρ ρ∂ ∂ ∂ ∂

+ + = ∇ + +∂ ∂ ∂ ∂

, (3.4)

( )2C C Cu v D C

t x y

∂ ∂ ∂+ + = ∇

∂ ∂ ∂ (3.5)

The symbol p indicates the pressure field, nfρ represents the constant density of the nano-

fluid, nfυ shows the kinematics viscosity, nfα signifies the thermal diffusivity and acts

as the strength of the magnetic field. Further, T , C and D are the temperature,

concentration and effective diffusion coefficient respectively.

Following the research contributions [101-102], we have:

( )

( )

( ) ( )( ) ( ) ( ) ( )( ) ( )

( )

2.5

3( 1)

1 , ,1( 2) ( 1)

1 , ,

2 21 , ,

2

,

s

nf f fnf

s sf

f f

nfnf f s nf

nf

s f f snfp p ps f nf

f s f f s

nfnf

p nf

k k k kkc c c

k k k k k

k

c

σφ

σ σ µµ

σ σσ ϕφσ σ

µρ ϕ ρ ϕρ υ

ρ

ϕϕ ρ ϕ ρ ρ

ϕ

αρ

− = + =−+ − −

= − + =

+ − − + − = = + + −

=

(3.6)

Where sρ and fρ are the densities, ( )nfpcρ is the heat capacitance, φ is the nano-particle

volume fraction parameter, nfk is the effective thermal conductivity, nfσ is the electrical

conductivity of the nano-fluid. Moreover, the electrical conductivities of the solid and the

base fluid are represented by sσ and fσ .

0B

37

The boundary conditions for the velocity, temperature and concentration at the lower and

upper walls of the channel may be expressed as

1 1

2 2

, 0, , ( ), ( )

, 0, , ( ), ( ).

T T u C C v Aa t at y a t

T T u C C v Aa t at y a t

′= = = = − = − ′= = = = =

(3.7)

HereA is the permeability of the wall, 1T and 2T are the fixed temperatures ( 21 TT > ) and

the fixed concentrations are C1 and C2 at the lower and upper walls respectively. After

eliminating the pressure field term from the flow equations and introducing the following

similarity variables

2 22

1 2 1 2

( , ), , ( , ), ( , ) , ( , ) ,( ) ( ) ( )

f fx C C T Tyu F t v F t t t

a t a t a t C C T Tη

υ υη η η χ η θ η

− − −= = = = =

− − (3.8)

we arrive at

2

(3 )

3( 1)

1 0,( 2) ( 1)

nft

f f

s

f f

s snf

f f

aF F F F FF F F

M F

ηηηη ηη ηηη ηη ηηη η ηη

ηη

υα η

υ υ

σφ

ρ σσ σρ φσ σ

+ + − + −

− − + = + − −

(3.9)

22.5 2

2

( ) Pr . . (1 )

3( 1)

.Pr. . 1 0,( 2) ( 1)

f ft

nf nf nf

s

f f

s snf

f f

kaF Ec F

k

kM Ec F

k

ηη η ηη

η

υθ ηα θ θ φ

α α

σφ

σσ σ

φσ σ

−+ + − + −

−

+ + =

+ − −

(3.10)

2 ( ) .t fa F Dη ηηχ υ ηα χ χ+ − − =

(3.11)

with the transformed boundary conditions as

1, , 1, 0, 1

0, , 0, 0, 1.

F R e F at

F Re F at

η

η

θ χ η

θ χ η

= = = = = −

= = − = = = (3.12)

Here the ratio of wall expansion isf

taa

υα

)(′= , the permeability Reynolds number is

2 f

AaaRe

υ′

= , the magnetic parameter is 2 2

0f

f

B aM

σ

µ= , the Eckert number is

38

2

41 2

( )

( )( )f

f

xEc

a T T cp

υ=

− , Prandtl number is

( )p f

f

cPr

k

µ= and the Schmidt number is

DSc fν= .

It is necessary to mention that the conservation equation (3.1) is identically satisfied by

the flow velocity field. It gives the evidence of the possible fluid motion and hence the

proposed velocity field is compatible with (3.1).

By lettingF

fR e

= and following [103], with α being assumed as a constant, we see that

( )f f η= and ( )θ θ η= , and hence 0=tθ , 0=tχ and 0tfηη = . In this way, the flow, heat

and concentration equations take the final form

( ) (3 )

3( 1)

1 0,( 2) ( 1)

nf

f

s

f f

s snf

f f

f Re ff f f f f

M f

ηηηη ηηη η ηη ηη ηηη

ηη

υα η

υ

σφ


+ − + +

− − + =

+ − −

(3.13)

2 2.5 2

2 2

( ) (1 )

3( 1)

1 0,( 2) ( 1)

f f

nf nf

s

f f

s snf

f f

kRe f Pr Ec Re f

k

kPr Ec Re M f

k

ηη η ηη

η

υθ ηα θ φ

α

σφ

σσ σ

φσ σ

−+ + + −

− + + =

+ − −

(3.14)

( ) 0,Re f Scηη ηχ χ ηα+ + = (3.15)

the boundary conditions are

1, 1, 1, 0, 1

0, 1, 0, 0, 1.

f f at

f f at

η

η

θ χ η

θ χ η

= = = = = −

= = − = = = (3.16)


The numerical.solution of Eqs. (3.13)-(3.15) can be obtained by constructing

the three sequences of vectors( ) ( ) ,m mf θ and ( ) mχ (these sequences of vectors

converge to the numerical solution of their respective equations) . We linearize Eq.

(3.13) to construct sequence of vectors( ) mf as given below:

39

We set:

( ) ( )( ) ( ) ( ) ( ) ( ) ( 1) ( ) ( 1) ( )( ) ( )

( , , , , ) (3 ) ( )

3( 1)

1 ,( 2) ( 1)

and

( , , , , )

nf

f

s

f f

s snf

f f

m m m m m m m m m

m m

G f f f f f f f f Re ff f f

M f

G GG f f f f f f f f f

f f

η ηη ηηη ηηηη ηηηη ηη ηηη ηηη η ηη

ηη

η ηη ηηη ηηηη η ηη

υα η

υ

σφ


+ +

≡ + + + −

−

− +

+ − −

∂ ∂+ − + −

∂ ∂

( ) ( ) ( )( 1) ( ) ( 1) ( ) ( 1) ( )( ) ( ) ( )

0,m m m m m m

m m m

G G Gf f f f f f

f f fηη ηη ηηη ηηη ηηηη ηηηηηη ηηη ηηηη

+ + +∂ ∂ ∂+ − + − + − =

∂ ∂ ∂

By simplifying:

( 1) ( ) ( 1) ( ) ( 1)

( ) ( 1) ( ) ( 1) ( ) ( ) ( ) ( )

3( 1)

(Re ) 3 Re 1( 2) ( 1)

Re Re Re Re .

s

nf f fm m m m m

s sf nf

f f

m m m m m m m m

f f f f M f

f f f f f f f f

ηηηη ηηη η ηη

ηη η ηηη ηηη η ηη

σφ

υ ρ σαη α

σ συ ρ φσ σ

+ + +

+ +

−

+ + + − − + + − −

− + = −

(3.17)

This represents the system of linear differential equations, where the thm equation has

numerical solution vector( )mf . In order to solve these linear ODEs, the derivatives are

replaced by the central difference approximations. The linear system may be

represented as follows:

( ) ( )( 1) ( ) ( )1 with and ,m m m

n n nBf C B B f C C f+× ×= ≡ ≡ (3.18)

where the number of grid points shown by n. On the other hand, the heat and

concentration equations (3.14)-(3.15) are linear in θ and χ respectively.

To construct the sequence of vectors( ) mθ and ( )mχ , we have

( 1) ( 1) ( 1) 2 2.5 ( 1)2

2 ( 1)2

( ) (1 )

3( 1)

1 0,( 2) ( 1)

f fm m m m

nf nf

s

f f m

s snf

f f

kRe f Pr Ec Re f

k

kPr Ec Re M f

k

ηη η ηη

η

υθ ηα θ φ

α

σφ

σσ σ

φσ σ

+ + + − +

+

+ + + −

− + + =

+ − −

(3.19)

40

( 1) ( 1) ( 1)( ) 0,m m mRe f Scηη ηχ χ ηα+ + ++ + = (3.20)

It is worth mentioning that ( 1)mf + is observed to be known and its derivatives are

replaced by approximations.

We list out the computational procedure as under:

o Provide the values of ( ) ( ) ( )0 0 0, &f θ χ as initial guess satisfying the boundary

conditions given in Eq. (3.16) ,

o The linear system (Eq. (3.18)) is solved to calculate ( )1f

o Using the value of ( )1f in Eqs. (3.19) - (3.20). After solving these equations,

we obtain ( ) ( )1 1&θ χ .

o The procedure is repeated by taking initial guess to generate the sequences

( ) ( ) ( )1 1 1, &f θ χ that respectively converge to , &f g θ which represent the

solution of Eqs. (3.13)-(3.15).

o The sequences of vectors are generated until

( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 6max , , 10 .m m m m m m

L L Lf f θ θ χ χ

∞ ∞ ∞

+ + + −− − − <

Further details about the computational procedure may be reviewed from Ali et al.

[104-106].


The Reynolds numbers Re, the Schmidt number Sc , the.magnetic.parameterM ,

nano-particle. volume fraction. parameterφ , the Eckert numberEc and the wall

expansion ratioα are parameters for the problem under investigation. The upper and

lower walls are expanding or contracting with time-dependent rate )(ta′ and having

same permeability. We suggest the nano-fluid based on water which contains 2 3Al O

nano-particle. For the flow, heat and concentration problem under consideration, the

following set of values are taken

( )12997.1, 10 , 3970, 25, 0.613, 0.05, 765,f s s s f f p sk k c andρ σ ρ σ−= = = = = = =

Pr 6.2.= The case 0=φ stands for pure water.

In case of both moving away or approaching walls 0>α or 0<α respectively. The

parameter 0R e > for injection. We are going to study the velocities ( )ηf & ( )ηf ′ ,

41

temperature field ( )ηθ and the concentration filed ( )χ η against set of values of some

parameters of physical nature. The flow stream lines are shown in Fig. 3.2.

In Figs. 3.3-3.6, influence of the applied magnetic field upon velocity and temperature

profiles is shown. The streamwise velocity is an increasing function whereas the

temperature is a decreasing function of applied magnetic field, irrespective of the

walls motion. The nano-particles volume fraction has negligible effect on the velocity

field. The temperature of the fluid is increasing for receding walls whereas an

opposite trend is observed for approaching walls. However, the change in the

temperature for approaching walls is significant as compare to that of receding walls.

These trends may be noted in Figs. (3.7-3.10).

The influence of the permeability Reynolds number on the flow velocity,

concentration and temperature distribution can be observed graphically from Figs.

3.11-3.16. Maximum velocity is obtained when both the walls are contracting in the

central region between the channel walls. However, this trend is reversed in case of

the expanding the walls. The increase in Re has the influence of rising temperature in

the whole domain for 0α > , whereas in case of the approaching walls, Re has non

uniform effect on the temperature. In both the cases whetherα is positive or negative,

the behaviour of Re on ( )χ η′ remains the same.

As α changes from positive to negative, there is a significant rise in highest value of

velocity while, on the other hand, temperature profiles are remarkably lowered.

Moreover, concentration profile is increasing in the lower half whereas it is

decreasing in the upper half of the channel. These may be observed in Figs. 3.17-3.19.

The Eckert number characterizes the viscous dissipation as shown in Figs. 3.20-3.21.

It has significant rise in the temperature profiles during the contraction or expansion

of the walls. Further, in approaching walls, the Eckert number has more remarkable

effect. Figures 3.22-3.23 clearly show that the concentration profiles have linear

variation against the Schmidt number.

Numerical results are well arranged as the step-size decreases. The convergence of the

results for the flow velocity is shown in Table 3.1. This convergence has significant

role in processing of our computational procedure. Shear stress along with heat and

mass transfer rates at the walls of the channel are increased as the external magnetic

field increases. This may be observed in Table 3.2. When we consider the problem of

approaching walls, it is noted that( 1)θ ′ − is more affected by magnetic field. This is

42

due to the Lorentz force which tends to drag fluid towards disk. The magnetic field

exerts a retarding force named as friction. This increases the shear stress at the walls

of the channel. Moreover, the fluid temperature and the temperature difference

increase due to the frictional force, between the walls and the .fluid. The heat transfer

rate, being directly .proportional to the. temperature difference, also increases.

The addition of the nano-particles to the water has the influence of decreasing the

shear stress as shown in Table 3.3. Further, the shear stress ( 1)f ′′ − and the mass

transfer rate ( 1)χ ′ − are shown in this table. Nano-particles volume fraction φ is more

influential at heat transfer rate ( 1)θ ′ − at the lower wall. Moreover, this transfer rate

has increasing trend as 0α > which is the case of expanding walls. A significant

decrease in ( 1)θ ′ − may be observed for approaching walls.

From Table 3.4, it is obvious that the heat transfer rate and shear stress both are

increased due to permeability Reynolds number when the walls are moving away in

the channel. However, this behaviour is reversed for the approaching walls. There is

always a decreasing trend in the mass transfer rate whether the walls are moving away

or tend to approach. Table 3.5 predicts that when the walls come closer to each other,

the fluid is pushed towards the boundaries. As a result the retarding forces at the walls

are increased. Similar behaviour can be seen in case of the heat as well as the mass

transfer rate. On the other hand, these trends are reversed when the walls are getting

farther to each other. Table 3.6 predicts that the viscous dissipation always increases

the heat transfer rate at the walls for 0<α as well as for 0>α . The increase in the

Eckert number changes the sign of ( )1−′θ when walls are approaching each other as it

may be seen Table 3.6. This heat transfer shows reverse phenomenon on the channel

walls. Thus, heat is flowing towards surface from fluid. An important result has been

drawn that viscous dissipation becomes the cause of change in the direction of heat

transfer on the walls of the channel. This was due to external magnetic field. As a

result, when channel flow is being studied, one must consider viscous dissipation and

its influence. Mass transfer rate at the walls of channel is also decreased which may

be noted in Table 3.7.

3.5. CONCLUSIONS

In the present chapter, all the parameters of interest namely the magnetic parameter M,

the nano-particle volume fraction parameter , the Reynolds number Re, the wall

expansion ratio , the Schmidt number Sc and the Eckert number Ec, effect the flow

φ

α

43

as well as the heat and mass transfer characteristics of the laminar incompressible

nano-fluid (based on water (containing 2 3Al O nano-particles)) through a channel. The

following conclusions may be drawn:

Forα > 0:

1) The shear stress is enhanced by M and Re while an opposite trend is observed

for and α .

2) The parameters M, Re and φ increase the heat transfer rate but this trend is

reversed in case of .

3) Re and Sc have the effect of decreasing the mass transfer rate while it is

increased by the applied magnetic field.

Forα < 0:

4) The magnetic field increases the heat transfer rate as well as the shear stress

whereas both the volume fraction and the permeability have opposite trend as

that of the external magnetic field.

5) Mass transfer is reduced by the Schmidt number, permeability Reynolds

number and volume fraction. On the other hand, both α and M make the shear

stress larger at the walls .

φ

α

44

Table 3.1. Dimensionless velocity ( )ηf on three grid sizes and extrapolated values

for 3, 0.06, 3Re φ α= = = − , 2M = , 0.5Ec = , and 1Sc = .

( )ηf

η 1st grid

( 0.02)h =

2nd grid

( 0.01)h =

3rd grid

( 0.005)h =

Extrapolated

values

0

0.2

0.4

0.6

0.8

0

0.2613

0.5135

0.7425

0.9218

0

0.2613

0.5136

0.7426

0.9219

0

0.2614

0.5136

0.7426

0.9219

0

0.2614

0.5136

0.7427

0.9219

Table 3.2. Effect of the magnetic parameter M on ( )1−′′f , ( )1−′θ , and ( )1' −χ for

0.06, 3Reφ = = , 2.0=Ec , and 1Sc = .

3α = 3α = −

M ( )1−′θ ( )1' −χ ( )1−′′f ( )1−′θ ( )1' −χ

0.5

1

1.5

2

2.5

1.5940

1.6408

1.6871

1.7330

1.7784

2.4363

2.5719

2.7097

2.8495

2.9914

-0.0335

-0.0337

-0.0338

-0.0339

-0.0340

4.4625

4.5047

4.5464

4.5877

4.6285

82.5751

87.7415

92.9614

98.2336

103.5568

-0.4251

-0.4255

-0.4260

-0.4264

-0.4269

( )1−′′f

45

Table 3.3.Effect of the nano-particle volume fraction parameter φ on ( )1−′′f ,

( )1−′θ , and ( )1' −χ for 2, 3M Re= = , 0.5Ec = , and 1Sc = .

3α = 3α = −

( )1−′θ ( )1' −χ ( )1−′′f ( )1−′θ ( )1' −χ

0

0.02

0.04

0.06

0.08

1.7911

1.7676

1.7483

1.7330

1.7213

2.5128

2.6048

2.7166

2.8495

3.0057

-0.0341

-0.0340

-0.0340

-0.0339

-0.0339

4.6227

4.6145

4.6027

4.5877

4.5695

99.2889

98.9108

98.5544

98.2336

97.9618

-0.4268

-0.4267

-0.4266

-0.4264

-0.4262

Table 3.4.Effect of the permeability Reynolds numberRe on ( )1−′′f , ( )1−′θ , and

( )1' −χ for 0.06,φ = 2,M = Pr 6.2, 0.5Ec= = , and 1Sc = .

3α = 3α = −

Re ( )1−′θ ( )1' −χ ( )1−′′f ( )1−′θ ( )1' −χ

1

2

3

4

5

1.4659

1.6183

1.7330

1.8218

1.8923

0.3934

1.3875

2.8495

4.6844

6.8140

-0.1004

-0.0588

-0.0339

-0.0193

-0.0109

5.0991

4.8272

4.5877

4.3778

4.1944

389.7383

164.9459

98.2336

89.1731

90.2037

-0.8924

-0.6287

-0.4264

-0.2793

-0.1772

φ ( )1−′′f

( )1−′′f

46

Table 3.5. Effect of the wall expansion ratioα on ( )1−′′f , ( )1−′θ , and ( )1' −χ for

0.06, 3, 2Re Mφ = = = , 0.5Ec = , and 1Sc = .

α ( )1−′θ ( )1' −χ

-4

-2

0

2

4

5.1827

4.0173

2.9751

2.0998

1.4145

307.2624

46.7843

14.6338

4.9287

1.6353

-0.6039

-0.2936

-0.1302

-0.0538

-0.0211

Table 3.6. Effect of the Eckert numberEc on ( )1−′θ for 3, 0.06Re φ= = , 2M = , and

1Sc = .

( )1−′θ

Ec 3α = 3α = −

0

0.1

0.2

0.3

0.4

0

0.5699

1.1398

1.7097

2.2796

-0.2001

19.4867

39.1734

58.8601

78.5468

( )1−′′f

47

Table 3.7.Effect of the Schmit number Sc on ( )1' −χ for 3, 0.06Re φ= = , 2M = ,

and 0.5Ec = .

( )1' −χ

Sc 3α = 3α = −

0

0.1

0.2

0.3

0.4

-0.5000

-0.3971

-0.3120

-0.2428

-0.1872

-0.5000

-0.4923

-0.4847

-0.4771

-0.4697

48

Figure 3.1 Physical model of the problem.

Figure 3.2 Streamlines for the problem forRe 6= , 0.06φ = , 3α =− and 2M = .

49

Figure 3.3 Streamwise velocity profiles forRe 3= , 0.06φ = , 3α = , 0.5Ec = and

1=Sc .

Figure 3.4 Temperature profiles forRe 3= , 0.06φ = , 3α = , 0.5Ec = and 1=Sc .

50

Figure 3.5 Streamwise velocity profiles forRe 3= , 0.06φ = , 3α = − , 0.5Ec = and

1=Sc .

Figure 3.6 Temperature profiles for Re 3= , 0.06φ = , 3α = − , 0.5Ec = and 1=Sc .

51

Figure 3.7 Streamwise velocity profiles for Re 3= , 3α = , 0.5Ec = , 1=Sc

and 2M = .

Figure 3.8 Temperature profiles for Re 3= , 3α = , 0.5Ec = , 1=Sc and 2M = .

52

Figure 3.9 Streamwise velocity profiles forRe 3= , 3α = − , 0.5Ec = , 1=Sc

and 2M = .

Figure 3.10 Temperature profiles forRe 3= , 3α = − , 0.5Ec = , 1=Sc and

2M = .

53

Figure 3.11 Streamwise velocity profiles for 3α = , 0.06φ = , 0.5Ec = , 1=Sc

and 2M = .

Figure 3.12 Temperature profiles for 3α = , 0.06φ = , 0.5Ec = , 1=Sc and

2M = .

54

Figure 3.13 Concentration profiles for 3α = , 0.06φ = , 0.5Ec = , 1=Sc and

2M = .

Figure 3.14 Streamwise velocity profiles for 3α = − , 0.06φ = , 0.5Ec = ,

1=Sc and 2M = .

55

Figure 3.15 Temperature profiles for 3α = − , 0.06φ = , 0.5Ec = , 1=Sc and

2M = .

Figure 3.16 Concetration profiles for 3α = − , 0.06φ = , 0.5Ec = , 1=Sc and

2M = .

56

Figure 3.17 Streamwise velocity profiles for 0.06φ = ,Re 3= , 0.5Ec = , 1=Sc

and 2M = .

Figure 3.18 Temperature profiles for 0.06φ = ,Re 3= , 0.5Ec = , 1=Sc and

2M = .

57

Figure 3.19 Concentration profiles for 0.06φ = ,Re 3= , 0.5Ec = , 1=Sc and

2M = .

Figure 3.20 Temperature profiles for 0.06φ = ,Re 3= , 3α = , 1=Sc and

2M = .

58

Figure 3.21 Temperature profiles for 0.06φ = ,Re 3= , 3α = − , 1=Sc and

2M = .

Figure 3.22 Concentration profiles for 0.06φ = ,Re 3= , 3α = , and 2M = .

59

Figure 3.23 Concentration profiles for 0.06φ = ,Re 3= , 3α = − , and 2M = .

60

CHAPTER NO 4

ON THE NON–NEWTONIAN FLUID FLOW DRIVEN BY

A CURVED STRETCHING SHEET

61

4.1. INTRODUCTION

MHD, Non-Newtonian and electrically conducting fluid flow subject to a magnetic

field applied transversely over a circular coiled curved stretching sheet is addressed in

the present chapter. For mathematical modelling, the well-known Casson fluid model

has been employed. The governing nonlinear differential equations are solved

numerically by employing an algorithm based on the Quasi-linearization method.

Physical parameters of the problem in terms of skin-friction coefficient, local Nusselt

number, flow velocity and thermal profiles are discussed through tables and graphs.


Let us consider the flow and heat transfer in 2-dimensional steady &

incompressible Casson fluid flow over a circular coiled curved stretching sheet of

radius R . The curved sheet is stretched with the fixed origin, due to the action of two

forces (equal in the magnitude but opposite in the direction) acting along the s -

direction. Further, the s andr -directions are perpendicular to each other. Where

u as= is velocity of the stretching surface & the strength of the stretching is 0a > .

The fluid has the property of electrical conduction & magnetic field applied in “r ”

direction having constant intensity 0B . The magnetic Reynolds number is assumed

really small with the intention that the induced magnetic field can be ignored. The

surface temperature is kept carefully atwT , keeping wT T∞> and T∞ is the uniform

temperature of the fluid far away. With these assumptions in addition to the

approximations of boundary layer, the governing equations of the Casson fluid flow

(taking into account the influence of the resistive Lorentz force) are given below:

The Continuity Equation

( ) 0u

r R v Rr s

∂ ∂+ + =

∂ ∂ (4.1)

s- Component of the Momentum Equation

2

.1u p

r R rρ∂

=+ ∂

(4.2)

r- Components of the Momentum Equation

2

2

20

2

1 1(1 )(

1) ,

( )

u Ru u uv uv

r r R s r R r r R

Bu u R pu

r r R r R s

υβ

σρ ρ

∂ ∂ ∂+ + = + +

∂ + ∂ + ∂ +

∂ ∂− − −

∂ + + ∂

(4.3)

62

The Heat Equation

2

0 2

22 20

1

1(1 )( ) ,

p

T uR T T Tc v K

r r R s r r R r

Buu

r

ρ

σµ

β ρ

∂ ∂ ∂ ∂ + = + ∂ + ∂ ∂ + ∂

∂− + +

∂

(4.4)

where the symbols u and v are the components of the velocity in s and r -directions,

correspondingly, ρ is the fluid density, µ is the fluid viscosity, p is the pressure, υ

is the kinematics viscosity of fluid, σ is the electrical conductivity, pc is the heat

(specific) keeping the pressure constant, thermal conductivity is 0k & T is the

temperature. It is noted that, for a stretching curved surface, inside the boundary layer

the pressure is no more constant (Sajid et al. [107]).

The suitable boundary conditions under are

0 , 0, ,

0, 0, .

wat r u as v T T

uas r u T T

r ∞

= = = =

∂→∞ → → →

∂

(4.5)

For simplifying and converting the flow and heat equations into ordinary ones, the

following dimensionless variables may be introduced

2 2

( ), ( ), ,

( ), ( ) .w

R au asf v a f r

r RT T

p a s PT T

η υ η ηυ

ρ η θ η ∞

∞

−′= = =

+−

= =−

(4.6)

Equation (4.1) is satisfied automatically whereas the equations (4.2) - (4.4) yield to

2p f

kη η′∂

=∂ +

, (4.7)

( )

( )

22

22

2 1(1 )

k k f fP f f

k k k k

k kff M f ff

k k

η η β η η

η η

′′ ′′ ′′′= − + + + −

+ + + +

′′ ′ ′+ − ++ +

, (4.8)

221 1

Pr 1 (1 ) Pr1

(1 )

k Mf Ec f

k kθ θ

η η ββ

′′ ′ ′= − − + − + + + +

, (4.9)

63

These equations include ak R υ= as radius of curvature and 2 20 /M B aσ ρ= is the

Hartmann number, 0

Pr pck

µ= is the Prandtl number,

2( )c

p

asE c T=

is the Eckert

number.

On exclusion of pressure term from equations (4.7) and (4.8), one can obtain

( ) ( ) ( )

( )( )

( )( )

( )

3 2

2

22 3

1 2(1 )

0

iv f f ff

k k k

f kM f f f f f

k k

k kf f f ff

k k

β η η η

η η

η η

′′′ ′ ′′+ + + −

+ + +

′′′ ′ ′′ ′′′− + − −

+ +

′′ ′ ′− − + − =+ +

, (4.10)

As the velocity ( )f η of the fluid is obtained, the pressure term can be determined

from equation (4.8) as given below:

( ) ( )

( )

2

2 2 22

1(1 )

2

k f fP f

k k k

k k kf M f f f f f

k k k

ηβ η η

η η η

′′ ′+ ′′′= + + − + +

′ ′ ′′ ′− − + ++ + +

, (4.11)

On the other hand, the corresponding boundary conditions at the surface and far from

the sheet become:

(0) 0, (0) 1, (0) 1,

( ) 0, ( ) 0, ( ) 0.

f f

f f

θθ

′= = =

′ ′′∞ = ∞ = ∞ = (4.12)

From engineering point of view, the physical quantities here are the local Nusselt

number in the s -direction and the skin friction coefficient which may be respectively

described, mathematically, as

2rs

fw

Cu

τρ

= , 1( )

ws

w

sqNu

k T T∞

=−

, (4.13)

These expressions have the shear stress at the wall as rsτ , and the thermal flux at. the

wall as wq which are specified as

1(1 )( )rs

u u

r r Rτ µ

β∂

= + −∂ +

at 0r =

1w

Tq k

r

∂=−

∂at 0r = (4.14)

64

Equation (4.13), after using equations (4.6) and (4.14), may be written as

12 1 (0)

Re (1 ) (0)f s

fC f

kβ′ ′′= + − +

,

1

2Re (0)s sNu θ−

′= − .

The local Reynolds number is 2

.es

asR

υ= .


The governing equations (4.9) and (4.10), on applying Quasi-linearization

(please see chapter 2), take the form:

21 1 12

1 1 1Pr 1 (1 ) Pr

1(1 )

j j jjk Mf Ec f

k kθ θ

η η ββ

+ + ++

′′ ′ ′= − − + − + + + +

(4.15)

( )

( )

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 1

1

1

22 2

23 2 3

2 3

1 1 11 2 1

1 11

1 1 1 21

j j

j

j

j j

iv j

j j

j j j

j

kf f f

k k

k kM f f f

kk k

k k kM f f f f

k kk k k

k k kf f f

k k k

β β η η

β ηη η

β η ηη η η

η η η

+ +

+

+

′′′+ + + + + +

′ ′′+ − + − − + ++ +

′ ′′ ′+ + − − − − + ++ + +

′′′ ′′ ′+ + −

+ + + ( )

( )( )

( )

1

2 3 0

j j j

j j j

j j

j j

kf f f f f

k

k kf f f f f

k k

η

η η

+ ′′′ ′ ′′= −+

′′ ′ ′+ − − =+ +

(4.16)

The approximations of central difference, when utilized for the derivatives in the

equations given above, yield the following systems of algebraic equations which may

then be solved by using some standard method for matrix inversion.

( ) ( ) ( )1 1

2

2 21 1

1 12 4 2

1Pr 1 2

2

j j ji i i

k ki i

k kh h

k k k k

f fh M Ec

θ θ θη η η η

β

− +

+ −

− − − + + − + + + +

− = + −

(4.17)

65

( ) ( 1) ( 1)2 1

( ) ( 1) 21

( ) ( )2 1 1

2 2

1 1 1 12 1 2 1 8 1

1 12 2 1 2

1 1 11 1

( ) ( ) 2 ( )

j j ji i i

j ji i

j ji i

i

kh f f f

k k

kh f f h

k k

f fk kM f

k k h k

β β η η β

β η η

β η β η η

+ +− −

+−

+ −

+ − + + − + + +

+ + + + + +

−− + − + − + + + +

( ) ( 1) 31

( ) ( ) ( ) ( ) ( )1 1 1 1

3 2 2

( 1)1( ) ( )

21 13

( 1) 2

21 21

( ) ( ) ( ) 2

1

( ) 2

112 1 4

11

j ji

j j j j ji i i i i

jij j

i i

ji

f h

f f f f fk k k

k k h k hf

f fkM

k h k

f h

β η η η

η η

β

+−

+ − + −

+−

+ −

+

−

− + −+ − − + + +

− − − + +

+ + −

− +( ) ( )

2 ( ) ( 1)1 12 2

( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1 2 1 1

3 2 2

4

( ) ( )1 1

3

1 11

( ) ( ) 2 ( )

2 2 2

( ) 2 ( )2

( )

j jj ji i

i i

j j j j j j ji i i i i i i

j ji i

f fk kM f f

k k h k

f f f f f f fk k

k h k hh

f fk

k

β η β η η

η η

η

++ −

+ + − − + −

+ −

− − + − + + + +

− + − − ++ + +

+−

−+

( 1)

( )

( 1) ( ) ( 1) 21 1

( ) ( )2 ( ) ( 1) 31 1

12 2

2

1 1 18 1 2 2 1 2

( ) ( )

1 11

( ) ( ) 2 ( )

11

ji

ji

j j ji i i

j jj ji i

i i

f

fh

kf h f f h

k k

f fk kM f f h

k k h k

β β η η

β η η η

β

+

+ ++ +

++ −+

− + − + + − + +

− − + − − + + + + +

+

( ) ( ) ( ) ( ) ( )1 1 1 1

3 2 2

( 1)1( ) ( )

21 13

( )

2 2

( ) ( ) ( ) 2

1

( ) 2 ( )

1 1 12 1 2 1

( ) ( )

j j j j ji i i i i

jij j

i i

ji i

f f f f fk k k

k k h k hf

f fkM

k h k

kh f f

k k

η η η

η η

β β η η

+ − + −

++

+ −

− + −− − + + +

− − − + +

+ + + + + + +

( 1)2 ( )j k

hkη

++ =

+

(4.18)

( ) ( )( )

( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1 2 1 1 1 1 2

22( ) ( ) ( ) ( ) ( ) ( ) ( )

2 1 1 2 1 1

3 ( ) ( ) ( )1 13

2 2 2( )

2 22

( )

j j j j j j j j j ji i i i i i i i i i

j j j j j j ji i i i i i i

j j ji i i

kf f f f f f f f f f

k

hh f f f f f f f

kh f f f

k

η

η

+ + − − + − + −

+ + − − + −

+ −

− + − − − − + + +

− + − − − −

− +

66


For understanding impact of various parameters to govern the coefficient of

skin-friction & heat transfer, numerical solution of the system of ODE’s (4.9) & (4.10)

along with the boundary conditions (4.12) has been obtained by employing the

numerical procedure mentioned above, for the various values of the governing

parameters (M, β , k , Pr, and Ec).

Table 4.1 shows the 1

2Res fC for the various values of the β . It is easy to see that, by

increasing the values of β , the magnitude of 1

2Res fC may decrease while increasing

the heat transfer at the curved sheet. It may be noted from the Table 4.2 that the

external magnetic field raises the skin-friction coefficient while lowering the heat

transfer at the sheet. Dependence of the heat transfer over the Prandtl number and the

Eckert number may be observed from the Tables 4.3 and 4.4. Obviously, both the

numbers tend to raise the heat transfer rate at the curved sheet. Table 4.5 shows how

the dimensionless curvature κ affects the skin-friction and the heat transfer. Clearly,

the parameter remarkably decreases both the physical quantities. However, the

dependence of the quantities on κ is much stronger when κ is nearer to unity.

We observe from the Figures 4.2 and 4.3 that, by raising values of the β , the fluid

velocity distribution decreases. With an increase in the β , the fluid yields stress

causing a resistance force which slows down the fluid motion. The temperature

profiles for different increasing values of the β for the curved sheet are shown in Fig

4.4. This figure shows that the Casson nature of the fluid lowers the fluid temperature

to such an extent that it goes beyond the temperature of the ambient fluid. Figures 4.5

and 4.6 show that M has the decreasing effect on both the velocity components while

raising its temperature as shown in Fig 4.7. It is because of the fact that the imposed

magnetic field acts normal to the direction of the flow, and thus induces the drag in

terms of the Lorentz force which retards the flow while increasing its temperature.

Hence, the external magnetic field bears the potential to enhance the temperature of

the fluid as it may be compared with literature work of Misra & Sinha [108]. Figure

4.8 reflects the influence of Pr on dimensionless temperature. As expected, the

increasing Pr leads to the reduction in the dimensionless temperature. As Pr raises, the

heat diffusivity reduces which leads to the decrease of energy ability that decreases

the thermal boundary layer (Ali et al. [109]). The effect of the Eckert number Ec on

67

the (0)θ ′− is shown in Fig. 4.9. It is clear that the thermal profiles decrease with an

increase in the viscous dissipation (quantified by the Eckert number).

Figures 4.10 and 4.11 depict the distribution of the velocity for different

values of the curvature parameter K. Here the fluid velocity is enhanced as K is

increased. In fact for higher values of curvature parameter, the stretching strength of

the sheet increases (for the fixed radius of the sheet), which enhances the fluid flow.

Fig. 4.12 shows that the dimensionless curvature parameter K has the ability to

balance the tremendous lowering of the thermal distribution due to the Casson nature

of the fluid with larger Pr.

4.5. CONCLUSIONS

The MHD flow and the (0)θ ′− of a Casson fluid passes through the curved sheet

coiled in a circle, subject to the governing parameters (namely M, β , k , Pr, and Ec)

is analysed here. Main findings of the present study can be summarized as:

1) The Casson parameter reduces the velocity as well as the temperature

distribution.

2) The magnetic parameter decreases both the velocity components while raising

the fluid temperature.

3) Effect of both the Prandtl number and the Eckert number is to raise the

temperature profile.

4) The curvature parameter may act as a neutralising agent to balance the

remarkable lowering of the temperature profile, due to the Casson nature of

the fluid at larger Prandtl number.

68

Table 4.1. Variation of 1

2Res fC and (0)θ ′− for 0.5M = , 7κ = , Pr 21= , and 1Ec =

against β .

Sr. No. β 1

2Res fC (0)θ ′−

1 0.2 -3.8400 3.4658

2 0.6 -2.3165 5.2456

3 0.9 -2.0183 5.9439

4 1.2 -1.8594 6.4161

5 1.4 -1.7888 6.6554

Table 4.2. Variation of1

2Res fC and (0)θ ′− for 2β = , 7κ = , Pr 21= and 1Ec =

against M .

Sr. No. M 1


1 0 -1.4748 8.2657

2 0.6 -1.7231 7.4712

3 1.2 -2.2397 5.5307

4 1.8 -2.8496 3.1137

5 2.4 -3.4949 0.4836

69

Table 4.3. Variation of (0)θ ′− for 2β = , 7κ = , 0.5M = and 1Ec = against Pr.

Sr. No Pr (0)θ ′−

1 6 3.2013

2 10 4.5714

3 14 5.7896

4 18 6.9118

5 22 7.9632

Table 4.4. Variation of (0)θ ′− for 2β = , 7κ = , 0.5M = and Pr 21= against Ec.

Sr. No Ec (0)θ ′−

1 0.4 5.4718

2 0.8 6.9613

3 1.2 8.4509

4 1.6 9.9404

5 2 11.4300

70

Table 4.5. Variation of1

2Res fC and (0)θ ′− for 0.5M = , 2β = ,Pr 21= and 1Ec =

against κ .

Sr. No. κ 1


1 1.0 -17.2013 90.9828

2 1.6 -3.8201 14.0949

3 2.2 -2.6712 9.9392

4 2.8 -2.2621 8.6969

5 3.4 -2.0513 8.1358

71

Figure 4.1. Geometry of the curved surface.

72

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

η

f

( η )

β = 0.2

β = 0.6

β = 0.9

β = 1.2

β = 1.4

Figure 4.2. Axial velocity profiles for 7κ = , 0.5M = ,Pr 21= , 1Ec = .

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f '

( η )

β = 0.2

β = 0.6

β = 0.9

β = 1.2

β = 1.4

Figure 4.3. Radial velocity profiles for 7κ = , 0.5M = ,Pr 21= , 1Ec = .

73

0 0.5 1 1.5 2 2.5 3 3.5 4-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

η

θ (

η )

β = 0.2

β = 0.6

β = 0.9

β = 1.2

β = 1.4

Figure 4.4. Temperature profiles for 7κ = , 0.5M = ,Pr 21= , 1Ec = .

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f

( η )

M = 0

M = 0.6

M = 1.2M = 1.8

M = 2.4

Figure 4.5. Axial velocity profiles for 7κ = , 2β = ,Pr 21= , 1Ec = .

74

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η

f '

( η )

M = 0

M = 0.6

M = 1.2M = 1.8

M = 2.4

Figure 4.6. Radial velocity profiles for 7κ = , 2β = ,Pr 21= , 1Ec = .

0 0.5 1 1.5 2 2.5 3 3.5 4-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

η

θ (

η )

M = 0

M = 0.6

M = 1.2M = 1.8

M = 2.4

Figure 4.7. Temperature profiles for 7κ = , 2β = ,Pr 21= , 1Ec = .

75

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

η

θ (

η )

Pr = 6

Pr = 10

Pr = 14Pr = 18

Pr = 22

Figure 4.8. Temperature profiles for 7κ = , 2β = , 0.5M = , 1Ec = .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

η

θ (

η )

Ec = 0.4

Ec = 0.8

Ec = 1.2Ec = 1.6

Ec = 2

Figure 4.9. Temperature profiles for 7κ = , 2β = , 0.5M = ,Pr 21= .

76

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

η

f

( η )

κ = 1.0

κ = 1.6

κ = 2.2

κ = 2.8

κ = 3.4

Figure 4.10. Axial velocity profiles for 0.5M = , 2β = ,Pr 21= , 1Ec = .

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

f '

( η )

κ = 1.0

κ = 1.6

κ = 2.2

κ = 2.8

κ = 3.4

Figure 4.11. Radial velocity profiles for 0.5M = , 2β = ,Pr 21= , 1Ec = .

77

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

η

θ (

η )

κ = 1.0

κ = 1.6

κ = 2.2

κ = 2.8

κ = 3.4

Figure 4.12. Temperature profiles for 0.5M = , 2β = ,Pr 21= , 1Ec = .

78

CHAPTER 5

ON THE SUITABILITY OF ROSSELAND

APPROXIMATION FOR THERMAL RADIATION IN

FLOW OVER A ROTATING CONE EMBEDDED IN

POROUS MEDIUM

79

5.1. INTRODUCTION

Flow, heat and mass transfer of MHD Casson fluid over a vertical rotating cone

through porous medium containing gyrotactic microorganisms in the presence of

Soret and Dufour effects are investigated numerically. A comparative study has been

taken into account between linear and nonlinear approaches of the Rosseland

approximation for the thermal radiations. We have noticed that the non linear form of

thermal radiation gives qualitatively right and quantitatively accurate results for the

mass transfer over the cone, under different values of the Soret parameter.


Consider two-dimensional, incompressible and steady boundary layer flow of a

Casson fluid over a vertically rotating cone with angular velocity Ω through porous

medium as shown in Fig. 5.1. Magnetic field having strength 0B is applied along z-

direction. The induced magnetic field is ignored. Effects of the Soret and Dufour are

considered. The material of the cone is assumed to be non-conducting. The equations

of the model under consideration may be written as under ([110]).

( ) ( )0

ru rw

x z

∂ ∂+ =

∂ ∂, (5.1)

2 22 1

2

20

1( ( )) (1 )

( ( ) ( ))cos ( / ) ,e T C

u u v uu w

x z x z

g T T C C u K u

δ ρ δ µβ

ρ β β α σβ µ

− −

∞ ∞

∂ ∂ ∂+ − = + +

∂ ∂ ∂

− + − − −

(5.2)

22 1 2

02

1( ( / )) (1 ) ( / ) ,

v v vu w uv x v K v

x z zδ ρ δ µ σβ µ

β− −∂ ∂ ∂

+ + = + − −∂ ∂ ∂

(5.3)

2 2 2 41

s2 2 21

41( / ) ( / c c ) ( )

3e p m T pp

T T T C Tu w k c D k

x z z z c K z

σρ

ρ∂ ∂ ∂ ∂ ∂

+ = + +∂ ∂ ∂ ∂ ∂

(5.4)

2 2

2 2( / )m m T m

C C C Tu w D D k T

x z z z

∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂ (5.5)

2

2( )c

n

bWN N N Cu w D N

x z z C z z

∂ ∂ ∂ ∂ ∂+ = −

∂ ∂ ∂ ∆ ∂ ∂ (5.6)

with the boundary conditions

0, , 0, , , 0

0, 0, 0, , ,

w w wu v r w T T C C N N at z

u v w T T C C N N as z∞ ∞ ∞

= = Ω = = = = = = = = = = = →∞

, (5.7)

80

where 1α = represents the case for the flow over a cone, µ is the viscosity

(dynamic), δ is parameter of the porosity, β is the Casson parameter, Tβ is the

volumetric expansion due to temperature differences, eg is the acceleration due to

gravity, ρ is the density of the fluid, Cβ are the volumetric expansion due to

concentration differences, σ is the electrical conductivity,K is the permeability of

the porous medium and α is the cone half angle, T is temperature of the fluid, wT is

the fluid temperature near the surface and T∞ is the fluid temperature far away, ek is

the effective thermal conductivity of the fluid, pc is the specific heat at constant

pressure, sc is the concentration susceptibility, Tk is the thermal diffusion ratio, mD

is the coefficient of mass diffusivity,N is the number density of motile

microorganisms, &wN N∞ are the respective uniform concentration of the

microorganisms near the surface and for away from the surface, nD is the diffusivity

of the microorganisms, C is the concentration of the fluid and &wC C∞ are the

concentrations near the wall and ambient respectively.

It is to mention that the last term in the equation 5.4 stands for the thermal

radiation based on the Rosseland approximation, with 1K and 1σ being the mean

absorption coefficient and Stefan-Boltzmann constant respectively. From now

onwards, there are two distinct (linear and nonlinear) approaches to handle the

thermal radiation term in the above equation.

5.2.1 Linear Form of Thermal Radiation

We assume the differences in temperature within the flow are such that 4T can be

expressed as a linear combination of the temperature. We expand 4T in Taylor’s

series about T∞ by neglecting the higher order terms as,

4 3 44 3T T T T∞ ∞= − (5.8)

Thus we have

2 4 23

2 24

T TT

z z∞

∂ ∂=

∂ ∂. (5.9)

Which turn Eq. (5.4) in to:

2 231

2 21 s

161( )

3 c ce m T

p p p

k D kT T T Cu w T

x z c c K z z

σρ ρ ∞

∂ ∂ ∂ ∂+ = + +

∂ ∂ ∂ ∂ (5.10)

81

5.2.2 Nonlinear Form of Thermal Radiation

For Non-linear Rosseland approximation we take

2 4 23 2 2

2 24 12 ( )

T T TT T

z z z

∂ ∂ ∂= +

∂ ∂ ∂ (5.11)

This yields:

2 23 2 21 1

2 21 s 1

16 481 1( ) ( ) ( ) ( )

3 c c 3e m T

p p p p

k D kT T T C Tu w T T

x z c c K z z c K z

σ σρ ρ ρ

∂ ∂ ∂ ∂ ∂ + = + + + ∂ ∂ ∂ ∂ ∂ (5.12)

We define the following similarity transformations

1 1

2 2

1

2

( sin ) , sin ( ), sin ( ), sin ,

( sin ) ( ), ( ) , ( ) , ( ) ,w w w

z u x f v x g r x

T T C C N Nw h

T T C C N N

η ν α α η α η α

ν α η θ η φ η ξ η

−

∞ ∞ ∞

∞ ∞ ∞

= Ω = Ω = Ω =

− − −= Ω = = =

− − −

(5.13)

By employing Eq. (5.13), into the equations (5.1)-(5.3), we get

1.

2f h′= − (5.14)

1 2 1

2 2 2

1(1 ) ( )

1( 2 ) 2 ( )cos 0,

2

h hh M Da h

h g

δ δβ

δ θ φ α

− − −

−

′′′ ′′ ′− + + + + +

′− + + Λ +Γ =

(5.15)

1 2 11(1 ) ( ) ( ) 0,g hg h g M Da gδ δ

β− − −′′ ′ ′+ − − − + = (5.16)

where Γ is the buoyancy ratio such that 0Γ < corresponds to opposing flow while

0Γ > corresponds to aiding flow, M is the magnetic parameter, ReL is the Reynolds

number, 2Ha is the Hartmann number, LGr is the Grashof number, Λ is the

buoyancy parameter, which are defined as

2 22 21 2 0

3

2 2

sin, , , Re ,

sin Re

( ) ( ), ,

Re ( )

LL

e T w C wLL

L T w

B LHa LDa M Ha

K

g T T L C CGrGr

T T

σν αα µ ν

β βν β

−

∞ ∞

∞

Ω= = = =

Ω

− −Λ = = Γ =

−

(5.17)

Equations (5.10) and (5.12), in view of Equation (5.13), reduce to

4Pr (1 ) Pr

3 rh N Duθ θ φ′ ′′ ′′= + + (5.18)

82

3 24(1 (1 ) ) (4 (1 ) Pr ) Pr 0

3 r rN N h Duεθ θ εθ εθ θ φ′′ ′ ′ ′′+ + + + − + = (5.19)

where Pr e

p

K

Cµ= is the Prandtl number,

s

( )

c c ( )m T L

p L

D k C CDu

T Tν∞

∞

−=

− is the Dufour number

and w

Twhere T T T

Tε ∞

∞

∆= ∆ = − is the temperature ratio parameter.

Using Eq. (5.13), in Eqs. (5.5) and (5.6) to obtain

1h Sr

Scφ φ θ′ ′′ ′′= + (5.20)

( ) ( ) 0Pe Sc h Pe Pe bξ φ ξ ξ σ φ′′ ′ ′ ′′− + − + = (5.21)

The corresponding boundary conditions are

0, 1, 1, 1, 1, 0

0, 0, 0, 0, 0,

h g at

h g as

θ φ ξ ηθ φ ξ η

′ = = = = = = ′ = = = = = →∞

(5.22)

where Sc is the Schmidt number, Sr is the Soret number, Pe is the Peclet number

and bσ is the dimensionless parameter, which are given by

m

ScD

ν= ,

( ),

( )m T w

m w

D k C CSr

T T Tν∞

∞

−=

− ,c

n

bWPe

D=

Nb

Nσ ∞=

∆ (5.23)

where fluid mean temperature is mT and cW is the maximum cell swimming speed

(the product cbW is assumed to be constant).

For physical quantities of interest, the friction factors along x and y directions, local

Nusselt, Sherwood numbers and the local density of motile micro organisms are given

by

1/2 1 1/21 1Re (1 ) (0), 2 Re (1 ) (0).fx fyC h C g

β β−′′ ′= − + = − + (5.24)

1/2 1/2 1/2Re (0), Re (0), Re (0)x x xNu Sh Nnθ φ ξ− − −′ ′ ′= − = − = − (5.25)

Where 2 sin

Rex α

υΩ

= , is the Reynolds number.


Numerical solution of the problem over the semi-infinite domain has been obtained by

following the domain truncation approach whereas the derivatives are discretized by

using the second order central differences. The details of the solution methodology

may be found in [111].


83

For numerical computations, we considered the non dimensional parametric values as

0.3β = , 3δ = , 1M = , 10A = , 1T = , 1α = , Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr =

0.5ε = , 1Pe = and 0.2bσ = .These values are as common in entire study except the

variation in respective figures and tables. In graphical results ( )f η is the tangential

velocity, ( )g η is the circumferential velocity, ( )h η is the normal velocity , ( )θ η is the

temperature,( )φ η is the concentration and ( )ζ η is the concentration of motile micro

organisms.

As mentioned in the previous section, the numerical results have been

obtained by considering the linear and the nonlinear approaches for the thermal

radiation. Before we give a description of our tabular results, it is important to

mention that the results obtained from employing the two approaches are qualitatively

same but differ quantitatively. Since the nonlinear radiation term is more accurate (as

it involves no truncation of any series), we will give the interpretation of the results

obtained without any linearization.

It is obvious from the Tables 5.1(a, b) that the magnetic field reduces the skin friction

in x-direction while raising it in the y-direction. On the other hand, the rest of the

physical quantities are decreasing with M. From the Tables 5.2 (a, b), it is interesting

to note that all the quantities of interest for the present study are supported by the

porosity parameter. It is noticed from the Tables 5.3 (a, b) that mass transfer decreases

when linearized model of thermal radiation is employed. The trend is, however,

entirely reversed if the linear form is not assumed. From the Tables 5.4 (a, b), we note

that Dufour effect quantified by the parameter Du has the ability to change the

direction of heat transfer at the cone (which is obvious due to a change in the sign of

(0)θ ′− . By comparing the numerical results of the Tables 5.5 (a, b), 5.6 (a, b) with

those of Tables 5.2 (a, b), it is obvious that the buoyancy, the porosity and the non-

Newtonian nature of the working fluid have similar effects on the physical quantities

considered in the present work. Figures 5.2(a, b)-5.7(a, b) depict the effect of

magnetic field parameter on momentum, thermal and concentration profiles of the

flow. It is obvious from Figs. 5.2(a, b)-5.5(a, b) that an increase in the magnetic field

parameter depreciates ( )f η , ( )g η as well as ( )h η , while increasing the temperature

distribution. Hence it shows that an increase in the magnetic field provides the drag

force which highly influences the velocity of the fluid. From Figs. 5.6(a, b) and 5.7(a,

b), it is seen that the two concentration profiles are boosted by the magnetic field.

84

Figs. 5.8(a, b)-5.13(a, b) illustrate the effect of the parameter of porosity on the

velocity, the temperature and the concentration profiles. It is seen that an increase in

the porosity parameter suppresses the momentum boundary layer while reducing the

rotation of the fluid. However, an increase in the tangential velocity is noticed near

the surface because of an increase in porosity parameter. It is clear from Figs. 5.11(a,

b)-5.13(a, b) that an increase in the porosity parameter decreases the temperature and

the concentration profiles of the fluid flow.

A boost in the concentration profiles with an increment in the Soret parameter has

been noticed from Figs. 5.15(a, b), which justifies that the Soret parameter has an

enhancing tendency to the concentration. On the other hand, a relatively small change

in the normal velocity has been observed in a region away from the rotating cone

(please see Fig. 5.14(a, b)). By increasing the Dufour number, a significant increase in

the tangential and normal velocities has been observed, which may be seen in Figs.

5.16(a, b) and 5.17(a, b). Further, no significant change is noted in concentration

distributions in this case. It is, however, the temperature profile which is remarkably

influenced (Fig. 5.18(a, b)). Figures 5.19(a, b)-5.24(a, b) represent the buoyancy

parameter effect on the velocity, temperature and concentration fields. By increasing

the buoyancy parameter an increase in the tangential velocity and the normal velocity

profiles is observed, whereas the circumferential velocity predicts a decreasing trend.

The buoyancy parameter, on the other hand, reduces the thermal as well as the

concentration distributions. Figures 5.25(a, b)-5.30(a, b) depict the effected velocity

temperature and concentration profiles due to Casson parameter. It is apparent that an

increment in the Casson parameter provides a hike near the rotating surface in the

tangential velocity field. Afterwards, a reverse action is taken at the free stream. A fall

is noticed in the circumferential and the normal velocities along with the temperature

and the concentration profiles.

5.5. CONCLUSIONS

The solutions provided in this study are related to the flow, heat and mass transfer of

MHD Casson fluid over a cone (vertical rotating) in the presence of Soret and Dufour

effects in the porous medium. This numerical study shows that the non-dimensional

governing parameters influence the fluid flow and the transfer of heat and mass.

It is noted that:

1) Magnetic field enhances the heat and mass transfer rates.

85

2) Soret parameter increases the concentration while Dufour parameter enhances

the velocity and heat transfer rate.

3) An increase in the buoyancy parameter raises the tangential velocity and

normal velocity profiles while reducing the thermal as well as the

concentration distributions.

4) An increase in the Casson parameters causes a hike in the tangential velocity

field near the rotating surface of the cone.

5) The non linear form of the thermal radiation gives qualitatively right and

quantitatively accurate results for the mass transfer over the cone, under

different values of the Soret parameter.

86

Table 5.1a Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ ′− and (0)ξ ′− for 0.3β = , 3δ = ,

10A = , 1T = , 1α = ,Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and

0.2bσ = against M in case of linear thermal radiation.

Sr. No. M (0)h′′− (0)g ′− (0)θ ′− (0)φ ′− (0)ξ ′−

1 3 -9.2582 -1.4948 -0.4064 -1.0359 -1.8612

2 6 -7.8315 -2.0464 -0.3395 -0.9312 -1.6843

3 9 -6.9181 -2.4752 -0.2963 -0.8566 -1.5574

4 12 -6.2607 -2.8353 -0.2655 -0.7987 -1.4583

5 15 -5.7558 -3.1496 -0.2422 -0.7517 -1.3772

Table 5.1b Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ ′− and (0)ξ ′− for 0.3β = , 3δ = ,


0.2bσ = against M in case of Non linear thermal radiation.

Sr. No. M (0)h′′− (0)g ′− (0)θ ′− (0)φ ′− (0)ξ ′−

1 3 -9.6567 -1.5010 -0.2644 -1.0746 -1.9193

2 6 -8.0795 -2.0541 -0.2186 -0.9619 -1.7293

3 9 -7.0977 -2.4873 -0.1894 -0.8825 -1.5945

4 12 -6.4047 -2.8528 -0.1688 -0.8214 -1.4903

5 15 -5.8795 -3.1732 -0.1534 -0.7722 -1.4057

87

Table 5.2a Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ ′− and (0)ξ ′− for 0.3β = , 1M = ,


0.2bσ = against δ in case of linear thermal radiation.

Sr. No. δ (0)h′′− (0)g ′− (0)θ ′− (0)φ ′− (0)ξ ′−

1 2 -8.1765 -0.8879 -0.4349 -1.0370 -1.8526

2 4 -13.3589 -1.0692 -0.5131 -1.2215 -2.1808

3 6 -17.6631 -1.2448 -0.5577 -1.3366 -2.3867

4 8 -21.4583 -1.4047 -0.5887 -1.4217 -2.5394

5 10 -24.9058 -1.5506 -0.6124 -1.4898 -2.6617

Table 5.2b Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ ′− and (0)ξ ′− for 0.3β = , 1M = ,


0.2bσ = against δ in case of Non linear thermal radiation.

Sr. No. δ (0)h′′− (0)g ′− (0)θ ′− (0)φ′− (0)ξ ′−

1 2 -8.62711 -0.9041 -0.2842 -1.0789 -1.9168

2 4 -14.1158 -1.0796 -0.3368 -1.2715 -2.2577

3 6 -18.6549 -1.2525 -0.3663 -1.3909 -2.4700

4 8 -22.6464 -1.4113 -0.3866 -1.4787 -2.6266

5 10 -26.2660 -1.5568 -0.4020 -1.5486 -2.7517

88

Table 5.3a Variation of (0)h′′− and (0)φ′− for 0.3β = , 3δ = , 1M = , 10A = , 1T = ,

1α = ,Pr 0.7= , 1Du = , 1Sc = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against .Sr in case

of linear thermal radiation.

Sr. No. .Sr (0)h′′− (0)φ′−

1 0.0 -10.9535 -1.1478

2 0.5 -11.0986 -1.1126

3 1 -11.2502 -1.0727

4 1.5 -11.4100 -1.0265

5 2 -11.5803 -0.9713

Table 5.3b Variation of (0)h′′− and (0)φ′− for 0.3β = , 3δ = , 1M = , 10A = , 1T = ,

1α = ,Pr 0.7= , 1Du = , 1Sc = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against Sr in case

of Non linear thermal radiation.

Sr. No. .Sr (0)h′′− (0)φ′−

1 0.0 -11.5572 -1.1871

2 0.5 -11.6398 -1.1963

3 1 -11.7208 -1.2096

4 1.5 -11.8000 -1.2278

5 2 -11.8770 -1.2519

89

Table 5.4a Variation of (0)h′′− and (0)θ ′− for 0.3β = , 3δ = , 1M = , 10A = , 1T = ,

1α = ,Pr 0.7= , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against Du in case

of linear thermal radiation.

Sr. No. Du (0)h′′− (0)θ ′−

1 1 -10.9181 -0.4810

2 2 -11.4008 -0.2905

3 3 -11.8883 -0.0848

4 4 -12.3810 0.1362

5 5 -12.8792 0.3734

Table 5.4b Variation of (0)h′′− and (0)θ ′− , for 0.3β = , 3δ = , 1M = , 10A = , 1T = ,

1α = ,Pr 0.7= , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against Du in case

of Non linear thermal radiation.

Sr. No. Du (0)h′′− (0)θ ′−

1 1 -11.5343 -0.3153

2 2 -11.8589 -0.2234

3 3 -12.1764 -0.1268

4 4 -12.4870 -0.0256

5 5 -12.7913 0.0800

90

Table 5.5a Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ′− and (0)ξ ′− for 0.3β = , 3δ = ,

1M = , 1T = , 1α = ,Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and

0.2bσ = against A in case of linear thermal radiation.

Sr. No. A (0)h′′− (0)g ′− (0)θ ′− (0)φ′− (0)ξ ′−

1 5 -4.5187 -0.9053 -0.3419 -0.8350 -1.4957

2 10 -7.9006 -0.9458 -0.4253 -1.0195 -1.8224

3 15 -10.9181 -0.9771 -0.4810 -1.1431 -2.0410

4 20 -13.7140 -1.0033 -0.5238 -1.2387 -2.2099

5 25 -16.3528 -1.0262 -0.5591 -1.3177 -2.3495

Table 5.5b Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ′− and (0)ξ ′− for 0.3β = , 3δ = ,

1M = , 1T = , 1α = ,Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and

0.2bσ = against A in case of Non linear thermal radiation.

Sr. No. A (0)h′′− (0)g ′− (0)θ ′− (0)φ′− (0)ξ ′−

1 5 -4.7416 -0.9117 -0.2234 -0.8683 -1.5463

2 10 -8.32786 -0.9559 -0.2785 -1.0609 -1.8859

3 15 -11.5343 -0.9899 -0.3153 -1.1898 -2.1129

4 20 -14.5089 -1.0184 -0.3436 -1.2894 -2.2881

5 25 -17.3188 -1.0432 -0.3669 -1.3716 -2.4327

91

Table 5.6a Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ′− and (0)ξ ′− for

3δ = , 1M = , 10A = , 1T = , 1α = , Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = ,

1Pe = and 0.2bσ = against β in case of linear thermal radiation.

Sr. No. β (0)h′′− (0)g ′− (0)θ ′− (0)φ′− (0)ξ ′−

1 0.1 -5.7734 -0.5907 -0.4092 -0.9385 -1.6723

2 0.2 -8.7562 -0.8192 -0.4559 -1.0683 -1.9059

3 0.4 -12.6037 -1.0971 -0.4972 -1.1940 -2.1333

4 0.6 -15.1082 -1.2715 -0.5173 -1.2609 -2.2547

5 0.8 -16.9038 -1.3943 -0.5294 -1.3038 -2.3328

Table 5.6b Variation of (0)h′′− , (0)g ′− , (0)θ ′− , (0)φ′− and (0)ξ ′− for

3δ = , 1M = , 10A = , 1T = , 1α = , Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = ,

1Pe = and 0.2bσ = against β in case of Non linear thermal radiation.

Sr. No. β (0)h′′− (0)g ′− (0)θ ′− (0)φ′− (0)ξ ′−

1 0.1 -6.1331 -0.5980 -0.2699 -0.9792 -1.7351

2 0.2 -9.2693 -0.8299 -0.2996 -1.1130 -1.9748

3 0.4 -13.2965 -1.1114 -0.3253 -1.2421 -2.2070

4 0.6 -15.9097 -1.2879 -0.3377 -1.3105 -2.3306

5 0.8 -17.7801 -1.4121 -0.3450 -1.3543 -2.4099

92

Figure No. 5.1 Physical configuration and coordinate system.

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

η

f

( η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(2a) Tangential velocity ( )f η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,

Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against various

M.

93

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

η

f

( η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No. (2b) Tangential velocity ( )f η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,


M.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(3a) Circumferential velocity ( )g η , for 0.3β = , 3δ = , 10Λ = , 1T = ,

1α = ,Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against

various M.

94

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(3b) Circumferential velocity ( )g η , for 0.3β = , 3δ = , 10Λ = , 1T = ,


various M.

0 1 2 3 4 5 6 7 8 9 10-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

η

h (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(4a) Normal velocity ( )h η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,


M.

95

0 1 2 3 4 5 6 7 8 9 10-6

-5

-4

-3

-2

-1

0

η

h

( η

)

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(4b) Normal velocity ( )h η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,


M.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(5a) Temperature ( )θ η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,


M.

96

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(5b) Temperature ( )θ η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,


M.

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(6a) Concentration ( )φ η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,


M.

97

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(6b) Concentration ( )φ η , for 0.3β = , 3δ = , 10Λ = , 1T = , 1α = ,


M.

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

η

ξ (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(7a) Concentration of motile micro organisms( )ζ η , for 0.3β = , 3δ = ,

10Λ = , 1T = , 1α = , Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and

0.2bσ = against various M.

98

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

η

ξ (

η )

M = 3

M = 6

M = 9M = 12

M = 15

Figure No.(7b) Concentration of motile micro organisms( )ζ η , for 0.3β = , 3δ = ,


0.2bσ = against various M.

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

η

f

( η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(8a) Tangential velocity ( )f η , for 0.3β = , 1M = , 10Λ = , 1T = ,


various δ .

99

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

3.5

η

f

( η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(8b) Tangential velocity ( )f η , for 0.3β = , 1M = , 10Λ = , 1T = , 1α = ,


δ .

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(9a) Circumferential velocity ( )g η , for 0.3β = , 1M = , 10Λ = , 1T = ,


various δ .

100

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(9b) Circumferential velocity ( )g η , for 0.3β = , 1M = , 10Λ = , 1T = ,


various δ .

0 1 2 3 4 5 6 7 8 9 10-7

-6

-5

-4

-3

-2

-1

0

η

h (

η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(10a) Normal velocity ( )h η , for 0.3β = , 1M = , 10Λ = , 1T = ,


various δ .

101

0 1 2 3 4 5 6 7 8 9 10-12

-10

-8

-6

-4

-2

0

η

h

( η

)

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(10b) Normal velocity ( )h η , for 0.3β = , 1M = , 10Λ = , 1T = , 1α = ,


δ .

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(11a) Temperature ( )θ η , for 0.3β = , 1M = , 10Λ = , 1T = , 1α = ,


δ .

102

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(11b) Temperature ( )θ η , for 0.3β = , 1M = , 10Λ = , 1T = , 1α = ,


δ .

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ

( η

)

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(12a) Concentration ( )φ η , for 0.3β = , 1M = , 10Λ = , 1T = , 1α = ,


δ .

103

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ

( η

)

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(12b) Concentration ( )φ η , for 0.3β = , 1M = , 10Λ = , 1T = , 1α = ,


δ .

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

η

ξ (

η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(13a) Concentration of motile micro organisms( )ζ η , for 0.3β = , 1M = ,


0.2bσ = against various δ .

104

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

η

ξ (

η )

δ = 2

δ = 4

δ = 6

δ = 8

δ = 10

Figure No.(13b) Concentration of motile micro organisms( )ζ η , for 0.3β = , 1M = ,


0.2bσ = against various δ .

0 1 2 3 4 5 6 7 8 9 10-8

-7

-6

-5

-4

-3

-2

-1

0

η

h (

η )

Sr = 0.2

Sr = 0.4

Sr = 0.6Sr = 0.8

Sr = 1

Figure No.(14a) Normal velocity ( )h η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,

1α = ,Pr 0.7= , 1Du = , 1Sc = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against various Sr .

105

0 1 2 3 4 5 6 7 8 9 10-9

-8

-7

-6

-5

-4

-3

-2

-1

0

η

h

( η

)

Sr = 0.2

Sr = 0.4

Sr = 0.6Sr = 0.8

Sr = 1

Figure No.(14b) Normal velocity ( )h η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ (

η )

Sr = 0.2

Sr = 0.4

Sr = 0.6Sr = 0.8

Sr = 1

Figure No.(15a) Concentration ( )φ η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


106

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ (

η )

Sr = 0.2

Sr = 0.4

Sr = 0.6Sr = 0.8

Sr = 1

Figure No.(15b) Concentration ( )φ η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

η

f

( η )

Du = 1

Du = 2

Du = 3Du = 4

Du = 5

Figure No.(16a) Tangential velocity ( )f η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,

1α = ,Pr 0.7= , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and 0.2bσ = against various

Du .

107

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

η

f

( η )

Du = 1

Du = 2

Du = 3Du = 4

Du = 5

Figure No.(16b) Tangential velocity ( )f η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


Du .

0 1 2 3 4 5 6 7 8 9 10-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

η

h (

η )

Du = 1

Du = 2

Du = 3Du = 4

Du = 5

Figure No.(17a) Normal velocity ( )h η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


Du .

108

0 1 2 3 4 5 6 7 8 9 10-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

η

h

( η

)

Du = 1

Du = 2

Du = 3Du = 4

Du = 5

Figure No.(17b) Normal velocity ( )h η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


Du .

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

η

θ (

η )

Du = 1

Du = 2

Du = 3Du = 4

Du = 5

Figure No.(18a) Temperature ( )θ η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


Du .

109

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

η

θ (

η )

Du = 1

Du = 2

Du = 3Du = 4

Du = 5

Figure No.(18b) Temperature ( )θ η , for 0.3β = , 3δ = , 1M = , 10Λ = , 1T = ,


Du .

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

η

f

( η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(19a) Tangential velocity ( )f η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

110

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

η

f

( η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(19b) Tangential velocity ( )f η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(20a) Circumferential velocity ( )g η , for 0.3β = , 3δ = , 1M = , 1T = ,


various Λ .

111

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(20b) Circumferential velocity ( )g η , for 0.3β = , 3δ = , 1M = , 1T = ,


various Λ .

0 1 2 3 4 5 6 7 8 9 10-9

-8

-7

-6

-5

-4

-3

-2

-1

0

η

h (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(21a) Normal velocity ( )h η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

112

0 1 2 3 4 5 6 7 8 9 10-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

η

h

( η

)

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(21b) Normal velocity ( )h η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(22a) Temperature ( )θ η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

113

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(22b) Temperature ( )θ η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(23a) Concentration ( )φ η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

114

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(23b) Concentration ( )φ η , for 0.3β = , 3δ = , 1M = , 1T = , 1α = ,


Λ .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ξ (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(24a) Concentration of motile micro organisms( )ζ η , for 0.3β = , 3δ = ,

1M = , 1T = , 1α = , Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and

0.2bσ = against various Λ .

115

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ξ (

η )

∧ = 5

∧ = 10

∧ = 15

∧ = 20

∧ = 25

Figure No.(24b) Concentration of motile micro organisms( )ζ η , for 0.3β = , 3δ = ,

1M = , 1T = , 1α = , Pr 0.7= , 1Du = , 1Sc = , 0.1Sr = , 1Nr = 0.5ε = , 1Pe = and

0.2bσ = against various Λ .

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

η

f

( η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(25a) Tangential velocity ( )f η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

116

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

η

f

( η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(25b) Tangential velocity ( )f η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(26a) Circumferential velocity ( )g η , for 3δ = , 1M = , 10Λ = , 1T = ,


various β .

117

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

g (

η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(26b) Circumferential velocity ( )g η , for 3δ = , 1M = , 10Λ = , 1T = ,


various β .

0 1 2 3 4 5 6 7 8 9 10-7

-6

-5

-4

-3

-2

-1

0

η

h (

η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(27a) Normal velocity ( )h η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

118

0 1 2 3 4 5 6 7 8 9 10-9

-8

-7

-6

-5

-4

-3

-2

-1

0

η

h

( η

)

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(27b) Normal velocity ( )h η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(28a) Temperature ( )θ η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

119

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

θ (

η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(28b) Temperature ( )θ η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ

( η

)

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.(29a) Concentration ( )φ η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

120

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

φ

( η

)

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No. 29(b) Concentration ( )φ η , for 3δ = , 1M = , 10Λ = , 1T = , 1α = ,


β .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ξ (

η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No.30(a) Concentration of motile micro organisms( )ζ η , for 3δ = , 1M = ,


0.2bσ = against various β .

121

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

η

ξ (

η )

β = 0.1

β = 0.2

β = 0.4

β = 0.6

β = 0.8

Figure No. 30(b) Concentration of motile micro organisms( )ζ η , for 3δ = , 1M = ,


0.2bσ = against various β .

122

POSSIBLE FUTURE WORK

In the chapter 2, we presented the numerical solution of the flow, heat and

concentration of an incompressible and electrically conducting nano-fluid

between the two permeable coaxial disks which move orthogonally, the nano-

particles 2TiO are inserted in the base fluid. The flow is laminar and unsteady.

The Joule heating and viscous dissipation are taken into account. The study

undertaken in the chapter 2 is intended to extend to the non-symmetric case

when one disk is porous while the other is non porous. We may further consider

the mass transfer of a chemically reactive species between the two disks. The

nano-fluid may be assumed to contain some chemically reactive species, and the

mixture thus formed is homogeneous. Moreover, 1st order homogeneous and

irreversible reaction may also assumed to be taking place in the fluid because of

chemical reactive nature of the species. The numerical investigation of the

problem of the flow, heat and concentration of an electrically conduction nano-

fluid through a semi-infinite open ended porous channel is carried out in

chapter 3 may be extended to the case of endothermic exothermic chemical

reactions. Physical characteristics of the momentum and thermal aspects of the

problem may be explored by analyzing the fluid velocity, temperature profile,

the skin-friction coefficient, couple wall stress and the local Nusselt number.

In chapter 4, we discussed the flow and heat transfer in 2-dimensional

steady and incompressible Casson fluid flow over a circular coiled curved

stretching sheet. The curved sheet is stretched with the fixed origin, due to the

action of two forces (equal in the magnitude but opposite in the direction). The

fluid has the property of electrical conduction and magnetic field. The magnetic

Reynolds number is assumed really small with the intention that the induced

magnetic field can be ignored. The numerical scheme used in this chapter may

be extended for the numerical solution of the complete Navier-Stokes equations,

to study the mixed convection in an enclosure with heat flux from some parts of

its vertical walls. We have already developed the computer code for the problem

123

in the MATLAB environment. As a sample output, the stream function and

temperature distribution for the problem (for certain values of the governing

parameters) are given below.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 1 Streamlines when only top lid is moving with Re = 500.

124

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2 Streamlines when both lids are moving in the same direction with Re = 500.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3 Streamlines when both lids are moving in the opposite direction with Re = 500.

125

In chapter 5, a comparative study has been carried out between linear and

nonlinear approaches of the Rosseland approximation for the thermal radiations

in the boundary layer flow of a Casson fluid over a vertically rotating cone.

We assumed the presence of an external magnetic field which was being

uniformly applied on the fluid, irrespective of its position w.r.t. the origin of the

coordinate system being used. In order to make the problem physically more

realistic, we intend to consider the spatially varying magnetic field. This will

remarkably alter the mathematical modeling of the problem, but the results thus

obtained will have more significant physical meanings. We have seen quite a

few references on this type of problems in different geometries.

126

LIST OF PUBLICATIONS

The contents of the thesis have led to the following papers which have been accepted / published

in international ISI indexed journals

1) Muhammad Farooq Iqbal, Kashif Ali and Muhammad Ashraf, 2014. Heat and mass

transfer analysis in unsteady titanium dioxide nanofluid between two orthogonally moving

porous coaxial disks-A numerical study. Canadian Journal of Physics (Canada), 93(3): 1-

10 (I.F=0.68).

2) Muhammad Farooq Iqbal, Shahzad Ahmad, Kashif Ali and Muhammad Ashraf, 2016.

Combined effect of heat and mass transfer in MHD flow of a nanofluid between

Expending/Contracting walls of a Porous Channel. Journal of Porous Media (USA), 19(9):

821-839 (I.F=1.05).

3) Muhammad Farooq Iqbal, Shazad Ahmad, Kashif Ali, M Zubair Akbar and M Ashraf,

2016. Heat and mass transfer analysis in unsteady nanofluid flow between moving disks with

chemical reaction-A numerical study. Heat Transfer Research (USA).

DOI: 10.1615/HeatTransRes.2018016244 (I.F=0.93)

4) Kashif Ali, Muhammad Zubair Akbar, Muhammad Farooq Iqbal and Muhammad Ashraf,

2014. Numerical simulation of heat and mass transfer in unsteady nanofluid between two

orthogonally moving porous coaxial disks. AIP Advances (USA), doi: 10.1063/1.4897947

(I.F=1.56).

5) Kashif Ali, Muhammad Farooq Iqbal, Muhammad Zubair Akbar and Muhammad Ashraf,

2014. Numerical simulation of unsteady water-based nanofluid flow and heat transfer

between two orthogonally moving porous coaxial disks. Journal of Theoretical and Applied

Mechanics (Poland), 52, 1033-1046 (I.F=0.68).

6) Muhammad Zubair Akbar, Muhammad Ashraf, Muhammad Farooq Iqbal, and Kashif Ali,

2016. Heat and mass transfer analysis of unsteady MHD nanofluid flow through a channel

with moving porous walls and medium. AIP Advances (USA) 6, doi.org/10.1063/1.4945440

(I.F=1.56).

127

7) M. Zubair Akbar Qureshi, Kashif Ali, Muhammad Farooq Iqbal, Muhammad Ashraf, and

Shazad Ahmad, 2017. Heat and mass transfer enhancement of nanofluids flow in thepresence

of metallic/metallic-oxides spherical nanoparticles. Eur. Phys. J. Plus, 132:57 (I.F=1.73).

8) M. Zubair Akbar Qureshi, Kashif Ali, Muhammad Farooq Iqbal and Muhammad Ashraf,

2017. Numerical modeling of non-newtonian fluid flow between porous disks in the presence

of nano particles. Nanoscience and Technology: AN International Journal (USA), 8(1), 67–

83 (I.F=0.0)

9) M Zubair Akbar, Kashif Ali, Muhammad Farooq Iqbal and M Ashraf, 2017. Heat and

mass transfer analysis of unsteady non-Newtonian fluid flow between porous surfaces in the

presence of magnetic nanoparticles. Journal of Porous Media (USA), 20(12):1137–1154

(I.F=1.05)

128

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