Flow and Heat Transfer Over Stretching and Shrinking...

Flow and Heat Transfer Over Stretching and

Shrinking Surfaces

By

Syed Muhammad Imran

CIIT/SP09-PMT-017/ISB

PhD Thesis

In

Mathematics

COMSATS Institute of Information Technology

Islamabad - Pakistan

Spring, 2013

ii

COMSATS Institute of Information Technology


Shrinking Surfaces

A Thesis Presented to

COMSATS Institute of Information Technology, Islamabad

In partial fulfillment

of the requirement for the degree of

PhD Mathematics

By

Syed Muhammad Imran


Spring, 2013

iii


Shrinking Surfaces

___________________________________________

A Post Graduate Thesis submitted to the Department of Mathematics as partial

fulfillment of the requirement for the award of Degree Ph D Mathematics

Name Registration Number

Syed Muhammad Imran


Supervisor

Prof. Dr. Saleem Asghar

Professor Department of Mathematics

Islamabad Campus

COMSATS Institute of Information Technology (CIIT)

Islamabad Campus

June, 2013

iv

Final Approval

________________________________________________

This thesis titled


Shrinking Surfaces

By

Syed Muhammad Imran


Has been approved

For the COMSATS Institute of Information Technology, Islamabad Campus

External Examiner 1: ________________________________________________

External Examiner 2: ________________________________________________

Supervisor: _________________________________________________________

Dr. Saleem Asghar

Professor CIIT, Islamabad

Head of Department: _________________________________________________

Dr. Moiz-ud-Din Khan


Chairperson Department: _______________________________________________

Prof. Dr. Tahira Haroon

Dean, Faculty of Science _______________________________________________

Prof. Dr. Arshad Saleem Bhatti

v

Declaration

I Syed Muhammad Imran, CIIT/SP09-PMT-017/ISB hereby declare that I have produced

the work presented in this thesis, during the scheduled period of study. I also declare that

I have not taken any material from any source except referred to wherever due that

amount of plagiarism is within acceptable range. If a violation of HEC rules on research

has occurred in this thesis, I shall be liable to punishable action under the plagiarism rules

of the HEC.

Date: _________________ Signature of the student:

____________________________

Syed Muhammad Imran


vi

Certificate

It is certified that Syed Muhammad Imran, CIIT/SP09-PMT-017/ISB has carried out all

the work related to this thesis under my supervision at the Department of Mathematics,

COMSATS Institute of Information Technology, Islamabad campus and the work fulfills

the requirement for award of PhD degree.

Date: _________________ Supervisor:

____________________________________

Dr. Saleem Asghar


Head of Department:

_____________________________

Prof. Dr. Moiz-ud-Din Khan

Department of Mathematics

vii

DEDICATED TO

MY PARENTS

viii

ACKNOWLEDGEMENTS

I start with the name of Almighty Allah, the most Beneficent and most Merciful, alone

without whose help we can do nothing. Who gave me strength, ability and courage to

fulfil requirement of the thesis. It is only His help and blessings that today I am able to

accomplish my PhD studies. I offer Darood pak to Holy Prophet Muhammad (peace be

upon Him) who showed the right path to the mankind. He is always a source of courage

and hope for me.

I express my gratitude to all my teachers whose teaching enabled me to reach at this

stage. Especially I am highly grateful to my kind natured, dedicated, sincere and

devoted supervisor Prof. Dr. Saleem Asghar, a great teacher and a great human, who

helped me with many stirring discussions and research. I am always inspired by his

dedication, his hard work and his style of research that would help me to conduct quality

research in the future as well. Without his generous help and guidance it was not possible

for me to complete this work.

I am grateful to Dr. S.M. Junaid Zaidi, the rector of COMSATS Institute of Information

Technology for providing access to facilities that ensured successful completion of my

work. I also wish to acknowledge HoD of Mathematics for providing good working

environment. Further, I am thankful to all my teachers especially Dr. Muhammad

Mushtaq for guidance and assistance during the research.

I want to extend my heartfelt thanks to my colleagues Dr. Dilnawaz Khan, Mr. Mudassar

Jalil, Mr. Muhammad Hussan, Mr. Aamir Ali and Dr. Adeel Ahmed. I am highly

indebted to Dr. Muhammad Saleem for his valuable discussion and guidance during

thesis writing.

Finally, it is a privilege for me to express my gratitude to my parents for their love and

great support. Very special thanks to my wife, one innocent daughter and two naughty

sons, who sacrificed during my studies, prayed for me and managed to support me.

Syed Muhammad Imran CIIT/SP09-PMT-017/ISB

ix

ABSTRACT

Flow and Heat Transfer Over Stretching and Shrinking Surfaces

Finding the numerical solutions of ordinary and partial differential equations of nonlinear nature

has become somewhat possible, during the last few decades, due to the evolution of efficient

computing. However, the governing equations for fluid flow are difficult to address in terms of

finding analytical solutions. This difficulty lies in the highly nonlinear nature of these equations.

Thus, it has always been a challenging task for mathematicians and engineers to find possible

exact/approximate analytical and numerical solutions of these equations. The analytical results

have great advantage in the sense that; it helps to make comparison with exact numerical

solution ensuring the reliability of the two results and also helps to explain the underlying

physics of fluid. The analytical solutions are further useful to develop an insight for the

development of new analytical techniques and for the modeling of new exciting fluid flow

problems in both Newtonian and non-Newtonian fluids.

There has been a continuously increasing interest of the researchers to investigate the boundary

layer fluid flow problems over a stretching/shrinking surface. It is now known that surface shear

stress and heat transfer rate for both viscous and non-Newtonian fluid are different. These

stretching and shrinking velocities can be of various types such as liner, power law and

exponential. Thus our main objective in this thesis is to analyze some boundary layer flow

problems due to stretching/shrinking sheet with different types of velocities analytically. Both

transient and steady forced and mixed convection flows are considered. The present thesis is

mainly structured in two parts. Chapters 3 to 5 consist of transient and steady mixed convection

boundary layer flow of Newtonian and some classes of non-Newtonian fluids with linear

stretching and shrinking cases. Chapters 6 to 8 present the investigation of exponential stretching

case. The chapters of the thesis are arranged in the following fashion.

Chapter 1 dealt with the previous literature related to boundary layer stretched flows of viscous

and non- Newtonian fluids. Chapter 2 includes the basic equations of fluid flow and heat

transfer. Definitions of dimensionless physical parameters are also presented here. Chapter 3

explores unsteady mixed convection flow of a viscous fluid saturating porous medium adjacent

x

to a heated/cooled semi-infinite stretching vertical sheet. Analysis is presented in the presence of

a heat source. The unsteadiness in the flow is caused by continuous stretching of the sheet and

continuous increase in the surface temperature. Both analytical and numerical solutions of the

problem are given. The effects of emerging parameters on field quantities are examined and

discussed. The magnetohydrodynamic boundary layer flow of Casson fluid over a shrinking

sheet with heat transfer is investigated in Chapter 4. Interesting solution behavior is observed

with multiple solution branches for a certain range of magnetic field parameter. Laminar two-

dimensional unsteady flow and heat transfer of an upper convected, an incompressible Maxwell

fluid saturates the porous medium past a continuous stretching sheet is studied in Chapter 5. The

velocity and temperature distributions are assumed to vary according to a power-law form. The

governing boundary layer equations are reduced to local non-similarity equations. The resulting

equations are solved analytically using perturbation method. Steady state solutions of the

governing equations are obtained using the implicit finite difference method and by local non-

similarity method. A good agreement of the results computed by different methods has been

observed.

Chapter 6 addresses the steady mixed convection boundary layer flow near a two-dimensional

stagnation- point of a viscous fluid towards a vertical stretching sheet. Both cases of assisting

and opposing flows are considered. The governing nonlinear boundary layer equations are

transformed into ordinary differential equations by similarity transformation. Implicit finite

difference scheme is implemented for numerical simulation. The features of the flow and heat

transfer characteristics for different values of the governing parameters are analyzed and

discussed. Chapter 7 examines the boundary layer flow of a viscous fluid. The flow is due to

exponentially stretching of a surface. Unsteady mixed convection flow in the region of

stagnation point is considered. The resulting system of nonlinear partial differential equations is

reduced to local non-similar boundary layer equations using a new similarity transformation. A

comparison of the perturbation solutions for different time scales is made with the solution

obtained for all time through the implicit finite difference scheme (Keller-box method).

Attention is focused to investigate the effect of emerging parameters on the flow quantities.

Chapter 8 aims to investigate the boundary layer flow of nanofluids. The flow here is induced

by an exponentially stretching surface with constant temperature. The mathematical formulation

xi

of this problem involves the effects of Brownian motion and thermophoresis. Numerical solution

is presented by two independent methods namely nonlinear shooting method, and finite

difference method. The effects of embedded parameters on the flow fields are investigated.

Chapter 9 summaries the research material presented in this thesis.

xii

TABLE OF CONTENTS

1. Introduction ....................................................................................... 1

2. Preliminaries .................................................................................... 11

2.1 Continuity equation ....................................................................... 12

2.2 Momentum equation for Newtonian fluid ...................................... 12

2.3 Energy equation in 2-Dimension .................................................... 13

2.4 Boundary layer approximation ....................................................... 14

2.4.1 Viscous boundary layer equation ........................................ 14

2.4.2 Thermal boundary layer equation ........................................ 16

2.5 Maxwell fluid ................................................................................ 17

2.6 Casson fluid ................................................................................... 18

2.7 Nanofluid ....................................................................................... 19

2.8 Some basic mechanism of fluid flow and heat transfer ................... 20

2.8.1 Forced, free and mixed convection ..................................... 20

2.8.2 Internal heat generation ....................................................... 20

2.9 Definition of some dimensionless parameters ................................ 20

2.9.1 Reynolds number ................................................................ 21

2.9.2 Prandtl number ................................................................... 21

2.9.3 Skin friction coefficient ...................................................... 21

2.9.4 Nusselt number ................................................................... 22

2.9.5 Sherwood Number .............................................................. 22

2.9.6 Mixed convection parameter ............................................... 23

3. Mixed Convection Flow over an Unsteady Stretching Surface in a

Porous Medium with Heat Source…. ............................................. 24

3.1 Mathematical formulation .............................................................. 25

xiii

3.2 Solution of problem ....................................................................... 28

3.2.1 Numerical solution .............................................................. 28

3.2.2 Perturbation solution for small parameter *α ....................... 28

3.3 Results and discussion ................................................................... 30

3.4 Conclusions ................................................................................... 31

4. Flow of Casson Fluid over a Shrinking Sheet with Heat

Transfer...…………………………………………………………....38

4.1 Mathematical analysis .................................................................... 39

4.2 Discussion ..................................................................................... 41

4.3 Conclusions ................................................................................... 43

5. Unsteady Mixed Convection Flow of Upper Convected Maxwell

Fluid over a Stretching Surface with Heat Transfer and Porous

Medium. ............................................................................................ 49

5.1 Solution methods for the problem .................................................. 52

5.1.1 Exact solution ..................................................................... 52

5.1.2 Perturbation solution for small ξ ........................................ 54

5.1.3 Local non-similarity (LNS) method .................................... 55

5.1.4 Implicit finite difference method (IFDM) ........................... 56


5.3 Conclusions ................................................................................... 59

6. MHD Mixed Convection Stagnation Point Flow and Heat Transfer

towards an Exponentially Stretching Sheet

…….……………………………………………….…………………65

6.1 Mathematical analysis of the problem ............................................ 66

6.2 Solution methodologies ................................................................. 67

6.2.1 Implicit finite difference method (IFDM) ........................... 67

xiv


6.4 Conclusions ................................................................................... 72

7. Unsteady Mixed Convection Stagnation Point Flow over an

exponentially stretching surface.. .................................................... 78

7.1 Problem formulation ...................................................................... 79

7.2 Solution procedures ....................................................................... 81

7.2.1 Perturbation solutions for small time ( 1)τ ≪ ...................... 82

7.2.2 Asymptotic solutions for large time τ ................................ 84

7.3 Discussion ..................................................................................... 85

7.4 Conclusions ................................................................................... 87

8. Boundary Layer Flow andHeat Transfer of Viscous Nanofluid over

an Exponentially Stretching Surface.. ............................................ 93

8.1 Problem formulation ...................................................................... 94


8.3 Concluding remarks ....................................................................... 97

9. Conclusions..................................................................................... 102

10. References…………………………………………………………105

xv

LIST OF FIGURES

Fig. 2.1: Boundary layer configuration in two dimensions. .............................................................. 14

Fig. 3.1: Physical model in coordinate system .................................................................................... 25

Fig. 3.2: Representation of (a) skin friction (b)Nusselt number as a function of λ for different

values of unsteadiness parameter*

α while Pr 0.72= , * * 0.1A B d= = = . ..................................... 34

Fig. 3.3: Graph of (a) velocity ( )f η′ (b) temperature distributions ( )θ η while Pr 0.72= ,

* * 0.1d A Bλ = = = = . For different values of unsteadiness parameter *α .................................. 35

Fig. 3.4: Effect of Prandtl number Pr for *

* * 0.1d A Bα λ= = = = = on (a) .................................. 35

Fig. 3.5: Effect of permeability d on (a) velocity ( )f η′ , (b) temperature ( )θ η while Pr 0.72= ,

** * 0.1.A Bα λ= = = = ....................................................................................................................... 36

Fig. 3.6: Effect of buoyancy parameter λ for *

* * 0.1d A Bα = = = = ,........................................... 36

Fig. 3.7: Effect of (a) *A on temperature ( )θ η at * 0.05B = (b) *B on temperature ( )θ η at

* 0.05.A = while Pr 0.72= ,*

0.3, 0.1α λ= = , 0.1d = ................................................................... 37

Fig. 4.1: Velocity profiles at M = 2 and 1.01β = under different mass suction parameters. ........ 44

Fig. 4.2: Velocity profiles at 0.5M = and 1 1.0β = under various mass suction parameters. ... 44

Fig. 4.3: Velocity profiles at 2M = and 2s = under different Casson fluid parameters. ............ 45

Fig. 4.4: The solution domain for skin friction coefficient (1

α ) for various M as a function s

when 1.0.1β = ....................................................................................................................................... 45

Fig. 4.5: Temperature profiles at 2M = and 1.01β = under different suction/injection

parameters. ........................................................................................................................................... 46

Fig. 4.6: Temperature profiles at 0.5M = and 1.01

β = under various suction/injection

parameters. ........................................................................................................................................... 46

Fig. 4.7: Temperature profiles at 0.5M = , Pr 1.0= and 3.7s = under different Casson fluid

parameter .1β ....................................................................................................................................... 47

Fig. 4.8: Temperature profiles at 0.5M = , 3.01β = and 3.7s = under different Prandtl

numbers Pr . . ......................................................................................................................................... 47

Fig. 5.1: The flow diagram. .................................................................................................................. 50

Fig. 5.2: Graph of 1 1/2

Re2

Cx f with ξ for different values of

*β while Pr 1.0, 1.0, 1.0.Dλ= = =

................................................................................................................................................................ 61

xvi

Fig. 5.3: Graph of (a) 1 1/2

Re2

Cx f(b)

1/2ReNux x

− against ξ for various Pr where

*1.0, 0.3, 1.0.Dλ β= = = ...................................................................................................................... 61

Fig. 5.4: (a) 1 1/2

Re2

Cx f (b)

1/2ReNux x

−for different values of λ when Pr=1.0, β*=0.3, D=1.0. 62

Fig. 5.5: Representation of (a) velocity ( , 0)f η′ (b) temperature ( , 0)θ η with η when 0ξ = . The

solid line shows the IFDM solution, and the symbols show the exact solution. ............................... 62

Fig. 5.6: (a) Velocity ( , )f η ξ′ (b) Temperature ( , )θ η ξ against η when Pr 1.0, 3.0=s

and 7.0 , 0.1, 0.4, 0.8,1.0ξ = while 1.0, 1.0Dλ = = and *

0.3.β = ..................................................... 63

Fig. 5.7: (a) Velocity ( , )f η ξ′ (b) Temperature ( , )θ η ξ against η for 0.0,1.0λ = and 2.0 ,

0.1, 0.3, 0.7,1.0ξ = while Pr 1.0, 1.0D= = and *

0.3.β = ................................................................. 63

Fig. 5.8: The effect of D on (a) Velocity ( , )f η ξ′ and (b) temperature ( , )θ η ξ for 0.5ξ = while

*Pr 1.0, 1.0, 0.3.λ β= = = ................................................................................................................... 64

Fig. 6.1: Graph of (a) Skin friction and (b)

12Re Nux x

− for Pr against λ when , / 1.0.M c a = ... 74

Fig. 6.2: Graph of (a)

11 2Re2

Cx f and (b)

12Re Nux x

− for /c a against λ when Pr, 1.0.M = ....... 75

Fig. 6.3: Graph of (a)

11 2Re2

Cx f and (b)

12Re Nux x

− for M against λ when Pr, / 1.0.c a = ........ 75

Fig. 6.4: Representation of (a) ( )f η′ and (b) ( )θ η for Pr against η when , / , 1.0.c a Mλ = .... 76

Fig. 6.5: Representation of (a) ( )f η′ and (b) ( )θ η for /c a against η when , Pr, 1.0.Mλ = ........ 76

Fig. 6.6: Representation of (a) ( )f η′ and (b) ( )θ η for Pr against η when , / , 1.0.c a Mλ = ........ 77

Fig. 6.7: Representation of (a) ( )f η′ and (b) ( )θ η for M against η when , / , Pr 1.0.c aλ = ......... 77

Fig. 7.1: The flow diagram and co-ordinate system. ......................................................................... 79

Fig. 7.2: Time evolution of (a) (0, )f τ′′ and (b) (0, )θ τ′− for Pr when 1λ = and / 1.c a = .......... 88

Fig. 7.3: Time evolution of (a) (0, )f τ′′ and (b) (0, )θ τ′− for λ when Pr 1= and / 1.c a = ......... 88

Fig. 7.4: Time evolution of (a) (0, )f τ′′ and (b) (0, )θ τ′− for /c a when Pr, 1.0.λ = ..................... 89

Fig. 7.5: (a) Velocity ( , )f η τ′ and (b) Temperature ( , )θ η τ for Pr when / 1c a = and 1.λ = ..... 89

Fig. 7.6: (a) Velocity ( , )f η τ′ (b) Temperature

( , )θ η τ for λ when / 1c a = and Pr 1.= ............ 90

Fig. 7.7: (a) Velocity ( , )f η τ′ (b) Temperature ( , )θ η τ for /c a when 1λ = and Pr 1.= ............ 90

Fig. 8.1: Physical model in coordinate system .................................................................................... 94

Fig. 8.2: Effect of Nt and Nbon ( )θ η Pr 10.Le= = ....................................................................... 99

Fig. 8.3: Effect of Nb

on ( )ϕ η Pr 10Le= = and 0.3.Nt = ............................................................ 100

xvii

Fig. 8.4: Effect of Pr on ( )θ η 0.3N Nt b= = , 10.Le = ................................................................ 100

Fig. 8.5: (a) Effect of Pr and Le on ( )θ η (b) Effect of Le on ( )ϕ η for fixed values of Nb

and .Nt

.............................................................................................................................................................. 101

xviii

LIST OF TABLES

Table 3.1: Comparison between analytical and numerical results for ( )f η′ and ............................ 32

Table 3.2: Values of (0)θ ′− when *

* * 0d A Bα λ= = = = = and comparison with previous work.

................................................................................................................................................................ 33

Table 3.3: The values of (0)f ′′ for various values of d and Pr when *

0, 1α λ= = , .................. 33

Table 3.4: The values of (0)θ ′− for various values of d and Pr when*

0, 1,α λ= = * * 0.A B= =

................................................................................................................................................................ 34

Table 4.1: Values of sc for different values of .M ............................................................................ 48

Table 4.2: Variation of Nusselt number (0)θ ′− when M=0.5, for various values of parameters. . 48

Table 5.1: (0, )f ξ′′ and (0, )θ ξ′− obtained through different methods for ξ when

Pr 1.0,= 3.0*

β = and 1.0.D = ............................................................................................................ 60

Table 6.1: Computed values of (0)θ ′− for different Pr when / 0M c aλ = = = . .......................... 73

Table 6.2: Computed values of (0)f ′′ for different M when 1λ = . ................................................ 73

Table 6.3: Computed values of (0)θ ′− for different M when 1λ = . ................................................ 74

Table 7.1: Skin friction computed using different methods at certain τ when Pr, / 1.0c a = . ..... 91

Table 7.2: Nusselt number computed using different methods for certain τ when Pr, / 1.0.c a =

................................................................................................................................................................ 92

Table 8.1: Comparison of different methods for Nur for N

b, 0.Nt = ............................................. 98

Table 8.2: Comparison of Nur for Nb

and Nt for Pr 10= , 10.Le = ................................................ 98

Table 8.3: Comparison of Shr for Nb

and Nt for Pr 10= , 10.Le = ................................................. 98

Table 8.4: Nur and Shr with 0.3N Nt b= = and for different Pr , and .Le ................................... 99

xix

LIST OF ABBREVIATIONS

a constant

1A Rivlin Ericksen kinenmatic tensor (1/ t)

b

constant

B

Magnetic field ( Tesla)

c constant

C ambient nanoparticle volume fraction

fC skin-friction coefficient

hC coefficient of heat transfer

wC nanoparticle volume fraction at the stretching surface

pc specific heat (L2

/ t2

T)

D

Dt

material time derivative (1/t)

BD Brownian diffusion coefficient

TD Thermophoretic diffusion coefficient

( )f η dimensionless stream function (L/t2)

g gravitational acceleration (L/t2)

Gr Grashof number

xGr local Grashof number

I identity tensor

k thermal conductivity of fluid (ML/ t3T)

xx

K permeability of the porous medium

L reference length (L)

Le

Lewis number

M magnetic parameter

Nu Nusselt number

xNu local Nusselt number

bN

Brownian motion parameter

tN thermophoresis parameter

( )O Order symbol: big O

p dimensionless fluid pressure (M/t2L)

Pr Prandtl number

q′′′ volume heat generation rate ( 3Wm− )

wq heat transfer for unit area at the boundary (M/t3)

Re Reynolds number

Re x local Reynolds number

Sh

Sherwood number

s suction parameter

t time (t)

T temperature of the fluid (T)

T Cauchy stress tensor (M/Lt2)

wT surface temperature (T)

xxi

T∞ ambient temperature (T)

0T reference temperature (T)

0u reference velocity (L / t)

wU velocity of the moving surface (L / t)

U∞ free stream Velocity (L / t)

u,v,w components of velocity in x, y and z direction (L/t)

u,v non-dimensional axial and normal velocity (L/t)

V velocity vector (L/t)

wv suction/injection velocity (L/t)

,x y non-dimensional axial and normal fluid velocity (L/t)

x y, dimensional axial and normal velocity (L / t)

Greek symbols

α thermal diffusivity (L2/t)

β coefficient of thermal expansion (1 / T)

*β elasticity parameter

µ viscosity of fluid (M/Lt)

ν kinematic viscosity (L2/t)

η similarity variable

δ thickness of boundary layer (momentum) (L)

Tδ thickness of boundary layer (thermal) (L)

T∆ temperature difference between surface and ambient fluid (T)

xxii

θ transformed fluid temperature

λ mixed convection parameter

ξ similarity variable

wτ wall shear stress (M/Lt2)

τ similarity variable

, , zzτ τ τxx yy normal stresses (M/Lt2)

, , ,

,

τ τ τ

τ τ τ

xy xz yx

yz zx zy

shear stresses (M/Lt2)

ρ fluid density (M/L3)

ρb body force for a unit volume (M/L2t2)

, , zb b bρ ρ ρx y body force per unit volume in x , y and z directions (M/L2t2)

( )ϕ η

rescaled nanoparticle volume fraction

ψ stream function (L2/t)

“Introduction”

1

Chapter 1

1. Introduction

“Introduction”

2

In 1904, Prandtl [1] introduced the concept of “Boundary Layer”, and devised an

approach to have a very precise analysis of “Viscous Fluid” flow with the most

prominent feature that the experimental results agree with it. Thus in the beginning of

the nineteenth century, the evolution of boundary layer theory has not only made it

possible to verify the experimental results but also successfully explained the

theoretical results which the classical theory of hydrodynamics for perfect fluids

failed to do. According to boundary layer theory, the fluid flow region around a solid

body can be divided into two regions: a very thin region in the immediate vicinity of

solid, called the Boundary Layer, where fluid friction cannot be neglected, and the

remaining region outside the boundary layer where the fluid friction is negligible. The

utility of boundary layer theory is twofold as it not only gives a physically viable

explanation of the real fluid flow but also proposes a very reasonable simplification to

the emerging mathematical difficulties arising in the solution of complete Navier

Stokes equations. Thus the boundary layer theory incorporates the presence of

tangential stress (fluid friction) between the fluid layers in the immediate

neighborhood of the surface, due to the effects of pressure and (or) gravity. The idea

of boundary layer theory achieved an immense popularity as it not only successfully

clarified the fluid flow phenomenon but it also made possible to address the problems

of ‘viscous fluid flow’ in theoretical analysis. This development eased the engineers a

lot as these analyses could be utilized in the practical applications. The benefits of

boundary layer theory are two-fold as it not only explains the phenomenon of viscous

fluid flow to a physically penetrating extent; it also poses a reasonable degree of

simplification on the mathematical difficulties that arise in complete Navier Stokes

Equations. An adequate explanation and application of boundary layer theory in

modeling fluid flow can be found in [2, 3].

One of the popular extensions of boundary layer theory is the study of flow with

stretching conditions. The study of flow pattern and heat transfer, driven by the

stretching of the surface and/or with a given temperature conditions has received

significant attention recently. Such interests arise because of its pertinent industrial

and engineering applications in a number of fields. The examples of such

phenomenon are the aerodynamics, extrusion of plastic sheets, extrusion of polymers

and process of condensation of metallic plates. The quality of the finished products

has a definite impact on the stretching/shrinking and heating and/or cooling of the

surface. The modeling of such processes is undertaken by using different stretching

“Introduction”

3

and/or shrinking velocities and temperature distributions. Few more examples are

wire drawing, cooling of electronic chips or metallic sheets, glass-fiber, heat treated

materials that travel between a feed and a wind-up roll, or manufacture of materials

by extrusion, filaments spinning, paper production, crystal growth, and food

processing etc. A great deal of research in this regard has been done in order to model

real world problems of such cases for better explanation of fluid behavior with an

adequate explanation of the results of the relevant experiments and to present

analytical and numerical results. Thus in this thesis we focus on analyzing boundary

layer flow driven by stretching/shrinking of surface. The present thesis is mainly

structured in two parts. Chapter 3 to 5 consist transient and steady mixed convection

boundary layer study of Newtonian and some classes of non-Newtonian fluids with

linear stretching and shrinking cases. Chapter 6 to 8 present the investigation of

exponential stretching case.

Some of the popular applications of stretching process are given in Fisher, [4] and

Altan et al.[5]. Sakiadis [6] investigated the flow over a constantly moving solid

surface. Tsou et al. [7] studied the flow as well as heat transfer past a stretching

surface. Analytical solution for steady flow over a stretching sheet is examined by

Crane [8]. Further studies of steady flow due to linear sstretching are also given [9-

13]. Physically the unsteadiness in mixed convection is also concerned with the

cooling during the steady fabrication processes (see also [2, 3]), and transient

crossover to steady state. These situations are explained through momentum and

thermal boundary layers, that is by taking the steady part of the stretching velocity

proportional to distance from the leading edge, whereas the unsteady part is inversally

proportional to time (which highlights cooling process). Further, similarity solution of

the time dependent Navier–Stokes equations for a thin film lying on a stretching

surface was investigated by Wang [14]. Andersson et al. [15] considered this problem

for heat transfer of a power law fluid. The solution for an impulsively started transient

flow past a stretching wall has been presented by Pop et al. [16]. The effect of

unsteadiness on flow pattern over a continuous stretching surface in the absence and

presence of internal heat generation was studied by Elbashbeshy and Bazid in [17,18].

Ishak et al. [19, 20] proposed numerical solutions of flow due to unsteady stretching

surface. Fang et al. [21] proposed a family of unsteady boundary layer past a

stretching surface.

“Introduction”

4

Flow and heat transfer characteristic of viscous fluids over a continuous stretching

sheet contained in a porous medium is of vital importance. There are a number of

mathematical models used to represent porosity of porous media. For instance, Darcy-

Brinkman, and Darcy-Brinkman-Forchheimer models. Nevertheless, Darcy-Brinkman

model is considered to be most appropriate in modeling the viscous flows. Reviews

on the topic of convection in porous media can be found in [22-24]. Mixed convection

regime emerges as the combined effect of free and force convection. The buoyancy

forces that appear due to heating / cooling of stretching surfaces significantly affect

the flow and thermal properties of the physical process. The study of mixed

convection flow in the boundary layer, past moving surfaces has been given in [25-

29].

The presence of a heat source or sink in the fluid is considered important due to the

sharp temperature fluctuations between solid surface and the ambient region may

ultimately influence heat transfer characteristics (see Vajravelu and Hadjinicolaou

[30]). Such sources can generally be space dependent and / or temperature dependent.

Thus in Chapter 3, we address the problem of an unsteady flow driven by mixed

convection in a porous medium saturated by a fluid, adjacent to a stretching vertical

surface, with heat source. Both cases of heated and / or cooled surface are undertaken.

We present both analytical as well as numerical solution to attain an appropriate

degree of confidence. This work aims to serve a number of purposes. The finding of a

reliable analytical solution for case of unsteady stretching, that can further be used in

the studies of unsteady problems, incorporating the heat source or sink in the form of

a source term in the energy transport equation, and taking into consideration the

presence of porous medium. Mathematically, appropriate similarity variables are

introduced to convert the governing system of nonlinear coupled differential

equations into self similar form. The system of equations thus obtained are solved

both numerically (using the shooting method) as well as analytically (using

perturbation method with Padé approximation). A reliable agreement has been

acheived between the two results thus found. Effect of the governing parameters on

the physical properties is investigated graphically explaining their physical meanings.

A comparison with existing literature is presented to support the results. The contents

of this chapter have been published in the journal: Mathematical Problems in

Engineering [31].

Existing literature on boundary layer flows shows that much attention has been given

“Introduction”

5

to the problems of stretching. However, although shrinking enjoys the same degree of

importance, nevertheless no or very little has been said for the shrinking of the sheets.

Extensive research on the topic of stretching has been undertaken since the work of

Sakiadis [6] and Crane [8]. For instance Banks [32] obtained the exact solution of the

stretching surface problem in self similar form. Rajagopal and Gupta [33] found the

exact solution for boundary layer flow of second grade fluid over stretching surface.

Some recent work on shrinking is done by Liao [34, 35]. Analytical solution of

Navier-Stokes equations satisfies the shrinking boundary conditions. Wang [36]

extended the shrinking sheet problem to power-law shrinking velocity. Ali [37]

investigated the effect of both suction and injection in the boundary layer flow over a

stretching surface. Miklavcic and Wang [38] discussed the shrinking sheet problems

through non-Newtonian fluids. An exact, closed form analytic solution for MHD

viscous fluid for shrinking sheet is considered by Fang and Jihang [39]. Fang [40]

also considered the power law velocity. Hayat et al. [41] considered MHD flow of

second grade fluid past a shrinking sheet. A series solution given by homotopy

analysis method was obtained by Sajid and Hayat [42].

Casson fluid model for non-Newtonian fluid is useful to represent rod like solids and

is often applied to molten chocolate and blood situations. We note that, Newtonian

fluids are important subclass of Casson fluid. There are only a few articles on Casson

fluid in the literature. For example very recently, Mustafa et.al [43] worked for the

transient flow of Casson fluid over a continuously moving solid surface. They also

addressed stagnation point flow and heat transfer towards stretching sheet in a Casson

fluid [44]. The magneto-hydrodynamic flow of Casson fluid by an exponentially

permeable shrinking sheet was investigated by Nadeem et al. [45] using Adomian

decomposition method. Very recently Bhattacharyya et al. [46] studied the steady

flow of Casson fluid over a porous stretching / shrinking sheet.

Chapter 4 extends the flow analysis of Fang and Jihang [39] to Casson fluid over the

surface with heat transfer. The effect of Casson fluid parameters on temperature and

velocity distributions, wall shear stress and Nusselt number is examined. The dual

solutions happen to exist for the velocity, temperature and wall shear stress for

particular values of magnetic parameter.

Maxwell fluid is a special kind of rate type fluid. This fluid predicts the relaxation

time effects. It is a simple subclass of rate type fluids which can describe the stress

relaxation effects. This model has been applied to the problems having small

“Introduction”

6

dimensionless relaxation time. However, in case of concentrated polymeric fluids it

can be useful to large dimensionless times as well. Hayat et al. [47] used homotopy

analysis method to study magneto-hydrodynamic stagnation-point flow of Maxwell

fluid along a stretched surface. They concluded that velocity shows a decreasing trend

with increasing elasticity parameter and there is an increase with the increase of

velocity ratio parameter. Aliakbar et al. [48] considered the radiation effects in MHD

flow of Maxwell fluid. They showed that there is a raise in the fluid temperature as

the value of elastic parameter is increased. Pahlavan et al. [49] extended the stretching

phenomenon to porous sheets. Karra et al. [50] studied on Maxwell fluids with

relaxation time and viscosity depending pressure. Heyhat and Khabazi [51] applied

non isothermal conditions for flow of a Maxwell fluid over a flat surface. For an

unsteady axial Couette flow, Ather et al. [52] obtained an exact solution of a

fractional Maxwell fluid. Recent contribution in unsteady mixed convection of

Maxwell fluid is made by Mukhopadhyay [53, 54]. This leads us to consider, in

Chapter 5, the problem of unsteady mixed convection flow with heat transfer in a

Maxwell fluid over vertical stretching is analyzed. The fluid saturates the porous

space. Series solution is obtained for small time regime whereas the solution for all

time regime is found numerically using the local non similarity method, (denoted by

LNS), and implicit finite difference method (denoted by IFDM). A good agreement

between the computed results of these methods has been observed. The effect of

different governing parameters is investigated on field variables with the help of

graphical solutions of velocity profiles, temperature profiles, heat transfer and skin

friction.

Stagnation point refers to a point in the domain of flow where local velocity becomes

zero in that region and static pressure has maximum value and is responsible for the

fluid flow. The importance and types of stretching velocities have been mentioned

earlier. To recall once again, these are: linear, power-law form of stretching and the

exponential stretching. Linear and power-law stretching have been treated

extensively. Although every possible type of stretching of sheets has significant

importance in the engineering processes; the exponential form of the stretching has

not been given the importance it deserves. For example, it is well known that

topological chaos depend on the periodic nature of motion of obstacles in a two-

dimensional flow to form nontrivial braids. This motion is responsible for the

exponential stretching of material lines which cause efficient mixing. Friedlander and

“Introduction”

7

Vishik [55] showed the existence of exponential stretching as a necessary condition

for the smooth flow in "fast" dynamo problem.

Chiam [56] studied the flow around the stagnation point over a stretching plate. He

concluded that no boundary-layer is formed near the plate when the stretching

velocity of the plate is equal to the stagnation flow velocity. The influence of heat

transfer in two dimensional steady stagnation point flow of the viscous fluid was

studied by Mahapatra and Gupta [57]. Nazar et al. [58] considered the normal

stagnation flow of a micro-polar fluid towards a stretching sheet. Layek et al. [59]

considered boundary layer stagnation point case with heat mass transfer past a

stretching porous plate, considering heat generation/absorption effects. Ishak et al.

[60] obtained dual solutions for mixed-convection case close to a stagnation point on

a porous solid surface. Gupta and Gupta [61] studied heat mass transfer over

stretching porous plate. Ali [62, 63] has examined boundary-layer flow with power

law stretching conditions with suction and injection. Magyari and Keller [64]

considered heat mass transfer within a boundary layer region due to exponentially

stretching surface. Elbashbeshy [65] examined heat transfer due to exponentially

stretching surface with suction. He considered an exponential stretching velocity

profile and an exponential similarity variable along the direction of stretching. The

influence of thermal radiation for such configuration has been investigated by Sajid

and Hayat [66], Bidin and Nazar [67]. MHD (magneto-hydrodynamic) effects in the

boundary layers due to exponential stretching were addressed by Al-Odat [68]. Also

Pal [69] performed an analysis to discuss MHD boundary layer mixed convection

flow by considering exponential stretching and temperature distribution in the

presence of internal heat generation / absorption and/or viscous dissipation.

The study of magneto-hydrodynamic flow problem is of great interest due to its

applications in designing cooling systems. It has an application in engineering

problems e.g MHD generators, geothermal energy extractions, nuclear reactors and

plasma studies. Pavlov [70] gave an exact similarity solution for MHD flow of an

over a stretching surface. Chakrabarti and Gupta [71] found the temperature

distribution in this flow when uniform suction is applied at the stretching surface.

Andersson [72] obtained an exact solution of the boundary layer equation for flow of

second grade fluid due to stretching of the surface in with transverse magnetic field.

Mahapatra and Gupta [73] found that the velocity profile at some fixed point

decreases / increases with an increase of magnetic field when the stretching velocity is

“Introduction”

8

greater / less than the free stream velocity. Liu [74] made an extension in Andersson’s

results by considering both the cases of thermal effects (prescribed temperature and

prescribed surface heat flux) on the stretching sheet. He considered both cases of.

Ishak et al. [75] concluded that when the free stream velocity is more than the

stretching velocity, the rate of heat transfer at the surface increases with the magnetic

parameter. Mixed convection boundary-layer flow over a stretching surface in

presence of magnetic field and taking constant wall temperature is discussed by Ishak

et al. [76]. Here dual solutions exist for some range of the mixed convection

parameter, he obtained dual solutions. Ishak [77] found that with an increase of

radiation and magnetic parameters the rate of heat transfer the surface decreases. In

We present an analysis for a MHD mixed-convection stagnation-point flow in a

viscous fluid over an exponentially stretching surface in Chapter 6. Representative

results for shear stress on the wall, rate of heat transfer, temperature and velocity

distributions are obtained for different values of the governing parameters.

The unsteady stagnation point flow along body has been actively studied during the

last few years because of it has many application in engineering such as convective

cooling and industrial manufacturing processes. The unsteady stagnation-point flow

towards a stationary plate was first investigated by Yang [78] and Williams [79].

Nazar et al. [80] reported the flow feature due to stretching sheet. Cheng and Dai [81]

studied flow over an impulsively stretching surface. They found a uniformly valid

series solution. The unsteady oblique stagnation-point flow was investigated by Wang

[82]. The unsteady stagnation-point flow of non-Newtonian fluid was investigated by

Xu et al. [83]. To the best of our knowledge, the case of mixed convection stagnation

point flow due to exponentially stretching surface with variable temperature is not

addressed yet.

Chapter 7 describes the unsteady mixed convection stagnation-point flow of viscous

fluid due to an exponentially stretching surface. The problem is solved for the initial

transient and the final steady state regimes. The initial transient regime in the

boundary layer has been solved analytically for small time scale. For the large time

the solution is obtained numerically. For the whole time regime, the finite difference

method is applied. Thus, both analytical and numerical results are obtained

independently and compared with each other giving a complete confidence and

reliability. Having obtained the results, effects of governing thermo-physical

parameters on the flow and heat transfer profiles are investigated. The contents of this

“Introduction”

9

Chapter are submitted for publication in the Journal of Applied Computational and

Mathematics.

There are many conventional ways for heat transfer enhancment within the fluids. For

instance changing the flow geometry, the flow domain, or changing the surface

heating conditions. Obviously the thermal conductivity of solids is greater than that of

the fluids and thus heat transfer in fluids can also be increased for thermal

conductivity enhancement of the fluid. This can be done by considering the nano size

solid particles of relatively larger thermal conductivity in ordinary fluids. In recent

these years, great interest has been generated in the study of heat transfer of

nanofluids. The importance of these fluids is their enhanced heat transfer properties

when compared with the ordinary fluids. This concept was first introduced by Choi

[84].

Wang and Mujumdar [85] gave a comprehensive review of the findings regarding the

nanofluids. They indicated that there are many factors that involve the formation of

nanofluids and these factors are responsible for the flow characterization of

nanofluids. Such factors include the manufacturing mechanism, the molecular nature

of the base fluid, the nature, size, surface area and clustering of solid particles. These

factors may be responsible for the poor characterization of such suspensions and lack

of agreement in the findings regarding the nanofluids. Buongiorno [86] found that

Brownian motion and thermophoresis are the vital parameters for the studies in the

study of nanofluids. A comprehensive review regarding recent developments in the

nanofluids is presented by in Das et al. [87]. Kakaç and Pramuanjaroenkij [88]

reviewed the efficacy of different thermal conductivity models proposed for the

nanofluids. A laminar free convection study of nanofluid is given by Polidori et al.

[89]. They used integral formulation approach for external boundary layer flows.

They indicated that other than thermal conductivity, the right choice of viscosity is

also a factor that causes the deteroiration of the thermo physical properties of

nanofluids. Natural convection boundary layer flow past a vertical surface for

nanofluid is investigated by Kuznetsov and Nield [90]. Bachok et al. [91] reported

the boundary layer study of nanofluid due to a moving surface. The boundary layer

flow of nanofluid over a linearly stretching sheet is presented by Khan and Pop [92].

Mustafa et al. [93] presented the stagnation-point flow of nanofluid over a linearly

stretching surface.

“Introduction”

10

Recently the effect of partial slip conditions in the boundary layer of nanofluid with

stretching conditions were studied by Noghrehabadi et al. [94]. They investigated the

effect of slip parameter, nanofluid parameter, Brownian motion and thermophoresis

parameter in the momentum and thermal boundary layers. They found that thickness

of the momentum boundary layer, and that of the thermal boundary layer increases

with an increase in velocity slip. Further the effect of velocity slip heat transfer is

increased when thermophoresis is larger. Haddad et al. [95] considered the classical

Rayleigh- Bénard problem to identify the physical significance of thermophoresis and

Brownian motion effects in the modeling of nanofluids. They concluded that

Brownian motion and thermophoresis effects result in higher heat transfer efficacy

when compared with the case where these effects are absent. Moreover neglecting

these effects may cause deterioration in heat transfer. However, thermophoresis and

Brownian motion effects do not significantly affect the patterns of isotherms and

streamlines.

With such considerations in view, Chapter 8 aims to discuss the flow of nanofluid

generated by exponential stretching of the surface in nanofluid. The model used for

the description of nanofluid includes Brownian and thermophoreosis effects. The

problem is devised in terms of the nonlinear governing equations and the appropriate

boundary conditions. Numerical solution of the problem is presented in terms of heat

and mass transfer rates. The effects of embedded parameters are plotted and analyzed.

Finally we made a comparison between the two different numerical solutions are

made to ensure the reliability of the results thus obtained. The solutions for nanofluid

and the base fluid are also compared. The contents of this chapter are submitted for

publication in the Journal of Mechanical Science and Technology.

“Preliminaries”

11

Chapter 2

2. Preliminaries

“Preliminaries”

12

This chapter presents some standard equations that directly govern the convective

flows by stretching surfaces, dimensionless numbers and mechanism for fluid flow

and heat transfer.

2.1 Continuity equation

Continuity equation represents the law of conservation of mass. In general the

continuity equation for an unsteady flow is

( ) 0,t

ρρ

∂+ ∇ ⋅ =

∂V (2.1)

where ρ is the fluid density, t the time and ( , , )u w=V V v the velocity vector in

three dimension. For the steady flow the continuity equation reduces to

( ) 0.ρ∇ ⋅ =V (2.2)

Continuity equation (2.2) for an incompressible fluid ( ρ constant) transforms to

0.∇ ⋅ =V (2.3)

2.2 Momentum equation for a Newtonian fluid

Fluids obeying Newton’s law of viscosity are called Newtonian fluids. There is linear

proportionality between shear stress and the rate of shear strain. i.e

,wdu

dyτ µ= (2.4)

where wτ is the shear stress applied by the fluid and µ is the dynamic viscosity which

is constant of proportionality.

The momentum equation governs the dynamic behavior of fluid motion. The equation

represent the conservation of momentum of the fluid. Its mathematical form is

,D

divDt

ρ ρ= +V

T b (2.5)

,p= − +T I S (2.6)

where ρb is body force per unit mass, T is Cauchy stress tensor, whereas p is the

scalar part of the pressure, I is the identity tensor and S is the extra stress tensor.

Here

“Preliminaries”

13

Du w

Dt t x y z

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂v (2.7a)

is the material time derivative. The Cauchy stress tensor is given by

xx xy xz

yx yy yz

zx zy zz

τ τ τ

τ τ τ

τ τ τ

=

T (2.7b)

where xxτ , yyτ and zzτ are the normal stresses along the coordinate axes whereas

xyτ ,

yxτ ,

zxτ ,

xzτ ,

yzτ and

zyτ are the shear stresses. Since the present thesis is focused

on the study of two dimensional boundary layer flows, we concentrate on two

dimensional form of Eq. (2.5). Thus using Eqs. (2.6) and (2.7), Eq. (2.5) becomes

,xyxx

x

u u uu

t x y x y

ττρ ρ

∂ ∂∂ ∂ ∂+ + = + + ∂ ∂ ∂ ∂ ∂

bv (2.8)

,yx yy

yut x y x y

τ τρ ρ

∂ ∂ ∂ ∂ ∂+ + = + + ∂ ∂ ∂ ∂ ∂

bv v v

v (2.9)

where ,u v are the velocity components, xρb and yρb are components of the body

force respectively along x and y directions. The term ρb in equation (2.5), represent

the body force, also referred to as the source term, represents a physical property that

may serve to modify the velocity field in a domain. However, if the flow is due to the

buoyancy only, which generally acts along the vertical, then for a Boussinesq type

fluid.

0, ( )x y

g T Tρ ρ β ∞= = −b b , (2.10)

where T represents the fluid temperature, T∞ is the ambient temperature, g is

gravitational acceleration and β is the coefficient of thermal expansion.

2.3 Energy equation in 2-Dimension

The energy equation is constituted from first law of thermodynamics. The transient

two dimensional form of thermal energy conservation of fluid with dissipation term is

given by

“Preliminaries”

14

2 2

2 2,p

T T T T Tc u k

t x y x yρ

∂ ∂ ∂ ∂ ∂+ + = + + Φ

∂ ∂ ∂ ∂ ∂ v (2.11)

in which p

c is specific heat, k is the coefficient of thermal conductivity, and Φ is the

source term in the energy transport. This term mathematically represents physical

factors that affect the thermal boundary layer. For instance, the term Φ may represent

the internal heat generation, as we consider in Chapter 5. It may stand for the viscous

dissipation function that affects the thermal energy.

2.4 Boundary layer approximation

-

Fig. 2.1: Boundary layer configuration in two dimensions.

Ludwig Prandtl introduced the boundary-layer concept for the first time in 1904.

According to Prandtl's boundary-layer concept, under certain conditions, viscous

forces are important only in the immediate vicinity of a solid surface where velocity

gradients are large. This region near the surface is termed as the boundary layer (see

Fig. 2.1). In the regions, far from the solid surface where there exist no large gradients

in fluid velocity, the fluid motion may be consider frictionless i.e., potential flow. We

now simplify the momentum equations employing the concept of boundary layer

approximation.

2.4.1 Viscous boundary layer equation

Consider an incompressible, viscous fluid over a surface stretched with

velocity ( )wU x , where the free stream velocity is ( )U x∞ . Neglecting body forces and

taking viscosity to be constant. The Eqs. (2.3), (2.8) and (2.9) may be written as:

“Preliminaries”

15

0,u

x y

∂ ∂+ =

∂ ∂

v (2.12)

2 2

2 2,

u u p u uu

x y x x yν ∂ ∂ ∂ ∂ ∂

+ = − + + ∂ ∂ ∂ ∂ ∂

v (2.13)

2 2

2 2.

pu

x y y x yν ∂ ∂ ∂ ∂ ∂

+ = − + + ∂ ∂ ∂ ∂ ∂

v v v vv (2.14)

To write the equations (2.12)-(2.14) in dimensionless form, taking 0u as reference

velocity and L as the reference length. Using the transformations

20 0 0

, , , , , ,x y u p

x y u pL L u u u

ν νρ

= = = = = =v

v (2.15)

The transformed equations are

2 2

2 20

,u u p u u

uu Lx y x x y

ν ∂ ∂ ∂ ∂ ∂ + = − + + ∂ ∂ ∂ ∂ ∂

v

(2.16)

2 2

2 20

.v p

uu Lx y y x y

ν ∂ ∂ ∂ ∂ ∂ + = − + + ∂ ∂ ∂ ∂ ∂

v v vv

(2.17)

Dropping bars and using 0Reu L

ν= , we have

2 2

2 2

1,

Re

u u p u uu

x y x x y

∂ ∂ ∂ ∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂ v (2.18)

2 2

2 2

1.

Re

pu

x y y x y

∂ ∂ ∂ ∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂

v v v vv (2.19)

The experimental observation shows that thickness of the boundary layer is very small

when it is compared with the length of the surface i.e

1L

δδ = ≪ . (2.20)

Using the order analysis for Eqs. (2.18)-(2.19) in boundary layer where (1)u O= ,

( )y O δ= and (1)x O= , we have

“Preliminaries”

16

2 2

2 2 2

1 1(1), , (1),

u u u uO O O O

x y x yδ δ

∂ ∂ ∂ ∂ = = = = ∂ ∂ ∂ ∂

. (2.21)

From continuity equation it follows that both u

x

∂

∂ and

y

∂

∂

v should be of same order.

Therefore

(1)Oy

∂=

∂

vwhich implies ( )O δ=v , ( )O

xδ

∂=

∂

v,

2

2( )O

xδ

∂=

∂

v and

2

2

1,O

y δ

∂ =

∂

v

which

shows that 2 2

2 2

u u

x y

∂ ∂

∂ ∂≪ . In Eqs. (2.18)-(2.19)

2

1Re ( )O

δ= . We see that in Eq (2.18) all

the terms are (1)O exceptp

x

∂

∂. Therefore

p

x

∂

∂= (1)O which implies

1pO

y δ

∂ =

∂ The

dominant term in Eq. (2.19) is

0,p

y

∂=

∂ (2.22)

showing that p is independent of y . Thus the governing equation finally reduces to

2

2

u u p uu

x y x yν

∂ ∂ ∂ ∂+ = − +

∂ ∂ ∂ ∂v . (2.23)

2.4.2 Thermal boundary layer equation

When the surface temperature is different from that of the ambient fluid then there is a

narrow thermal boundary layer near the surface. For the steady incompressible

laminar flow of viscous fluid with constant thermo-physical properties, the energy

equation is

2 2

2 2.

T T T Tu

x y x yα ∂ ∂ ∂ ∂

+ = + ∂ ∂ ∂ ∂

v (2.24)

Using the transformation (2.15) along with w

T TT

T T

∞

∞

−=

−, the energy equation has the

dimensionless form

2 2

2 2

1.

Pr Re

T T T Tu

x y x y

∂ ∂ ∂ ∂ + = + ∂ ∂ ∂ ∂

v (2.25)

Omitting bars we have

“Preliminaries”

17

2 2

2 2

1,

Pr Re

T T T Tu

x y x y

∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂ v (2.26)

where Prν

α= is the Prandtl number and

0Reu L

ν= is the Renoylds number. We

observe that in boundary layer, (1)T O= because it varies from the surface

temperature to zero at Ty δ= and also ( )Ty O δ= . We note that the order of each of

terms2

2, ,

T T Tu

x y x

∂ ∂ ∂

∂ ∂ ∂v is one and

2

2 2

1

T

TO

y δ

∂=

∂ . To balance the above equation (2.26),

the term Pr Re should be of order 2

1

Tδ i.e Pr

T

Oδ

δ

=

. It can also be seen

that2 2

2 2

T T

x y

∂ ∂

∂ ∂≪ . Thus the thermal boundary layer equation becomes

2

2.

T T Tu

x y yα

∂ ∂ ∂+ =

∂ ∂ ∂v (2.27)

2.5 Maxwell fluid

The Cauchy stress tensor for Maxwell fluid is defined as [52]

1 1 ,D

D tλ µ+ = A

SS (2.28)

in which the Rivlin-Ericksen tensor 1 ( )T= ∇ + ∇A V V and

,TD d

L LD t d t

= − −S S

S S (2.29)

,d

udt t x y

∂ ∂ ∂= + +

∂ ∂ ∂

S S S Sv (2.30)

and 1λ is the relaxation time.

The x and y components of equation of motion are

,xx xyu u u p S S

ut x y x x y

ρ ∂ ∂ ∂ ∂ ∂ ∂

+ + = − + + ∂ ∂ ∂ ∂ ∂ ∂

v (2.31)

“Preliminaries”

18

.yx yyp S S

ut x y y x y

ρ ∂ ∂ ∂ ∂ ∂ ∂

+ + = − + + ∂ ∂ ∂ ∂ ∂ ∂

v v vv (2.32)

In component form Eq. (2.28) takes the form

12 ,

xx xxxx

xx

xy xx xy

S S uu S

ux y xS

xu u uS S S

y x y

λ µ

∂ ∂ ∂+ −

∂∂ ∂ ∂ + = ∂∂ ∂ ∂

− − − ∂ ∂ ∂

v

(2.33)

1 2 ,

xy yyxy

xy

yy xx xy

S S uu S

ux y xS

x xS S S

y x y

λ µ

∂ ∂ ∂+ −

∂ ∂∂ ∂ ∂ + = + ∂ ∂∂ ∂ ∂ − − −

∂ ∂ ∂

vv

v v v (2.34)

1 2 ,

yy yyxy

yy

yy yx yy

S Su S

x y xS

xS S S

y x y

λ µ

∂ ∂ ∂+ −

∂∂ ∂ ∂ + = ∂∂ ∂ ∂

− − − ∂ ∂ ∂

vv

v

v v v (2.35)

The boundary layer flow equation is given by

2 22 2

2 2 2

1 22

1.

2

u uu

x yu u u p uu

t x y x yuu

x y

λ νρ

∂ ∂+ +

∂ ∂∂ ∂ ∂ ∂ ∂ + + + = − + ∂ ∂ ∂ ∂ ∂∂

∂ ∂

v

v

v

(2.36)

2.6 Casson fluid

Casson fluid is a plastic fluid that exhibits the shear thinning characteristics. Casson

fluid model for non-Newtonian fluid was developed to represent rod like solids and is

often applied to molten chocolate and blood. Newtonian fluids are important subclass

of Casson fluid.The mathematical model for Casson fluid is [43]

1 1

*2 2 n

w yτ τ µ γ= + ɺ (2.37)

where wτ is the shear stress and yτ yielding stress. *µ is called the consistency co-

efficient and n is a power.

The boundary layer equation of motion for Casson fluid is [43]

“Preliminaries”

19

2

2

1

11 ,

u u u uu

t x y yν

β

∂ ∂ ∂ ∂+ + = +

∂ ∂ ∂ ∂ v (2.38)

where 1β is the Casson fluid parameter.

2.7 Nanofluid

Colloids made up by mixing nanometer sized solid particles into a base fluid are

called Nanofluids. The diameter of these small solid particles may vary between 1nm

- 100 nm in. They are studied due to the reason that such materials serve to enhance

thermal conductivity, heat transfer and convective properties of the base fluid. To see

the effective heat transfer increase, nanofluids are commonly composed of using 5%

volume fraction of nanoparticles. Nanoparticles have a very stable suspension with

little gravitational settling over a long period of time. Some commonly used

nanoparticles are alumina oxide, silica oxide, titania copper oxide and some metals

(like copper and gold).

The flow equations for nanofluid in boundary layer form [90]

2

2,

u u uu

x y yν

∂ ∂ ∂+ =

∂ ∂ ∂v (2.39)

22*

2,T

B

DT T T C T Tu D

x y y y y T yα τ

∞

∂ ∂ ∂ ∂ ∂ ∂ + = + +

∂ ∂ ∂ ∂ ∂ ∂ v (2.40)

2 2

2 2,TB

DC C C Tu D

x y y T y∞

∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂v (2.41)

where α is the thermal diffusivity, ν is kinematic viscosity, BD is the Brownian

diffusion coefficient, TD is the thermophoretic coefficient of diffusion. Further,

*( )

( )

p

f

c

c

ρτ

ρ= is the ratio between the heat capacity of nanoparticle’s material to the

heat capacity of the base fluid.

“Preliminaries”

20

2.8 Some basic mechanism of fluid flow and heat transfer

2.8.1 Forced, free and mixed convection

Heat transfer due to motion of fluid is called convection. Convection is divided into

two categories:

(a) Free convection

(b) Forced convection

The flow induced by the body forces that occur due to the differences between fluid

densities resulting from a temperature gradient in the flow field is called free

convection. Generations of these body forces are actually due to pressure gradients

applied on the whole fluid. The most common source of this imposed pressure field is

gravity. The body forces in this case are usually termed buoyancy forces. Examples of

free convection are heat rejection to atmosphere, the buoyancy flow originating from

heating of rooms, fires and many other such processes.

In forced convection, the flow of fluid is caused by some external agency e.g. the

wind, a blower, a fan or in the absence of an external agent by the movement of

heated body. There is a little effect of buoyancy on the direction of flow in forced

convection.

If there exist forced velocity and the buoyancy forces are not negligible then such

flows are termed as combined- or mixed free and forced convective flows.

2.8.2 Internal heat generation

Internal heat generation is the heat energy evolved because of the internal energy of

the molecules of the fluid. If the thermal energy of the system is induced by internal

heat generation then the source term Φ in equation (2.6) for uniform heat generation

rate within a control volume takes the form [18].

,q′′′Φ = (2.42)

where the term q′′′ is called the uniform volumetric heat generation rate.

2.9 Definition of some dimensionless parameters

Dimensionless numbers are the measure of relative importance of different aspects of

the flow. These variables also reduce the number of experiments as different

“Preliminaries”

21

properties of the flow or fluid are combined together through these dimensionless

numbers to see their cumulative effect.

2.9.1 Reynolds number

The inertial to viscous forces ratio is termed as Reynolds number (Re) and

mathematically we write:

2

Re/

ρ

µ ν= =R R

R

U U L

U L

(2.43)

Here , , ,RU L ρ µ and ν represent characteristic velocity, reference length, density,

viscosity and kinematic viscosity respectively. This number plays very important role

in solving the Navier Stokes equations. If Re is small, the viscous forces are large

compared to the inertial forces, so the viscous forces will reduce any disturbance that

arise in the flow and the flow will tend to be laminar. In contrast, if Re is large, the

inertial forces would be dominating the viscous forces i.e., the disturbances that arise

in the flow will grow and the turbulent flow will tend to develop.

2.9.2 Prandtl number

It represents a parameter which approximates the kinematic viscosity to thermal

diffusivity ratio. It is written in the form

Prν

α= . (2.44)

Here α represents thermal diffusivity and ν signifies fluid kinematic viscosity.

Prandtl number explains the influence of fluid thermo-physical characteristics on

transfer of heat and is combined with other dimensionless numbers for determining

coefficient of heat transfer between two layers. It also shows the relation of fluid

viscosity to that of heat conduction.

2.9.3 Skin friction coefficient

The skin friction co-efficient is defined by

2

,1

2

wfC

U

τ

ρ ∞

= (2.45)

“Preliminaries”

22

where wτ is the local shear stress from the wall, ρ the fluid density and U∞ is

velocity of free stream (velocity taken outside the boundary layer).

2.9.4 Nusselt number

Convection is one of the basic mechanisms of heat transfer and we can quantify it

using the coefficient of heat transfer. This quantity is then converted to a

dimensionless parameter known as Nusselt number Nu . If convection occurs then it

serves to measure the improvement of heat transfer.

If hC represents the coefficients of heat transfer, wT is the wall temperature whereas

T denotes the fluid temperature, then the convective heat transfer from the wall

depends depend upon the magnitude of ( ) −h wC T T . Whereas if heat transfer is

because of conduction solely, the Fourier's law implies that heat transfer rate

measures the quantity( ) wk T T

L

−. The Nusselt number is described mathematically as

follows:

( )( )

.

h w h

w

C T T C LNu

k T T k

L

−= =

−

(2.46)

It quantifies the expression for proportion of convection to conduction heat transfer

rates.

2.9.5 Sherwood Number

The dimensionless number in mass convection is known as Sherwood number. It is

defined by

massh L

ShD

= (2.47)

where massh is the mass transfer coefficient and D is the mass diffusivity. The

Sherwood numbers is the measure of effectiveness mass transfer at the surface

respectively.

“Preliminaries”

23

2.9.6 Mixed convection parameter

In mixed convection flows, the parameter λ is defined as the ratio of buoyancy forces

to the inertial forces.

2 2

( ).

Re

w

w

g T T LGr

U

βλ

−= = (2.48)

This is significant to note here that for natural convection flows, it is the buoyancy

force which provides the inertial force but for the case of mixed convection, the

inertial force exists in the form of free stream or that in the form of movement of the

boundary.

“Mixed Convection Flow over an Unsteady Stretching…”

24

Chapter 3

3. “Mixed Convection Flow over an Unsteady Stretching Surface in a Porous Medium with Heat Source”


25

In this chapter we discuss the unsteady mixed convection of a viscous fluid. The flow

is due to heated / cooled semi-infinite solid surface. An incompressible fluid saturates

the porous medium with heat source. Numerical and series solutions are obtained and

compared. Impacts of various parameters on the flow quantities are discussed.

3.1 Mathematical formulation

We consider an incompressible viscous fluid in a half space 0y > . The sheet is

assumed to be placed vertically in porous medium which follows the Darcy Brinkman

model. The flow geometry is shown in Fig 3.1.

Fig. 3.1: Physical model in coordinate system

Employing the boundary layer as well as Boussinesq approximations, we can express

the unsteady two-dimensional Navier-Stokes together with the energy equations as

∂ ∂+ =∂ ∂u vx y

0 , (3.1)

2

2( ),

u u u uu u g T T

t x y y K

νν β ∞

∂ ∂ ∂ ∂+ + = − + −

∂ ∂ ∂ ∂v (3.2)

2

2,

p

T T T T qu

t x y y cα

ρ

′′′∂ ∂ ∂ ∂+ + = +

∂ ∂ ∂ ∂v (3.3)

Subject to the conditions

O

O

x

y

y

x

Uw

Uw

(a) Flow configuration of

assisting/aiding flow

(b) Flow configuration of

opposing flow

T∞ T∞

Tw Tw

Momentum

boundary layer Thermal

boundary layer

Momentum

boundary layer

g

g

Thermal

boundary layer


26

U ( , ), 0, = ( , ), at 0,

0, as .

w wu x t T T x t y

u T T y∞

= = =

→ → → ∞

v (3.4)

Here u , v being velocity components, T is the temperature of the fluid, K is

permeability of porous medium, t is the time,α is the thermal diffusivity and ν the

kinematic viscosity. q′′′ is the rate of heat generation / absorption per unit volume. The

form of q′′′ is taken as

( , )[ ( ) ( )],w w

kU x tq A T T B T T

xν∗ ∗

∞ ∞′′′ = − + − (3.5)

where A∗ is a space-dependent, whereas B∗ is a temperature-dependent heat

generation and / or absorption parameters. Mathematically, both are positive for heat

source and negative for heat sink. Further we assume that stretching

velocity ( , )wU x t and surface temperature ( , )wT x t are given by

2( , ) , ( , ) ,

1 (1 )w w

ax bxU x t T x t T

ct ct∞= = +

− − (3.6)

where 0a > and ( 0)c > are constants having dimension 1/time (t), where t here stands

for time such that the product 1ct < . The dimension of b is Temperature (T) / L,

where L is the reference length. It is noted that 0b > and 0b < respectively

correspond to the aiding and opposing flow whereas 0b = shows the forced

convection limit.

We now define

1 12 2( ) ( ), ( ) , ( ) ,

1 (1 )w

T Ta axf y

ct T T ct

νψ η θ η η

ν∞

∞

−= = =

− − − (3.7)

where

, .uy x

ψ ψ∂ ∂= = −

∂ ∂v (3.8)

Substituting Eq. (3.7) into Eqs. (3.2) and (3.3) we get

2 * 1( ) 0,2

f ff f f f dfα η λθ′′′ ′′ ′ ′ ′′ ′+ − − + + − = (3.9)


27

* * *1 1 ( ) (2 ) 0,2

A f B f fpr

θ θ θ θ α θ ηθ′′ ′ ′ ′ ′+ + + − − + = (3.10)

together with the boundary conditions

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

( ) 0, ( ) 0.

f f

f

θ

θ

′= = =

′ ∞ = ∞ =

(3.11)

The primes represent the differentiation of the functions with respect to η . 3

2

( )wx

g T T xGr

β

ν∞−= and Re wx

U x

ν= are respectively the local Grashof and Reynolds

numbers, λ is the buoyancy/mixed convection parameter 2 .Rex

x

Grλ =

0λ > and 0λ < correspond to the assisting and opposing flow. 1

dD

= , where

Rex xD Da= in which 1

2 2

(1 )x

K ctKDa

x x

−= = is the local Darcy parameter and 1K

being the initial permeability. *c

aα = is the dimensionless parameter to representing

the unsteadiness effects and Prν

α= the Prandtl number. The skin friction denoted

byfC whereas local Nusselt number xNu defined as

2

2,wf

w

Cu

τ

ρ=

(3.12)

,( )

wx

w

xqNu

k T T∞=

−

(3.13)

where wτ and wq from the surface are

0

,w

y

u

yτ µ

=

∂=

∂

(3.14)

0

,wy

Tq k

y=

∂= −

∂

(3.15)

with µ the dynamic viscosity, k is the thermal conductivity. From transformations

(3.7), we get

12

1Re (0),

2f xC f ′′=

(3.16)


28

12Re (0).x xNu θ

− ′= − (3.17)

3.2 Solution of problem

3.2.1 Numerical solution

Eqs. (3.9) and (3.10) can be written as

2 * 1( ( ) ),2

f ff f f f dfα η λθ′′′ ′′ ′ ′ ′′ ′= − − − + + − (3.18)

*1 1( ) Pr( ) (2 ),Pr 2

A f B f fθ θ θ θ α θ ηθ∗ ∗′′ ′ ′ ′ ′= − + + − + + (3.19)

and the corresponding conditions are

1 2(0) 0, (0) 1, (0) , (0) 1, (0) ,f f f α θ θ α′ ′′ ′= = = = = (3.20)

where1α and 2α being the unknown initial conditions. These missing conditions are

obtained using the nonlinear shooting method.

3.2.2 Perturbation solution for small parameter *α

We take both λ and α small, and let mλ ε= and (1)m O= , *α ε= .The Eqs. (3.9)

and (3.10) yield

2 1( ) 0,2

f ff f f f m dfε η εθ′′′′′ ′′ ′ ′′+ − − + + − =

(3.21)

1 1( ) (2 ) 0.

Pr 2A f B f fθ θ θ θ ε θ ηθ∗ ∗′′ ′ ′ ′ ′+ + + − − + =

(3.22)

Expanding f andθ in the powers ofε as follows:

( ) ( ),nn

f fη ε η=∑ (3.23)

( ) ( ),n

nθ η ε θ η=∑ (3.24)

the zeroth order system is


29

2

0 0 0 0 0 0f f f f df′′′ ′′ ′ ′+ − − = (3.25)

0 0 0 0 0 0 0

1( ) 0,

PrA f B f fθ θ θ θ∗ ∗′′ ′ ′ ′+ + + − =

(3.26)

with

0 0 0

0 0

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

( ) 0, ( ) 0.

f f

f

θ

θ

′= = =

′ ∞ = ∞ =

(3.27)

The exact analytic solution of Eq. (3.25) satisfying the given condition is

0

1( ) (1 ),cf e

c

ηη −= − (3.28)

where 1 .c d= + Substituting Eq. (3.28) in Eq. (3.26) and making use of the Padé approximation, the

temperature 0θ is

2

0 2 3 4 5 6

1.0 0.60 0.21( ) .

1.0 1.23 0.71 0.22 0.04 0.008 0.001

η ηθ η

η η η η η η

+ +=

+ + + + + +

(3.29)

First order systems are

1 1 1 1 0

1 1(1 ) ( 2 ) 0,

2

c c c c cf e f d e f ce f m e c e

c

η η η η ηθ η− − − − −′′′ ′′ ′+ − − + − + − + = (3.30)

1 1 1 0 1 1 0 0

1 1 1(1 ) ( ) ( ) ( ) 2 0.

Pr Pr Pr 2

c cB Ae e f f

c

η ηθ θ θ θ η θ θ∗ ∗

− −′′ ′ ′ ′+ − + − + − + − − = (3.31)

The resulting expressions of 1f and 1θ are

2 4 5 6 7

1( ) 0.13 0.27 0.009 0.001 0.0002 ,f η η η η η η= − + − + (3.32)

2

1 2 3

1.35 0.30( ) ,

1.0 0.31 0.07 0.03

η ηθ η

η η η

− +=

+ + +

(3.33)

Finally, the two term perturbation solution of Eqs. (3.21) and (3.22) are written as


30

0 1( ) ( ) ( ),f f fη η ε η= + (3.34)

0 1( ) ( ) ( ),θ η θ η εθ η= + (3.35)

or

2 4 5 6 71( ) (1 ) (0.13 0.27 0.009 0.001 0.0002 ),cf ec

ηη ε η η η η η−= − + − + − + (3.36)

2

2 3 4 5 6

2

2 3

1.0 0.60 0.21( )

1.0 1.23 0.71 0.22 0.04 0.008 0.001

1.35 0.30( ).1.0 0.31 0.07 0.03

η ηθ η

η η η η η η

η ηε

η η η

+ += +

+ + + + + +

− +

+ + +

(3.37)

3.3 Results and discussion

The effects of various governing parameters on the velocity and temperature profile,

skin friction and Nusselt number are depicted. In Table 3.1, the results of analytical

and numerical solutions are compared, showing agreement. In order to compare the

results of existing literature, we take * * * 0d A Bα λ= = = = = in our Eqs. (3.9) and

(3.10) the comparison with the works given in references [9, 10, 20, 29] is made (see

Table 3.2). In Tables 3.3 and 3.4, the skin friction Nusselt number, for different values

of Pr and d , are shown and is also compared with [20]. The comparisons shown in

the Tables 2-4 now clearly indicates an excellent agreement. The results discussed are

obtained through shooting technique (Figs. 3.2-3.7).

The graphs of skin friction and Nusselt number are depicted in the Figs. 3.2 (a)

and 3.2 (b). From the graph one can see that there is an increase in the skin friction for

the case of aiding flow ( 0)λ > and the behavior is opposite for opposing flow ( 0)λ < .

Flow and Heat Transfer Over Stretching and Shrinking...

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