by ahmer mehmood

PURELY ANALYTIC SOLUTIONS TO SOME

MULTIDIMENSIONAL VISCOUS FLOWS WITH HEAT

TRANSFER

By

AHMER MEHMOOD

DEPARTMENT OF MATHEMATICS

QUAID-I-AZAM UNIVERSITY

ISLAMABAD, PAKISTAN

2010

PURELY ANALYTIC SOLUTIONS TO SOME

MULTIDIMENSIONAL VISCOUS FLOWS WITH HEAT

TRANSFER

By

AHMER MEHMOOD

Supervised By

DR ASIF ALI



ISLAMABAD, PAKISTAN

2010

Purely Analytic Solutions to Some Multidimensional Viscous

Flows with Heat Transfer

by

Ahmer Mehmood

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DEPARTMENT OF MMAATTHHEEMMAATTIICCSS


ISLAMABAD, PAKISTAN



ISLAMABAD, PAKISTAN

(2010)

DECLARATION

I Ahmer Mehmood s/o Namet Ali Registration No. 1669-MA/Ph.D. 2005, a student of

Doctor of Philosophy at Quaid-i-Azam University, Islamabad Pakistan, do hereby solemnly

declare that the thesis entitled “Purely Analytics Solutions to Some Multidimensional Viscous

Flows with Heat Transfer” submitted by me in partial fulfillment of the requirements for Doctor

of Philosophy degree in mathematics is my original work and has not been submitted and shall

not, in future, be submitted by me for obtaining any degree from this or any other University or

Institution.

Date: ____________ Signature________________

(Ahmer Mehmood)

APPROVAL SHEET

The thesis attached hereto, titled “Purely Analytics Solutions to Some Multidimensional

Viscous Flows with Heat Transfer” proposed and submitted by Mr. Ahmer Mehmood in partial

fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics is hereby

accepted.

Dr Asif Ali

Supervisor

Viva Voce Committee

___________________ ________________________

External Examiner External Examiner

___________________

Prof. Dr Muhammad Ayub

Chairman

Date: _________________

Dedicated to

My Parents, who have always been a source of Inspiration, Zeal

and Strength for me.

Memories of my late Sister.

All members of my family.

ACKNOWLEDGEMENTS

Praise is to almighty Allah, creator of the heavens and earth and Lord of lords, who gave

me the potential and ability to complete this dissertation. All of my respect goes to the

holy prophet Muhammad (Peace be upon him) who emphasized the significance of

knowledge and research.

Academic research involves not only the efforts of the researcher but it also requires

generous and sincere cooperation of many other accessories like different departments,

research institutes, libraries and consulting personalities. The present Ph. D Thesis is

likewise the result of valuable input from various quarters.

First of all, I would like to pay tributes and thanks to my parents and family whose

constant, unending and countless prayers buoyed me up.

I am deeply indebted of my all teachers at QAU whose teachings have brought me to this

stage of academic zenith. In particular I am very thankful and obliged to my supervisor

Dr Asif Ali for his dynamic supervision during the course of my research work. I am

grateful to Prof. Dr Muhammad Ayub, chairman department of mathematics QAU, for

providing a conducive and pleasant atmosphere for studies and research. I would like to

thank Dr. Zawar Hussain, Dr. Muhammad Riaz, Mr. Nasir Abbass, Mr. Muhammad

Akbar, and Mr. Muhammad Saleem for always being ready to listen me and provide

favor. I am also thankful to Dr Iftikhar, Dr Akhlaq, Dr Sajid, Dr Nasir, Dr Tariq, Dr

Faisal, Dr Mazhar, Dr Sherbaz, Dr Raheel, Dr. Zaheer, Mr. Niaz, Mr. Amanullah dar, Mr.

Muhammad Shah, Mr. Abdu-r-Rahman, Mr. Amir Mann, and Mr. Hamid Rasool for their

good wishes and moral support. I offer tribute to all of my teachers and friends

throughout my academic career especially Ali, Murtaza, Imran, Ghazanfar, Ishfaq,

Shakoor, Ajmal, Shafqat, Aqeel, and Sher. I would further like to extend my thanks to a

bunch of students from statistics and social sciences that had always been around me with

love and respect. Though last but not the least, I would like to acknowledge the moral

support and encouragement from my worthy teacher Dr Qari Naseer Ahmad (late), may

his soul rest in heaven.

AAHHMMEERR MMEEHHMMOOOODD

AbstractExact solution of Navier-Stokes equations is possible only for very simple flow situations

such as unidirectional flows. Due to the nonlinear nature of these equations their analytic

solutions are rare and the situation gets worse in the case of unsteady and multidimensional

flow problems.

In this thesis we report highly accurate and purely analytic solutions to some steady/unsteady

multidimensional viscous flows over flat surface. Heat transfer analysis has also been carried

out where the flat surface is considered as a stretching sheet. In each case the skin friction

and the rate of heat transfer has been reported. The issue of cooling of stretching sheet in the

presence of viscous dissipation has been discussed in detail.

We have considered multidimensional flows of viscous fluid over a flat plate in different

flow situations such as flow over an impulsively started moving plate; flow over a stretching

sheet, viscous flow in a channel of lower stretching wall, and the channel flow with lower wall

as a stretching sheet in a rotating frame. In all the above mentioned flow situations similarity

transformations have been used in order to normalize the problem. The reduced governing

equations are then solved analytically.

We have used homotopy analysis method to solve the governing nonlinear differential equa-

tions. The results are purely analytic and highly accurate. The accuracy of results has been

proved by calculating the residual errors and (or) giving the comparison with the existing re-

sults. For unsteady flows it is worthy to mention here that our analytic solutions are uniformly

valid for all time in the whole spatial domain.

1

Contents

1 Introduction 5

1.1 Introduction and brief history . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.1 Fundamental laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.2 Boundary-layer governing equations . . . . . . . . . . . . . . . . . 18

1.2.3 Important definitions . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.4 Homotopy analysis method . . . . . . . . . . . . . . . . . . . . . 22

2 Unsteady boundary-layer flow over an impulsively started surface 26

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Unsteady flow past a moving rigid plate . . . . . . . . . . . . . . . . . . 27

2.2.1 Mathematical description of the problem . . . . . . . . . . . . . . 27

2.2.2 Analytic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.3 Discussion on results . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Unsteady flow past an impulsively started porous plate . . . . . . . . . . 41

2.3.1 Mathematical description of the problem . . . . . . . . . . . . . . 41



2.3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2

3 Unsteady boundary-layer flow over an impulsively stretching surface

with heat and mass transfer 52

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Mathematical description of the problem . . . . . . . . . . . . . . . . . . 53

3.3 Solution by homotopy analysis method . . . . . . . . . . . . . . . . . . . 55

3.3.1 Convergence and accuracy of results . . . . . . . . . . . . . . . . 59


3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Generalized 3D viscous flow and heat transfer over a stretching plane

wall 67

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . 68

4.2.2 Complete analytic solution . . . . . . . . . . . . . . . . . . . . . . 70

4.2.3 Convergence and accuracy of results . . . . . . . . . . . . . . . . 74

4.3 Heat transfer analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


4.4 Discussion on results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Heat transfer analysis of 3D viscous flow in a channel of lower stretching

wall 87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.1 Dynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.2 Analytic homotopy solution . . . . . . . . . . . . . . . . . . . . . 90

5.2.3 Convergence of HAM solution . . . . . . . . . . . . . . . . . . . . 94

5.2.4 Graphical results and discussion . . . . . . . . . . . . . . . . . . . 96

3



5.3.2 Convergence of HAM solution . . . . . . . . . . . . . . . . . . . . 106


5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Unsteady 3DMHDboundary-layer flow over impulsively started stretch-

ing sheet 115

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Analytic HAM solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.1 Convergence and validity of HAM solution . . . . . . . . . . . . . 124


6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Rotating flow in a channel of lower stretching wall with heat transfer 132

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2 Flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2.2 Analytic solution by HAM . . . . . . . . . . . . . . . . . . . . . . 136

7.2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 139


7.3.1 Energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.3.2 HAM solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145


7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8 Conclusion 156

4

Chapter 1

Introduction

This chapter is devoted to the introduction of the thesis. Introduction

is accompanied by a brief historical background of the problem. Some basic

governing laws for viscous fluid and an introduction to the homotopy analysis

method are also appended to this chapter.

1.1 Introduction and brief history

From the beginning of human history man had been solving the problem of fluid resis-

tance, for example, by introducing the streamlined weapons and fin-stabilized arrows.

Even the introduction of arrow-shaped weapons can be regarded as the foundation of

modern aerodynamics. The earliest contributions in the development of fluid mechanics

were made by Archimedes (287 B.C− 212 B.C) in the form of postulates of buoyance.

With the passage of time the development in fluid mechanics turned more and more sci-

entific till Leonardo da Vinci who in 1500 introduced a mass conservation law for incom-

pressible fluids in one-dimensional motion. In 1644 Evangelista Torricelli (1608− 1647)came forward with his famous theorem stating that the efflux velocity of a fluid flowing

from a hole in a tank is equal to the velocity which a fluid particle could attain in free

fall from its surface. This theorem is of fundamental importance in inviscid theory. The

5

above developments are not directly related to the viscous theory which reflects that the

researchers of that time had been considering the fluid as inviscid or ideal. However,

in 1687 Newton introduced his famous law of viscosity by stating that "the resistance

which arises from the lack of lubricity in the parts of a fluid -other things being equal-

is proportional to the velocity which the parts of the fluid are being separated from each

other". Therefore, the fluids owing such kinds of relationship are termed as Newtonian,

in his honour. But unfortunately, Newtons theory of viscous flow failed to receive a gen-

eral acceptance from the researchers in the field and the people kept on working on the

inviscid flows. In 1738 Daniel Bernoulli proposed a relationship between pressure and

acceleration of the fluid which was latter modified by Leonhard Euler in 1755 commonly

known as Bernoulli equation. Bernoulli equation is still regarded as fundamental to the

theory of hydrodynamics. Later on, Navier in 1827 [1] and Stokes in 1845 [2] introduced

the frictional resistance term to the Euler’s equation of inviscid fluid. In 1883 [3] Osborn

Reynolds categorized the viscous flows in to two types, namely, the flows with slow mo-

tion of fluid and the other with very high velocity of the fluid. Reynold also predicted

that the classification of viscous flows in to two types does not depend upon the fluid ve-

locity only but it actually depends upon the parameter V d/ν (= Re, named as Reynolds

number in his regard). Reynolds number is still regarded as a fundamental parameter in

viscous flow theory.

Even after the development of Navier-Stokes theory and the birth of Reynolds num-

ber the world was still unable to resolve the d’Alembert paradox showing that a body

immersed in frictionless flow would have zero drag. The d’Alembert paradox may be

rephrased in a descriptive manner as, that ignoring the small air resistance in theory re-

sulted in the prediction of zero resistance to the motion, which is against the experiment.

Prandtl himself assigned the task to resolve the d’Alembert paradox. He started com-

puting the motion of fluid by assuming the fluid friction so small that its effect could be

neglected every where except for the regions where large velocity differences are present.

After having completed his study Prandtl reached a conclusion that in such type of flu-

6

ids there is a very thin layer adjacent to the solid boundary where the viscous effects

are prominent (not to be negligible). Prandtl termed this thin region as boundary-layer,

or transition-layer. Prandtl presented his first paper on boundary-layer concept at the

Third International Mathematical Congress at Heidelberg in 1904 [4] . Prandtl’s concept

of boundary-layer enabled theory to meet the experiment. Till the 20th century there

had been a great schism present between the two important branches of fluid mechan-

ics, namely, the theoretical hydrodynamics and the hydraulics. Prandtl has the credit of

establishing the high degree of correlation between these two divergent branches of fluid

mechanics based on theory and experiment respectively. After Prandtl, it had become

clearer that the gap between the results of theory and experiment was basically due to

the ignorance of fluid friction in the theory. A number of text books have been written

on boundary-layer theory but the most remarkable one is by Schlichting [5] .

The concept of boundary-layer in viscous flows is fundamental for the understanding

of aerodynamical properties of the flow such as the wall friction or the dynamic drag.

Due to the boundary-layer theory proposed by Prandtl it became possible to calculate

the wall drag accurately. Blasius [6] in 1908 investigated the boundary-layer flow past a

flat plate at zero incidence. Boltze [7] investigated the boundary-layer flow around a body

of revolution and Hiemenz [8] studied separation on a circular cylinder in 1911. Later in

1912 Töpfer [9] refined the computations of Blasius. Blasius [6] for his two-dimensional

boundary-layer flow past a flat plate calculated the wall friction to be 0.332057. Boundary-

layer flow over a moving surface finds important industrial applications such as in the

manufacturing of food and paper; plastic sheets extrusion, application of coating of paints

layers on surface etc. Keeping this fact in mind Sakiadis in 1961 [10] considered the similar

two-dimensional steady boundary-layer flow over a moving flat plate in a still fluid at

zero incidence. The skin fiction as reported by Sakiadis was 0.445. Sakiadis predicted

that the drag force for his flow problem was 34% greater than the drag force for the

Blasius flow [6] . The theoretical results of Sakiadis [10] were experimentally confirmed

by Tsou et al. [11] . In both ([10] and [6]) either fluid or plate is assumed to be at

7

rest. However, more practical situation might be the one in which both the fluid and

the plate are moving . Such a flow situation seems to be more practical in studying

the aerodynamical properties of the flow and one may assume the moving plate as a

wing of an aircraft. Being motivated from these applications, Klemp and Acrivos [12]

considered two-dimensional boundary-layer flow of a viscous fluid over a flat surface

moving in a stream of constant velocity. Hussaini et al. [13, 14] determine similarity

solution of the boundary-layer equations with upstream moving wall. Later on, in 2003

Fang [15, 16] studied the similarity solution of boundary-layer flow and heat transfer for

steady case. A most important phenomenon in the study of boundary-layer flows is the

mass suction/injection at the solid surface. The mass transfer phenomenon including

the suction and injection to the Blasius flow was studied by Schlichting and Bussmann

in 1943 [17] . Afterwards in 1982, Dye [18] investigated the similarity solution for the

case of suction/injection. In 1990 Vajravelu and Mohapatra [19] studied the effect of

mass injection on the moving wall. In [19] the authors reported that the dynamical drag

reduces by introducing the wall injection. Recently, in 2003 Fang [16] extended the work

of previous investigators to general situations which include the mass suction/injection

to the case when wall and the free stream have the same direction of motion. All the

above mentioned studies deal with the steady flow, however, there are situations in which

the flow is essentially unsteady. The unsteadiness might be due to impulsive start or due

to time dependent oscillations. In such situations it is quite difficult to get an analytic

solution which is uniformly valid for all time. However, there are few studies in which

authors have considered viscous flow over an impulsively started flat plate [20− 24] , buttheir solutions are not uniformly valid for all time. To the best of our knowledge, there

had been reported no purely analytic solution for unsteady boundary-layer flow over a

moving wall which is uniformly valid for all time in the whole spatial domain. We have

extended the two-dimensional boundary-layer flow of viscous fluid over a moving wall in

a free stream of constant velocity to the unsteady case. We have considered two cases,

namely, the flow over a rigid flat surface and the flow over a porous moving wall. The

8

work has been published in two separate papers [25, 26] published in different journals.

We have presented this work in chapter 2 of this thesis.

Another important class of boundary-layer flows over flat surface is one in which the

moving plate is replaced by a stretching sheet. The flow of a viscous fluid over a stretching

sheet has important technological applications in metallurgical and chemical industries

such as; polymer extrusion, drawing of copper wires, continuous stretching of plastic

films, artificial fiber, hot rolling, wire drawing, glass fiber, metal extrusion and metal

spinning. In the manufacturing of the polymer sheets the melt issues from a slit and is

subsequently stretched to achieve the desired thickness. The achievement of the desired

characteristics by the final product depends strictly on the rate of stretching and the

rate of cooling. Industrially, the study of heat transfer phenomenon from a stretching

surface to the ambient fluid is of great importance in several engineering applications

such as materials manufactured by extrusion processes and heated materials travelling

between a feed roll and a wind up roll or on a conveyer belt. In the manufacturing of

the polymer sheets the sheets are some times passed through an ambient fluid in order

to cool them. In such processes the study of heat transfer phenomenon is important in

order to control the rate of cooling. Being inspired from such an important applications

Crane [27] initiated the study of viscous flow over a stretching surface. Crane happened

to be lucky enough in reporting a closed form solution for his problem. Crane considered

a steady two-dimensional boundary-layer flow over a uniformly stretching sheet. It was

later observed that the Crane’s problem admits a closed form solution even if other

physical features such as suction, MHD, viscoelasticity etc. are taken into account (see

for instance Andersson [28] , Troy et al. [29] , Ariel [30]). In his paper [27] , Crane also

reported the heat transfer analysis for his steady flow. After Crane the problem enjoyed

a great attention from researchers in the field. Several extensions have been made by

introducing many interesting features such as suction/blowing; MHD, viscoelasticity of

the fluid, etc. Gupta and Gupta [31] investigated heat and mass transfer on the stretching

surface in the presence of suction or blowing. Further, in 1982 Carragher and Crane [32]

9

investigated heat transfer in the flow over a surface being stretched in its own plane with a

velocity varying linearly with distance from a fixed point in the case when the temperature

difference between the surface and the ambient fluid is proportional to the power of the

distance from the fixed point. Banks [33] investigated the similarity solution to the

boundary-layer equations for a stretching wall in 1983. Further, Banks and Zaturska

[34] reported the eigensolutions to the boundary-layer flow over a stretching surface.

Chen and Char [35] investigated the heat transfer over a stretching sheet with suction or

blowing. Dandapat and Gupta [36] extended the heat transfer analysis over a stretching

sheet to the non-Newtonian fluids by considering a viscoelastic fluid over a stretching

wall. Further contributions on the flow and heat transfer over a stretching sheet can be

found in (Dutta et al. [37] , Magyari and Keller [38] , Rollin and Vajravelu [39] , Ali [40] ,

Elbashbeshy [41] , Grubka and Boba [42]). In all above mentioned studies authors have

considered steady flow. Unsteady tow-dimensional boundary-layer flow over a stretching

sheet has also been considered by some authors, such as; Seshadri et al. [43] , Pop and

Na [44] , Wang et al. [45] , Nazar et al. [46] etc. In [46] , Nazar et al. studied unsteady

boundary-layer flow over an impulsively started stretching surface and obtained a first

order perturbation solution. There had been no purely analytic solution available for such

kind of flows which are uniformly valid for all time till Liao [47] . In [47] Liao presented

a purely analytic solution for unsteady flow over an impulsively stretching sheet which is

uniformly valid for all time. Following Liao we have studied the cooling of a stretching

sheet in unsteady boundary-layer flow in the presence of mass transfer phenomenon [48] .

In our paper [48] we have shown our solution to be uniformly valid for all time in the

whole spatial domain. We present this problem in chapter 3 of this document.

In practical applications MHD is observed to be useful in controlling the boundary-

layer thickness. In industrial applications the cooling of continuous strips is controlled

by drawing them in an electrically conducting fluid subject to a magnetic field. Heat

transfer analysis of MHD fluid over a uniformly stretching sheet was investigated by

Chakrabarti and Gupta [49] . Andersson [50] studied the MHD flow of a viscoelastic fluid

10

past a stretching sheet. Further, studies on two-dimensional MHD boundary-layer flow

can be seen in the references [51− 54] and the references there in.The study of boundary-layer flows over a flat stretching surface was extended to the

three-dimensional case byWang in 1984 [55] .Wang considered a steady three-dimensional

boundary-layer flow over a stretching sheet where the sheet was assumed to be stretching

in two lateral directions. Wang introduced similarity transformations to normalize the

governing Navier-Stokes equations and determined an approximate solution and termed

it as exact solution. In his paper Wang [55] determined a numerical solution (for three-

dimensional flow) guided by a perturbation solution obtained around the two-dimensional

flow. The same problem [55] was reconsidered by Ariel in 2003 [56] in which he also

reported a numerical and an analytic solution to the three-dimensional boundary-layer

equations. Recently, Hayat et al. [57] extended the Wang’s three-dimensional boundary-

layer flow over a stretching sheet for the MHD fluid and obtained an analytic solution

for it. The problem has also been extended to the non-Newtonian case by Hayat and his

co-worker [58] . In both papers [57, 58] the authors have reported purely analytic solutions

using homotopy analysis method. Heat transfer analysis of three-dimensional boundary-

layer flow over a stretching sheet (as considered by Wang [55]) has been performed

by Mehmood and Ali [59] . In [59] we have investigated the process of cooling of the

plate in the presence of viscous dissipation. The contents of the paper [59] have been

presented in chapter 4 of this thesis. Three-dimensional boundary-layer flow over a

stretching sheet has also been extended to the unsteady case by some researchers such

as: Lakshmisha et al. [60] , Aboeldahad and Azzam [61] , Takhar et al. [62] etc. In [62] ,

Takhar et al. used the similarity transformations obtained by Williams and Rhyne [63] to

normalize the problem and solved the resulting system of ordinary differential equations

numerically. In [62] the authors claimed that their results are uniformly valid for all time

but unfortunately the observations collected from their numerical results are contrary

to the actual situation. Therefore, we regard the results reported in [62] as incorrect

which need to be corrected. We have corrected the results of [62] by reporting a purely

11

analytic solution to the considered problem which is uniformly valid for all time. We

have published the corrigendum [64] in the same journal i.e. Acta Mechanica in 2008.

We present this problem in chapter 6 of this thesis.

Almost in all the above mentioned studies the authors have considered the flow over a

flat surface with unbounded domain. Very little attention has been given to the channel

flows where the lower plate is a stretching sheet. However, there are few studies available

in literature in which the authors have considered channel flows over a stretching sheet. A

most remarkable contribution in this category was made by Borkakoti and Bharali [65] in

1983. They considered the hydrodynamic flow and heat transfer in viscous fluid bounded

by two parallel plates where the lower plate was assumed to be stretching at a constant

rate. Further, the lower plate was assumed to be heated at a constant temperature so that

heat can flow form plate to the surrounding fluid. To control the flow and rate of heat

transfer in the channel they also introduced constant suction/injection at the upper wall.

To enhance the flow in the channel rotation was introduced by Banerjee [66] to the flow

considered by Borkakoti and Bharali [65] . Banerjee [66] used perturbation technique and

obtained the results up to first order of approximation. His perturbation results are only

valid for small values of the Reynolds number Re (= ah2/ν) . Vajravelu and Kumar [67]

obtained analytic (perturbation) as well as numerical solution of the coupled nonlinear

system arising in MHD viscous flow between two horizontal plates in a rotating system

where the lower plate being a stretching sheet. Being inspired from these studies we

have investigated steady three-dimensional flow of a viscous fluid in a channel with lower

plate as a stretching sheet being stretched in two lateral direction [68] . The heat transfer

analysis in the presence of viscous dissipation of the flow studied in [68] is given in [69] .

In [69] detailed analysis of heat transfer phenomenon has been given with an emphasis

on the cooling of stretching sheet. We have presented these studies [68, 69] in chapter 5

of this thesis. To investigate the effect of rotation on the heat transfer phenomenon we

have studied three-dimensional flow in channel of lower stretching wall in the presence of

viscous dissipation where the plates and the fluid are assumed to be rotating in unison

12

[70] . This paper [70] has been published in Journal of Heat Transfer, Transactions of the

ASME. We have included this paper as seventh chapter in this thesis.

It is globally accepted that the Navier-Stokes (NS) equations are nonlinear in nature

and there is no general analytic solution available for them. The nonlinearity in NS-

equations is due to the presence of inertial terms at the left hand side of the momentum

conservation law. It is observed that the boundary-layer assumption leads to the simplifi-

cation of the NS-equations but even then the nonlinearity survives which create troubles

for the investigator. That’s why people use to look for some kind of approximate solu-

tions for their nonlinear problems. Perturbation techniques [71, 72, 73] have largely been

used by engineers and scientists. Numerical analysts are engaged in introducing new nu-

merical techniques by minimizing the errors [74] . Different analytic techniques have also

been devised for nonlinear differential equations [75] . Among all the available analytic

techniques for nonlinear equations the homotopy analysis method [76] is the most power-

ful analytic technique and has the potential to deal with complicated nonlinear problems

arising in science and engineering. Before going further in homotopy analysis method it

is worthy to mention here the limitations and problems with the perturbation techniques.

Certainly, perturbation methods are applicable only to those problems where a small or

large parameter is available. Therefore, the problems with the absence of any small or

large parameter could not be dealt with perturbation techniques. Further, for unsteady

problems perturbation is usually applied for small or large values of time meaning that

the perturbation methods fail to give an analytic solution which is uniformly valid for all

values of time. To cope with these problems Liao stood with his new analytic technique,

namely, homotopy analysis method (HAM) [76] . Liao used the idea of homotopy [77] , a

branch of topology [78] and introduced his analytic technique in his doctoral thesis [79]

in 1992. Based on the idea of homotopy, Liao named it as homotopy analysis method.

Different from perturbation techniques homotopy analysis method provides a great free-

dom to adjust and control the errors and the convergence of the solution. The basic idea

of HAM is given in section 3 of this chapter.

13

Liao himself applied his newly developed analytic technique to many nonlinear prob-

lems such as; nonlinear water waves [80] , similarity boundary-layer flows [81] , Cheng-

Chang equation [82] , Von Karman swirling flow [83] , Blasius problem [84] , and so on

(for further applications of HAM by Liao the reader is referred to see the references

[85− 92]). The other pioneering contributions in the work with HAM are made by Ayub

and Rasheed [93] , Hayat et al. [94− 96] , Asghar et al. [97] etc. Recently Liao appliedthe homotopy analysis method to the unsteady boundary-layer flow over an impulsively

stretching sheet [47] . The solution obtained by Liao is uniformly valid for all time in

the whole spatial domain. In [47] Liao claimed that there had been no such analytic

solution available in literature which is uniformly valid for all time. Further studies

on unsteady flows dealt with HAM can be found in [98− 102] . In all these studies thesolutions are uniformly valid for all time in the whole spatial domain. The literature

on homotopy analysis method is growing day by day. Therefore, it is not possible to

cite all the papers in this document but some significant contributions can be found in

references [103− 117] . We would like to mention here that we have applied homotopyanalysis method to all the problems presented in this thesis. For unsteady problems it

is shown that the solutions are uniformly valid for all time. We have divided the thesis

into eight chapters. The layout of the thesis is as follows:

In chapter 2 we have presented unsteady two-dimensional boundary-layer flow past a

moving plate, in chapter 3 unsteady two-dimensional boundary-layer flow of an incom-

pressible viscous fluid over a stretching sheet has been studied, chapter 4 consists of the

study of three-dimensional steady flow over a stretching sheet, channel flow of a viscous

fluid over a stretching sheet has been discussed in chapter 5, unsteady three-dimensional

boundary-layer flow over a stretching sheet has been discussed in chapter 6, in chapter

7 we have studied a three-dimensional channel flow over a stretching sheet in a rotating

frame, and finally in chapter 8 we have given the conclusion of the thesis. In all the above

mentioned studies heat transfer analysis has also been carried out in the cases where the

flat surface is assumed to be a stretching sheet. It is worthy to mention here that the

14

work presented in chapters 2−7 has been published in scientific journals of internationalrepute. The chapter wise details of the published work are given as under:

Chapter 2 :

• Unsteady boundary-layer flow due to an impulsively started moving plate, Proc.IMechE. Part G: J. Aerospace Engineering.

• Unsteady boundary-layer viscous flow due to an impulsively started porous plate,Canadian J. Physics.

Chapter 3 :

• Heat transfer analysis of unsteady boundary-layer flow by homotopy analysis method,Commun. Nonlinear Sci. Numer. Simulat..

Chapter 4 :

• Analytic solution of generalized three-dimensional flow and heat transfer over a

stretching plane wall, Int. Commun. Heat and Mass Transfer.

Chapter 5 :

• Analytic homotopy solution of generalized three-dimensional channel flow due touniform stretching of the plate, Acta Mechanica Sinica.

• Heat transfer analysis of three-dimensional flow in a channel of lower stretchingwall, J. Taiwan Institute of Chemical Engineering.

Chapter 6 :

• Corrigendum to: “Unsteady three-dimensional MHD boundary-layer flow due to

the impulsive motion of a stretching surface (Acta Mech. 146, 59−71(2001))” ActaMechanica.

Chapter 7 :

• Analytic solution of three-dimensional viscous flow and heat transfer over a stretch-ing flat surface by homotopy analysis method, ASME-J. Heat Transfer.

15

1.2 Preliminaries

1.2.1 Fundamental laws

The basic governing equations for the flow of an incompressible viscous fluid are well

known, but the exact number of fundamental equations for a particular problem depends

upon the assumptions made by the investigator. In most of the texts the authors prefer

to call the three conservation laws of physical systems as the fundamental laws [118] (for

incompressible fluid), namely,

Conservation of mass

∇ ·V = 0,

Conservation of momentum (Newtons second law)

ρDV

Dt=∇ ·T+ f ,

Conservation of energy (First law of thermodynamics)

ρCpDθ

Dt= T · L+ k∇2θ,

where ρ is the density, V is velocity vector, ∇ is the vector operator, D/Dt denotes the

substantive derivative which consists of local contribution (in non steady flow) ∂/∂t, and

the convective contribution (due to translation), T is the stress tensor, L is the velocity

gradient, f is the body force, Cp is the specific heat, θ is the temperature, k is the thermal

conductivity of the fluid, and T · L is defined as

T · L = tr (TL) ,

where tr denotes the trace of the matrix.

If one considers the magnetohydrodynamic case then the momentum conservation law

16

modifies as

ρDV

Dt=∇ ·T+ J×B,

and the number of conservation laws increases from three to seven by appending the four

Maxwell’s equations [119] , namely,

Gauss’s law

∇ ·E = 1

0ρ,

Faraday’s law

∇×E = −∂B∂t

,

Ampere’s law with Maxwell’s correction

∇×B = µ0J+ µ0 0∂E

∂t,

and

∇ ·B = 0,

which possesses no name. Here, J is the current density, B is the total magnetic field,

i.e. B = B0 + b0 where B0 is the applied magnetic field and b0 is the induced magnetic

field, µ0 is the magnetic permeability, 0 is the permittivity of free space, and E is the

electric field.

In our study we will also be going for rotating flow, therefore, it is useful to give the

governing law in this situation too. In the case of rotating flows the Newton’s second

law modifies due to the addition of Centripetal acceleration (Ω×Ω× r) and the Coriolisacceleration (2Ω×V) to the inertial part of the momentum conservation law ( for detailssee [120]). The momentum conservation law for a rotating flow can be written as

ρDV

Dt+Ω×Ω× r+ 2Ω×V =∇ ·T+ f ,

17

where Ω is the angular velocity vector and r is the position vector.

1.2.2 Boundary-layer governing equations

In developing a mathematical theory of boundary-layers, the first step is to show the

existence, as the Reynolds number Re tends to infinity, or the kinematic viscosity ν

tends to zero, of a limiting form of the equations of motion, different from that obtained

by putting ν = 0 in the first place. A solution of these limiting equations may then

reasonably be expected to describe approximately the flow in a laminar boundary-layer

for which Re is large but not infinite. This is the basis of the classical theory of laminar

boundary-layers.

Consider the Navier-Stokes equations for a two-dimensional flow

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −1

ρ

∂p

∂x+ ν

µ∂2u

∂x2+

∂2u

∂y2

¶, (1.1)

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y= −1

ρ

∂p

∂y+ ν

µ∂2v

∂x2+

∂2v

∂y2

¶, (1.2)

and the continuity equation∂u

∂x+

∂v

∂y= 0, (1.3)

where x and y are the Cartesian coordinates, u and v the velocity components in x−,and y−directions respectively and p is the pressure. Assuming that the wall is located

at y = 0. We introduce the dimensionless variables

x =x

L, y =

y

δ, u =

u

U, v =

vL

Uδ, p =

p

ρU2, t =

tU

L, (1.4)

where L is the horizontal length scale, δ is the boundary-layer thickness at x = L, which

is unknown. We will obtain an estimate for it in terms of the Reynolds number Re . U is

the flow velocity, which is aligned in the x−direction parallel to the solid boundary. The

18

non-dimensional form of the governing equations (after dropping the bars) is given by

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −∂p

∂x+

ν

UL

∂2u

∂x2+

ν

UL

L2

δ2∂2u

∂y2, (1.5)

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y=

L2

δ2∂p

∂y+

ν

UL

∂2v

∂x2+

ν

UL

L2

δ2∂2v

∂y2, (1.6)

and the continuity equation (1.3) remains unchanged. In this problem the Reynolds

number is given by

Re =UL

ν. (1.7)

Inside the boundary-layer, viscous forces balance inertia and pressure gradient forces.

In other words, inertia and viscous forces are of the same order, so

ν

UL

L2

δ2= O (1) =⇒ δ = O

³Re−1/2 L

´, (1.8)

Due to equation (1.8) the governing boundary-layer equations are

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −∂p

∂x+1

Re

∂2u

∂x2+

∂2u

∂y2, (1.9)

1

Re

µ∂v

∂t+ u

∂v

∂x+ v

∂v

∂y

¶= −∂p

∂y+

1

Re2

µ∂2v

∂x2+

∂2v

∂y2

¶. (1.10)

In the limit Re→∞, the equations (1.3) , (1.9) , and (1.10) reduce to:

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −∂p

∂x+

∂2u

∂y2, (1.11)

−∂p∂y= 0, (1.12)

∂u

∂x+

∂v

∂y= 0. (1.13)

From eq. (1.12) it is inferred that the pressure is constant across the boundary-layer.

Equations (1.11) − (1.13) govern the two-dimensional boundary-layer flow of a viscous

19

fluid. Using similar analogy one can easily derive the three-dimensional boundary-layer

equations from the full Navier-Stokes equations. The stuff presented in this subsection

is taken form the link: web.mit.edu/fluids-modules/www/highspeed_flows.

1.2.3 Important definitions

Skin friction

The viscous drag (skin friction) at the surface of a flat plate in two-dimensional motion is

obtained by simple process of integrating the shearing stress τw at the wall. The shearing

stress at the wall is given by

τw (x) = µ

µ∂u

∂y

¶y=0

,

therefore, the viscous drag D is given by

D =

Z l

0

τw (x) dx,

where l is the length of the fluid layer. In the study of boundary-layer flows a most

important quantity of physical interest is the coefficient of skin friction Cf defined by

(for two-dimensional case)

Cf (x) =τw (x)12ρU2

,

where U is the free stream velocity, and ρ is the density of the fluid. It is important

to note that the skin friction is not a fundamental quantity but simply a part of the

solution.

Nusselt number

It is a dimensionless number important in the study of convective heat transfer phenom-

enon. It is defined as the rate of convective heat transfer to the conductive heat transfer,

i.e.

Nu =qw(convective)qw(conductive)

,

20

where

qw(convective) = h (x) (θw − θ∞) ,

and

qw(conductive) =k (θw − θ∞)

x,

in which h (x) is the rate of heat transfer and x is the length of the fluid layer. Using

the Fourier’s law of heat flux the Nusselt number in the form of temperature gradient is

given by (for flat plate case)

Nu (x) = −pRexT

0 (0) ,

where T is the dimensionless temperature and Rex is the local Reynolds number. Further

details can be found in a nice text by White [121] .

Prandtl number

Prandtl number is a non-dimensional parameter of a convecting system that characterizes

the regime of convection. It is defined as the ratio of viscous diffusion rate to the thermal

diffusion rate, i.e.

Pr =viscous diffusion ratethermal diffusion rate

=µCp

k.

Eckert number

In the dissipative systems Eckert number is used to characterize the dissipation. It is a

dimensionless number defined as the ratio of kinetic energy to the enthalpy, i.e.

Ec =V 2

Cp∆T.

21

1.2.4 Homotopy analysis method

Homotopy

LetX and Y be two continuous functions of real variables R, then the family of continuous

maps fp (x) : [0, 1]×R→ R, (0 ≤ p ≤ 1) defined by

fp (x) = (1− p)X (x) + pY (x) ,

for all x ∈ R and p ∈ [0, 1] is called a homotopy. Obviously, fp (x) being a combination ofcontinuous functions is continuous. Notice that for p = 0, f0 (x) = X (x) , and for p = 1,

f1 (x) = Y (x) , thus as p varies from 0 to 1 the homotopy continuously deforms the one

function into the other.

Basic idea of homotopy analysis method

Consider a differential equation A (h (x)) = 0, where A is a nonlinear operator, x denotesthe independent variable and h (x) is an unknown variable. Let h0 (x) denotes an initial

approximation of h (x) and L denotes an auxiliary linear operator with the property

Lf = 0, when f = 0.

We then define a function Φ (x; p) such that

Limp→0

Φ (x; p) = h0 (x) ,

Limp→1

Φ (x; p) = h (x) .

Liao [76] improved the traditional homotopy by introducing a non-zero auxiliary para-

meter ~ and a non-zero auxiliary function H (x) . In this way the new homotopy results

22

in the following form:

(1− p)L [Φ (x; p,~,H)− h0 (x)] = p~H (x)A [Φ (x; p, ~, H)] , (1.14)

where p ∈ [0, 1] is the embedding parameter. Clearly, from eq. (1.14) , at p = 0 and

p = 1, we respectively have the initial guess

Φ (x; 0) = h0 (x) , (1.15a)

and the final solution

Φ (x; 1) = h (x) . (1.15b)

Thus as p varies from 0 to 1 the solution Φ (x; p) continuously deforms from initial

approximation h0 (x) to the final solution h (x) . In homotopy analysis method eq. (1.14)

is usually called as zero-order deformation equation. Let us assume that the solution

function Φ (x; p) can be expanded in the form of Taylor’s series with respect to p at

p = 0, given by

Φ (x; p) = h0 (x) ++∞Xm=1

hm (x) pm, (1.16)

where

hm (x) =∂mΦ (x; p)

∂pm|p=0 .

Notice that if the series (1.16) converges as p → 1 then the function Φ (x; 1) represents

a solution of the considered differential equation. The functions hm (x) (m ≥ 1) denotethe solutions of the mth-order deformation equation obtained by differentiating the zero-

order deformation equation m−times with respect to p at p = 0 and then dividing the

resulting equation by m! :

L £h (x)− χm−1h0 (x)¤= ~H (x)Rm (x) , (1.17)

23

where

Rm (x) =mXk=1

αkδm−k(x), (1.18)

under the definition

δn(x) =1

n!

∂nA (Φ (x; p))∂pn

|p=0, (1.19)

and

χk = 0, for k ≤ 1, (1.20)

= 1, for k > 1.

Equation (1.17) represents a system of linear differential equations which can easily be

solved for all m ≥ 1. In this way the final solution can be written with the help of eqs.(1.15b) and (1.16) in the form of an infinite series of functions, i.e.

h (x) = h0 (x) ++∞Xm=1

hm (x) . (1.21)

While using homotopy analysis method it is very much important to ensure the con-

vergence of the solution series (1.21). The convergence of the solution series (1.21) can be

controlled or made faster by the selection of suitable initial guess approximation; the lin-

ear operator, the auxiliary function, and the auxiliary parameter. Once the initial guess;

the linear operator, and the auxiliary function have been selected the convergence of the

solution series then strongly depends upon the auxiliary parameter ~. The suitable values

of ~ are determined by plotting the so-called ~−curves. The intervals on the ~−axis forwhich the ~−curve is flat are recognized as the intervals of admissible values of ~. Ourexperience shows that for all admissible values of ~ the series (1.21) is convergent but

the rate of convergence is different for different values. In order to find the appropriate

value of ~ for which the convergence of the solution series is faster can be determined

by trying different values of ~ (from the intervals of admissible values) and calculating

24

the corrections to the solution series at successive orders of approximation. The value of

~ for which the corrections become negligible for minimum number of terms of (1.21) is

the value for which the series converges rapidly.

Once the convergence of the solution series (1.21) is achieved then it will serve as a

solution of the considered problem due to the following theorem by Liao [76] :

As long as the series

h0 (x) ++∞Xm=1

hm (x)

is convergent, where hm (x) is governed by the high-order deformation equation (1.17) un-

der the definitions (1.18), (1.19), and (1.20), it must be a solution of equation A (h (x)) =

0.

At the end we would like to mention here that in homotopy analysis method the

series (1.21) is not presumed to be convergent. Therefore, for a particular problem under

consideration it is necessary to ensure the convergence of the solution series (1.21).

25

Chapter 2

Unsteady boundary-layer flow over

an impulsively started surface

In this chapter we present a purely analytic solution to unsteady boundary-

layer flow of an incompressible viscous fluid over a moving plate. The plate is

started impulsively to move in a uniform stream of viscous fluid. The solution

is uniformly valid for all time.

2.1 Introduction

Analytic solutions to boundary-layer flows are rare in literature, even for the simplest

case, namely, the Blasius flow [6] . A purely analytic solution to Blasius flow was reported

by Liao [84, 85] in 1999. The scarcity grows considerably when one looks for the un-

steady boundary-layer flows. Blasius [6] considered the simplest two-dimensional steady

boundary-layer flow of an incompressible viscous fluid when a semi infinite flat plate is

fixed in a free stream (of constant velocity) at zero incidence. The same problem was also

considered by Sakiadis [10] by interchanging the character of fluid and the plate. The

problem becomes more complicated if one assumes the situation in which both the fluid

and the plate are moving simultaneously. Such a situation seems to be more practical for

26

understanding the aerodynamical properties of the flow phenomenon. Such kind of flow

situation was considered by Klemp and Acrivos [12] for the steady case. In all three situ-

ations [6, 10, 12] the governing equation is the same with different boundary restrictions.

In all the above mentioned studies the solution reported by the authors are not analytic.

The situation turns out to be more complicated if one considers the unsteady flow. There

is no purely analytic solution available for an unsteady boundary-layer flow started due

to impulsive motion of the flat plate which is uniformly valid for all time. However, we

do present in this chapter. The chapter is distributed in the following sections:

We consider two cases, namely, the flow over a non porous and porous wall in sections

2 and 3 respectively. In section 2 we consider the unsteady boundary-layer flow of a

viscous fluid moving with constant free stream velocity past a flat rigid surface. Similarity

transformations have been used to reduce the number of variables involved. In section

3 we have investigated the effect of suction/injection on the same flow by replacing the

rigid plate by a porous one. The solutions are purely analytic and are uniformly valid

for all time. The accuracy and validity of present analytic solutions has been proved by

giving comparison with the existing results.

2.2 Unsteady flow past a moving rigid plate

2.2.1 Mathematical description of the problem

We consider an incompressible hydrodynamic viscous fluid bounded by a semi-infinite

plate situated at y = 0 and y−axis goes deep into the fluid. The fluid at infinity isassumed to be flowing with a constant free stream velocity U∞. Suddenly at time t > 0,

the plate starts moving with some constant velocity Uw. Under the above assumptions,

the boundary-layer equations (in the laboratory frame) governing the unsteady laminar

flow due to an impulsive motion of the plate are given by

∂u

∂x+

∂v

∂y= 0, (2.1)

27

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2, (2.2)

subject to the boundary conditions (for t > 0)

u = Uw = λU∞, v = 0, at y = 0, (2.3)

u → U∞, as y → +∞,

where U∞ is the constant free stream velocity and λ = Uw/U∞ is the ratio of plate

velocity to the free stream velocity. The initial condition for (t ≤ 0) is given by

u = v = 0, at all points (x, y) . (2.4)

Fig. 2.1 : Schematic diagram of the flow situation.

For a moving plate problem the similarity solution is possible if the plate velocity is

proportional to the power distributions along the distance from the leading edge. For

example, Dey [18] investigated similarity solution for the case of suction and (or) injection

28

by taking Uw ∼ x−1/2. Therefore, it is reasonable to assume Uw to be a function of x.We

introduce the following similarity variables in order to non-dimensionalize the equations

and boundary conditions. To see (how these transformations are obtained) the reader is

referred to follow the reference [63]

η =

sU∞νxξ

y, ψ =pνU∞xξf (η, ξ) , ξ = 1− e−τ , τ =

U∞x

t, (2.5)

where ψ denotes the stream function and f is the reduced (dimensionless) stream func-

tion. Let us assume that in time t (required to reach the steady state) the particle

reaches far from the leading edge i.e. x > 1, therefore we can ignore the derivative

∂τ/∂x = −U∞t/x2. Therefore, the governing system (2.1)− (2.4) readily transforms to

∂3f

∂η3+1

2(1− ξ) η

∂2f

∂η2+1

2ξf

∂2f

∂η2− ξ (1− ξ)

∂2f

∂η∂ξ= 0, (2.6)

with the transformed boundary conditions

f (0, ξ) = 0,∂f

∂η|η=0= λ, and

∂f

∂η|η=+∞= 1. (2.7)

Notice that for λ = 0, and ξ = 1, the above solution reduces to the classical Blasius flow

problem [6] .

2.2.2 Analytic solution

In most of the technological applications, engineers and scientists frequently use pertur-

bation techniques to reduce a nonlinear problem to an infinite number of linear problems.

Perturbation methods fail if there is no small or large parameter at all. In such situations

homotopy analysis method gets very much important. Using one interesting property of

homotopy [77] , one can transform any nonlinear problem into an infinite number of linear

problems, no matter whether or not there exists a small or large parameter. In order to

solve eq. (2.6) with boundary conditions (2.7) we use homotopy analysis method [76].

29

Keeping in view the boundary conditions (2.7) it is reasonable to assume that f (η, ξ)

can be expressed by the following set of base functions

©ξrηqe−nη/r ≥ 0, q ≥ 0, n ≥ 0ª ,

such that

f (η, ξ) =+∞Xn=0

+∞Xq=0

+∞Xr=0

Arn,qξ

rηqe−nη, (2.8)

where Arn,q are the coefficients of the series. Due to the boundary conditions (2.7) we

choose the initial guess of the form

f0 (η, ξ) = η − 1− e−γη

γ+ ληe−γη, (2.9)

and correspondingly the linear operator is given by

L ≡ ∂3

∂η3+ γ

∂2

∂η2, (2.10)

with the property

L·C1 + C2η +

1

γ2C3e

−γη¸= 0, (2.11)

where C 0i s, (i = 1, 2, 3) and γ are arbitrary constants.

Based on the idea of homotopy we construct the so-called zero-order deformation

equation

(1− p)L [F (η, ξ; p)− f0 (η, ξ)] = pN [F (η, ξ; p)] , (2.12)

subject to the boundary conditions

F (0, ξ; p) = 0,∂F (η, ξ; p)

∂η|η=0= λ,

∂F (η, ξ; p)

∂η|η=+∞= 1, (2.13)

where p ∈ [0, 1] is an embedding parameter, and the nonlinear operator N [F (η, ξ; p)] is

30

defined through

N [F (η, ξ; p)] =∂3F

∂η3+1

2(1− ξ) η

∂2F

∂η2+1

2ξF

∂2F

∂η2− ξ (1− ξ)

∂2F

∂η∂ξ. (2.14)

Clearly, when p = 0, and p = 1, the above zero-order deformation eqs. (2.12) and (2.13)

have the solutions

F (η, ξ; 0) = f0 (η, ξ) , (2.15)

and

F (η, ξ; 1) = f (η, ξ) , (2.16)

respectively. Thus as p varies from 0 to 1, F (η, ξ; p) varies from initial guess f0 (η, ξ) to

the final solution f (η, ξ) of the problem under consideration. By Taylor’s theorem we

have

F (η, ξ; p) = F (η, ξ; 0) ++∞Xm=1

fm (η, ξ) pm, (2.17)

where

fm (η, ξ) =1

m!

∂mF (η, ξ; p)

∂pm|p=0 . (2.18)

Assume that the auxiliary parameter ~ is so properly chosen that the series (2.17) is

convergent at p = 1, we get from (2.15) and (2.16) that

f (η, ξ) = f0 (η, ξ) ++∞Xm=1

fm (η, ξ) . (2.19)

Differentiating the zero-order deformation equations (2.12)& (2.13) , m−times (m ≥ 1)with respect to the parameter p at p = 0 and dividing them by m!, we obtain the mth-

order deformation equation

L [fm (η, ξ)− χmfm−1 (η, ξ)] = ~Rm (η, ξ) , (2.20)

31


fm (0, ξ) =∂fm (η, ξ)

∂η|η=0= ∂fm (η, ξ)

∂η|η=+∞= 0, (2.21)

where

Rm (η, ξ) =∂3fm−1∂η3

+1

2(1− ξ) η

∂2fm−1∂η2

−ξ (1− ξ)∂2fm−1∂η∂ξ

+1

2ξm−1Xk=0

fm−1−k∂2fk∂η2

, (2.22)

and

χk = 0, for k ≤ 1, (2.23)

= 1, for k > 1.

Notice that eq. (2.20) is a linear differential equation whose left hand side is governed

by the same linear operator L (for m ≥ 1). These linear equations can easily be solved,especially by means of symbolic computation software such as MATHEMATICA, MAT-

LAB, MAPLE and so on. By solving the first few deformation equations we find that

fm (η, ξ) can be expressed in the general form

fm (η, ξ) =2m+2Xn=0

Ψm,n (η, ξ) e−nη, (2.24)

where Ψm,n (η, ξ) is defined by

Ψm,n (η, ξ) =2m+2−nX

q=0

2m+2−nXr=0

aq,rm,nξrηq, (2.25)

where the constants aq,rm,n appearing in the series (2.25) can be obtained through the

recurrence relations given by:

32

a0,rm,0 = χmχ2m+1χ2m+1−ra0,rm−1,0 −

2m+2Xq=0

δq,rm,0µq0,1 −

2m+1Xq=0

χ2m+3−rδq,rm,1µ

q1,1 (2.26)

−2m+2−rXn=2

2m+2−nXq=0

χ2m+2−r (1− n) δq,rm,nµqn,0 −

2m+2−rXn=2

2m+2−nXq=1

χ2m+2−rδq,rm,nµ

qn,1,

0 ≤ r ≤ 2m+ 2,

ak,rm,0 = χmχ2m+1χ2m+1−kχ2m+1−rak,rm−1,0 +

2m+2Xq=k−1

δq,rm,0µq0,k, (2.27)

0 ≤ r ≤ 2m+ 2, 1 ≤ k ≤ 2m+ 2,

a0,rm,1 = χmχ2mχ2m−ra0,rm−1,1 +

2m+2Xq=0

δq,rm,0µq0,1 +

2m+1Xq=0

δq,rm,1µq1,1 (2.28)

+2m+2−rXn=2

2m+2−nXq=1

χ2m+2−rδq,rm,nµ

qn,1 −

2m+2−rXn=2

2m+2−nXq=0

χ2m+2−rnδq,rm,nµ

qn,0,

0 ≤ r ≤ 2m+ 1,

ak,rm,1 = χmχ2mχ2m−kχ2m−rak,rm−1,1 +

2m+1Xq=k−1

δq,rm,1µq1,k, (2.29)

0 ≤ r ≤ 2m+ 1, 1 ≤ k ≤ 2m+ 1,

ak,rm,n = χmχ2m+1−nχ2m+1−n−kχ2m+1−n−rak,rm−1,n +

2m+1−nXq=k

δk,rm,nµqn,k, (2.30)

2 ≤ n ≤ 2m+ 2, 0 ≤ k ≤ 2m+ 2− n, 0 ≤ r ≤ 2m+ 2− n,

33

where

δq,rm,n = ~

χ2m+1−nχ2m+1−n−qχ2m+1−n−r

Cq,rm−1,n +

12χq+1

¡Bq−1,rm−1,n − χr+1B

q−1,r−1m−1,n

¢−χr+1Dq,r−1

m−1,n + χrDq,r−2m−1,n

+12χr+1σ

q,r−1m,n

,(2.31)

in which

Aq,rm,n = (q + 1) a

q+1,rm,n − naq,rm,n, (2.32)

Bq,rm,n = (q + 1)A

q+1,rm,n − nAq,r

m,n, (2.33)

Cq,rm,n = (q + 1)B

q+1,rm,n − nBq,r

m,n, (2.34)

Dq,rm,n = (r + 1)A

q,r+1m,n , (2.35)

σq,rm,n =m−1Xk=0

Minn,2k+2Xp=Max0,n−2m+2k

Minq,2k+2−pXs=Max0,q−2m+2k+n−p

Minr,2k+2−pXl=Max0,r−2m+2k+n−p

Bs,lk,pa

q−s,r−lm−1−k,n−p,

and µq0,k, µq1,k, µ

qn,k, are defined through

µq0,k =q! (−1)k+q

k!, q ≥ 0, 2 ≤ k ≤ q + 2, (2.36)

µq1,k =q! (q + 2− k)

k!, q ≥ 0, 0 ≤ k ≤ q + 1, (2.37)

µqn,k =

q−kXp=0

−q! (p+ 1)k!(n− 1)q+1−k−pnp+2 , q ≥ 0, 0 ≤ k ≤ q, n ≥ 2, (2.38)

to see how these coefficients have been calculated the reader is referred to see the reference

[84] .

Convergence and accuracy of results

Homotopy solution always results in the form of an infinite series of functions (see eq.

(2.19)). Before we move further it is important to ensure the convergence of the series

34

(2.19). In homotopy analysis method the convergence of the solution series strongly de-

pends upon the choice of the initial guess approximations; the linear operator, and the

auxiliary parameter ~. Once the linear operator and the initial guess have been selected

the convergence then strongly depends upon the auxiliary parameter ~. The values of

the auxiliary parameter ~ are determined through the ~−curve [76] . In fig. 2.2 we haveplotted the ~−curve at 10th-order of approximation. From figure one can see that the

graph is extremely stable at ~ = −1.0. Further our analysis shows that the series con-verges more rapidly at ~ = −1.0 as shown in table 2.1. From table it is clear that by

increasing the order of approximations the corrections to the solution become negligible,

which proves the convergence of the solution series.

Fig. 2.2 : ~−curve at 10th-order approximation at γ = 4.0.

35

γ = 4.0, λ = 0.2, ξ = 0.5, η = 0.0

Orders of app. f 00(η, ξ)

5th 0.402921

10th 0.460573

15th 0.450296

20th 0.451360

25th 0.451382

30th 0.451343

Table 2.1: HAM solution at different orders.

Fig. 2.3a : Comparison between analytic and numerical solutions at 10th-order approximation.

36

Fig. 2.3b : Comparison between analytic and numerical solutions at 20th-order approximation.

Fig. 2.3c : Comparison between analytic and numerical solutions at 40th-order approximation.

As we have already mentioned that for λ = 0 and ξ = 1 the present problem reduces

to the Blasius flow problem [6] . Liao [84] studied the Blasius problem and presented a

37

uniformly valid analytic solution for it. Liao proved the accuracy of his analytic solu-

tion by giving a comparison with the numerical results of Howarth [122] . It is worth

mentioning here that the results obtained by Liao [84] can easily be recovered from the

present solution by letting λ = 0 and ξ = 1. A comparison of the present solution with

the analytic results of Liao [84] and the numerical results of Howarth [122] is given in

figs. 2.3a− 2.3c. Clearly, the analytic solution is in good agreement with the numericaldata of Howarth which proves the accuracy and validity of our HAM solution.

2.2.3 Discussion on results

A remarkable property of the present HAM solution is that it is uniformly valid for all

values of the dimensionless time τ . The velocity profile for different values of τ has been

plotted in figure 2.4.

Fig. 2.4 : Velocity profile at different time.

38

Clearly the viscosity induces in the fluid with the passage of time and after some time

the steady state is achieved which proves the uniform validity of the solution for all time

in the whole spatial region 0 ≤ η <∞. From figure it is also observed that the vorticity

induces rapidly as it takes very short time to reach the steady state. The effect of λ on

velocity fη is shown in fig. 2.5.From the graph it is clear that by increasing the values

of λ the velocity at the plate increases rapidly and attains the free stream velocity as

we go in increasing direction of η. From figure it is also clear that the velocity gradient

at the wall decreases with the increase in velocity of the plate. Further it can easily

be observed that the velocity gradient increases at the plate if the plate moves in the

opposite direction to that of the direction of the free stream. This causes to increase

the skin friction at the wall, however, the skin friction at the wall decreases if the plate

moves parallel to the free stream.

Fig. 2.5 : Velocity profile for different values of λ.

39

In fig. 2.6 we have plotted the coefficient of skin friction against the dimensionless time

τ at different values of λ. It is observed that the skin friction at the plate is large at very

initial time and decreases with the passage of time. This is because of the reason that the

viscous effects are stronger at initial time. From figure it can also be seen that the skin

friction also decreases by increasing the values of the parameter λ that is by increasing

the velocity of the plate (parallel to the free stream) the skin friction decreases.

Fig. 2.6 : Coefficient of skin friction for different values of λ at γ = 4.0.

2.2.4 Concluding remarks

In this section we dealt with the unsteady boundary-layer flow over a suddenly moved

flat plate in a stream of fluid moving at constant velocity. Homotopy analysis method

has been used to solve the governing nonlinear problem and purely analytic solution valid

for all time has been obtained. The accuracy and validity of the solution has been proved

through table and by giving comparison with previous results. The solution is valid for

both the cases, namely, when the plate moves parallel or anti parallel to the free stream.

40

It is observed that the skin friction at the wall decreases if λ increases (keeping λ > 0),

but the effect is reversed if plate increases its velocity in reverse direction. The coefficient

of skin friction is observed to decay rapidly (for λ > 0) for initial time and then becomes

uniform for large time values.

2.3 Unsteady flow past an impulsively started porous

plate

This section is devoted to examine the flow phenomenon of a viscous fluid when the

impulsively started moving wall is subjected to uniform suction/injection. Our primary

objective is to investigate the effect of the wall suction/injection and the parameter λ on

skin friction at the wall for unsteady case. The problem is solved analytically and the

solution is highly accurate. Also the present analytic solution is uniformly valid for all

dimensionless time in the entire flow regime. The solution is presented in the form of

infinite series and the convergence of the solution series is discussed in detail.

2.3.1 Mathematical description of the problem

We consider an incompressible viscous fluid bounded by a semi-infinite plate situated at

y = 0 and y-axis goes deep into the fluid. The fluid at infinity is assumed to be flowing

with a constant free stream velocity U∞. The plate is assumed to be uniformly porous

so that suction or injection is possible at the wall. Initially the plate is at rest, suddenly

at time t ≥ 0, the plate is started to move with a constant velocity Uw (see fig. 2.7).

The boundary layer equations (in laboratory frame) for the present case are the same

as given in eqs. (2.1) and (2.2) . However, the present problem assumes the boundary

conditions of the form (for t ≥ 0)

u = Uw = λU∞, v = −V ∗0 (x, t), at y = 0, (2.39)

u → U∞, as y → +∞,

41

where λ is the same as defined in previous section, V ∗0 (x, t) is the suction/injection

velocity (V ∗0 > 0 corresponds to suction and V ∗0 < 0 corresponds to injection). The

initial condition (for t < 0) is given by

u = v = 0, at all points (x, 0) .

Due to the similarity transformations (2.5) the equation (2.39) takes the form

f (0, ξ) = w0,∂f

∂η|η=0= λ, and

∂f

∂η|η=+∞= 1, (2.40)

where w0 =px/νU∞ξV ∗0 is the dimensionless suction and (or) injection velocity. Thus

our mathematical model describing the considered flow comprises of the equations (2.16)

and (2.40) . Notice that for w0 = λ = 0 and ξ = 1 the present problem (2.16) and (2.40)

reduces to the classical Blasius flow problem [6].

Fig. 2.7 : Schematic diagram of the flow situation.

42


To start with the homotopy analysis method it is very much important to choose an initial

guess approximation and a linear operator. Therefore, due to the boundary conditions

(2.40) it is reasonable to choose the initial guess approximation

f0 (η, ξ) = w0 + η − 1− e−γη

γ+ ληe−γη, (2.41)

and the linear operator

L ≡ ∂3

∂η3+ γ

∂2

∂η2, (2.42)

which satisfies the following property:

L·C4 + C5η +

1

γ2C6e

−γη¸= 0, (2.43)

where C 0i s, (i = 4, 5, 6) and γ are arbitrary constants. To avoid the repetition of same

steps (as performed in section 2.2.2) we directly write themth-order (m ≥ 1) deformationequation for the present case (see for instance section 2.2.2)

L [fm (η, ξ)− χmfm−1 (η, ξ)] = ~Qm (η, ξ) , (2.44)


fm (0, ξ) =∂fm (η, ξ)

∂η|η=0= ∂fm (η, ξ)

∂η|η=+∞= 0, (2.45)

where

Qm (η, ξ) =∂3fm−1∂η3

+1

2(1− ξ) η

∂2fm−1∂η2

−ξ (1− ξ)∂2fm−1∂η∂ξ

+1

2ξm−1Xk=0

fm−1−k∂2fk∂η2

, (2.46)

43

and

χk = 0, for k ≤ 1, (2.47)

= 1, for k > 1.

Let f∗m (η, ξ) denotes a special solution of eqs. (2.44) and (2.45) , then with the help

of eq. (2.43), we have its general solution

fm (η, ξ) = f∗m (η, ξ) + C4 + C5η + C61

γ2e−γη (2.48)

where the constant coefficients C4, C5, and C6 are determined through the boundary

conditions (2.45) in the following manner:

C5 = 0, C4 = −1γ

∂f∗m (η, ξ)∂η

|η=0 −f∗m (0, ξ) , C6 = γ∂f∗m (η, ξ)

∂η|η=0 . (2.49)

In this way one can easily solve the linear equations (2.44) and (2.45) for allm = 1, 2, 3, ...

with the help of some symbolic computation software such as MATHEMATICA. There-

fore, the final solution fm (η, ξ) may be written in the following form of an infinite series

f (η, ξ) = f0 (η, ξ) ++∞Xm=1

fm (η, ξ) . (2.50)

Convergence and accuracy of results

Before going to the discussion of results it is necessary to ensure the convergence of

present analytic solution. As mentioned by Liao [76] that whenever the solution series

obtained by homotopy analysis method is convergent it will be one of the solutions of

the considered problem. After the appropriate choice of the initial guess approximation

and the auxiliary linear operator, the convergence of the solution series strongly depends

upon the (non-zero) auxiliary parameter ~. The admissible values of the parameter ~ are

44

determined by the so-called ~−curve. For the present analysis, in order to determine theconvergence of the solution series we have plotted the ~−curve in fig. 2.8. The admissiblevalues of ~ are those for which the ~−curve has the zero gradient. From fig. 2.8 one caneasily observe that the interval of allowed values of ~ is [−1.2,−0.8] (roughly speaking).However, our analysis shows that the rapid convergence is assured at ~ = −0.9. In orderto show the convergence and accuracy of the solution series (2.50) we define the absolute

difference between two successive terms of the infinite solution series in the following

way:

∆f 00i =¯f 00i − f 00i−1

¯,

where the subscript i represents the order of approximation and 0 represents differentiation

with respect to η. In table 2.2 we have listed the HAM solution at different orders of

approximation. From table it is clear that for higher order approximations the corrections

to the solution become so small to be negligible. This proves the convergence of present

HAM solution.

Fig. 2.8 : ~−curve at 15th-order of approximation.

45

γ = 3.0, λ = 0.2, w0 = 0.5, ξ = 0.5, η = 0.0

Orders of app. f 00(η, ξ) ∆f 00i =¯f 00i − f 00i−1

¯5th 0.423436 0.0421065000

10th 0.455099 0.0026972000

15th 0.450903 0.0001689800

20th 0.451382 0.0000173022

25th 0.451347 1.11979× 10−6

30th 0.451352 8.2303× 10−7

Table 2.2: HAM solution at different orders.


To understand the flow phenomenon and the effect of the parameters w0, and λ, the

graphs are plotted in figs. 2.9− 2.13. Note that all the figures 2.9− 2.13 are plotted forγ = 3.0. In fig. 2.9 we have plotted the skin friction coefficient against the dimensionless

time τ for different values of the suction/injection parameter w0. Clearly, an increase in

suction velocity causes to increase the skin friction at the plate. Further, it can easily

be observed that for small values of the suction velocity w0, the skin friction curves take

more time to become stable as compared with the curves drawn at some higher values of

the suction velocity w0, which shows the strong dependence of skin friction on w0. From

fig. 2.9 it can also be noted that the skin friction changes rapidly for small values of time

and then becomes stable. The interval of time in which the skin friction changes rapidly

is relatively small which means that the transition of flow from unsteady to steady takes

place in a short interval of time.

The mass injection at the solid boundary causes to reduce the wall drag and it grows

in the case of suction, this fact can easily be observed from fig. 2.10. Figure 2.10 also

depicts that by increasing the suction velocity the layer thickness decreases whereas it

increases for the case of injection. The parameter λ is found to reduce the drag at the

wall when both the fluid and the plate move in same direction and for λ = 1 when the

46

plate and the free stream move with same velocity then the drag at the wall becomes zero

(as shown in fig. 2.11). As we have already mentioned that the present analytic solution

is uniformly valid for all values of the dimensionless time τ ∈ [0,+∞) in the entire spatialdomain η ∈ [0,+∞). In fig. 2.12 the velocity fη is plotted against η for different values ofthe time τ . From figure it is clearly seen that the velocity changes uniformly by changing

the time and attains its steady state after some time which proves the uniform validity

of our present solution for all time. In fig. 2.13, f (η, ξ) is plotted against η for different

values of the suction parameter w0 in order to see the effect of suction on the normal

component of velocity. Clearly, by increasing the values of the suction parameter the

normal velocity also increases.

Fig. 2.9 : Effect of suction on skin friction coefficient.

47

Fig. 2.10 : Effect of suction/injection on velocity.

Fig. 2.11 : Velocity profile for different λ.

48

Fig. 2.12 : Velocity plotted at different values of dimensionless time τ .

Fig. 2.13 : Effect of suction on normal component of velocity.

49

2.3.4 Concluding remarks

In this section the unsteady boundary-layer flow of a viscous fluid past an impulsively

started porous plate moving in same and (or) opposite direction to the free stream has

been studied. Purely analytic solution which is uniformly valid for all time throughout the

spatial domain is obtained by a newly developed analytic technique, namely, homotopy

analysis method. The effect of wall suction is to increase the dynamic drag and the effect

of injection is to decrease the drag at the wall. Also the suction is observed to decrease

the boundary-layer thickness. The effect of parameter λ (> 0) , (the ratio of plate velocity

to that of free stream velocity) is to reduce the skin friction and to increase the velocity

at the plate. The effect is reversed for the case when λ (< 0) . It is important to remark

here that the homotopy analysis method is a very useful analytic technique to solve such

kind of unsteady nonlinear problems analytically.

2.4 Conclusion

In this chapter we investigated the unsteady boundary layer flow of an incompressible

viscous fluid over an impulsively started plate moving in a free stream of constant velocity

for both the cases, namely, for rigid plate, and for porous plate. Purely analytic solution

valid for all time have been obtained by homotopy analysis method. It is observed that

in both the cases the skin friction at the plate increases when the fluid and the plate

move in opposite directions but the situation is reversed for the case when fluid and the

plate move parallel to each other. The suction is observed to control the layer thickness

whereas the injection causes to increase the boundary-layer thickness. But it is also

observed that injection happens to reduce the skin friction at the solid wall whereas the

strong suction results in large skin friction at the wall. Thus injection helps to reduce the

dynamic drag whereas the suction increases the drag at the moving surface. In both the

cases it is also observed that the vorticity diffuses rapidly as the steady state is reached

after a small interval of time.

50

In this chapter we considered unsteady boundary-layer flow over a moving

wall. In the coming chapter we are going to consider the unsteady boundary-

layer viscous flow over a stretching wall with heat transfer phenomenon.

51

Chapter 3

Unsteady boundary-layer flow over

an impulsively stretching surface

with heat and mass transfer

In this chapter we investigate the heat transfer phenomenon from stretch-

ing sheet to the ambient fluid. The flow is assumed to be unsteady two-

dimensional started due to impulsive motion of the stretching sheet. Effects

of suction/injection on thermal boundary-layers have been studied.

3.1 Introduction

Heat transfer analysis from a stretching sheet to the ambient fluid is important in in-

dustrial point of view. The study of thermal boundary-layers enables one to control the

cooling process of the sheet which in turn helps to obtain the final product of desired

characteristics. The effect of Prandtl number on the thermal boundary-layers helps to de-

cide the coolant with desired rate of heat exchange. Further the mass transfer at the wall

also effects the rate of heat transfer to some extent. In this regard the present analysis

is an important case of our study in which we investigate the heat transfer phenomenon

52

at the stretching wall in the presence of mass transfer at the wall. The heat transfer

analysis in a viscous fluid over a stretching sheet for steady case had been investigated

by Crane [27] in which Crane presented analytic solution to the considered problem.

However, the situation becomes worse when one considers the unsteady flow. It turns

out to be very difficult to obtain purely analytic solution which is uniformly valid for all

time. To the best of our knowledge literature lacks in analytic solutions of such kind of

flows. In our present analysis we have succeeded in finding the purely analytic solution

to the unsteady flow problem over a stretching sheet in the presence of heat transfer.

The problem is solved analytically and the solution is highly accurate. The problem is

dealt with a purely analytic technique “homotopy analysis method”. The accuracy of the

HAM solution has been proved by giving a comparison with the results already present in

literature [27] . Also the present analytic solution is uniformly valid for all dimensionless

time in the entire domain.

3.2 Mathematical description of the problem

We consider an incompressible hydrodynamic viscous fluid bounded by an infinite plate

situated at y = 0 and the fluid occupies the region y > 0. The plate is assumed to

be heated at constant temperature θw. The plate is a highly elastic membrane and is

assumed to be uniformly porous so that suction or injection is possible. Suddenly two

equal and opposite forces are applied along x−axis so that the wall is stretched keepingthe origin fixed. For two-dimensional unsteady flow the velocity vector must be of the

form V = [u (x, y, t) , v (x, y, t) , 0] . Therefore, it is reasonable to assume the temperature

function of the same form i.e. θ = θ (x, y, t) . The governing equations are the continuity

equation; the momentum equation, and the energy equation for the considered flow.

After applying the boundary-layer approximation the governing equations in the absence

of viscous dissipation are given be:

53

∂u

∂x+

∂v

∂y= 0, (3.1)

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2, (3.2)

∂θ

∂t+ u

∂θ

∂x+ v

∂θ

∂y= κ

∂2θ

∂y2, (3.3)

subject to the boundary conditions (for t ≥ 0)

u = ax, v = −V ∗0 (t), at y = 0, (3.4)

u → 0, as y → +∞,

and

θ = θw, at y = 0, θ → θ∞, as y → +∞, (3.5)

where a > 0 is a constant, V ∗0 (t) is the time dependent wall suction/injection velocity, κ

is the thermal diffusivity, θw is the wall temperature, and θ∞ is the temperature of the

fluid. The initial condition on velocity components for (t < 0) is given by

u = v = 0, at all points (x, y) . (3.6)

In order to non-dimensionalize the problem we introduce the similarity transforma-

tions

η =

ra

νξy, u = ax

∂f (η, ξ)

∂η, v = −

paνξf (η, ξ) , ξ = 1− e−τ , (3.7)

τ = at, T (η, ξ) =θ − θ∞θw − θ∞

,

where f (η, ξ) is the dimensionless stream function and T (η, ξ) is the dimensionless tem-

perature function. Due the above similarity transformations (3.7) the continuity equation

54

is satisfied identically and eqs. (3.2) and (3.3) take the forms

∂3f

∂η3+1

2(1− ξ) η

∂2f

∂η2+ ξ

Ãf∂2f

∂η2−µ∂f

∂η

¶2!− ξ (1− ξ)

∂2f

∂η∂ξ= 0, (3.8)

∂2T

∂η2+Pr

·ξf

∂T

∂η− ξ (1− ξ)

∂T

∂ξ+1

2(1− ξ) η

∂T

∂η

¸= 0, (3.9)

respectively, with transformed boundary conditions

f (0, ξ) = w0,∂f

∂η|η=0= 1, ∂f

∂η|η=+∞= 0, (3.10)

T (0, ξ) = 1, T (η, ξ)→ 0, as η → +∞, (3.11)

where Pr = ν/κ is the Prandtl number, and w0 = V ∗0 /√aνξ is the dimensionless wall

suction/injection velocity.

3.3 Solution by homotopy analysis method

In view of boundary conditions (3.10) and (3.11) the solution expressions for f (η, ξ) and

T (η, ξ) are respectively given by

f (η, ξ) =+∞Xn=0

+∞Xq=0

+∞Xr=0

Arn,qξ

rηqe−nη, (3.12)

and

T (η, ξ) =+∞Xn=0

+∞Xq=0

+∞Xr=0

Brn,qξ

rηqe−nη, (3.13)

in the form of base functions

©ξrηqe−nη/r ≥ 0, q ≥ 0, n ≥ 0ª ,

55

where Arn,q and Br

n,q are constant coefficients. They provide us with the so-called rule of

solution expressions for f (η, ξ) and T (η, ξ) . According to the rule of solution expressions

(3.12) , (3.13) and from eqs. (3.10) , (3.11) it is straight forward to choose the initial

approximations

f0 (η, ξ) = w0 + 1− e−η, (3.14)

and

T0 (η, ξ) = e−η, (3.15)

with the linear operators

Lf ≡ ∂3

∂η3− ∂

∂η, (3.16)

LT ≡ ∂2

∂η2+

∂

∂η, (3.17)

which satisfy the properties

Lf

£C1 + C2e

η + C3e−η¤ = 0, (3.18)

and

LT

£C4 + C5e

−η¤ = 0, (3.19)

where C 0i s, (i = 1, 2, ..., 5) are arbitrary constants. From eqs. (3.8) and (3.9) we define

the nonlinear operators N1 [F (η, ξ; p)] and N2 [F (η, ξ; p) ,Λ (η, ξ; p)]

N1 [F (η, ξ; p)] =∂3F

∂η3+1

2(1− ξ) η

∂2F

∂η2+ξ

ÃF∂2F

∂η2−µ∂F

∂η

¶2!−ξ (1− ξ)

∂2F

∂η∂ξ, (3.20)

N2 [F (η, ξ; p) ,Λ (η, ξ; p)] =∂2Λ

∂η2+Pr

·ξF

∂Λ

∂η− ξ (1− ξ)

∂Λ

∂ξ+1

2(1− ξ) η

∂Λ

∂η

¸, (3.21)

respectively.

Let ~ denotes the nonzero auxiliary parameter. We construct the so-called zero-order

56

deformation equations

(1− p)Lf [F (η, ξ; p)− f0 (η, ξ)] = p~N1 [F (η, ξ; p)] , (3.22)

(1− p)LT [Λ (η, ξ; p)− T0 (η, ξ)] = p~N2 [F (η, ξ; p) ,Λ (η, ξ; p)] , (3.23)

with their respective boundary conditions

F (0, ξ; p) = w0,∂F (η, ξ; p)

∂η|η=0= 1, ∂F (η, ξ; p)

∂η|η=+∞= 0, (3.24)

and

Λ (0, ξ; p) = 1, Λ (+∞, ξ; p) = 0, (3.25)

where 0 ≤ p ≤ 1 is an embedding parameter. Clearly, when p = 0 and p = 1 we have

from eqs. (3.22) and (3.23) that

F (η, ξ; 0) = f0 (η, ξ) , Λ (η, ξ; 0) = T0 (η, ξ) , (3.26)

and

F (η, ξ; 1) = f (η, ξ) , Λ (η, ξ; 1) = T (η, ξ) . (3.27)

So, as p increases from 0 to 1 our initial guess approximations f0 (η, ξ) and T0 (η, ξ) move

to the final solutions f (η, ξ) and T (η, ξ) of equations (3.8) and (3.9) respectively.

To get the mth-order deformation equations, we first differentiate the zero-order de-

formation equations (3.22)− (3.25) m−times (m = 1, 2, 3, ...) with respect to p at p = 0

and then divide the resulting expression by m! to get

Lf [fm (η, ξ)− χmfm−1 (η, ξ)] = ~Rm (η, ξ) , (3.28)

LT [Tm (η, ξ)− χmTm−1 (η, ξ)] = ~Qm (η, ξ) , (3.29)

57

with boundary conditions

fm (0, ξ) =∂fm (η, ξ)

∂η|η=0= ∂fm (η, ξ)

∂η|η=+∞= 0, (3.30)

and

Tm (0, ξ) = 0, Tm (+∞, ξ) = 0, (3.31)

where

Rm (η, ξ) =∂3fm−1∂η3

+1

2(1− ξ) η

∂2fm−1∂η2

− ξ (1− ξ)∂2fm−1∂η∂ξ

(3.32)

+ξm−1Xk=0

·fm−1−k

∂2fk∂η2− ∂fm−1−k

∂η

∂fk∂η

¸,

Qm (η, ξ) =∂2Tm−1∂η2

+Pr

12(1− ξ) η ∂Tm−1

∂η− ξ (1− ξ) ∂Tm−1

∂ξ

+ξm−1Xk=0

hfm−1−k ∂Tk∂η

i , (3.33)

and

χk = 0, for k ≤ 1, (3.34)

= 1, for k > 1.

Let f∗m (η, ξ) and T ∗m (η, ξ) denote the special solutions of eqs. (3.28) − (3.31). Thenone can find the general solutions fm (η, ξ) , and Tm (η, ξ) for any m (m = 1, 2, 3, ...) as

under

fm (η, ξ) = f∗m (η, ξ) + C1 + C2eη + C3e

−η, (3.35)

Tm (η, ξ) = T ∗m (η, ξ) + C4 + C5e−η, (3.36)

respectively, where the constant coefficients C 0i s, (i = 1, 2, ..., 5) are determined through

58

the boundary conditions (3.30) and (3.31) in the following manner:

C2 = C4 = 0, C1 = −∂f∗m (η, ξ)

∂η|η=0 −f∗m (0, ξ) , C3 =

∂f∗m (η, ξ)∂η

|η=0, C5 = −T ∗m (0, ξ) .(3.37)

In this way we can solve the linear system (3.28)− (3.31) for all m = 1, 2, 3, ... with the

help of some symbolic computation software such as MATHEMATICA. Therefore, the

final solution f (η, ξ) , and T (η, ξ) may respectively be written in the following form of

infinite series

f (η, ξ) = f0 (η, ξ) ++∞Xm=1

fm (η, ξ) , (3.38)

T (η, ξ) = T0 (η, ξ) ++∞Xm=1

Tm (η, ξ) . (3.39)

3.3.1 Convergence and accuracy of results

Liao, in his book [76] proved that whenever the solution series converges it will be one of

the solutions of the considered problem. The convergence of the solution series strongly

depends upon the auxiliary parameter ~. The admissible values of ~ for which the solution

series converge are determined by drawing the so-called ~−curves [76] . The problemunder consideration was solved by Liao [47] for velocity field in the case of impermeable

wall and by Ali and Mehmood [98] for permeable wall in a porous medium. In [47]

and [98] the authors have shown that the solution series (4.38) converges at ~ = −0.25.Therefore, in the present analysis we need only to investigate the convergence of the

series (3.39). In order to make the series (3.39) convergent the admissible values of ~

are determined by drawing the ~−curves in figure 3.1 for different values of Prandtlnumber Pr . It is important to mention here that the auxiliary parameter ~ may be

a function of the physical parameters involved in the problem (see for instance [116]).

In the problem under discussion it is found that the auxiliary parameter ~ is strongly

dependent upon the Prandtl number Pr. The ~−curves drawn for different values ofthe Prandtl number Pr are shown in fig. 3.1. Figure 3.1 shows that by increasing the

59

values of Pr the intervals of allowed values of ~ become smaller and shift towards zero.

The convergence of the solution series (3.38) and (3.39) has been proved by calculating

HAM solution at different orders of approximation in table 3.2. Clearly, by increasing the

orders of approximation the corrections to the solutions become negligible. This allows

us to truncate the series (3.38) and (3.39) at some appropriate value of m. To ensure the

accuracy of present results a comparison between our steady results (obtained at ξ = 1)

and the exact (steady state) solution presented by Crane [27] is given in table 3.1. From

table it is clear that our present analytic results are in good agreement with those given

by Crane [27] for the steady state case.

Pr = 1, ~ = −0.65 Pr = 2, ~ = −0.44 Pr = 3, ~ = −0.31η Exact HAM Exact HAM Exact HAM

0.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1 0.9418 0.9418 0.9091 0.9091 0.8840 0.8837

0.2 0.8843 0.8843 0.8200 0.8200 0.7713 0.7711

0.3 0.8278 0.8278 0.7340 0.7340 0.6645 0.6642

0.4 0.7727 0.7727 0.6523 0.6523 0.5656 0.5652

0.5 0.7194 0.7194 0.5759 0.5758 0.4761 0.4766

0.6 0.6687 0.6681 0.5052 0.5050 0.3965 0.3962

0.7 0.6191 0.6191 0.4406 0.4406 0.3269 0.3271

0.8 0.5725 0.5725 0.3822 0.3822 0.2671 0.2677

0.9 0.5284 0.5285 0.3298 0.3294 0.2163 0.2165

1.0 0.4869 0.4869 0.2833 0.2833 0.1739 0.1739

Table 3.1 : Comparison of HAM sol. with exact sol. by Crane [27] for T (η) .

60

Order −f 00(0, ξ) −T 0(0, ξ) ¯f 00i (0, ξ)− f 00i−1(0, ξ)

¯ ¯T 0i (0, ξ)− T 0i−1(0, ξ)

¯5th 1.20214 0.944207 0.0230656 0.00515291

10th 1.25944 0.933426 0.00642422 0.00113968

15th 1.27514 0.930708 0.00173447 0.000292528

20th 1.27932 0.930023 0.000455713 0.0000721966

25th 1.28041 0.929858 0.000116888 0.0000170651

30th 1.28069 0.929819 0.0000293498 3.84885× 10−6

35th 1.28075 0.929811 7.2329× 10−6 8.22306× 10−7

40th 1.28077 0.929809 1.75389× 10−6 1.64186× 10−7

Table 3.2 : HAM solution at different orders of approximation.

Fig. 3.1: ~−curve at 10th-order of approximation.

61


Graphical representation of results is very useful in the discussion of physical features

presented by the solutions. The effects of wall suction/injection on the velocity com-

ponents fη and f have been discussed in detail in [98] . Therefore, we will confine our

attention only to the temperature distribution in the flow field and the effects of different

parameters on it. In [98] it is observed that increasing the suction velocity the skin fric-

tion at the plate increases and the layer thickness decreases, but the effect is observed to

be totally reversed in the case of injection. In figs. 3.2−3.5 the non dimensional temper-ature T (η, ξ) is plotted for different values of the parameters w0 (the suction/injection

parameter); the Prandtl number Pr, and the dimensionless time τ respectively. From

fig. 3.2 it is clear that by increasing the injection velocity the heat spreads in the fluid

due to which the thermal boundary-layers grow in the fluid. However, an increase in the

suction velocity causes to increase the rate of heat transfer at the wall which causes to

reduce the thermal boundary-layer thickness.

Fig. 3.2 : Temperature profile for different values of suction parameter w0.

62

The temperature profile for different values of the Prandtl number Pr is plotted in fig. 3.3.

Clearly, by increasing the values of Pr the temperature and the thermal layer thickness

decrease which is in accordance with the existing results. Further, an increase in Pr

results in increasing the rate of heat transfer at the wall, which suggests that the fluids

with large Prandtl numbers must be selected in order to enhance the cooling process.

The validity of our HAM solution for all values of the dimensionless time τ is shown in

fig. 3.4 by drawing T (η, ξ) for different values of the dimensionless time τ . It is clear

from the graph that as time increases the solution travels to reach its steady state. Such

type of analytic solution for heat transfer analysis has never been reported before to the

best of our knowledge.

Fig. 3.3 : Temperature profile for different values of the Prandtl number Pr .

In figure 3.5 the Nusselt number is plotted against the dimensionless time τ for different

values of suction parameter w0. Clearly, rate of heat transfer increases by increasing the

suction velocity which means that the cooling process can be catalyzed by introducing

63

the suction at the surface of the stretching sheet. But in such a situation one must

keep in mind that strong suction may damage the surface of the stretching sheet as it

is observed in [98] that the skin friction at the plate increases by increasing the suction

velocity. Further, from figure 3.5 it is also observed that at very initial time the rate of

heat transfer decreases rapidly and then becomes stable after some time. This is due to

the fact that at very initial stage there is a large difference between the plate and the fluid

temperatures, that’s why at the very initial time the rate of heat transfer is maximum

and this difference in temperatures becomes small with the passage of time resulting in

decay of rate of heat exchange.

Fig. 3.4 : Temperature profile at differet time τ .

64

Fig. 3.5: Nusselt number plotted against τ at different valeus of suction parameter w0.

3.4 Conclusion

In this chapter the unsteady boundary-layer viscous flow over a stretching sheet with

heat transfer analysis is studied. Highly accurate and complete analytic solutions which

are uniformly valid for all dimensionless time 0 ≤ τ < +∞ in the whole spatial domain

0 ≤ η < +∞ are obtained by a purely analytic technique, namely, the homotopy analysis

method. With the help of graphs it is shown that as time increases the solution travels to-

wards its steady state. Further, the validity and accuracy of the present analytic solution

for temperature distribution is proved by giving a comparison between the present results

and the results already available in literature. It is observed that our present analytic

results are in good agreement with the exact solution given by Crane [27], which provides

a useful mathematical check. The effect of the wall suction and the Prandtl number on

temperature field is to decrease the thermal boundary-layers at the wall. The rate of heat

transfer decreases by increasing the injection at the wall whereas it increases in the case

65

of suction. Therefore, in order to expedite the cooling process suction is recommended to

be introduced at the stretching surface. Also, for large Prandtl number the rate of heat

transfer is also very large. Therefore, the cooling process can be improved if one selects

the fluid with high Prandtl number as a coolant. It is also observed that there are rapid

changes in the Nusselt number in very short interval of time at very initial stage.

Hear we dealt with the unsteady two-dimensional boundary-layer flow over

a stretching sheet with heat and mass transfer. The situation gets more com-

plicated if one considers three-dimensional flow instead of two-dimensional.

In the coming chapter we are going to deal with three-dimensional flow of a

viscous fluid over a stretching sheet.

66

Chapter 4

Generalized 3D viscous flow and

heat transfer over a stretching plane

wall

In last two chapters we studied two-dimensional boundary-layer flows of

viscous fluid over a flat plate. One can consider more general case when the

flow is essentially three-dimensional. This chapter is devoted to investigate

such kind of generalized three-dimensional viscous flow with heat transfer

analysis in the presence of viscous dissipation. The flow is generated due to

uniform stretching of the plane wall.

4.1 Introduction

In the manufacturing of polymer sheets it is sometimes recommended to stretch the sheet

in two lateral directions in order to attain the desired thickness. This gives rise to the

study of three-dimensional flow over a stretching sheet. Such kind of problems are rare in

literature. This is because of their high difficulty level due to the presence of strong non-

linearities in the governing equations. Wang [55] considered a steady three-dimensional

67

boundary-layer flow of an incompressible viscous fluid over a flat plate. Wang [55] as-

sumed uniform stretching of the sheet in two lateral directions at different rates. The

problem was reconsidered by Ariel [56] in 2003. In [56] , Ariel determined numerical and

an approximate analytic solution to the same problem as considered by Wang [55] . How-

ever, the perturbation solution presented by Ariel is not uniformly valid for all values of

the parameter β and the approximate analytic solution does not give good approxima-

tion as it does not match exactly with the numerical solution of Ariel (see for instance

table 4.2). We consider the same problem as considered by Wang [55] to compute the

purely analytic solution of the problem that is uniformly valid for all values of β with

the emphasis on heat transfer analysis from plate to fluid. Energy losses due to viscous

resistance have been taken into account. The effect of Prandtl number and the Eckert

number(s) on heat transfer rate have been studied which guides us to select the coolant

for cooling process.

4.2 Flow analysis

4.2.1 Mathematical formulation

We choose the Cartesian coordinates as our reference axes. Assume that the fluid is

bounded by an infinite flat plate. The plate coincides with the xy−plane and z−axisis taken perpendicular to it and the fluid occupies the space z > 0. The plate is being

stretched in two lateral directions by introducing two equal and opposite forces in x−and y−directions so that the point (0, 0, 0) remains unchanged. The governing equationsare the continuity equation, and the momentum equation for considered flow. Since the

flow is three-dimensional, therefore, the velocity field must be depending upon the three

space variables, i.e. V = [u (x, y, z) , v (x, y, z) , w (x, y, z)] . Thus the continuity equation

for incompressible viscous flow is given by

∂u

∂x+

∂v

∂y+

∂w

∂z= 0, (4.1)

68

and the momentum equation (in component form) after applying the boundary-layer

approximation are given by

u∂u

∂x+ v

∂u

∂y+ w

∂u

∂z= ν

∂2u

∂z2, (4.2)

u∂v

∂x+ v

∂v

∂y+ w

∂v

∂z= ν

∂2v

∂z2, (4.3)


u = ax, v = by, w = 0, at z = 0, (4.4)

u → 0, v → 0, as z → +∞,

where a > 0, and b ≥ 0 are the velocity gradients, ν is the kinematic viscosity of thefluid.

Fig. 4.1 : Schematic diagram.

Making use of suitable similarity transformations (as introduced by [55])

69

η =

ra

νz, u = axf 0 (η) , v = ayg0 (η) , w = −√aν (f + g) , (4.5)

the continuity equation is satisfied identically, where as equations (4.2) and (4.3) trans-

form to

f 000 − f 02 + (f + g) f 00 = 0, (4.6)

g000 − g02 + (f + g) g00 = 0, (4.7)

respectively. The boundary date (4.4) take the form

f 0 (η) = 1, g0 (η) = β, f (η) + g (η) = 0, at η = 0, (4.8)

f 0 (η) → 0, g0 (η)→ 0, as η → +∞,

where 0 denotes differentiation with respect to η, and β = b/a (a 6= 0) is the ratio ofvelocity gradients. We will restrict ourselves for the values of β to the range 0 ≤ β ≤ 1,because for β > 1, one can interchange the axes (x, and y).

4.2.2 Complete analytic solution

In order to solve the reduced nonlinear system (4.6) − (4.8) we employ the homotopyanalysis method [76] . According to the boundary conditions (4.8) , it is reasonable to

assume that f (η) and g (η) can be expressed by the following set of base functions

©ηke−nη/k ≥ 0, n ≥ 0ª , (4.9)

such that

f (η) =+∞Xn=0

+∞Xk=0

an,kηke−nη, (4.10)

g (η) =+∞Xn=0

+∞Xk=0

bn,kηke−nη, (4.11)

70

where an,k and bn,k are constant coefficients. It provides us with the so-called rule to

solution expressions for f (η) and g (η) respectively. In eq. (4.8) the condition f (0) +

g (0) = 0 can also be considered as f (0) = g (0) = 0. Therefore, due to the solution

expressions (4.10) , (4.11) and the boundary conditions (4.8) , it is straightforward to

choose the initial guess approximations

f0 (η) = 1− e−η, (4.12)

g0 (η) = β¡1− e−η

¢. (4.13)

We consider the linear operator

LV ≡ d3

dη3+

d2

dη2, (4.14)

for equations (4.6) and (4.7) satisfying the property

LV

£C1 + C2η + C3e

−η¤ = 0, (4.15)

where Ci‘s, (i = 1, · · · , 3) are arbitrary constants and the subscript ‘V ’ denotes the cor-respondences of the linear operator to velocity equations.

We construct the so-called zero-order deformation equations

(1− p)LV [F (η; p)− f0 (η)] = p~N1 [F (η; p) , G (η; p)] , (4.16)

(1− p)LV [G (η; p)− g0 (η)] = p~N2 [F (η; p) , G (η; p)] , (4.17)


∂F (η; p)

∂η| η=0 = 1,

∂G (η; p)

∂η|η=0= β, F (0; p) +G (0; p) = 0, (4.18)

∂F (η; p)

∂η| η=+∞ =

∂G (η; p)

∂η|η=+∞= 0,

71

where p ∈ [0, 1] is the embedding parameter, ~ is the nonzero auxiliary parameter, andthe nonlinear differential operators N1 and N2 are defined by

N1 [F (η; p) , G (η; p)] =∂3F

∂η3−µ∂F

∂η

¶2+ (F +G)

∂2F

∂η2, (4.19)

N2 [F (η; p) , G (η; p)] =∂3G

∂η3−µ∂G

∂η

¶2+ (F +G)

∂2G

∂η2. (4.20)

Notice that when p = 0 we have (from eqs. (4.16) and (4.17)) the initial approximations

F (η; 0) = f0 (η) , and G (η; 0) = g0 (η) , (4.21)

and for p = 1, eqs. (4.16) and (4.17) are same as the original eqs. (4.6) and (4.7)

respectively, so that

F (η; 1) = f (η) , and G (η; 1) = g (η) . (4.22)

So, as p varies from 0 to 1, F (η; p) moves from initial guess f0 (η) to the exact solution

f (η) and similarly G (η; p) from g0 (η) to g (η) . By Taylor’s theorem and eq. (4.21) we

have

F (η; p) = f0 (η) ++∞Xm=1

fm (η) pm, (4.23)

G (η; p) = g0 (η) ++∞Xm=1

gm (η) pm, (4.24)

where

fm (η) =1

m!

∂mF (η; p)

∂pm|p=0, and gm (η) =

1

m!

∂mG (η; p)

∂pm|p=0 . (4.25)

Assume that the auxiliary parameter ~ is so properly chosen that the series (4.23) and

72

(4.24) are convergent at p = 1, then using (4.22) we have

f (η) = f0 (η) ++∞Xm=1

fm (η) , (4.26)

g (η) = g0 (η) ++∞Xm=1

gm (η) . (4.27)

For the sake of simplicity, we define the vectors

−→f m (η) = f0, f1, ..., fm ,

−→g m (η) = g0, g1, ..., gm .

Differentiating m times (m = 1, 2, 3, · · · ) the zero-order deformation equations (4.16) −(4.18) with respect to p and then dividing them by m! and finally setting p = 0, we get

the mth-order deformation equations given below:

LV [fm (η)− χmfm−1 (η)] = ~Rm

³−→f m−1 (η) ,−→g m−1 (η)

´, (4.28)

LV [gm (η)− χmgm−1 (η)] = ~Qm

³−→f m−1 (η) ,−→g m−1 (η)

´, (4.29)


dfm (η)

dη=

dgm (η)

dη= fm (η) + gm (η) = 0, at η = 0, (4.30)

dfm (η)

dη→ 0,

dgm (η)

dη→ 0, as η → +∞,

where

Rm

³−→f m−1 (η) ,−→g m−1 (η)

´= f 000m−1 +

m−1Xk=0

£fm−1−kf 00k + gm−1−kf 00k − f 0m−1−kf

0k

¤, (4.31)

73

Qm

³−→f m−1 (η) ,−→g m−1 (η)

´= g000m−1 +

m−1Xk=0

£fm−1−kg00k + gm−1−kg00k − g0m−1−kg

0k

¤, (4.32)

and

χm = 0, for m ≤ 1, (4.33)

= 1, for m > 1.

Let f∗m (η) and g∗m (η) represent special solutions of eqs. (4.28)− (4.29) , respectively.From eq. (4.15) their general solutions read (for m ≥ 1)

fm (η) = f∗m (η) + C1 + C2η + C3e−η, (4.34)

and

gm (η) = g∗m (η) + C4 + C5η + C6e−η, (4.35)

where the constants C 0i s, (i = 1, ..., 6) are determined through the boundary conditions

(4.30) and are given by

C2 = C5 = 0, C3 =df∗m (η)dη

|η=0, C1 = −df∗m (η)

dη|η=0 −f∗m (0) , (4.36)

C6 =dg∗m (η)dη

|η=0, C4 = −dg∗m (η)

dη|η=0 −g∗m (0) .

In this way it is easy to solve the linear system (4.28)− (4.30) successively by means ofa symbolic computing software such as MATHEMATICA.

4.2.3 Convergence and accuracy of results

Liao [76] showed that whenever a solution series (obtained by HAM) converges it will

be one of the solutions of considered problem. As pointed out by Liao [76] the rate of

convergence of the approximations obtained by homotopy analysis method is controlled

by the auxiliary parameter ~. The admissible values of ~ for which the solution series

74

converges can be determined by drawing the so-called ~−curves. The interval on ~−axisfor which the ~−curve becomes parallel to the ~−axis is recognized as the set of admissiblevalues of ~ for which the solution series converges. For the present problem we have drawn

the ~−curves for f and g (see fig. 4.2) to find the suitable values of ~ in order to get

the convergent solution. Notice that we have plotted 9th-order approximations and the

interval of values of ~ can further be refined by drawing higher order approximations.

It is noted that our analytic solution converges at ~ = −0.373. In Table 4.1, we havelisted the HAM solution at different orders of approximation by augmenting the columns

that show difference between two successive approximations defined by4f 00i (0) =¯f 00i (0)− f 00i−1(0)

¯,

and 4g00i (0) =¯g00i (0)− g00i−1(0)

¯for f and g respectively. Clearly, at higher orders of ap-

proximations the corrections to the solution series are negligible and we can truncate the

series (4.26) and (4.27) . In Table 4.2, comparison of present analytic solution with the

numerical and approximate solutions obtained by Ariel [56] is presented. The present

analytic solution is calculated at the above mentioned value of ~. From Table it can be

observed that our analytic solution is in close agreement with numerical solution given

by Ariel [56] as compared to the approximate solution of Ariel [56], which proves the

accuracy of present analytic HAM solution.

Order of App. −f 00(0) −g00(0) 4f 00i (0) 4g00i (0)

5th 1.08163 0.466084 0.00620821 0.00145801

10th 1.09195 0.465009 0.000707992 0.0000374668

15th 1.09301 0.465176 0.0000623396 0.0000177485

20th 1.09309 0.465204 3.99697× 10−6 1.37478× 10−6

25th 1.09309 0.465205 2.64457× 10−7 4.23281× 10−8

30th 1.09309 0.465205 4.0245× 10−8 6.79564× 10−9

40th 1.03931 0.465205 2.5754× 10−10 3.78763× 10−10

50th 1.03931 0.465205 1.22591× 10−12 8.09908× 10−12

Table 4.1 : HAM solution at different orders of approximation.

75

Fig. 4.2 : ~−curves for velocity components.

Num. sol by Ariel [56] App. sol. by Ariel [56] HAM Sol.

β −f 00 (0) −g00 (0) −f 00 (0) −g00 (0) −f 00 (0) −g00 (0)0 1.00000 0.00000 1.00000 0.00000 1.00000 0.000000

0.1 1.02025 0.06684 1.01952 0.06796 1.02026 0.0668471

0.2 1.03949 0.14873 1.03827 0.15018 1.03949 0.148735

0.3 1.05795 0.24335 1.05642 0.24476 1.05795 0.243359

0.4 1.07578 0.34920 1.07406 0.35039 1.07579 0.349209

0.5 1.09309 0.46520 1.09129 0.46606 1.09309 0.465205

0.6 1.10994 0.59052 1.10814 0.59101 1.10995 0.590529

0.7 1.12639 0.72453 1.12466 0.72460 1.12640 0.724532

0.8 1.14248 0.86668 1.14089 0.86634 1.14249 0.866682

0.9 1.15825 1.01653 1.15683 1.01577 1.15825 1.01654

1.0 1.17372 1.17372 1.17253 1.17253 1.17372 1.17372

Table 4.2 : Comparison of results by Ariel [56] with present HAM solution.

76

4.3 Heat transfer analysis

We assume that the plate is heated at a constant temperature θw and fluid is fixed at

temperature θ∞ such that θw > θ∞. Due to the presence of temperature difference heat

flows form plate to fluid. In order to make the heat transfer analysis we assume the

temperature function to depend upon the same variables as does the velocity vector, i.e.

θ = θ (x, y, z) . Therefore, the energy equation in the presence of viscous dissipation is

given by

u∂θ

∂x+ v

∂θ

∂y+ w

∂θ

∂z= κ

µ∂2θ

∂x2+

∂2θ

∂y2+

∂2θ

∂z2

¶(4.37)

+µ

ρCp

³∂u∂y+ ∂v

∂x

´2+¡∂u∂z+ ∂w

∂x

¢2+³∂v∂z+ ∂w

∂y

´2+2

µ¡∂u∂x

¢2+³∂v∂y

´2+¡∂w∂z

¢2¶ ,


θ = θw, at z = 0, and θ → θ∞, as z → +∞, (4.38)

where ρ is density, µ is coefficient of viscosity, Cp is the specific heat and κ is the thermal

diffusivity. We introduce the dimensionless temperature

T (η) =θ − θ∞θw − θ∞

, (4.39)

where η is defined in eq. (4.5) . Therefore, the energy equation (4.37) in dimensionless

form can be written as

T 00 +Pr£(f + g)T 0 +Ecxf

002 +Ecyg002 + 4Ec

¡f 02 + g02 + f 0g0

¢¤= 0, (4.40)


T (0) = 1, and T → 0, as η → +∞, (4.41)

77

where Pr = ν/κ is the Prandtl number, Ec = µa/ρCp (θw − θ∞) is the Eckert number,

Ecx = a2x2/Cp (θw − θ∞) and Ecy = a2y2/Cp (θw − θ∞) are the local Eckert numbers

based on the variables x and y respectively.


In order to find a complete analytic solution of eqs. (4.40) and (4.41) we follow the same

procedure as performed in section 4.2.2 of the current chapter. To avoid the repetition

we omit the details of the solution procedure. Due to the boundary conditions (4.41) we

choose the initial guess of the form

T0 (η) = e−η, (4.42)

and the corresponding linear operator

LT ≡ d2

dη2+

d

dη, (4.43)

which satisfies the property

LT

£C7 + C8e

−η¤ = 0, (4.44)

where C7 and C8 are arbitrary constants.

The zero-order deformation equation for the present case is

(1− p)LT [Λ (η; p)− T0 (η)] = p~N3 [Λ (η; p) , F (η; p) , G (η; p)] , (4.45)


Λ (0; p) = 1, and Λ (η; p)→ 0, as η → +∞, (4.46)

78

where the nonlinear differential operator N3 [Λ (η; p) , F (η; p) , G (η; p)] is defined by

N3 [Λ (η; p) , F (η; p) , G (η; p)] =∂2Λ

∂η2+Pr

(F +G) ∂Λ∂η+ Ecx

³∂2F∂η2

´2+Ecy

³∂2G∂η2

´2+4Ec

µ³∂F∂η

´2+³∂G∂η

´2+ ∂F

∂η∂G∂η

¶ ,(4.47)

and correspondingly the mth-order deformation equation (m ≥ 1) is given by

LT [Tm − χmTm−1] = ~Pm

³−→f m−1 (η) ,−→g m−1 (η) ,

−→T m−1 (η)

´, (4.48)


Tm (0) = 0, and Tm (η)→ 0, as η → +∞, (4.49)

where

Pm

³−→f m−1 (η) ,−→g m−1 (η) ,

−→T m−1 (η)

´= T 00m−1+Pr

m−1Xk=0

(fm−1−k − gm−1−k)T 0k

+Ecxf00m−1−kf

00k +Ecyg

00m−1−kg

00k

+4Ec

f 0m−1−kf0k + g0m−1−kg

0k

+f 0m−1−kg0k

.(4.50)

Let T ∗m (η) represents a special solution of eq. (4.48) subject to the boundary condi-

tions (4.49) , then the general solution Tm (η) can be written as:

Tm (η) = T ∗m (η) + C7 + C8e−η, (4.51)

where C7 and C8 are arbitrary constants which can be determined using (4.49) as under

C7 = 0, C8 = −T ∗m (0) . (4.52)

79

Therefore, the complete analytic solution for temperature distribution is given by

T (η) = T0 (η) ++∞Xm=1

Tm (η) . (4.53)

4.4 Discussion on results

Before going to the graphical representation of results it is important to ensure the

convergence of the solution series (4.53) .As discussed in previous section, the convergence

is strongly dependent upon the auxiliary parameter ~. To find the admissible values of ~

we have plotted the ~−curve given in fig. 4.3. The interval of admissible values of ~ iscan be seen in fig. 4.3. Roughly it can be shown by the interval [−0.8,−0.4] . However,we have plotted all the graphs at ~ = −0.58. The convergence of solution series (4.53)has been shown in Table 4.3 by giving the solution at different orders of approximation.

Clealy, at higher orders of approximation the corrections to the solution T (η) become so

small to be negligible.

In figs. 4.4− 4.9 the dimensionless temperature function T (η) is plotted against the

dimensionless space variable η for different values of the parameters β, Pr, Ec, Ecx and

Ecy respectively. In fig. 4.4 the effect of the parameter β on T is shown. It is observed

that by increasing β the temperature increases in the neighborhood of stretching wall

and then starts decaying as we go in increasing direction of η. This unexpected behavior

happened due to the presence of the viscous dissipation terms in the temperature equation

(4.40). By increasing the values of β the temperature profiles shoot up near the plate, this

is because of the reason that by increasing the values of parameter β the velocity increases

which in turn causes the frictional heating. In fig. 4.5 the effect of the Prandtl number Pr

on temperature T is shown. Clearly, by increasing the Prandtl number the temperature

distribution near the wall increases and the thermal boundary-layer thickness decreases.

Again this unexpected shoot up in temperature near the wall is due to the presence of

viscous dissipation. Further, an increase in the Prandtl number results in increasing the

80

rate of heat transfer and this rate of heat transfer is positive which is not favorable for

the process of cooling of the sheet. Therefore, due to this fact it is recommended that

in the presence of viscous dissipation the fluids with small Prandtl number can serve as

good coolant. Further discussion on this issue is coming later. In fig. 4.6 temperature T

is plotted for different values of Eckert number Ec. Figure 4.6 depicts that by increasing

Ec the temperature increases near the wall. This is due to the fact that heat energy is

stored in the fluid due to the frictional heating. Thus the strong frictional heating slows

down the cooling process and in this case the study suggests that the rapid cooling of the

plate can be made possible if the viscous dissipation can be made as small as possible.

Similar effects of the local Eckert numbers Ecx and Ecy are observed which are shown in

figs. 4.7 and 4.8 respectively. It is noted that for Ecy the effect of local viscous dissipation

strongly depends on the parameter β. Notice that the small values of β correspond to

the situation in which the velocity along x direction is greater than the velocity in y

direction and therefore, in this case the viscous dissipation in y direction is small as

shown in fig. 4.8. The difference in the effects of Ecx and Ecy can be seen from figs. 4.7

and 4.8. In fig. 4.9 we have plotted the Nusselt number against the Prandtl number Pr

for different values of Ec. In this study fig. 4.9 is very much important. Clearly, for very

small values of Ec the cooling of the plate is possible and in such situations the fluids

with high Prandtl number are recommended. From figure it is clearly seen that for some

large values of Ec (> 0.4) the cooling of the plate is almost impossible.

81

Fig. 4.3 : ~−curve for temperature function.

Order of App. T 0(0) 4T 00i (0)

5th 0.369726 0.0495773

10th 0.429771 0.00922957

15th 0.438373 0.00105009

20th 0.441366 0.000381785

25th 0.442432 0.000134773

30th 0.442815 0.0000496583

35th 0.442960 0.0000193183

40th 0.443018 7.78973× 10−6

45th 0.443042 3.2184× 10−6

50th 0.443051 1.35575× 10−6

Table 4.3 : HAM sol. at different orders.

82

Fig. 4.4 : Effect of β on temperature profiles.

Fig. 4.5 : Temperature graphs for different Prandtl numbers.

83

Fig. 4.6 : Temperature graphs at different Eckert number.

Fig. 4.7 : Effect of loca Eckert number on temperature profiles.

84

Fig. 4.8 : Effect of local Eckert unmber on temperature profiles.

Fig. 4.9 : Nusselt number plotted against Pr at different Eckert number.

85

4.5 Conclusion

We have presented complete and highly accurate analytic solution to three-dimensional

flow of a viscous fluid over a stretching sheet with heat transfer analysis in the presence

of viscous dissipation. The solution is uniformly valid for all values of the parameter β

in the whole spatial domain 0 ≤ η < ∞. It is observed that the cooling process can bemade faster if one uses the fluids with small Prandtl numbers in the presence of viscous

dissipation. The effect of viscous dissipation is to increase the temperature near the

stretching wall. To make the cooling process efficient it is strongly recommended that

the viscous dissipation must be made as small as possible. However, in the presence of

viscous dissipation it is useful to plot the Nusselt number against Prandtl number for

fixed Eckert number. It helps in the selection of coolant for the cooling of the sheet. It

is also observed that the local Eckert numbers depend upon the parameter β.

Generalized three-dimensional steady flow over a stretching sheet with

heat transfer analysis has been studied in this chapter. Here we considered

the flow in an unbounded domain. The situation gets more interesting if one

considers the same flow in a channel of finite width. The coming chapter

deals with this case.

86

Chapter 5

Heat transfer analysis of 3D viscous

flow in a channel of lower stretching

wall

In this chapter we consider generalized three-dimensional channel flow of a

viscous fluid in the presence of heat and mass transfer. The system is assumed

to be dissipative so that the effect of viscous dissipation can be studied.

5.1 Introduction

The cooling of the stretching sheet is one of the major issue in polymer industry. Crane’s

[27] study of thermal boundary-layers over stretching sheet has been extended in different

directions by number of researchers. Abundant of literature is available in which authors

have studied the heat transfer phenomenon by making different flow assumptions in

order to achieve the optimum cooling of the sheet. In all such studies there is one thing

common, namely, the authors had been considering the flow in an unbounded domain.

However, there is a provision of considering the flow in a bounded domain (i.e. channel)

also, in which the lower wall of the channel can be considered as a stretching sheet.

87

Although, there are few studies available in which authors have considered such kind of

flow situations but they are very few in number [65− 67] and literature lacks in providingpurely analytic solutions to such kind of flows to the best of our knowledge. In such kind

of flows people [65, 67] use to apply perturbation on the Reynolds number due to which

the obtained solutions are no more valid for large Reynolds numbers. To make a complete

analysis of the problem, a solution is required which must be valid for small as well as

for large values of the Reynolds number. In this study we consider the full Navier-Stokes

equations governing a three-dimensional stead flow of a viscous fluid. Purely analytic

solution is obtained by using homotopy analysis method.

5.2 Flow analysis

5.2.1 Dynamic equations

We consider an incompressible viscous fluid bounded by two infinite parallel plates sepa-

rated by a distance h and the lower plate is a highly elastic membrane situated at z = 0.

The upper plate is uniformly porous subjected to constant injection in the channel fixed

at z = h. In order to make the fluid to flow, two equal and opposite forces are applied

along x−axis, and y−axis so that the plate is stretched in both directions keeping theorigin fixed. Therefore, the equations governing this type of steady flow are the full

Navier-Stokes equations

u∂u

∂x+ v

∂u

∂y+ w

∂u

∂z= −1

ρ

∂p

∂x+ ν

µ∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2

¶, (5.1)

u∂v

∂x+ v

∂v

∂y+ w

∂v

∂z= −1

ρ

∂p

∂y+ ν

µ∂2v

∂x2+

∂2v

∂y2+

∂2v

∂z2

¶, (5.2)

u∂w

∂x+ v

∂w

∂y+ w

∂w

∂z= −1

ρ

∂p

∂z+ ν

µ∂2w

∂x2+

∂2w

∂y2+

∂2w

∂z2

¶, (5.3)

88

along with the continuity equation

∂u

∂x+

∂v

∂y+

∂w

∂z= 0, (5.4)


u = ax, v = by, w = 0, at z = 0, (5.5)

u = 0, v = 0, w = −w∗0, at z = h,

where u (x, y, z) , v (x, y, z) , and w (x, y, z) are the velocity components, p is the pressure,

x, y, and z are the space coordinates, ρ is the density and ν is the dynamic viscosity

of the fluid, a and b are velocity gradients and w∗0 > 0 is constant injection velocity

at the upper plate. Notice that in comparison to the case of semi-infinite domain; eq.

(5.3) , the pressure terms in eqs. (5.1) and (5.2), and the terms ∂2u∂x2+ ∂2u

∂y2and ∂2v

∂x2+ ∂2v

∂y2

in the viscosity parts of eqs. (5.1) and (5.2) respectively are the additional terms that

came into consideration due the existence of a channel. This is simply because of the

non applicability of boundary-layer assumption on a channel flow. Introducing the non-

dimensional variables

η =1

hz, u = axf 0 (η) , v = ayg0 (η) , w = −ah (f + g) , (5.6)

the system (5.1)− (5.3) reduces to the following system of ordinary differential equations

a2x£f 000 −Re ¡f 02 − (f + g) f 00

¢¤=1

ρ

∂p

∂x, (5.7)

a2y£g000 −Re ¡g02 − (f + g) g00

¢¤=1

ρ

∂p

∂y, (5.8)

a2h [f 00 + g00 −Re ((f + g) (f 0 + g0))] =1

ρh

∂p

∂η, (5.9)

89

the continuity equations (5.4) is satisfied identically and the boundary conditions (5.5)

transform as

f 0 = 1, g0 = β, , f + g = 0, at η = 0, (5.10)

f 0 = 0, g0 = 0, , f + g = w0, at η = 1,

where Re = ah2/ν is the Reynold’s number, β = b/a (a 6= 0) is the ratio of the velocitygradients, w0 = w∗0/ah is the dimensionless injection parameter. Equation (5.7) with the

help of eq. (5.9) can be written as

f 000 − Re ¡f 02 − (f + g) f 00¢= A, (5.11)

where A is a constant. On differentiating the above equation with respect to η, one gets

f 0000 − Re (f 0f 00 − ff 000 − gf 000 − g0f 00) = 0. (5.12)

Similarly, in the same fashion eq. (5.8) takes the following form

g0000 −Re (g0g00 − fg000 − f 0g00 − gg000) = 0. (5.13)

Finally, our reduced system comprises of the equations (5.10) , (5.12) , and (5.13) .

5.2.2 Analytic homotopy solution

Our boundary conditions suggest the expressions for f (η) and g (η) of the form

f (η) =+∞Xi=1

Aiηi, (5.14)

g (η) =+∞Xj=1

Bjηj, (5.15)

90

where Ai and Bj are the coefficient of the solution series. In order to satisfy the boundary

conditions (5.10) we choose the initial approximations for f (η) and g (η) of the form

f0 (η) = η +

µ3w02− 2¶η2 + (1− w0) η

3, (5.16)

g0 (η) = βη +

µ3w02− 2β

¶η2 + (β − w0) η

3, (5.17)

respectively, and the corresponding auxiliary linear operator (for two equations (5.12)

and (5.13)) is given by

LV ≡ d4

dη4, (5.18)

which satisfies

LV

£C1 + C2η + C3η

2 + C4η3¤= 0 (5.19)

where C 0i s, (i = 1, ..., 4) are arbitrary constants.

We construct the zero-order deformation equations

(1− p)LV [F (η; p)− f0 (η)] = p~VN1 [F (η; p) , G (η; p)] , (5.20)

(1− p)LV [G (η; p)− g0 (η)] = p~VN2 [F (η; p) , G (η; p)] , (5.21)

with boundary conditions in the following form

∂F (η; p)

∂η= 1,

∂G (η; p)

∂η= β, F (η; p) +G (η; p) = 0, at η = 0, (5.22)

∂F (η; p)

∂η= 0,

∂G (η; p)

∂η= 0, F (η; p) +G (η; p) = w0 at η = 1,

where ~V is the non zero auxiliary parameter, F (η; p) andG (η; p) denote our required so-

lutions, p ∈ [0, 1] is an embedding parameter, and the nonlinear operatorsN1 [F (η; p) , G (η; p)]

and N2 [F (η; p) , G (η; p)] are defined through

N1 [F (η; p) , G (η; p)] =∂4F

∂η4−Re

µ∂F

∂η

∂2F

∂η2− F

∂3F

∂η3−G

∂3F

∂η3− ∂G

∂η

∂2F

∂η2

¶, (5.23)

91

N2 [F (η; p) , G (η; p)] =∂4G

∂η4− Re

µ∂G

∂η

∂2G

∂η2− F

∂3G

∂η3−G

∂3G

∂η3− ∂F

∂η

∂2G

∂η2

¶. (5.24)

Notice that, when p = 0 and p = 1, we respectively have (from eqs. (5.20) and (5.21))

F (η; 0) = f0 (η) , F (η; 1) = f (η) , (5.25)

G (η; 0) = g0 (η) , G (η; 1) = g (η) . (5.26)

Thus, as p varies from 0 to 1, F (η; p) and G (η; p) move to the final solutions f (η) and

g (η) respectively. To get the mth-order deformation equations, we first differentiate the

zero-order deformation equations (5.20) − (5.22) m−times (m = 1, 2, 3, ...) with respect

to p at p = 0 and then divide the resulting expressions by m! and find that

LV [fm (η)− χmfm−1 (η)] = ~VRm (η) , (5.27)

LV [gm (η)− χmgm−1 (η)] = ~VQm (η) , (5.28)


f 0m (η) = g0m (η) = fm (η) + gm (η) = 0, at η = 0, (5.29)

f 0m (η) = g0m (η) = fm (η) + gm (η) = 0, at η = 1,

where

Rm (η) = f 0000m−1 − Rem−1Xk=0

£f 0m−1−kf

00k − fm−1−kf 000k − gm−1−kf 000k − g0m−1−kf

00k

¤, (5.30)

Qm (η) = g0000m−1 −Rem−1Xk=0

£g0m−1−kg

00k − fm−1−kg000k − f 0m−1−kg

00k − gm−1−kg000k

¤, (5.31)

92

and

χk = 0, for k ≤ 1, (5.32)

= 1, for k > 1.

The above system (5.27) − (5.29) is a linear system and could easily be solved by

some symbolic computing software such as MATHEMATICA, MAPLE, MATLAB etc.

We have used MATHEMATICA to solve it. Let f∗m (η) and g∗m (η) represent special

solutions of eqs. (5.27) and (5.28) respectively then their general solutions read as:

fm (η) = f∗m (η) + C1 + C2η + C3η2 + C4η

3, (5.33)

gm (η) = g∗m (η) + C5 + C6η + C7η2 + C8η

3, (5.34)

where the constants C 0i s (i = 1, ..., 8) are determined through the boundary conditions

(5.29) as follows:

C1 = −f∗m (0) , C2 = −f∗0m (0) , C3 = f∗0m (1)− 3f∗m (1) + 3f∗m (0) + 2f∗0m (0) ,C4 = 2f∗m (1)− f∗0m (1)− 2f∗m (0)− f∗0m (0) , (5.35)

C5 = −g∗m (0) , C6 = −g∗0m (0) , C7 = g∗0m (1)− 3g∗m (1) + 3g∗m (0) + 2g∗0m (0) ,C8 = 2g∗m (1)− g∗0m (1)− 2g∗m (0)− g∗0m (0) , (5.36)

in this way, the complete analytic solution can be written in the form of infinite series of

function as under

f (η) =+∞Xm=0

fm (η) , (5.37)

and

g (η) =+∞Xm=0

gm (η) . (5.38)

93

5.2.3 Convergence of HAM solution

It is important to ensure that the series (5.37) and (5.38) are convergent. As mentioned

by Liao [76], the convergence of the solution series is directly determined through the

auxiliary parameter ~V . The values of ~V for which the solution series are convergent can

be found by drawing the so-called ~−curves [76]. The interval on the ~−axis for whichthe ~−curve is parallel to the ~−axis is considered as the allowed region for the valuesof ~V in order to get the convergent solution series. In the following figures (figs. 5.1

and 5.2) the ~−curves of f 00 (0) and g00 (0) are drawn at 15th-order of approximation. Infigs. 5.1 and 5.2 the intervals on ~−axis for which the ~−curves are parallel to ~−axisare recognized as the intervals of the admissible values of ~V for which the solution series

(5.37) and (5.38) converge. However, our analysis shows that a rapid convergence is

assured at ~V = −0.6. In order to show the convergence of the series (5.37) and (5.38)at ~V = −0.6 we have displayed the HAM results at different orders of approximation

in table 5.1. From table it is clear that after 20th-order approximation the corrections to

the solution become very small which proves the convergence of the solution series (5.37)

and (5.38) .

β = 0.5, w0 = 0.5, ~V = −0.6Orders of app. −f 00 (0) −g00 (0) 4f 00i (0) 4g00i (0)

5th 2.98703 0.503716 0.00675094 0.000182495

10th 2.99051 0.503764 0.0000278694 1.47762× 10−6

15th 2.99052 0.503763 3.02578× 10−7 5.51239× 10−9

20th 2.99053 0.503763 3.60449× 10−9 1.52553× 10−10

25th 2.99053 0.503763 2.38671× 10−11 1.20703× 10−12

Table 5.1 : HAM solutions at different orders.

94

Fig. 5.1 : ~−curve corresponding to f 0 (η) at 15th-order approximation.

Fig. 5.2 : ~−curve corresponding to g0 (η) at 15th-order approximation.

95

5.2.4 Graphical results and discussion

To see the effects of the physical parameters β, Re, and w0 on the dimensionless functions

f 0 (η) , g0 (η) , f (η) , and g (η) graphs are plotted for different values of these parameters.

The effect of the dimensionless parameter β on f 0 (η), g0 (η) , f (η) , g (η) are shown in

figs. 5.3 − 5.6 respectively. The effect of β on f 0 and f is very little (as shown in figs.

5.3 and 5.5), but there is a reasonable effect on g0 (η) and g (η) . By increasing β, f 0

decreases near the stretching plate and increases almost from the middle of the channel

whereas f (η) decreases by increasing β. The effect of β on g0 (η) and g (η) is shown

in figs. 5.4 and 5.6 respectively. From fig. 5.4 it is observed that by increasing β the

velocity increases at certain height in the channel and then goes on decreasing but g (η)

increases by increasing β throughout the channel width as shown in fig. 5.6.

.

Fig. 5.3 : f 0 (η) plotted at different values of β.

The effect of the Reynold’s number Re on f 0 (η), g0 (η) , f (η) and g (η) is shown in figs.

5.7 − 5.10. The velocity functions f 0 (η) and f (η) are plotted for different values of Re

96

in figs. 5.7 and 5.9 respectively. From fig. 5.7 it is clearly seen that the behavior of the

velocity changes from almost middle of the channel, that is by increasing the values of

Re the velocity decreases in the lower half of the channel and increases in the upper half

of the channel, but in fig. 5.9 the effect of Re is seen to decrease f (η) across the channel.

In fig. 5.8 the effect of Re on g0 (η) is shown for different values of Re . The effect of Re

on g0 (η) is reverse to that of f 0 (η) . By increasing Re the velocity in the lower half of the

channel increases and decreases in the upper half. Also in the case of g (η) the effect of

Re is reverse to that of f (η) , such kind of behavior is expected because the upper plate

is stationary. Figure 5.10 depicts that g (η) increases by increasing the values of Re and

these effects are more pronounced near the upper wall, this is because of the presence

of injection at the upper wall. The effects of constant injection on f 0 (η) and f (η) are

shown in figs. 5.11 and 5.13. Figure 5.11 shows that the velocity near the upper plate

decrease for 0 ≤ w0 ≤ 0.5 and for w0 > 0.5 the velocity increases.

Fig. 5.4 : g0 (η) plotted at different values of β.

97

Further, for values w0 > 0.5 the velocity increases as we move from upper plate to the

lower plate and this increase attains its maximum near the stretching wall as shown in

fig. 5.11. From here we infer that the injection opposes the reverse flow appearing in the

upper half of the channel. Figure 5.12 shows the effect of w0 on g0 (η) . The effect of w0

on g0 (η) is similar to that on f 0 (η) . From figs. 5.13 and 5.14 it is observed that the effect

of w0 on f (η) and g (η) are also same. An increase in the value of w0 causes to increase

the functions f (η) and g (η) on the upper plate as shown in figs. 5.13 and 5.14.

Fig. 5.5 : f (η) plotted at different values of β.

98

Fig. 5.6 : g (η) plotted at different values of β.

Fig. 5.7 : Velocity profile for different Reynolds number.

99

Fig. 5.8 : Velocity profile for different Reynolds number.

Fig. 5.9 : f (η) plotted at different values of Reynolds number.

100

Fig. 5.10 : g (η) plotted at different values of Reynolds number.

Fig. 5.11 : Effect of wall injection on f 0 (η) .

101

Fig. 5.12 : Effect of wall injection on g0 (η) .

Fig. 5.13 : Effect of wall injection on f (η) .

102

Fig. 5.14 : Effect of wall injection on g (η) .


In this section we study the heat transfer analysis in the viscous flow as considered

in section 5.2. The lower plate of the channel is a uniformly stretching sheet heated

at constant temperature θw and the upper plate and the intermediate fluid is assumed

to have a constant temperature θh. Since the flow is three-dimensional therefore it is

reasonable to assume the temperature profile of the form θ (x, y, z) . Thus, the energy

equation for such type of flow in the presence of viscous dissipation is given by

u∂θ

∂x+ v

∂θ

∂y+ w

∂θ

∂z= κ

µ∂2θ

∂x2+

∂2θ

∂y2+

∂2θ

∂z2

¶(5.39)

+µ

ρCp

³∂u∂y+ ∂v

∂x

´2+¡∂u∂z+ ∂w

∂x

¢2+³∂v∂z+ ∂w

∂y

´2+2

µ¡∂u∂x

¢2+³∂v∂y

´2+¡∂w∂z

¢2¶ ,

103

and the boundary data is described as

θ = θw, at z = 0, and θ = θh, at z = h, (5.40)

where ρ is density, µ is coefficient of viscosity, Cp is the specific heat, and κ is the thermal

diffusivity of the fluid. It is reasonable to mention here that due the assumption of channel

flow eq. (5.39) considers all of its terms in the governing system. However, on the other

hand if one considers an unbounded domain he always has the facility to ignore some

terms by applying the boundary-layer assumption. In such situation only the terms ∂2θ∂z2

,

∂u∂z, and ∂v

∂zon right hand side of eq. (5.39) along with left hand side will constitute the

governing equation. Therefore, all the terms of eq. (5.39) other than above mentioned

terms consider the existence of a channel. We introduce the dimensionless temperature

T (η) =θ − θhθw − θh

, (5.41)

where η is defined in eq. (5.6) . Therefore, the energy equation in dimensionless form can

be written as

T 00 +Pr£Re (f + g)T 0 + 4EcRe

¡f 02 + g02 + f 0g0

¢+Ecxf

002 +Ecyg002¤ = 0, (5.42)


T (0) = 1, T (1) = 0, (5.43)

where Re = ah2/ν, is the Reynolds number, Pr = ν/κ is the Prandtl number, Ec =

µa/ρCp (θw − θh) is the Eckert number, Ecx = a2x2/Cp (θw − θh) , andEcy = a2y2/Cp (θw − θh)

are the local Eckert numbers based on the lateral directions x and y respectively.

104


We repeat the same procedure as we did in section 5.2.2 by reporting the important steps

only. We choose our initial guess approximation satisfying the boundary data (5.43) of

the form

T0 (η) = 1− η, (5.44)

and the corresponding linear operator

LT ≡ d2

dη2, (5.45)


LT [C9 + C10η] = 0, (5.46)


The zero-order deformation equation for the present case is

(1− p)LT [Λ (η; p)− T0 (η)] = p~TN3 [Λ (η; p) , F (η; p) , G (η; p)] , (5.47)


Λ (0; p) = 1, Λ (1; p) = 0, (5.48)

where p ∈ [0, 1] is the embedding parameter, ~T is the non zero auxiliary parameter, andthe nonlinear differential operator N3 [Λ (η; p) , F (η; p) , G (η; p)] is defined by

N3 [Λ (η; p) , F (η; p) , G (η; p)] =∂2Λ

∂η2+Pr

Re (F +G) ∂Λ∂η+Ecx

³∂2F∂η2

´2+Ecy

³∂2G∂η2

´2+4EcRe

µ³∂F∂η

´2+³∂G∂η

´2+ ∂F

∂η∂G∂η

¶ ,(5.49)

105

correspondingly, the mth-order deformation equation (m ≥ 1) is given by

LT [Tm − χmTm−1] = ~TPm (η) , (5.50)


Tm (0) = 0, and Tm (1) = 0, (5.51)

where

Pm (η) = T 00m−1 +Prm−1Xk=0

(fm−1−k − gm−1−k)T 0k +Ecxf

00m−1−kf

00k

+Ecyg00m−1−kg

00k

+4Ec¡f 0m−1−kf

0k + g0m−1−kg

0k + f 0m−1−kg

0k

¢ . (5.52)

Let T ∗m (η) denotes a special solution of eq. (5.50) subjected to (5.51) , then the general

solution Tm (η) can be obtained as

Tm (η) = T ∗m (η) + C9 + C10η, (5.53)

where C9 and C10 are arbitrary constants which can be determined using (5.51)

C9 = −T ∗m (0) , C10 = T ∗m (0)− T ∗m (1) . (5.54)

Therefore, the complete analytic solution for temperature distribution can be written as

T (η) = T0 (η) ++∞Xm=1

Tm (η) . (5.55)

5.3.2 Convergence of HAM solution

In order to make the series (5.55) convergent one needs to determine the suitable values

of the auxiliary parameter ~T . For this purpose we have plotted the so-called ~−curves

106

(in figure 5.15) at different orders of approximation. From figure the interval of allowed

values of ~T seems to be −1.3 ≤ ~T ≤ −0.3. Further, our analysis shows that the rapidconvergence is achieved at ~T = −0.6.

Orders of app. −T 0 (0) 4T 0i (0) =¯T 0i (0)− T 0i−1(0)

¯5th 0.686344 0.0041355

10th 0.683748 0.0000389243

15th 0.683721 4.63236× 10−7

20th 0.683721 5.94452× 10−9

25th 0.683721 7.2112× 10−11

30th 0.683721 8.00804× 10−13

Table 5.2 : HAM solutions at different orders.

Fig. 5.15 : ~−curves at different orders of approximation.

107

However, large changes in the values of involved parameters can affect this particular

value of the auxiliary parameter ~T because it is observed that in many problems (such

as Mehmood and Ali [116]) the auxiliary parameter strongly depends upon the physical

parameters involved in the problem. In such cases one can modify the value of the

auxiliary parameter ~T for a set of chosen values of the involved parameters. In table 5.2

the convergence of the HAM solution has been proved by listing the solution at different

orders of approximation at ~T = −0.6.

Fig. 5.16 : Residual errors plotted against ~T .

Clearly, after 15th-order of approximation there is no correction in the solution up to the

acceptable number of decimal places which proves the convergence of the solution series

at ~T = −0.6. Table 5.2 is calculate for β = w0 = 0.5, Pr = 0.71, ~V = ~T = −0.6,Re = 2.0, Ec = Ecx = Ecy = 0.2. In order to prove the accuracy of our present analytic

solution we have also calculated the residual errors. In fig. 5.16 the residual errors are

plotted against ~T . Clearly, error is minimum at ~T = −0.6. This proves the accuracyand validity of our HAM solution.

108


In fig. 6.17 temperature graph is plotted for different values of the Prandtl number Pr in

the absence of viscous dissipation. From figure we see that the thermal boundary-layer

thickness decreases with the increase in Prandtl number which in turn shows that the

Prandtl number discourages the heat spread in the channel which is favorable for cooling

of the sheet. Further from graph it is clear that an increase in the value of Prandtl

number increases the heat transfer rate which expedites the cooling of the sheet. Also,

the heat transfer rate is small for air (Pr = 0.7) as compared to the water (Pr = 7.0).

It suggests that to enhance the heat transfer phenomenon water is far better than air

(when there is no viscous dissipation), however, other fluids with further higher Prandtl

numbers may also be selected in cooling the heated sheet. The effect of Prandtl number

on temperature profile in the presence of viscous dissipation is shown in fig. 5.18.

Fig. 5.17 : Effect of Prandtl number in the absence of viscous dissipation.

109

The effect of Prandtl number is the same as in fig. 5.17 but due to the presence of viscous

dissipation the rate of heat transfer near the wall is not that strong as it was in the absence

of viscous dissipation. In the presence of viscous dissipation the water no more serves as

a good coolant. We shall discuss this fact later in our discussion.In fig. 5.19 the effect of

the Eckert number Ec has been shown on the temperature profiles. From the graph it

is clear that the presence of viscous resistance increases the fluid temperature near the

solid wall which in turn depreciates the heat transfer from plate to the fluid. Thus in

order to enhance the cooling process the fluid is expected to have small Eckert number.

Similar effect of the local Eckert numbers have been observed in figures 5.20 and 5.21. A

comparison of figures 5.20 and 5.21 reveals that the effect of Ecx is stronger than that of

Ecy. This is due the reason that the fluid motion in x−direction is faster than that of iny−direction at β = 0.5. Our analysis shows that the effect of local Eckert number Ecydepends strongly on the values of the stretching ratio β. This is because of the presence

of Ecy as a coefficient of the term g002 in eq. (5.42) .

Fig. 5.18 : Effect of Prandtl number in the presence of viscous dissipation.

110

Thus, in order to increase the cooling process the fluids with large Prandtl number but

small Eckert number are suggested. In fig. 5.22 the Nusselt Number is plotted against

the Eckert number Ec for air and water at different values of local Eckert numbers Ecx

and Ecy. From figure it is clear that the cooling of plate is possible only for small values

of the Eckert number in both the cases, namely, when the local dissipation is present or

not. However, even in the absence of local Eckert numbers the cooling of the sheet is

possible only for very small values of Ec; for such small values of Ec water serves as a

good coolant but at relatively higher values of Ec the air is preferable, whereas in the

presence of local Eckert numbers the air is preferred over water to serve as a coolant.

Fig. 5.19 : Effect of Eckert number on temperature profile.

111

Fig. 5.20 : Effect of local Eckert number on temperature profile.

Fig. 5.21 : Effect of local Eckert number on temperature profile.

112

Fig. 5.22 : Nusselt number plotted against Ec.

5.4 Conclusion

In this chapter we considered a steady three-dimensional channel flow of an incompressible

viscous fluid with heat transfer phenomenon from the heated sheet to the ambient fluid.

Purely analytic solutions which are uniformly valid for all values of the parameters β,Re,

w0, Re, Ec, Ecx, and Ecy is obtained. Homotopy analysis method has been used to

get the analytic solutions. It is observed that for small values of Re the effect of Re on

f 0 (η) and g0 (η) is very little, whereas for values Re > 30 the effects are considerable.

It is observed that the constant injection at the upper plate increases the fluid velocity

and this increase in velocity is maximum near the lower (stretching) plate. Heat transfer

phenomenon has been studied in detail. It is observed that in the absence of viscous

dissipation the water is preferred over air as a coolant whereas in the presence of viscous

dissipation the air is preferred over water. Further the effect of local Eckert number Ecy

depends strongly on the values of β (stretching ratio).

113

After having a detailed analysis of generalized three-dimensional steady

flow over a stretching sheet with heat transfer analysis in both the cases,

namely, in unbounded and bounded domain it is natural to move to the

unsteady case of three-dimensional flow. This is what we are going to do in

the next chapter.

114

Chapter 6

Unsteady 3D MHD boundary-layer

flow over impulsively started

stretching sheet

In this chapter the study of heat transfer phenomenon in generalized three-

dimensional viscous flow over a stretching sheet has been extended to un-

steady case. The flow is developed due to an impulsively started stretching

sheet.

6.1 Introduction

In the continuation of the last two chapters it is natural to consider the unsteady case for

three-dimensional flow over a stretching sheet. This problem has already been considered

by Takhar et al. [62] , but unfortunately the results reported in [62] are incorrect. In [62]

authors used numerical method to integrate the reduced system of differential equations

subjected to the specified boundary conditions. The numerical results were presented

and discussed through graphs. In our opinion the flow behavior as shown in figures 8

and 9 of [62] is not in accordance with the actual situation. In figs. 8 and 9 authors

115

have shown that their numerical results are valid for all time 0 ≤ τ < +∞ over the

entire spatial domain 0 ≤ η < +∞. In [62] the authors made the assumptions (for t < 0)

that u = v = 0, in y ≥ 0, and for t ≥ 0, u and v are assumed to be non-zero. This

means that at t ≥ 0, the disturbance starts to penetrate in the fluid and the velocityof the fluid will start from zero to its maximum with the passage of time. The figures

8 and 9 in reference [62] depict that at very initial time the velocity is maximum and

with the passage of time the velocity reduces to its minimum which is totally against the

assumptions of the problem. We therefore, consider the same problem in order to make

the correct analysis. It is worthy to mention here that our results are uniformly valid for

all time τ . The effect of the MHD parameter on the velocity and the temperature profiles

is also studied. It is observed that the skin friction and the rate of cooling is effected by

the magnetic field.

6.2 Problem formulation

Consider an incompressible electrically conducting viscous fluid bounded by an infinite

flat plate z = 0. The fluid occupies the region z > 0. A uniform magnetic field of

strength B0 is applied in z−direction. The magnetic Reynolds number is assumed tobe small so that the induced magnetic field is smaller than the applied magnetic field.

Under this assumption one can neglect the induced magnetic field as it is so small to

be negligible. Further the plate is assumed to be heated at constant temperature θw

and the fluid has temperature θ∞ such that θw > θ∞. The three-dimensional nature of

flow suggests the velocity vector and temperature function of the form V = V (x, y, z, t)

and θ = θ (x, y, z, t) respectively. The governing system for this flow are the three laws,

namely,

Law of conservation of mass:

∂u

∂x+

∂v

∂y+

∂w

∂z= 0, (6.1)

116

Law of conservation of momentum (in component form after applying the boundary-

layer assumption):

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z= ν

∂2u

∂z2− σB2

0

ρu, (6.2)

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ w

∂v

∂z= ν

∂2v

∂z2− σB2

0

ρv, (6.3)

Law of energy conservation:

∂θ

∂t+ u

∂θ

∂x+ v

∂θ

∂y+ w

∂θ

∂z= κ

µ∂2θ

∂z2

¶, (6.4)

in the absence of viscous dissipation. According to flow assumptions the boundary con-

ditions for t ≥ 0 are given by

u = ax, v = by, w = 0, θ = θw, at z = 0, (6.5)

u → 0, v → 0, θ → θ∞, as z → +∞,

and for t < 0, the initial conditions are given by

u = v = w = 0, at all points (x, y, z), (6.6)

where u (x, y, z, t) , v (x, y, z, t) , andw (x, y, z, t) are the velocity components, and θ (x, y, z, t)

denotes the temperature which are functions of x, y, z and t, ρ is the density and ν is

the kinematic viscosity of the liquid, σ is the electrical conductivity, B0 is the strength

of the magnetic field, a (> 0) , and b (≥ 0) are the constants, Cp is the specific heat and

κ is the thermal diffusivity of the fluid.

In order to get the solution of the above system valid for all time we introduce the

117

following non-dimensional variables [62]

η =

ra

νξz, u = axf 0 (η, ξ) , v = ayg0 (η, ξ) , w = −

paνξ (f + g) ,

T (η, ξ) =θ − θ∞θw − θ∞

, ξ = 1− e−τ , τ = at, a > 0, (6.7)

due to which the continuity equation (6.1) is satisfied identically and the system (6.2)−(6.4) transforms to

∂3f

∂η3+1

2η (1− ξ)

∂2f

∂η2+ ξ (f + g)

∂2f

∂η2− ξ

µ∂f

∂η

¶2− ξ (1− ξ)

∂2f

∂η∂ξ−Mξ

∂f

∂η= 0, (6.8)

∂3g

∂η3+1

2η (1− ξ)

∂2g

∂η2+ ξ (f + g)

∂2g

∂η2− ξ

µ∂g

∂η

¶2− ξ (1− ξ)

∂2g

∂η∂ξ−Mξ

∂g

∂η= 0, (6.9)

∂2T

∂η2+Pr

·1

2η (1− ξ)

∂T

∂η+ ξ (f + g)

∂T

∂η− ξ (1− ξ)

∂T

∂ξ

¸= 0, (6.10)

respectively, where M = σB20/ρ is the magnetic parameter, and Pr = ν/κ is the Prandtl

number. The boundary data (6.5) also normalize to

∂f (η, ξ)

∂η| η=0 = 1,

∂g (η, ξ)

∂η|η=0= β, f (0, ξ) + g (0, ξ) = 0, T (0, ξ) = 1,

∂f (η, ξ)

∂η| η=+∞ = 0,

∂g (η, ξ)

∂η|η=+∞= 0, T (+∞, ξ) = 0, (6.11)

where β = b/a (a 6= 0) is the ratio of the velocity gradients.

6.3 Analytic HAM solution

We use homotopy analysis method to solve the system (6.8)−(6.11) .Due to our boundaryconditions (6.11) we give the solution expressions of f (η, ξ) , g (η, ξ) , and T (η, ξ) by the

following set of base functions

©ξkηme−nη/k ≥ 0,m ≥ 0, n ≥ 0ª ,

118

in the form

f (η, ξ) =+∞Xn=0

+∞Xq=0

+∞Xr=0

Arn,qξ

rηqe−nη, (6.12)

g (η, ξ) =+∞Xn=0

+∞Xq=0

+∞Xr=0

Brn,qξ

rηqe−nη, (6.13)

T (η, ξ) =+∞Xn=0

+∞Xq=0

+∞Xr=0

Crn,qξ

rηqe−nη, (6.14)

where Arn,q, B

rn,q, and C

rn,q are the constant coefficients to be determined. They provide us

with the so-called rule of solution expression for f (η, ξ) , g (η, ξ) and T (η, ξ) respectively.

Keeping in mind the boundary conditions (6.11) and the rule of solution expression it is

straightforward to choose the following initial guess approximations for f (η, ξ) , g (η, ξ)

and T (η, ξ)

f0 (η, ξ) = 1− e−η, (6.15)

g0 (η, ξ) = β¡1− e−η

¢, (6.16)

T0 (η, ξ) = e−η, (6.17)

respectively. Correspondingly we choose the auxiliary linear operators (for velocity equa-

tions we choose the same linear operator)

LV ≡ ∂3

∂η3− ∂

∂η, (6.18)

LT ≡ d2

dη2− 1, (6.19)

which have the following properties

LV

£C1e

−η + C2eη + C3

¤= 0, (6.20)

LT

£C4e

−η + C5eη¤= 0, (6.21)

119

where C 0i s, (i = 1, ..., 5) are constants.

Let ~ denotes the nonzero auxiliary parameter. We construct the so-called zero-order

deformation equations

(1− p)LV [F (η, ξ; p)− f0 (η, ξ)] = p~N1 [F (η, ξ; p) , G (η, ξ; p)] , (6.22)

(1− p)LV [G (η, ξ; p)− g0 (η, ξ)] = p~N2 [F (η, ξ; p) , G (η, ξ; p)] , (6.23)

(1− p)LT [Λ (η, ξ; p)− T0 (η, ξ)] = p~N3 [Λ (η, ξ; p) , F (η, ξ; p) , G (η, ξ; p)] , (6.24)


∂F (η, ξ; p)

∂η| η=0 = 1,

∂G (η, ξ; p)

∂η|η=0= β, F (0, ξ; p) +G (0, ξ; p) = 0, Λ (0, ξ; p) = 1,

∂F (η, ξ; p)

∂η| η=+∞ = 0,

∂G (η, ξ; p)

∂η|η=+∞= 0, Λ (+∞, ξ; p) = 0, (6.25)

where F (η, ξ; p) , G (η, ξ; p) , and Λ (η, ξ; p) denote our required solutions, p ∈ [0, 1] is anembedding parameter and the nonlinear operatorsN1 [F (η, ξ; p) , G (η, ξ; p)],N2 [F (η, ξ; p) , G (η, ξ; p)] ,

and N3 [Λ (η, ξ; p) , F (η, ξ; p) , G (η, ξ; p)] are defined through

N1 [F (η, ξ; p) , G (η, ξ; p)] =∂3F

∂η3+1

2η (1− ξ)

∂2F

∂η2+ ξ (F +G)

∂2F

∂η2− ξ

µ∂F

∂η

¶2−ξ (1− ξ)

∂2F

∂η∂ξ−Mξ

∂F

∂η, (6.26)

N2 [F (η, ξ; p) , G (η, ξ; p)] =∂3G

∂η3+1

2η (1− ξ)

∂2G

∂η2+ ξ (F +G)

∂2G

∂η2− ξ

µ∂G

∂η

¶2−ξ (1− ξ)

∂2G

∂η∂ξ−Mξ

∂G

∂η, (6.27)

N3 [Λ (η, ξ; p) , F (η, ξ; p) , G (η, ξ; p)] =∂2Λ

∂η2+Pr

12η (1− ξ) ∂Λ

∂η+ ξ (F +G) ∂Λ

∂η

−ξ (1− ξ) ∂Λ∂ξ

.(6.28)

120

Clearly, for p = 0, and p = 1 we have from eqs. (6.22)− (6.24)

F (η, ξ; 0) = f0 (η, ξ) , F (η, ξ; 1) = f (η, ξ) , (6.29)

G (η, ξ; 0) = g0 (η, ξ) , G (η, ξ; 1) = g (η, ξ) , (6.30)

Λ (η, ξ; 0) = T0 (η, ξ) , Λ (η, ξ; 1) = T (η, ξ) , (6.31)

respectively. So, as p increases from 0 to 1 the initial approximations move to the final

solutions. Assume that the auxiliary parameter ~ is so properly chosen that the Taylor’s

series of F (η, ξ; p) , G (η, ξ; p) , and Λ (η, ξ; p) expanded with respect to the embedding

parameter p, i.e.

F (η, ξ; p) = F (η, ξ; 0) ++∞Xm=1

fm (η, ξ) pm, (6.32)

G (η, ξ; p) = G (η, ξ; 0) ++∞Xm=1

gm (η, ξ) pm, (6.33)

Λ (η, ξ; p) = Λ (η, ξ; 0) ++∞Xm=1

Tm (η, ξ) pm, (6.34)

where

fm (η, ξ) =1

m!

∂mF (η, ξ; p)

∂pm|p=0, (6.35)

gm (η, ξ) =1

m!

∂mG (η, ξ; p)

∂pm|p=0, (6.36)

Tm (η, ξ) =1

m!

∂mΛ (η, ξ; p)

∂pm|p=0, (6.37)

converge at p = 1. Then we have from eqs. (6.32)− (6.34) after using (6.29)− (6.31) that

f (η, ξ) = f0 (η, ξ) ++∞Xm=1

fm (η, ξ) , (6.38)

g (η, ξ) = g0 (η, ξ) ++∞Xm=1

gm (η, ξ) , (6.39)

121

T (η, ξ) = T0 (η, ξ) ++∞Xm=1

Tm (η, ξ) , (6.40)

where fm (η, ξ) , gm (η, ξ) , and Tm (η, ξ) (m ≥ 1) can be determined by solving the mth-order deformation equations. To get themth-order deformation equations, we first differ-

entiate the zero-order deformation system (6.22)− (6.25) m−times (m = 1, 2, 3, ...) with

respect to p at p = 0 and then divide the resulting expressions by m! to find that

LV [fm (η, ξ)− χmfm−1 (η, ξ)] = ~Rm (η, ξ) , (6.41)

LV [gm (η, ξ)− χmgm−1 (η, ξ)] = ~Qm (η, ξ) , (6.42)

LT [Tm (η, ξ)− χmTm−1 (η, ξ)] = ~Pm (η, ξ) , (6.43)


∂fm (η, ξ)

∂η=

∂gm (η, ξ)

∂η= fm (η, ξ) + gm (η, ξ) = 0, Tm (η, ξ) = 0, at η = 0,

∂fm (η, ξ)

∂η=

∂gm (η, ξ)

∂η= Tm (η, ξ) = 0, at η = +∞, (6.44)

where

Rm (η, ξ) =∂3fm−1∂η3

+1

2η (1− ξ)

∂2fm−1∂η2

− ξ (1− ξ)∂2fm−1∂η∂ξ

−Mξ∂fm−1∂η

(6.45)

+ξm−1Xk=0

·fm−1−k

∂2fk∂η2

+ gm−1−k∂2fk∂η2− ∂fm−1−k

∂η

∂fk∂η

¸,

Qm (η, ξ) =∂3gm−1∂η3

+1

2η (1− ξ)

∂2gm−1∂η2

− ξ (1− ξ)∂2gm−1∂η∂ξ

−Mξ∂gm−1∂η

(6.46)

+ξm−1Xk=0

·fm−1−k

∂2gk∂η2

+ gm−1−k∂2gk∂η2− ∂gm−1−k

∂η

∂gk∂η

¸,

122

Pm (η, ξ) =∂2Tm−1∂η2

+Pr

12η (1− ξ) ∂Tm−1

∂η− ξ (1− ξ) ∂Tm−1

∂ξ

+ξm−1Xk=0

³fm−1−k

∂Tm−1∂η

+ gm−1−k∂Tm−1∂η

´ , (6.47)

and

χk = 0, for k ≤ 1, (6.48)

= 1, for k > 1.

The above system (6.41)−(6.44) is a linear system which can easily be solved by somesymbolic computing software such as MATHEMATICA, MAPLE, MATLAB etc.. We

have used MATHEMATICA to solve it. Let f∗m (η, ξ) , g∗m (η, ξ) and T ∗m (η, ξ) represent

special solutions of eqs. (6.41)− (6.43) respectively, then their general solutions read

fm (η, ξ) = f∗m (η, ξ) + C1e−η + C2e

η + C3, (6.49)

gm (η, ξ) = g∗m (η, ξ) + C4e−η + C5e

η + C6, (6.50)

Tm (η, ξ) = T ∗m (η, ξ) + C7e−η + C8e

η, (6.51)


(6.44) , given by

C2 = 0, C1 =∂f∗m (η, ξ)

∂η|η=0, C3 = −C1 − f∗m (0, ξ) , (6.52)

C5 = 0, C4 =∂g∗m (η, ξ)

∂η|η=0, C6 = −C4 − g∗m (0, ξ) ,

C8 = 0, and C7 = −T ∗m (0, ξ) .

In this way it is easy to solve the linear system (6.41) − (6.44) one after the other inorder.

123

6.3.1 Convergence and validity of HAM solution

As mentioned by Liao [76] that whenever the solution series obtained by homotopy analy-

sis method converges, it must be one of the solutions of the considered problem. The

convergence of the solution series (6.38)− (6.40) strongly depends upon the auxiliary pa-rameter ~, for which we can choose a proper value by plotting the so-called ~−curves toensure the convergence of the solution series. In order to determine the region of allowed

values of ~, we have plotted the ~−curves for our present HAM solution in figs. 6.1 and

6.2. Clearly, the interval of values of ~ can be guessed as −0.7 ≤ ~ ≤ −0.5. Our analysisshows that the solution series (6.38)− (6.40) converge at ~ = −0.6. Notice that in HAMsolution the higher order solutions (for m ≥ 1) are corrections to the initial approxima-tion and on increasing the order of approximation the errors must went on decreasing.

Table 6.1 show that by increasing the order of approximation the error decays, even at

15th-order of approximation the corrections to the solution are so small to be negligi-

ble. The present problem was first studied by Wang [55] for steady flow. In his study

Wang considered the hydrodynamic case without heat transfer analysis. Wang presented

his results for different values of the parameter β in tabular form. It is important to

mention here that the Wang’s results can easily be recovered from our present results at

~ = −0.6 by substituting ξ = 1 and M = 0 which proves the validity and accuracy of

our present results. A comparison between the numerical values obtained by Wang [55]

and those obtained by present analytic solution is given in table 6.2. Table 6.2 shows

that our present analytic solution is in good agreement with Wang’s solution [55] which

proves the accuracy and validity of our present results.

124

Fig.6.1 : ~−curve corresponding to f (η, ξ) .

Fig.6.2 : ~−curve corresponding to g (η, ξ) .

125

β 5th-order app. 10th-order app. 15th-order |15th− 14th|0.25 −fηη (0, ξ) 1.047540 1.048800 1.048810 1.11994× 10−6

0.25 −gηη (0, ξ) 0.195169 0.194548 0.194564 2.48154× 10−6

0.50 −fηη (0, ξ) 1.091800 1.093090 1.093090 1.21772× 10−7

0.50 −gηη (0, ξ) 0.464788 0.465209 0.465204 3.83057× 10−6

0.75 −fηη (0, ξ) 1.133350 1.134470 1.134490 0.000013514

0.75 −gηη (0, ξ) 0.793858 0.794612 0.794621 5.60505× 10−6

1.00 −fηη (0, ξ) 1.172640 1.173710 1.173720 3.04646× 10−6

1.00 −gηη (0, ξ) 1.112640 1.173710 1.173720 3.67987× 10−6

Table 6.1 : HAM solutions at different orders at ξ = 0.5.

Present HAM sol. for ~ = −0.6, ξ = 1 Wang’s [55] sol.

β −fηη (0) −gηη (0) f (+∞) g (+∞) −fηη (0) −gηη (0) f (+∞) g (+∞)0.00 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00

0.25 1.04881 0.194564 0.907151 0.257991 1.048813 0.194564 0.907075 0.257671

0.50 1.09309 0.465204 0.842388 0.451697 1.093097 0.465205 0.842360 0.451671

0.75 1.13449 0.794621 0.792299 0.612135 1.134485 0.794622 0.792308 0.612049

1.00 1.17372 1.173720 0.751485 0.751500 1.17320 0.173720 0.751527 0.751527

Table 6.2 : Comparison between present HAM solution and Wang’s [55] results.


The effect of the magnetic parameter M on velocity and temperature profiles has been

shown in figures 6.3− 6.5. From figures 6.3 and 6.4 it is clear that the velocity decreaseswith the increase in the magnetic field and as a consequence the layer thickness decreases.

Further it is also evident from figures 6.3 and 6.4 that by increasing the magnetic field

strength the skin friction at the wall increases and large skin friction has the capability

126

to damage the solid surface. Thus the magnetic field is fruitful in controlling the layer

thickness but on the other hand a strong magnetic field may result in increasing the

roughness of the surface. In figure 6.5 temperature profile is plotted for different values

of the magnetic parameterM. Clearly, the magnetic field causes to raise the temperature

in the vicinity of the hot surface. It can also be seen from fig. 6.5 that the temperature

gradients at the wall decrease with the increase in magnetic field strength which in turn

shows the decrease in rate of heat transfer at the wall and as a consequence the cooling

of the stretching sheet can be delayed. Therefore, to expedite the cooling process one

need to make the magnetic field as weak as possible.

Fig. 6.3 : Velocity component f(η, ξ) plotted at different M.

In figures 6.6− 6.8 we have plotted the velocity and temperature graphs against thespace variable for different time values to show the validity of our HAM solution for all

time. From figures 6.6 and 6.7 it is clear that by increasing the time variable the veloc-

ity increases and after some particular value of time τ the velocity remains unchanged

showing that the steady state has reached. Similar behavior of the temperature function

127

is shown for different time in figure 6.8. This proves the uniform validity of our analytic

solution for all time in the whole spatial domain. From all these figures viz. 6.6 − 6.8one can easily observe that the fluid attains its steady state in a short interval of time

which means that the time dependence of the flow is confined in very initial moments of

the flow phenomenon.

Fig. 6.4 : Velocity component g(η, ξ) plotted at different M.

128

Fig. 6.5 : Temperature profiles at different values of M.

Fig. 6.6 : Velocity component f(η, ξ) plotted at different time.

129

Fig. 6.7 : Velocity component g(η, ξ) plotted at different time.

Fig. 6.8 : Temperature profiles plotted at different time.

130

6.4 Conclusion

In this chapter we considered an unsteady three-dimensional boundary-layer flow of an

incompressible MHD viscous fluid over a stretching sheet. Heat transfer analysis from

plate to the fluid has also been investigated. Magnetic field is observed to control the

layer thickness but the skin friction increases by increasing the magnetic field which may

cause to damage the surface of the sheet. To avoid the roughness of the surface the

magnetic field must be controlled. Further, strong magnetic field results in depreciating

the cooling process. Again, magnetic field needs to be controlled in order to expedite the

cooling process. The present analytic solution is uniformly valid for all time in the whole

spatial domain which proves the usefulness of the homotopy analysis method. Incorrect

results of Takhar et al. [62] have been corrected.

Up till now we have considered three problems of generalized three-dimensional

flow, namely, the steady flow, the channel flow and the unsteady flow. How-

ever, a three-dimensional viscous flow may be considered in a rotating fluid

over a stretching sheet. In the coming chapter we deal with such type of flow.

131

Chapter 7

Rotating flow in a channel of lower

stretching wall with heat transfer

This chapter deals with heat transfer analysis in three-dimensional channel

flow of an incompressible MHD viscous fluid in a rotating frame. The flow is

developed due to the continuous stretching of the lower wall of the channel.

7.1 Introduction

In previous chapters we have studied the heat transfer phenomenon in multidimen-

sional viscous flows under varied physical assumptions such as suction/injection; MHD,

steady/unsteady etc. However, an interesting flow situation is yet missing which is of

great importance in engineering point of view, namely, the flow in a rotating frame. In

this chapter we study heat transfer analysis in an electrically conducting fluid bounded

by two parallel plates in the presence of viscous dissipation. The flow problem (without

considering the heat transfer) was studied by Vajravelu and Kumar [67] . In [67] authors

reported analytic (perturbation) and numerical results. Their analytic solution is valid

only for small values of the Reynold number as they expanded the velocity in powers

of Re (assuming Re as small parameter). In this study we report a purely analytic so-

132

lution for velocity and temperature functions valid for small as well as large values of

the parameter Re . Further we investigate the heat transfer phenomenon in the channel

which is important in controlling the process of cooling of the stretching sheet. The gov-

erning partial differential equations are transformed to a system of ordinary differential

equations with the help of similarity transformations. Homotopy analysis method is used

to get complete analytic solution for velocity and temperature profiles. The effects of

different parameters are discussed through graphs.

7.2 Flow analysis

7.2.1 Governing equations

We consider a fully developed three-dimensional hydromagnetic flow of an incompressible

viscous fluid between two parallel plates. One is situated at y = 0 and the other is situated

at y = h. The flow is developed by uniform stretching of the lower sheet. Two equal and

opposite forces are applied in x−direction in order to stretch the plate. The upper plateis uniformly porous subjected to constant injection. A magnetic field of strength B0 is

applied perpendicular to the plates along y−axis. The plates and the fluid rotate inunison about y−axis with an angular velocity Ω =(0,Ω∗, 0) . Due the uniform stretchingin one direction the velocity vector for this flow suits V = [u (x, y) , v (x, y) , w (x, y)] .

Therefore, the mass and momentum conservation laws (see section 1.2.1) read as

∂u

∂x+

∂v

∂y= 0, (7.1)

u∂u

∂x+ v

∂u

∂y+ 2Ω∗w = −1

ρ

∂p

∂x+ ν

·∂2u

∂x2+

∂2u

∂y2

¸− σB2

0

ρu, (7.2)

u∂v

∂x+ v

∂v

∂y= −1

ρ

∂p

∂y+ ν

·∂2v

∂x2+

∂2v

∂y2

¸, (7.3)

133

u∂w

∂x+ v

∂w

∂y− 2Ω∗u = ν

·∂2w

∂x2+

∂2w

∂y2

¸− σB2

0

ρw, (7.4)

where p is pressure, ν is kinematic viscosity, ρ is density, B0 is the strength of the magnetic

field, and σ is the electrical conductivity.

Fig. 7.1 : Schematic diagram of flow situation.

134

Our flow assumptions suggest the boundary conditions of the form

u = ax, v = 0, w = 0, at y = 0, (7.5)

u = 0, v = −V ∗0 , w = 0, at y = h,

where a (> 0) is a constant, V ∗0 denotes the normal velocity at upper plate.

Before going to the solution of the problem we first normalize the problem using the

following dimensionless variables

η =1

hy, u = axf 0 (η) , v = −ahf (η) , w = axg (η) , (7.6)

due to which the continuity equation (7.1) is satisfied identically and the system (7.2)−(7.4) transforms to

−1ρ

∂p

∂x= a2x

·f 02 − ff 00 − 1

Ref 000 +

M

Ref 0 + 2

Ω

Reg

¸, (7.7)

− 1ρh

∂p

∂η= a2h

·ff 0 +

1

Ref 00¸, (7.8)

g00 −Re [f 0g − fg0] + 2Ωf 0 −Mg = 0, (7.9)

respectively, where M = σB20/ρ is the magnetic parameter, Re = ah2/ν is the Reynold

number, and Ω = Ω∗h2/ν is the rotation parameter. Cross differentiation of (7.7) and

(7.8) gives

f iv −Re (f 0f 00 − ff 000)− 2Ωg0 −Mf 00 = 0, (7.10)

and the boundary conditions (7.5) in the form of new variables are given by

f = 0, f 0 = 1, g = 0, at η = 0, (7.11)

f = w0, f 0 = 0, g = 0, at η = 1,

where w0 = V ∗0 /ah (a 6= 0) is the constant injection parameter. Thus our governing

135

system in normalized form comprises of eqs. (7.9)− (7.11) .

7.2.2 Analytic solution by HAM

We use HAM to solve the nonlinear system (7.9) − (7.11) analytically. Due to the

boundary conditions (7.11) it is reasonable to choose the initial approximations of the

form

f0 (η) = (1− 2w0)η3 + (3w0 − 2) η2 + η, (7.12)

g0 (η) = 0, (7.13)

where the initial approximations are required to satisfy the boundary data (7.11). The

corresponding linear operators are given by

Lf ≡ d4

dη4, (7.14)

Lg ≡ d2

dη2, (7.15)

which satisfy the properties

Lf

£C1η

3 + C2η2 + C3η + C4

¤= 0, (7.16)

Lg [C5η + C6] = 0, (7.17)

where C 0i s, (i = 1, ..., 6) are arbitrary constants.

We construct the so-called zero-order deformation equations

(1− p)Lf [F (η; p)− f0 (η)] = p~N1 [F (η; p) , G (η; p)] , (7.18)

(1− p)Lg [G (η; p)− g0 (η)] = p~N2 [F (η; p) , G (η; p)] , (7.19)

136


F (η; p) = 0,∂F (η; p)

∂η= 1, G (η; p) = 0, at η = 0, (7.20)

F (η; p) = w0,∂F (η; p)

∂η= 0, G (η; p) = 0, at η = 1,

where p ∈ [0, 1] is the embedding parameter, ~ is the nonzero auxiliary parameter, and thenonlinear operators N1 [F (η; p) , G (η; p)] and N2 [F (η; p) , G (η; p)] are defined through

N1 [F (η; p) , G (η; p)] =∂4F

∂η4−Re

·∂F

∂η

∂2F

∂η2− F

∂3F

∂η3

¸− 2Ω∂G

∂η−M

∂2F

∂η2, (7.21)

N2 [F (η; p) , G (η; p)] =∂2G

∂η2−Re

·∂F

∂ηG− F

∂G

∂η

¸+ 2Ω

∂F

∂η−MG. (7.22)

Notice that for p = 0 and p = 1 we respectively have

F (η; 0) = f0 (η) , G (η; 0) = g0 (η) , (7.23)

and

F (η; 1) = f (η) , G (η; 1) = g (η) . (7.24)

So, as p varies from 0 to 1, F (η; p) and G (η; p) move from initial guess approximations

f0 (η) and g0 (η) to the final solutions f (η) and g (η) respectively. By Taylor’s theorem

and eq. (7.23), we have

F (η; p) = f0 (η) ++∞Xm=1

fm (η) pm, (7.25)

G (η; p) = g0 (η) ++∞Xm=1

gm (η) pm, (7.26)

where

fm (η) =1

m!

∂mF (η; p)

∂pm|p=0, (7.27)

137

gm (η) =1

m!

∂mG (η; p)

∂pm|p=0 . (7.28)

We assume that the series (7.25) and (7.26) are convergent at p = 1 (with a suitable

choice of ~) then we have from eq. (7.24) that

f (η) = f0 (η) ++∞Xm=1

fm (η) , (7.29)

g (η) = g0 (η) ++∞Xm=1

gm (η) . (7.30)

Differentiating the zero-order deformation equations (7.18)−(7.20)m−times (m = 1, 2, 3...)

with respect to p at p = 0 and then dividing the resulting expressions by m! we get the

mth-order deformation equations

Lf [fm (η)− χmfm−1 (η)] = ~Rm (η) , (7.31)

Lg [gm (η)− χmgm−1 (η)] = ~Qm (η) , (7.32)


fm (η) = f 0m (η) = gm (η) = 0, at η = 0, (7.33)

fm (η) = f 0m (η) = gm (η) = 0, at η = 1,

where

Rm (η) = f ivm−1 − 2Ωg0m−1 −Mf 00m−1 −Rem−1Xk=0

£f 0m−1−kf

00k − fm−1−kf 000k

¤, (7.34)

Qm (η) = g00m−1 + 2Ωf0m−1−k −Mgm−1 −Re

m−1Xk=0

£f 0m−1−kgk − fm−1−kg0k

¤, (7.35)

138

and

χk = 0, for k ≤ 1, (7.36)

= 1, for k > 1.

The linear system (7.31)− (7.33) can easily be solved by using some symbolic computingsoftware such as MATHEMATICA. Let f∗m (η) , and g∗m (η) represent special solutions of

eqs. (7.31) and (7.32) respectively, then their general solutions read (for m ≥ 1)

fm (η) = f∗m (η) + C1η3 + C2η

2 + C3η + C4, (7.37)

gm (η) = g∗m (η) + C5η + C6, (7.38)


(7.33) , given by

C4 = −f∗m (0) , C3 = −f∗0m (0) , C1 = C3 + 2C4 − f∗0m (1) + 2f∗m (1) , (7.39)

C2 = −2C3 − 3C4 + f∗0m (1)− 3f∗m (1) , C5 = −g∗m (1) , C6 = −g∗m (0) .

In this way one can easily find fm (η) , and gm (η) (for m ≥ 1) and the final solutionsf (η) , and g (η) can be obtained by substituting fm (η) , and gm (η) in (7.29) and (7.30)

respectively.

7.2.3 Results and discussion

Before going to the discussion of results it is necessary to prove the convergence of the

solution series (7.29) and (7.30) and to determine the value of the auxiliary parameter

~. As mentioned by Liao [76] that whenever the solution series obtained by homotopy

analysis method converges, it must be one of the solutions of the considered problem.

The convergence of the solution series depends strongly upon the auxiliary parameter ~.

139

The values of ~ for which the solution series converge are determined by drawing the so-

called ~−curve. In our problem, to find the values of ~ we have plotted the ~−curves infig. 7.2. From figure the interval of allowed values of ~ is observed to be −1.6 ≤ ~ ≤ −0.2(roughly). Our analysis shows that a fast convergence is achieved at ~ = −0.8. Theconvergence of our solution series (7.29) and (7.30) is shown in table 7.1 where we have

reported the HAM solution at different orders of approximation. From table it is clear

that the corrections to the solution become negligible at higher orders even after the

10th-order approximation (up to six decimal places).

The effect of constant injection is shown in figs. 7.3 and 7.4 on velocity functions

f 0 (η) and g (η) respectively. Clearly, injection promotes flow in the channel. The effect

of the magnetic filed is to reduce the velocity f 0 (η) in the lower half of the channel but

in the upper half of the channel f 0 (η) increases by increasing M. This is because of the

presence of constant injection at the upper wall (see fig. 7.5). However, g (η) decreases

across the channel by increasing the parameter M as shown in fig. 7.6. The effect of

Ω on the velocity f 0 (η) is shown in fig. 7.7. Clearly, in the presence of rotation, the

velocity f 0 (η) oscillates across the channel. But the flow is developed in z−direction byincreasing the rotation parameter as shown in fig. 7.8. A comparison of figs. 7.6 and

7.8 reveals that the magnetic field counters the effect of rotation on g (η) . Therefore, the

effects of rotation can be controlled with the help of magnetic field.

M = 1.0, Re = 2.0, w0 = 0.5, Ω = 0.5

Orders of app. −f 00 (0) g0 (0) 4f 00i (0) 4g00i (0)

5th-order 1.178060 0.144786 0.000128816 0.0000504036

10th-order 1.178080 0.144798 2.37667× 10−8 1.12773× 10−8

15th-order 1.178080 0.144798 7.17715× 10−12 2.88× 10−12

Table 7.1 : HAM solutions at different orders at ~ = −0.8.

140

Fig. 7.2 : Velocity ~−curves.

Fig. 7.3 : Effect of constnat suction on veclocity component f 0 (η) .

141

Fig. 7.4 : Effect of constant suciton velocity on g (η) .

Fig. 7.5 : Veclocity f 0 (η) plotted at different values of the magnetic parameter.

142

Fig. 7.6 : Effect of magnetic field on velcoity component g (η) .

Fig. 7.7 : Velocity graphs at diffenet values of the rotation parameter.

143

Fig. 7.8 : Velocity graphs at diffenet values of the rotation parameter.


7.3.1 Energy equation

In the preceding section we considered the velocity vector as a function of x, and y. It

is therefore, reasonable to consider the temperature θ also as a function of x, and y, i.e.

θ = θ (x, y) .We assume that the lower plate is heated at a constant temperature θw and

the upper plate has temperature θh such that (θw > θh) . The energy equation for this

type of flow in the presence of viscous dissipation is given by

u∂θ

∂x+ v

∂θ

∂y= κ

µ∂2θ

∂x2+

∂2θ

∂y2

¶+

µ

ρCp

"4

µ∂u

∂x

¶2+

µ∂u

∂y+

∂v

∂x

¶2+

µ∂w

∂x

¶2+

µ∂w

∂y

¶2#,

(7.40)

with conditions at the boundary

θ (0) = θw, and θ (h) = θh. (7.41)

144

In order to normalize the eqs. (7.40) and (7.41) we introduce the normalized temperature

T (η) =θ − θhθw − θh

, (7.42)

where η is the dimensionless space variable defined in eq. (7.6). The normalized velocity

components are defined in eq. (7.6) . Therefore, the energy equation in non-dimensional

form is given by

T 00 +Pr£Re fT 0 +Ec

¡4f 02 + g2

¢+Ecx

¡f 002 + g02

¢¤= 0, (7.43)

with dimensionless boundary data

T (0) = 1, and T (1) = 0, (7.44)

where Pr = ν/κ, is the Prandtl number, Ec = a2h2/Cp (θw − θh) is the Eckert number,

and Ecx = a2x2/Cp (θw − θh) is the local Eckert number.

7.3.2 HAM solution

To find the analytic solution of eq. (7.43) subject to the boundary conditions (7.44), we

follow the same procedure as performed in section 7.2.2 of this chapter. To avoid the

repetition, we omit the details of the method and give the main steps only. Due to the

boundary conditions (7.44) we choose the initial approximation

T0 (η) = 1− η, (7.45)

with linear operator

LT ≡ d2

dη2, (7.46)


LT [C7η + C8] = 0, (7.47)

145


The zero-order deformation equation is then given by

(1− p)LT [Λ (η; p)− T0 (η)] = p~N3 [F (η; p) , G (η; p) ,Λ (η; p)] , (7.48)

with subjected boundary conditions

Λ (0; p) = 1, and Λ (1; p) = 0, (7.49)

where the nonlinear operator N3 [F (η; p) , G (η; p) ,Λ (η; p)] is defined through

N3 [F (η; p) , G (η; p) ,Λ (η; p)] =∂2Λ

∂η2+Pr

ReF∂Λ∂η+Ec

½4³∂F∂η

´2+G2

¾+Ecx

½³∂2F∂η2

´2+³∂G∂η

´2¾ . (7.50)

For p = 0, and p = 1, we respectively have

Λ (η; 0) = T0 (η) , and Λ (η; 1) = T (η) . (7.51)

The mth-order deformation equation is then given by

LT [Tm (η)− χmTm−1 (η)] = ~Pm (η) , (7.52)


Tm (0) = 0, and Tm (1) = 0, (7.53)

where

Pm (η) = T 00m−1 +Prm−1Xk=0

Re fm−1−kT 0k +Ec¡4f 0m−1−kf

0k + gm−1−kgk

¢+Ecx

¡f 00m−1−kf

00k + g0m−1−kg

0k

¢ . (7.54)

146

Notice that the system (7.52) and (7.53) is a system of linear ODE’s which one can

easily solve by using a symbolic computing software to find all Tm (η) (m ≥ 1) . Let T ∗m (η)represents special solution of eqs. (7.52) and (7.53) , then (due to (7.47)) their general

solution reads (for m ≥ 1)

Tm (η) = T ∗m (η) + C7η + C8, (7.55)

where the constants C7 and C8 are determined through the boundary conditions (7.53) ,

i.e.

C8 = −T ∗m (0) , and C7 = T ∗m (0)− T ∗m (1) . (7.56)

In this way we can get a complete analytic solution in the form of infinite series of

functions given by

T (η) = T0 (η) ++∞Xm=1

Tm (η) . (7.57)

While using the homotopy analysis method the convergence of the solution series must

be assured. As mention in the preceding section the convergence of the HAM solution

series strongly depends upon the auxiliary parameter ~. To determine the interval of

admissible values of ~ we have plotted the so-called ~−curve for the temperature functionT (η) in fig. 7.9. The interval of admissible values of ~ seems to be −1.6 ≤ ~ ≤ −0.3.However, our analysis shows that a rapid convergence is assured at ~ = −0.8. At thisparticular value of ~ we have obtained a table of values of T 0 (0) at different orders of

approximation. From table 7.2 it is clear that by increasing the orders of approximation

147

the corrections to the solution become so small to be negligible.

M = 1.0, Re = 2.0, w0 = Ω = Ec = Ecx = 0.5, Pr = 0.7

Orders of approximation T 0 (0) 4T 0i (0)

5th-order app. 0.658585 0.000409368

10th-order app. 0.658474 1.80598× 10−7

15th-order app. 0.658474 6.38385× 10−11

Table 7.2 : HAM solutions at different orders at ~ = −0.8.

Fig. 7.9 : ~−curve corresponding to temperature.


To see the effects of different parameters on temperature distribution across the channel

we have plotted the temperature T against η for different sets of values of the involved

parameters. The effects of the parameter w0 which corresponds to the constant injection

at the upper plate are shown graphically in fig. 7.10. It is observed that by increasing

148

the injection at the wall the temperature increases and this increase is maximum near

the stretching wall. This is due to the reason that the injection increases the velocity in

the channel and the effect of viscous dissipation gets stronger and as a consequence the

temperature rises near the solid wall. Similar effect of rotation parameter is observed

on temperature profile (see fig. 7.11) bearing the same physical reasoning. Thus the

presence of strong injection or rotation in the flow situation causes to weaken the cooling

process. Therefore, it is suggested that to enhance the cooling process (in the presence

of viscous dissipation) the injection and the rotation must be controlled. In fig. 7.12 the

effect of the Reynolds number is shown on the temperature graphs. Clearly, an increase

in the Reynolds number increases the heat transfer rate from sheet to the fluid. Thus

high Reynolds number flow (in the presence of rotation) expedites the cooling process.

In figure 7.13 temperature function is plotted for different values of the Prandtl num-

ber in the presence of viscous dissipation. Obviously, on increasing the values of the

Prandtl number the temperature shoots up near the solid surface. This is due to the

presence of viscous dissipation. The effect of Prandtl number in the absence of viscous

dissipation (Ec = Ecx = 0) is shown in fig. 7.14. Clearly, by increasing the values of

Pr the temperature in the channel decreases and the rate of heat transfer at the plate

increases. Thus from figs. 7.13 and 7.14 we conclude that in the presence of viscous dis-

sipation the cooling of the plate is possible if one uses the fluids with very small Prandtl

number, however, in the absence of viscous dissipation the fluids with higher Prandtl

numbers are preferred to cool the sheet quickly.

The effect of viscous dissipation on temperature profile is shown in figs. 7.15 and 7.16.

Figure 7.15 is plotted for different values of the parameter Ecx. From figure it is clear

that increasing Ecx the temperature increases in the neighborhood of the plate. Similar

effect of the Eckert number Ec is observed in fig. 7.16. Therefore, in order to cool the

stretching sheet it is necessary to minimize the viscous dissipation in the flow. The effect

of applied magnetic field on the temperature profile is shown in fig. 7.17. From figure

it can be seen that on increasing the magnetic field the temperature decreases across

149

the channel and as a consequence the rate of heat transfer from plate to fluid increases.

Thus the temperature shoot up caused due to the viscous dissipation can be controlled

by applying strong magnetic field. Numerical values of heat transfer rate at the plate are

reported in table 7.3 for different values of the involved parameters.

Pr Ecx Ec w0 Ω M −T 0 (0)1.0 0.5 0.5 0.5 0.5 1.0 0.5153791

2.0 0.5 0.5 0.5 0.5 1.0 0.0539821

0.7 1.0 0.5 0.5 0.5 1.0 0.4802890

0.7 2.0 0.5 0.5 0.5 1.0 0.1239200

0.7 0.5 1.0 0.5 0.5 1.0 0.3178090

0.7 0.5 0.5 0.5 0.5 2.0 0.6614910

0.7 0.5 0.5 0.5 0.5 3.0 0.6597170

0.7 0.5 0.5 0.5 1.0 1.0 0.6453420

0.7 0.5 0.5 0.5 2.0 1.0 0.4375950

Table 7.3 : 15th-order HAM sol. at Re = 2.0,~ = −0.8.

150

Fig. 7.10 : Temperature distribution for different values of the section parameter.

Fig. 7.11 : Effect of rotation on temperature distribution.

151

Fig. 7.12 : Temperature profile at different Reynolds number

Fig. 7.13 : Effect of Prandtl number in the presence of viscous dissipation

152

Fig. 7.14 : Effect of Prandtl number in the absence of viscous dissipation.

Fig. 7.15 : Temperature profile at different values of the local Eckert number.

153

Fig. 7.16 : Temperature profile at different values of Eckert number.

Fig. 7.17 : Effect of magnetic field on temperature distribution.

154

7.4 Conclusion

Heat transfer in a rotating electrically conducting fluid bounded by two parallel plates

is studied. The upper plate is porous and the lower plate is a stretching sheet. Ana-

lytic solution to velocity and temperature distribution is obtained by a purely analytic

technique, namely, homotopy analysis method. The convergence of the solution series

is shown by giving HAM solutions at different orders. It is observed that injection at

the upper plate causes to increase the temperature in the channel and the magnetic field

decreases the temperature across the flow field. It is also observed that in the presence

of viscous dissipation the Prandtl number causes to enhance the temperature whereas in

the absence of viscous dissipation an increase in the Prandtl number causes to reduce

the temperature across the channel. In the presence of viscous dissipation it is recom-

mended that the fluids with low Prandtl number must be selected as coolant whereas

in the absence of viscous dissipation the fluid with large Prandtl number can serve as

good coolant. It is also observed that the cooling process can be enhanced by increas-

ing the Reynolds number of the flow. Further an increase in the angular velocity slows

down the cooling process. To enhance the cooling process viscous dissipation needs to

be minimized and the magnetic field is found to be the controlling agent in this regard.

155

Chapter 8

Conclusion

Multidimensional viscous flows in the vicinity of uniformly moving plate or

uniformly stretching sheet have been discussed in this thesis. The governing

nonlinear differential equations for both steady and unsteady flows have been

solved analytically by means of homotopy analysis method (HAM). In this

chapter we shall be discussing the strength and limitations of the method

used (i.e. HAM) in addition to the physics observed during this study.

Homotopy analysis method is a useful analytic technique devised for the solution of

strongly nonlinear problems. This is a generalization of the previously known pertur-

bation and non-perturbation analytic techniques. In the previous techniques there is

no freedom to choose different base functions for a given nonlinear problem. However,

this limitation is removed in homotopy analysis method. Because HAM provides us

a flexibility to use different base functions, therefore a suitable approximation for the

considered nonlinear problem can be reached by using proper set of base functions. In

previous analytic techniques the convergence of the series solution for all range of pa-

rameter values is a serious question. Most of the times the solutions obtained are valid

only in a very small range of the parametric values. This issue has also been resolved by

the introduction of an auxiliary parameter in the generalized deformation equation. The

convergence region and rate of approximation is adjusted by choosing a suitable value of

156

the auxiliary parameter from the valid range estimated through the so-called ~−curves.Unlike perturbation method, HAM is independent of small or large parameter. Hence

this method is applicable to all types of equations whether they involve a small/large

parameter or not. In his book [76] Liao has explicitly proved that HAM contains as a

special case the Lypanov’s artificial parameter method, the δ−expansion method, andthe Adomian decomposition method. Unlike previous analytic techniques HAM provides

us the opportunity of selecting the four fundamental quantities, namely, the linear op-

erator; the initial guess, the auxiliary function, and the auxiliary parameter. All these

quantities collectively provide us a more versatility to approximate a solution which is

very close to the exact one.

Provided with all freedom and versatility as pointed out in the previous paragraph,

still there is a lot of criticism to be addressed. The choice of initial guess (base function),

and appropriate linear operator is itself a big question. There does not exist a theory that

can exactly tell us that which particular choice of base function is a good one. The second

and more important drawback is the computational time for achieving the higher order

approximations. For strongly nonlinear problems usually it takes too much time to find

the higher order approximations. Although one can use a high performance machine to

handle such kind of problems but still in some cases one needs approximation beyond 100

which are hard to achieve. Since homotopy analysis method is based on the Taylor series

expansion therefore it finds only analytic unknown solution with the help of analytic

known solution. However, if some problem has non-analytic solution this method may

not work. This particular problem has yet to be addressed for further improvement and

generalization of HAM. Further the problem having chaotic behavior cannot be solved

using HAM to date and this may also be considered as a drawback.

Boundary-layer flow due to the impulsive motion of a semi-infinite flat plate in a free

stream of uniform velocity has been studied in chapter 2. Two cases, namely, when the

plate is rigid and when the plate is assumed to be porous have been discussed. In the

case when plate and free stream move in same direction the skin friction at the wall

157

decreases by increasing the velocity of the plate in both cases however, in the case of

uniform suction the plate faces more resistance as compared to the case when there is

no suction at the plate. In contrast, the blowing at the moving boundary appreciates

the boundary-layers to grow and as a consequence the skin friction at the plate decreases

by increasing the injection velocity. On the other hand, when the plate and free stream

move in opposite directions, shear stress at the plate increases significantly. In this case

too the suction strengthens the viscous drag at the wall by thinning the boundary layers

and injection causes to reduce the viscous drag. It is worthy to mention here that the

present results are uniformly valid for all time 0 ≤ τ <∞ in the entire flow regime. It is

also observed that for initial time the skin friction is very large and decreases with the

passage of time.

Heat transfer phenomenon in unsteady boundary-layer flow over an impulsively started

uniformly stretching surface has been investigated in chapter 3. Accuracy and validity

of HAM solution has been proved. The effect of suction/injection on the cooling process

has been studied. It is observed that thermal boundary layers grow in the presence of

wall injection and as a consequence temperature of the fluid neat the wall increases but

the effect of suction is reverse to it. Thus the rate of heat exchange can be increased by

increasing the suction velocity at the stretching surface. It is also observed that the fluids

of large Prandtl number have the ability to serve as good coolant as compared to those

with low Prandtl number. Further, our analysis reveals that the rate of heat exchange

decays with the passage of time.

Generalized three-dimensional boundary-layer flow over a flat surface stretching in two

lateral directions has been considered in chapters 4 and 6 for the steady and unsteady

cases respectively. In both the cases heat transfer phenomenon has also been studied.

In steady case, it is observed that in the presence of viscous dissipation the plate can

be cooled rapidly if the fluids of small Prandtl number are used where as in the absence

of viscous dissipation the fluids of large Prandtl number are recommended. In order to

maintain the process of cooling of the stretching sheet the viscous dissipation must be

158

controlled and it can be done by the appropriate selection of coolant. In the unsteady case

the cooling process depends upon time and decays with the passage of time. It is observed

that strong magnetic field causes to damage the stretching surface as a consequence of

large skin friction produced by strong magnetic field. It is also observed that in this case

increasing magnetic field enhances the thermal boundary layers and as a consequence

the rate of heat exchange decreases. Thus in order to avoid the roughness of the surface

and to enhance the process of cooling it is recommended that the magnetic field must be

controlled.

Generalized three-dimensional flow as discussed in chapter 4 has been considered in

a channel of lower stretching wall with the consideration of heat transfer phenomenon.

It is observed that the injection at the upper plate increases the flow in channel. The

cooling of plate is impossible in the presence of strong viscous dissipation; however, it

is possible if the viscous dissipation is weak enough. It is noted that in the absence of

viscous dissipation water serves as good coolant instead of air and in the presence of

viscous dissipation air is preferred over water. In chapter 7 effect of rotation has been

studied on the heat transfer phenomenon from a stretching surface to the ambient fluid

in a three dimensional channel flow. Again it is observed that the presence of viscous

dissipation causes to enhance the temperature in the channel. Also the effect of rotation

is discouraging in this case because an increase in angular velocity decreases the rate of

heat exchange between the plate and fluid. However, in the presence of viscous dissipation

the magnetic field is observed to be a controlling agent.

159

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