Fluctuating Hydromagnetic Flow of Viscous Incompressible

Fluctuating Hydromagnetic Flow of Viscous

Incompressible Fluid past a Magnetized Heated

Surface

By

Muhammad Ashraf

CIIT/FA08-PMT-007/ISB

PhD Thesis

In

Doctor of Philosophy in Mathematics

COMSATS Institute of Information Technology

Islamabad-Pakistan

Spring, 2012

ii

COMSATS Institute of Information Technology



Surface

A Thesis Presented to

COMSATS Institute of Information Technology, Islamabad in partial fulfillment

of the requirement for the degree of

PhD in Mathematics

By

Muhammad Ashraf


Spring, 2012

iii



Surface

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

fulfillment of the requirement for the award of Degree of PhD.

Name Registration No.

Muhammad Ashraf CIIT/FA08-PMT-007/ISB

Supervisor

Professor Dr. Md. Anwar Hossain

Department of Mathematics

Islamabad Campus

COMSATS Institute of Information Technology (CIIT)

Islamabad

Co-Supervisor

Professor Dr. Saleem Asghar

Department of Mathematics

Islamabad Campus

COMSATS Institute of Information Technology (CIIT)

Islamabad May, 2012

iv

Final Approval

This thesis titled



Surface

By

Muhammad Ashraf


Has been approved

For the COMSATS Institute of Information Technology, Islamabad

External Examiner 1:

Professor Dr. Tahir Mahmood, IU, Bahawalpur

External Examiner 2:

Dr. M. Masood, UoS, Sargodha

Supervisor:

Professor Dr. Md. Anwar Hossain CIIT, Islamabad

Co-Supervisor:

Professor Dr. Saleem Asghar CIIT, Islamabad

Head of the Department:

Dr. Moiz ud Din Khan HOD Mathematics, CIIT, Islamabad

Dean Faculty of Science:

Professor Dr. Arshad Saleem Bhatti

v

Declaration

I, Muhammad Ashraf registration# FA08-PMT-007/ISB, hereby declare that I have

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

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

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

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

plagiarism rules of the HEC.

Date: ____________ Signature of student:

Muhammad Ashraf CIIT/FA08-PMT-007/ISB

vi

Certificate

It is certified that Muhammad Ashraf registration# FA08-PMT-007/ISB has carried out

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

COMSATS Institute of Information Technology, Islamabad and the work fulfills

the requirement for award of PhD degree.

Date: _______________

Supervisor:

Professor Dr. Md. Anwar Hossain CIIT, Islamabad

Co-Supervisor:

Professor Dr. Saleem Asghar CIIT, Islamabad

Head of Department:

Dr. Moiz ud Din Khan Associate Professor CIIT, Islamabad Department of Mathematics

vii

DEDICATION

I dedicate this thesis to my loving grandmother and my

genius maternal grandfather

Fazal Bibi (Late)

Muhammad Hussain (Late)

&

My Parents

viii

Acknowledgements

O Allah! Lord of power (and Rule) You give power to whom You will and You take away power from whom You will, and you endue with honor whom you will, and humiliate whom you will. In Your hand is the good. Verily, You are able to do all things." {Soorah al-Imran (3):26} First and foremost, I would like to thanks to my supervisor of this project, Professor Dr. Md. Anwar Hossain for the valuable guidance and advice. He inspired me greatly to work in this project. His willingness to motivate me contributed tremendously to my project. I wish to express my sincere gratitude to my Co-Supervisor Professor Dr. Saleem Asghar for his valuable suggestions and guidance. I have definite conclusions that he is the pioneer of fluid mechanics in Pakistan. I would like to pay special thanks to Dr. Moiz-ud-Din Khan (HoD) for providing conducive research environment at Department of Mathematics COMSATS. I would like to show my very special thanks to competent authorities of my parent Department (CENUM) Dr. Nasir Mehmood (Director), Mr. Zulqarnain Syed (PS) to provide me a chance and moral support to complete this project. Here, I would like pay special thanks Dr. Shan Elahi (SS) for his useful discussions and accompanying me in his office during my stay at CENUM. At last, I would like to pay thanks to the people working in establishment (Mr. M. Usman Khan and his team)/administration (Mr. S. M. A Butt and his team) and accounts branch (CENUM) for their nice cooperation during my study period. I would like to gratefully acknowledge my friends Dr. Muhammad Mushtaq, Mr. Manshoor Ahmad, Mr. Amir Ali, Mr. Adeel Ahmmad, Mr. M. Saleem, Mr. Imran Shah, Mr. Muddasar Jalil and Mr. Muhammad Akram Butt (PU) for their support at all stages of this project. I would like to pay special thanks of my friend Mr. Muhammad Tariq and his family for providing me a hospitality and moral support during my stay at Islamabad for this project. I can’t acknowledged the prayers and concerns of my parents, my uncle Malik Muhammad Hanif, my mother in law, my brothers, sisters, and all of my cousins throughout my life. Much of what I have learned over the years came as the result of being a father to six wonderful and delightful children, Muneeb, Najeeb, Adeeb, Adeel, Naqeeb and Raheel all of whom, in their own ways inspired me and, subconsciously contributed a tremendous amount to the content of this project. A little bit of each of them including their mother will be found here weaving in and out of the pages – thanks my wife and kids!!

ix

Muhammad Ashraf

ABSTRACT

The phenomena of convective heat transfer between an ambient fluid and a body

immersed in it, stems give a better insights into the nature of underlying physical

processes such as processing with high temperature, space technology, engineering and

industrial areas such as propulsion devices for missiles, aircraft, satellites and nuclear

power plants. With this understanding, in the present work, an immense research effort

has been expended in exploring and understanding the convective heat transfer between

fluid and submerged vertical plate. In practice, we are interested in the full details of

velocity, temperature and transverse component of magnetic field profiles, boundary

layer thickness and some other quantities at the surface of the vertical plate such as the

heat transfer from liquid to the plate or from plate to the liquid, frictional drag exerted by

the fluid on the surface and current density for the case of magnetohydrodynamics

(MHD) flow field. For this purpose, the boundary layer equations are transformed into

convenient form by introducing independent variables such as primitive variables for

finite difference method and stream function formulation for asymptotic series solutions

to calculate the above mentioned quantities.

For the development of the topic, an extensive literature survey is outlined in Chapter 1

with appropriate references well targeted to the title of the problem. The purpose of the

Chapter 2 of this thesis is to introduce the boundary layer concepts and to show how the

equations of viscous flow are simplified hereby. The standard boundary layer parameters

and boundary layer equations are introduced in more general form in this chapter.

Chapter 3 deals with the thermal radiation effects on hydromagnetic mixed convection

laminar boundary layer flow of viscous, incompressible, electrically conducting and

optically dense grey fluid along a magnetized vertical plate. The solution of transformed

boundary layer equations are then simulated by employing two methods (i) finite

x

difference method for entire values of ξ and (ii) asymptotic series solution for small and

large values of transpiration parameter ξ . The physical parameters that dominate the

flow and other quantities such as the local skin friction, rate of heat transfer and current

density at the surface of the plate has been discussed. The effect of magnetic force

parameter S, conduction radiation parameter dR , Prandtl number Pr, magnetic Prandtl

number mP and mixed convection parameter λ with surface temperature wθ in terms of

local skin friction, rate of heat transfer and current density at the surface have been shown

graphically and in tabular form. The material used in Chapter 3 is modified in Chapter 4

and reformulated to calculate the effects of conduction-radiation on hydromagnetic

natural convection flow by using the same numerical techniques as used in Chapter 3.

The material has been divided into two parts. The first part Chapters 3 and 4 presents

steady part of the problem for mixed and natural convection flow. The second part of the

thesis is the Chapters 5 and 6 which is devoted to find the numerical solution of the

problem for unsteady part of mixed and natural convection flow. Chapter 5 describes the

effect of conduction radiation on fluctuating hydromagnetic mixed convection flow of

viscous, incompressible, electrically conducting and optically dense grey fluid past a

magnetized vertically plate. The effects of different values of the mixed convection

parameterλ , the conduction radiation parameter dR , Prandtl number Pr, the magnetic

Prandtl number mP , the magnetic force parameter S and the surface temperature wθ , are

discussed in terms of amplitudes and phases of shear stress, rate of heat transfer and

current density at the surface. The effects of these parameters on the transient shear

stress, rate of heat transfer and current density have also been discussed in detail. The

finite difference method for the entire values of local frequency parameterξ and

asymptotic series solution for small and large values of local stream wise parameter ξ

have been implemented in this study. In Chapter 6, we extended the Chapter 4 into

unsteady form and find the numerical solutions of the effects of conduction radiation on

fluctuating hydromagnetic natural convection flow of viscous, incompressible,

electrically conducting and optically dense grey fluid past a magnetized vertically plate.

xi

CONTENTS

1. Introduction 1

2. Fundamental equations along with boundary layer theory 12

2.1 Fundamental equations 13

2.2 Dimensionless boundary layer equation 15

2.2.1 Prandtl number 16

2.2.2 Reynolds number 16

2.2.3 Grashof number 16

2.2.4 Mixed convection parameter 17

2.2.5 Radiation parameter 17

2.2.6 Magnetic force parameter 17

2.2.7 Magnetic Prandtl number 18

2.3 Mechanism of heat transfer 18

2.3.1 Conduction 18

2.3.2 Radiation 19

2.3.3 Convection 19

2.3.3.1 Natural convection 19

2.3.3.2 Forced convection 20

2.3.3.3 Mixed convection 20

2.4 Computational techniques 20

2.4.1 Finite difference method 21

2.4.2 Asymptotic method 21

3. Radiative magnetohydrodynamic mixed convection flow past a magnetized vertical permeable heated plate 23

3.1 Formulation of the mathematical model 24

3.2 Methods of solution 26

xii

3.3 Results and discussion 29

3.3.1 Effects of different parameters on skin friction, magnetic intensity and rate of heat transfer 30

3.3.2 Effects of different parameters on velocity, temperature and magnetic field profiles 35

3.4 Asymptotic solutions for small and large local transpiration parameter ξ 37

3.4.1 When local transpiration parameter ξ is small 38

3.4.2 When local transpiration parameter ξ is large 43

3.5 Conclusion 45

4. Radiative magnetohydrodynamic natural convection flow past a magnetized vertical heated plate 47

4.1 Mathematical analysis and governing equations 48


4.2.1 Primitive variable formulation 51

4.3 Asymptotic solutions for small and large local transpiration parameter ξ 52

4.3.1 When local transpiration parameter ξ is small 53

4.3.2 When local transpiration parameter ξ is large 55


4.4.1 The effects of physical parameters on skin friction, current density and rate of heat transfer 58

4.4.2 The effects of physical parameters on velocity, temperature and transverse component of magnetic field 61

4.5 Conclusion 64

5. Radiative fluctuating magnetohydrodynamic mixed convection flow past a magnetized vertical heated plate 67

5.1 Basic equations and the flow model 68


5.2.1 Primitive variable formulation 72

5.2.2 Asymptotic solutions for small and large local Parameter ξ 74

5.2.2.1 When parameter ξ is small 75

5.2.2.2 When parameter ξ is large 77


xiii

5.3.1 Effects of physical parameters upon amplitude and phase of rate of heat transfer, shear stress and current density 82

5.3.2 Effects of physical parameters upon transient rate of heat transfer, shear stress and current density 87

5.4 Conclusion 90

6. Radiative fluctuating magnetohydrodynamic natural convection flow past a magnetized vertical heated plate 93

6.1 Mathematical analysis and governing equations 94

6.2 Solution methodology 98

6.2.1 Primitive variable n 98

6.2.1.1 1Transformation for steady case 98

6.2.1.2 2Transformation for unsteady case 99

6.2.2 Asymptotic solution for small and parameter ξ frequency 101

6.2.2.1 When parameter ξ is small 101

6.2.2.2 When parameter ξ is large 107

6.3 Results and Discussion 109

6.3.1 Effects of physical parameters upon amplitude and phase of heat transfer, coefficient of skin friction and current density

110

6.3.2 Effects of physical parameters upon transient rate of heat transfer, shear stress and current density 114

6.4 Conclusion 116

7. References 118

xiv

LIST OF FIGURES Figure 3.1 The coordinate system and flow configuration 25

Figure 3.2 Numerical solution of (a) coefficient of skin friction (b) rate of heat

transfer (c) magnetic intensity for different values of radiation parameter dR 32


transfer (c) magnetic intensity for different values of mixed convection

parameter λ

33

transfer (c) magnetic intensity for different values of Prandtl number Pr 34



transfer (c) magnetic intensity for different values of magnetic Prandtl

number mP

34

Figure 3.6 (a) Velocity (b) temperature (c) transverse component of magnetic

field profile for different values of mixed convection parameterλ 35

Figure 3.7(a) Velocity (b) temperature (c) transverse component of magnetic

field profile for different values of magnetic force parameter S 36

Figure 3.8(a) Velocity (b) temperature (c) transverse component of magnetic

field profile for different values of transpiration parameter ξ 36

Figure 3.9 (a) Velocity (b) temperature (c) transverse component of magnetic

field profile for different values of radiation parameter dR 37

Figure 4.1 The coordinate system and flow configuration 49

Figure 4.2 The behavior of coefficients of (a) skin friction (b) rate of heat

transfer (c) current density for different values of radiation parameter dR 59


transfer (c) current density for different values of magnetic force parameter S 59

Figure 4.4 The behavior of coefficients of (a) skin friction (b) rate of heat 60

xv

transfer (c) current density for different values magnetic Prandtl number mP


transfer (c) current density for different values of Prandtl number Pr 61

Figure 4.6 (a) Velocity (b) temperature distribution (c) transverse component of

magnetic field for different values of radiation parameter dR 62

Figure 4.7 Velocity (b) temperature distribution (c) transverse component of

magnetic field for different values of magnetic force parameter S 63


magnetic field for different values of Prandtl number Pr 63


magnetic field for different values of magnetic Prandtl number Pm 64


magnetic field for different values of transpiration parameterξ 64


Figure 5.2 Numerical solution of amplitude of phase angle of heat transfer for

different values of radiation parameter dR 83

Figure 5.3 Numerical solution of amplitude of phase angle of shear stress for


Figure 5.4 Numerical solution of amplitude of phase angle of current density for


Figure 5.5 Comparison of numerical solutions of finite difference method with

asymptotic method for amplitude and phase of current density for different

values of magnetic Prandtl number mP

84

Figure 5.6 Comparison of numerical solutions of finite difference method with

asymptotic method for amplitude and phase of shear stress for different values of

magnetic force parameter S

85

Figure 5.7 Numerical solution of amplitude of phase angle of rate of heat transfer

for different values of Prandtl number Pr

85

Figure 5.8 Numerical solution of amplitude of phase angle of shear stress for 86

xvi

different values of Prandtl number Pr

Figure 5.9 Numerical solution of amplitude of phase angle of current density for

different values of Prandtl number Pr

86

Figure 5.10 Numerical solution of amplitude of phase angle of rate of heat

transfer for different values of surface temperature wθ

87

Figure 5.11 Numerical solution of amplitude of phase angle of shear stress for

different values of surface temperature wθ

87

Figure 5.12 Numerical solution of amplitude of phase angle of rate of current

density for different values of surface temperature wθ

88

Figure 5.13 Solution for transient (a) heat transfer (b) shear stress (c) current

density for different values of magnetic force parameter S

88

Figure 5.14 Solution for transient (a) heat transfer (b)shear stress (c) current

density for different values of magnetic force parameter S

89


density for different values of magnetic Prandtl number mP

89


density for different values of Prandtl number Pr

90


density for different values of radiation parameter dR

90


density for different values of mixed convection parameter λ

91


density for different values of surface temperature wθ

91


Figure 6.2 Numerical solution of amplitude of phase angle of heat transfer for

different values of radiation parameter dR

110

Figure 6.3 Numerical solution of amplitude of phase angle of coefficient of skin

friction for different values of radiation parameter dR

110

Figure 6.4 Numerical solution of amplitude of phase angle of coefficient of 111

xvii

current density for different values of radiation parameter dR


for different values of magnetic Prandtl number mP

111


friction for different values of magnetic Prandtl number mP

112

Figure 6.7 Numerical solution of amplitude of phase angle of coefficient of

current density for different values of magnetic Prandtl number mP

112

for different values of surface temperature wθ 113



friction for different values of surface temperature wθ

113

Figure 6.10 Numerical solution of amplitude of phase angle of coefficient of

current density for different values of surface temperature wθ

114

Figure 6.11 Numerical solution of transient (a) rate of heat transfer (b)

coefficient of skin friction for different values of radiation parameter dR

115


coefficient of skin friction for different values of magnetic force parameter S

115

Figure 6.13 Numerical solution of transient (a) coefficient of skin friction (b)

coefficient of current density for different values of magnetic Prandtl number mP

116


coefficient of skin friction for different values of local frequency parameter ξ

116

xviii

LIST OF TABLES

Table 3.1 Numerical values of coefficient of skin friction obtained for dR = 1.0,

10.0 against ξ by two methods

30

Table 3.2 Numerical values of magnetic intensity obtained for dR = 1.0, 10.0

against ξ by two methods.

31

Table 3.3 Numerical values of rate of heat transfer obtained for dR = 1.0, 10.0

against ξ by two methods

31

Table 3.4 Values of skin friction and magnetic intensity obtained by present

author and Glauert [2] for mP = 1.0 and 10.0

41

Table 3.5 Values of skin friction obtained by present author, Glauert [2] and

Davies [3] for S= 0.1 and 0.05 against ξ =0.0

42

Table 3.6Values of rate of heat transfer obtained by present author,

Ramamoorthy [6] for different S

42

Table 4.1 Numerical values of coefficient of skin friction obtained for surface

temperature wθ by two methods

57

Table 4.2 Numerical values of coefficient of heat transfer obtained for surface


57

Table 4.3 Numerical values of coefficient of current density obtained for surface


57

Table 6.1 Numerical values of amplitude and phase angle of heat transfer for

different values of S obtained by two methods

105

Table 6.2 Numerical values of amplitude and phase angle of coefficient of skin

friction for different values of S obtained by two methods

106

Table 6.3 Numerical values of amplitude and phase angle of coefficient of 106

xix

current density for different values of S obtained by two methods

Notations S Magnetic force parameter

u Velocity along x-axis

v Velocity along y-axis

u Nondimensional velocity along x-axis

v Nondimensional velocity along y-axis

f Transformed stream function

T Dimensioned temperature

T Dimensionless temperature

mP Magnetic Prandtl number

Rex Local Reynolds number

xGr Local Grashof number

xCf Skin friction

xB Dimensionless magnetic field along the surface

yB Dimensionless magnetic field normal to the surface

xNu Local Nusselet number

Pr Prandtl number

dR Thermal Radiation parameter

Greek letters

ψ Fluid stream function, [m2.s-1]

φ Transformed stream function for magnetic field

α Thermal diffusivity, [m2.s-1] µ Dynamic viscosity, [Kg.m-1.s-1] η Similarity transformation

xx

ν Kinematic viscosity, [m2.s-1]

θ Dimensionless temperature function

wθ Surface temperature ratio to the ambient fluid

ρ Density of the fluid, [Kg.m-1.s-1]

σ Electrical conductivity, [S(Siemens).m-1]

sσ Stefan-Boltzman constant γ Magnetic diffusion

β Coefficient of cubical expansion

µ Magnetic permeability

Subscripts

w Wall condition

∞ Ambient condition

1

Chapter 1

Introduction

Chapter 1: Introduction

1.1 Introduction to the problem

The theory of fluid mechanics is the foundation for literally dozens of fields within

science and engineering. Its uses and branches are limitless. The understanding of

fluid mechanics is essential to model the complex physical problems in meteorol-

ogy, oceanography, astronomy, aerodynamics, propulsion, combustion, bio-fluids,

acoustics and particle physics. This thesis highlights about the solutions of the

laminar boundary layer equations. The concept of boundary layer, one of the

corner stones of modern fluid dynamics, was introduced by Prandtl (1904) in

an attempt to account for the sometimes considerable discrepancies between the

predictions of classical inviscid incompressible fluid dynamics and the results of

experimental observations. He further reported that in flow past a streamlined

body, the region in which viscous forces are important is often confined to a thin

layer adjacent to the body and to a thin wake behind it. This thin layer is referred

to as the boundary layer. When this condition holds, the equations governing the

motion of the fluid within the boundary layer take a form considerably simpler

than the full viscous flow equations and it is the solution of these equations with

which we shall be presently concerned. In present work, the heat transfer phenom-

ena with inclusion of radiation term in energy equation is examined in the presence

of incompressible, viscous, electrically conducting and optically dense grey fluid

for steady and unsteady cases. At this stage, it is necessary to highlights some ba-

sic concepts of magnetohydrodynamics and thermal radiation to couple magnetic

field and radiation terms in momentum and energy equations. Magnetohydrody-

namics is a combination of fluid mechanics and electromagnetism, we may say that

it is electrically conducting fluid in the presence of electric and magnetic fields.

The equations governing the flow are well known Navier Stokes equations and the

terms appear in the equations due to magnetohydrodynamics effects and their

simplification via the boundary layer approximation which are used to solve many

fluids problems. These equations represent the differential form of the conserva-

tion of linear momentum and are applicable in describing the motion of a fluid

particle at an arbitrary location in the flow field at any instant of time. In many

2


engineering and industrial problems such as glass industry, combustion chambers,

atmospheric phenomena, shock wave problems and industrial furnaces in which

some situations arises where heat is transported within a medium by radiation

and conduction. The conduction radiation parameter is introduced and the en-

ergy equation is formulated in such a way that the emission or absorption can

effect the heat transfer in a convection boundary layer.

From the above two paragraphs, we can estimate how will be the model of

the problem and its depth and importance in real life problems. Due to combined

mode problems treated in this thesis, we have to give the survey of the work which

had done by other authors in the field of magnetohydrodynamics for magnetized

and non magnetized surface, convective heat transfer in the absence and pres-

ence of thermal radiation for steady and unsteady/fluctuating cases in the field of

boundary layer theory.

1.1.1 Introduction to the Chapter-3

”In the part of magnetohydrodynamics, Greenspan and Carrier [1] have discussed

the flow of viscous incompressible electrically conducting fluid of constant prop-

erties by applying uniform magnetic field in the free stream direction parallel to

the rigid plate. They analyzed the magnetic and velocity fields explicitly and

accurately the functions of parameters to achieve some insight of the nature of

magnetohydrodynamics flow by using a direct extension of asymptotic method.

They observed that a formal perturbation series expansion in magnetic Prandtl

number Pm, can not succeed in two dimensional problems. The boundary layer

flow of a viscous electrically conducting liquid in the neighborhood of semi-infinite

unmagnetized plate has been investigated by Davies [2] and [3] theoretically. He

analyzed that the flow opposed by magneto-dynamic pressure gradient by placing

a parallel magnetic field well away from the plate and he obtained results by using

the method of iteration and the first approximation technique. Gribben [4] then

considered an axisymmetric magnetohydrodynamic flow of an incompressible, vis-

cous, electrically conducting fluid near a stagnation point considering that the

magnetic field lines are circles and parallel to the surface. Gribben [5] considered

3


the magnetohydrodynamics boundary layer in the presence of an external magne-

tohydrodynamic pressure gradient by using series expansion in magnetic Prandtl

number Pm in each layer and their boundary conditions are satisfied by their coef-

ficients and determined by matching principle in two layers. He also presented the

results of physical quantities in terms of skin friction and tangential component of

the magnetic field at the wall. Ramamoorthy [6] extended the classical theory of

heat transfer in boundary layer to the hydromagnetic case where he considered the

flow of viscous electrically conducting fluid past a insulated semi-infinite plate in

the presence of magnetic field parallel to the plate. He examined that the temper-

ature distribution in the boundary layer is reduced by applying the magnetic field

which slow down the fluid movement in flow domain. He also examined that the

dissipation of current due to Joule heating is very small. The case of heat transfer

in an aligned flow past a semi-infinite flat plate, when the flow velocity and mag-

netic field are considered at some distance from the plate has been studied by Tan

and Wang [7]. They concluded that the increase in magnetic field increase the

viscous, magnetic and thermal boundary layer thicknesses, and the rate of heat

transfer reduces for Eckeret number Ek ≤ 0. Hildyard [8] extended the problem

for numerical integration is used to establish the validity of the series solution.

The magnetohydrodynamics boundary layer flow of thermally conducting plate

with a aligned magnetic field placed at large distance from the plate has been

discussed by Ingham [9]. He carried out this study by using series expansion and

integral approximation to find the numerical solution of the problem for different

parameters in terms of coefficients of skin friction and rate of heat transfer.

The effects of thermal radiation in different geometries have been discussed by

several authors. Ali et al. [10] illustrated the boundary layer flow over a horizon-

tal flat plate with cold and hot ends of grey fluid and Roseland approximation is

used to calculate the radiative heat transfer.The radiation effects of free convec-

tion boundary layer flow past a vertical plate which is immersed in an emitting,

absorbing and isotropic scattering grey fluid has been calculated by solving non

similar energy and momentum equation by Arpaci [11], Sparrow and Cess [12].

Soundalgekar et al [13] have examined the effects of forced and free convection

4


flow with variable surface temperature. In this study they analyzed the effects of

different physical parameters in terms of velocity and temperature function with

the help of series solutions in powers of mixed convection parameter. Hossain and

Takhar [14] have been investigated the radiation effects on mixed convection flow

of an optically dense, viscous and incompressible fluid flow along a vertical plate

with uniform surface temperature. In this investigation they used the implicit fi-

nite difference method together with Keller-Box scheme and presented results for

local shear stress and rate of heat transfer for different ranges of the values of per-

tinent parameters. Aboeldahab and Gendy [15] considered the effects of radiation

on the convective boundary layer in the presence of a uniform transverse mag-

netic field. They have examined the effects of temperature ratio, magnetic field

and thermal conductivity and radiation parameter of the coefficients of heat flux,

shear stress and as well as on velocity and temperature distribution profiles. Ef-

fects of thermal radiation on unsteady free convection flow past a vertical porous

plate with Newtonian heating have recently been demonstrated by Mebine and

Adigio [16], who obtained the analytical results by using the Laplace transforma-

tion technique. Palani and Abbas [17] have been investigated the effects of MHD

flow of viscous, compressible and electrically conducting fluid on the free convec-

tion flow past a semi-infinite impulsively started vertical plate in the presence of

thermal radiation.

Convective boundary layer flows are often controlled by injecting (blowing) or

suction (withdrawing) fluid through porous bounding heating surface. This can

lead to enhanced heating or cooling of system and can help to delay the tran-

sition from laminar to turbulent flow. With this understanding, Eichhorn [18]

obtained those power law variations in surface temperature and transpiration ve-

locity which give rise to a similarity solution for the flow from a vertical surface.

The case of uniform suction and blowing through an isothermal vertical wall was

investigated first by Sparrow and Cess [19], they obtained a series solution which

is valid near the leading edge. The numerical solutions of the effects of blowing

and suction on free convection boundary layer for a horizontal circular cylinder

have been computed by Merkin [40]. He concluded that for both the suction and

5


blowing, the asymptotic solutions are valid at large distances from the leading

edge. Using the method of matched asymptotic expansions, the next order cor-

rection to the boundary layer solution for this problem was obtained by Clarke

[20], who obtained the range of applicability of the analysis by not invoking the

Boussinesq approximation. The effect of strong suction and blowing from general

body shapes which admit a similarity solution has been given by Merkin [40]. A

transformation which allows arbitrary distribution of both wall temperature and

blowing has been carried out by Vedhanayagam et al.[22]. The study of low speed

natural convection motion over a hot horizontal porous surface by assuming that

the thermal conductivity and dynamic viscosity are proportional to temperature

has been carried out by Clarke and Riley [23]. Lin and Yu [24] theoretically studied

the effects of blowing and suction on free convection boundary layer flow.

In view of the above literature survey the magnetohydrodynamics boundary

layer and heat transfer past a magnetized heated surface in the presence of ther-

mal radiation has not been yet studied to the best of our knowledge. The aim

of the present thesis is to establish some mathematical models to study the mag-

netohydrodynamic boundary layer and heat transfer past a magnetized heated

surface and their numerical results to gain knowledge which can leads a deeper

understanding of fluid dynamical problem. As a first step, Chapter 2 discusses

the basic model of the problem and some numerical methods which are carried

out throughout the thesis and some basic definitions of some parameters and their

mathematical formulations which are closely related and frequently used in this

thesis. In Chapter 3, thermal radiation effects on hydromagnetic mixed convec-

tion flow along a magnetized vertical porous plate has been discussed. Finite

difference method along with assymptotic series solution for small and large val-

ues of transpiration parameter ξ = (V0x/ν)/Rex1/2 has been implemented to find

the numerical solution of the above mentioned problem. In this chapter, the im-

portant results which show the behaviour of conduction radiation parameter Rd,

magnetic force parameter S, Prandtl number Pr, magnetic Prandtl number Pm,

mixed convection parameter λ and surface temperature ratio θw on the local skin

6


friction Cfx, rate of heat transfer and magnetic intensity at the surface are pre-

sented. Moreover, the effects of these parameters on velocity and temperature

profile along with transverse component of magnetic filed are also discussed in

detail. This part of the thesis work has been published in Mathematical prob-

lems in engineering (2010) Volume 2010, Article ID 686594, 29 pages

doi:10.1155/2010/686594.


The simultaneous convection and radiation boundary layers with prescribed heat

flux applications to determine the surface temperature distribution along a flat

plate has been simulated by Sparrow and Cess [12], given both the exact and series

solution for the problem. Lin and Cebeci [25] studied the effects of radiation for

laminar boundary layer flow alon a flat plate. Perlmutter and Siegel[26], Siegel and

Keshock [27] investigated the radiation effects for flow inside circular tubes. Chen

[33] introduced radiation effects by assuming that the heat transfer to the gas at

the tube wall is proportional to the fourth power of the wall temperature, but this

is physically unrealistic boundary conditions since it neglects radiation incident

on the surface from other surface element. Dussan and Irvine [28] calculated

heat transfer by assuming linearized radiation and using an exponential Kernal

approximation. A more complete and realistic model that did not require a priori

knowledge of the heat transfer coefficient was used by Thorsen [29]and Thorsen

and Kachanagom [30] to investigate the effects of radiation on heat transfer for

flow inside the circular tube and by Liu and Thorsen [31] for flow between parallel

channels. The hydromagnetic steady shear flow along an electrically insulating

porous plate has been studied by Gupta et al. [32] and observed that the velocity

at given point increases with the increase in either the magnetic field or or suction

velocity. Chen [33] studied the response of Nusselt number to the magnetophysical

parameter of anisotropic radiative heat transfer on steady magnetohydrodynamic

natural convection boundary layer flow from a horizontal plate. Fazalina and Anur

[34] have been studied the similarity solutions for the problem of mixed convection

boundary layer flow adjacent to streching vertical sheet in a compressible electrical

7


conducting fluid in the presence of transverse magnetic field. Recently, MHD

boundary layer flow and heat transfer over streching sheet with induced magnetic

field have been studied by Fadzillah et al [35].

From the motivation of the above survey, in Chapter 4, we have investigated

the combined effects of radiation and hydromagnetics on natural convection flow

along a magnetized vertical permeable plate. The effects of varying the radiation

parameter Rd, Prandtl number, Pr, magnetic Prandtl number Pm, magnetic force

parameter S and surface temperature θw on coefficients of skin friction, rate of

heat transfer and current density are shown. The effects of above mentioned

parameters on velocity profile, temperature distribution and transverse component

of magnetic field are also examined. The numerical solutions for intermediate

range of transpiration parameter ξ are obtained by using finite difference method.

Asymptotic solutions are obtained both near and away from the leading edge

and compared with the numerical solutions that are obtained by finite difference

method and found to be in good agreement. The contents of this work has been

published in Appl. Math. Mech. Engl. Ed. 33(6), 731-750, (2012).


Helmy [55] studied two dimensional unsteady magnetohydrodynamics flow past

a vertical porous plate by considering a uniform magnetic field act prependicular

to the plate which absorbs the fluid with suction and velocity varied periodically

with time about a constant non zero mean. Further, he observed the effect of

Prandtl number, suction parameter, magnetic parameter, on the angular velocity

and temperature distribution. Kuiken [56] has been investigated the problem of

magnetohydrodynamic free convection flow of an electrically conducting fluid in

strong cross field. He solved this problem by using singular perturbation tech-

nique and presented the solution for the different range of Prandtl number Pr

from zero to order 1. The effect of oscillation on the time mean heat transfer

of the incompressible laminar boundary layer flow on a flat plate has been as-

sumed by Hiroshi [57] analytically. He concluded that the oscillation is of high

frequency and the time dependent mean heat at the surface of the wall can be

8


several times as large that without oscillation. Coenen and Riley [58] et al have

focused the study of oscillatory flow about a pair of two identical cylinders by

varying the distance between them and calculated the effect of oscillation in fluid

flow domain. Priestley [59] has made a study of the magnitude of temperature

fluctuations in the atmospheric boundary layer which covers the wide range of

height thermal stratification and wind speed. He used regression relations to cal-

culate the fluctuations in terms of heat flux and vertical temperature gradient.

The self induced oscillations in the flow over a circular cylinder at Re = 1.06×105

have been studied by Dwyer and McCorkkey [60] et al up to the point of separa-

tion. An investigation regarding to the influence of surface temperature and mass

concentration fluctuating has carried out from a vertical flat plate by Hossain et

al [61] in terms of amplitude and phase angles. Moreover, finite difference method

and perturbation theory is used to calculate the numerical solutions. Hiroshi [62]

studied the time mean skin friction in terms of flow oscillation amplitude outside

the boundary layer for low and high frequency ranges, and concluded that the

approximate formoulae are valid for small amplitude and high frequency. Pedeley

[63] calculated the numerical solution of the two dimensional boundary layer flow

which is generated on a semi-infinite plane boundary when a viscous incompress-

ible fluid flows over it in such a way that free stream velocity oscillated without

reversing. Theoretical analysis of heat transfer fluctuation in a periodic boundary

layer near two dimension stagnation point has been made by Hiroshi [64], when

the velocity of coming stream relative to the body oscillated and body is heated

at constant temperature. Sears [65] undertook a systematically boundary layer in

cross field magnetohydrodynamics flow at large Reynolds number ReL, magnetic

Reynolds number Rm and thickness of the boundary layer is equal to the product

of inverse square root of Re and Rm i.e. (Re×Rm)−1/2. Rotem [66] investigated

a class of magneto free convective flow over a horizontal surface in the presence

of strong magnetic field and concluded that a class of similarity solutions is valid

within the domain of multiple parameter space. The magnetohydrodynamic flow

of viscous fluid past a sphere has been discussed Richard et al. [67] with the as-

sumption that the ambient fluid flow field is collinear with the ambient magnetic

9


field.

The effect of conduction radiation on fluctuating hydromagnetic mixed con-

vection flow past a magnetized vertical heated plate, when the magnetic field,

surface temperature and free stream velocity oscillates in magnitude about a con-

stant non-zero mean have not been considered simultaneously. In Chapter 5, we

purpose the study of the effect of conduction radiation on oscillatory hydromag-

netic mixed convection flow past a magnetized plate and highlights the effects of

varying, the mixed convection parameter λ, the conduct radiation parameter Rd,

the Prandtl number, Pr, the magnetic Prandtl number Pm, the magnetic force

parameter S and wall temperature θw in terms of amplitudes and phases of shear

stress, rate of heat transfer and current density. The part of this work of the thesis

has been submitted for publication in Journal of Heat Transfer (2011).


Takhar et al. [68] studied the heat transfer case for free convection flow due to

buoyancy, radiation and transverse magnetic field over a semi-infinite vertical plate

derived by expanding stream function and temperature in terms of pseudosimi-

larity variable ξ by using series solution method. Chamkha [69] discussed the two

dimensional free convection flow of water up in the presence of a uniform trans-

verse magnetic field and solar radiation. In the presence of magnetic field, the

effect of temperature dependent viscosity on free convection flow past a vertical

porous plate with first order homogeneous chemical reaction has been studied by

Makinde and Ogulu [70]. The numerical solutions of coupled second order dif-

ferential equations to calculate the heat and mass transfer from vertical porous

medium has been carried out by Makinde and Aziz [71] with detail. The principle

of natural convection flow in a convergent channel with finite wall thickness has

been investigated by Bianco et al. [72] numerically. Seth et al. [73] studied the

MHD, couette flow of a viscous, incompressible, electrically conducting fluid in a

rotating system and obtained the exact solution of the problem with the help of

Laplace transformed technique. Makinde [74] studied the MHD mixed convection

heat and mass transfer flow of Boussinesq fluid past a vertical porous plate with

10


radiative heat transfer of an nth order homogeneous chemical reaction between

fluid and the diffusing species”. By following the above literature survey in Chap-

ter 6, the effect of radiation on fluctuating hydro-magnetic natural convection flow

of viscous, incompressible, electrically conducting fluid past a magnetized verti-

cal plate is studied; when the magnetic field and surface temperature oscillate in

magnitude about a constant non zero mean. The numerical solutions of different

values of radiation parameter Rd, magnetic Prandtl number Pm, magnetic force

parameter S, Prandtl number Pr and surface temperature θw in terms of amplitude

and phase of coefficients of skin friction, rate of heat transfer and current density

at the surface of the plate. Moreover, the effects of these parameters on transient

coefficients of skin friction, rate of heat transfer and current density have been

discussed. The finite difference method for primitive variable formulation and as-

ymptotic series solution for stream function formulation have been used to obtain

the numerical solution of the boundary layer flow field. The part of this work of

the thesis has been published in Thermal Science Journal Vol. 16, issue 4,

pp. 1081-1096,(2012).

11

12

Chapter 2

Fundamental equations along with boundary layer theory

Chapter 2: Fundamental equations along with boundary layer theory

In this chapter, we proceed further with our discussion and precisely develop the

underlying theory which governs the motion of a fluid. The primary field equa-

tions of fluid mechanics are represented consciously in terms of a set of partial

differential equations in the physically important unknown parameters such as

velocity, pressure and/or an appropriate energy variable. Such partial differential

equations naturally arises when some physical phenomena are formulated mathe-

matically. The most familiar examples for linear cases are wave, heat and Laplace

equations whereas for the nonlinear case, the most important example is the set

of Navier-Stokes equations which are observed in many physical problems such

as in the study of aeronautical sciences, thermo-hydraulics, meteorology, plasma

physics, petroleum industry etc.

2.1 Fundamental equations

The Navier-Stokes equations are the fundamental equations which govern the mo-

tion of the fluids like air, blood, water, oil etc subject to some general conditions

and they happen to appear either alone or coupled with other equations. It seems

to be clumsy to explicitly take into account all of the terms in the general discus-

sion of these equations. However, its more suitable to introduce the equations in

their most primitive form in which the terms are all-embracing but very general.

Flow configuration with conjugate effect of buoyancy force in presence of external

magnetic field, following equations for conservation of mass, momentum, energy

and magnetic field are obtained Conservation of Mass equation:

∇ · v = 0 (2.1.1)

Momentum equation:

∂v

∂t+ (v · ∇)v = ν∇2v +

1

ρ(j ×B) + gβ(T − T∞) (2.1.2)

Energy equation:∂T

∂t+ (v · ∇)T = α∇2T − 1

κ∇.qr (2.1.3)

13


It should be noted that the term qr in the energy equation represents radiative

heat flux term in the y direction and finally given by the following expression

qr = − 4σ

3αR

∇(T 4) (2.1.4)

The Lorentz force for two dimensional problem is given by

F = j ×B (2.1.5)

Ohm’s law:

j = σ (E + u×B) (2.1.6)

Maxwell’s equations:

∇×E = 0 , ∇.B = 0 , ∇×B = µ0j (2.1.7)

where ∇ =(

∂∂x

, ∂∂y

, ∂∂z

), u = (u, v, w) is the velocity vector, T the temperature

field in the boundary layer, p the pressure, j the electric current density, B =

(Bx, By, Bz) represents the magnetic induction vector, E the applied electric field,

ρ the density of the fluid, ν the kinematic viscosity, g = (−g, 0, 0) identifies the

gravitational vector,κ the thermal conductivity, α the thermal diffusivity, σ the

electrical conductivity and D the diffusion coefficient.

Under the above assumptions the two dimensional Navier-Stokes Eqns. (2.1.1)and

(2.1.2) coupled with energy and magnetic field Eqns. (2.1.3) and (2.1.4) for the

steady state, along with the thermal radiation by following Davies [2] becomes

∂u

∂x+

∂v

∂y= 0 (2.1.8)

u∂u

∂x+ v

∂v

∂y= ν

∂2u

∂y2+

µ

ρ

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)+ gβ(T − T∞) (2.1.9)

∂Bx

∂x+

∂By

∂y= 0 (2.1.10)

u∂Bx

∂x+ v

∂Bx

∂y−Bx

∂u

∂x−By

∂u

∂y=

1

γ

∂2Bx

∂y2(2.1.11)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2− ∂qr

∂y(2.1.12)

We now discuss the boundary layer theory and afterwards apply the Prandtl

approximations on the set of Eqns. (2.1.8)-(2.1.12).

14


2.2 Dimensionless Boundary Layer Equations

The nonlinearity of the momentum, hydromagnetic and energy equation makes it

difficult to obtain a closed mathematical solution to the problem. However, by

introducing the following non-dimensional dependent and independent variables

we have,

u = U0u, v =ν

LRe

1/2L v, x =

x

L, y =

y

LR

12eL, Bx = B0Bx,

By =B0

Re1/2L

By, θ =T − T∞4T

(2.2.1)

where 4T is the temperature difference. By using expression (2.2.1) in Eqns.

(2.1.8)-(2.1.12), we have∂u

∂x+

∂v

∂y= 0 (2.2.2)

u∂u

∂x+ v

∂v

∂y=

∂2u

∂y2+ S

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)+

GrL

Re2L

θ (2.2.3)

∂Bx

∂x+

∂By

∂y= 0 (2.2.4)

u∂Bx

∂x+ v

∂Bx

∂y− Bx

∂u

∂x− By

∂u

∂y=

1

Pm

(∂2Bx

∂x2+

∂2Bx

∂y2

)(2.2.5)

u∂θ

∂x+ v

∂θ

∂y=

1

Pr

[∂2θ

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θ)3 ∂θ

∂y}]

(2.2.6)

The equations (2.2.2)-(2.2.6) are the governing boundary layers equations that

can be used to analyze the different physical quantities which very important due

to their wide range of applications in different engineering fields. The parameter

arises when we make dimensionless the problem, are given as below:

ReL =U0L

ν, GrL =

gβ4TL3

ν2, Pr =

ν

α, λ =

GrL

Re2L

, Pm =υ

γ,

α =K

ρCp

, S =µB2

0

ρU20

, Rd =KαR

4σT 3∞

(2.2.7)

Further, we will define these parameters as follows:

15


2.2.1. Prandtl number

It is defined as the ratio of the kinematic viscosity of the fluid to its thermal

diffusivity and mathematically we can write as below:

Pr =ν

α

where ν and α are kinematic viscosity and thermal diffusivity of the fluid respec-

tively. Thus Prandtl number is the thermophysical property of the fluid and have

different numerical values for different fluids e.g. for water (7.0) and for air (0.71)

and for liquid metals, the range of Prandtl number is (0.1-0.001).

2.2.2. Reynolds number

The Reynolds number parameter may be considered as the ratio of inertial to

viscous forces and mathematical form of this parameter is given as

ReL =ρUL

µ

Here ρ, U , µ and L are fluid density, stream velocity, fluid viscosity and charac-

teristic length respectively. The nature of a given flow of an incompressible fluid

is characterized by its Reynolds number. For large values of Reynolds number

the inertial force is larger than the viscous force. This implies a large expanse

of fluid, high velocity, great density extremely small viscosity, or combinations of

these extremes.

2.2.3. Grashof number

This parameter plays a significant role in convective flows and simply defined

as the ratio of buoyancy and viscous force. The mathematical form of this key

parameter is:

GrL =gβ4TL3

ν2

Here g is acceleration due to gravity, 4T is the temperature difference and β is

known as volumetric expansion coefficient.

16


2.2.4. Mixed convection parameter

It is the ratio of buoyancy force to the inertial force and mathematically we can

write it as:

λ =GrL

Re2L

For (λ >> 1), the buoyancy force is dominant in the flow regime and thus the

trend is marked as natural or free convection flow but when the mixed convection

parameter (λ << 1) this implies that the inertial force is dominant in the flow

regime and the flow pattern is named as forced convection flow. Generally, the

mixed convection regime is occured for the middle range of mixed convection

parameter λ.

2.2.5. Radiation parameter

It is the ratio of Roseland mean absorption coefficient to ambient fluid temperature

which can be defined as follows:

Rd =KαR

4σT 3∞

It is pertinent to mention that with the increase of radiation parameter Rd,

the rate of heat transfer increases.

2.2.1 2.2.6. Magnetic force parameter

The magnetic force parameter is defined as the ratio of magnetic energy to kinetic

energy.

S =µB2

0

ρU20

where B0 and U0 are reference magnetic field and reference velocity respectively

and µ is magnetic permeability.

17


2.2.7. Magnetic Prandtl number

It is defined as a measure of viscous force to magnetic diffusion and mathematically

it can be expressed as follows

Pm =ν

γ

Here γ is magnetic diffusion and can be defined as follows

γ =1

µσ

where σ is the electrical conductivity.

ReL is the Reynolds number, GrL the Grashof number, Rd is the Planck num-

ber (radiation conduction parameter), L the reference length, λ is the mixed con-

vection parameter, Pr the Prandtl number and S the magnetic force parameter

(also known as Chandrasekhar number), Pm is the magnetic Prandtl number, and

α is the thermal diffusivity.

2.3 Mechanism of heat transfer

Heat transfer is energy in transit, which occurs as a result of a temperature gra-

dient or difference. This temperature difference is thought of a driving force that

causes heat to flow. Heat transfer occurs by three mechanism or mode: conduc-

tion, radiation and convection.

2.3.1 Conduction

Conduction is the transmission of heat through a substance without perceptible

motion of the substance itself. Heat can be conducted through gases, liquids and

solids. In the case of fluids in general, conduction is the primary mode of heat

transfer when the fluid has zero bulk velocity. In opaque solids, conduction is the

only mode by which heat can be transferred. The kinetic energy of the molecules of

a gas is associated with property we call temperature. In high temperature region,

gas molecules have higher velocities than those in low temperature region. The

random motion of molecules results in collisions and an exchange of momentum

18


and energy. When the random motion exists and temperature gradient presents

in the gas, molecules in high temperature region transfer some of their energy

through collisions, to molecules in the low temperature region. We identify this

transport of energy as a heat transfer via the diffusive or conductive mode.

Conduction of heat in solids is thought to be due to motion of free electrons,

lattice waves, magnetic excitations and electromagnetic radiation. The motion

of electrons occurs only in substances that are considered to be a good electrons

conductors. The theory is that the heat can be transported by electrons which are

free to move through the lattice structure of conductors, in the same way those

electrons is conducted. This is usually the case for metals.

2.3.2 Radiation

Radiation is the transfer of energy by electromagnetic radiation having defined

range of wavelength. One common example of radiant heat transfer is that the

energy transport between the sun and the earth. It is observed that all the sub-

stances emit radiant heat but that the net flow of heat is from the high temperature

to low temperature region. So the coolant substances will absorb more radiant

energy than it emits.

2.3.3 Convection

Convection is the term applied to heat transfer due to bulk motion of the fluid.

The study of fluid mechanics plays an important role in the analysis of convection

problems. The mechanism of convection is categorized as natural convection,

forced convection and mixed convection.

2.3.3.1. Natural convection

The phenomena of natural convection is observed due to fluid motion in which

changes arise from heating or cooling process. It is very important mode of heat

transfer with many engineering and industrial applications. The examples of nat-

ural heat transfer are heating systems in buildings, steam heated coils and electric

19


immersion heaters in process vessels, heat loss from process piping, heat removal

from electric conductors and electronic components, heat removal from spent nu-

clear fuel bundles and cooling of nuclear reactor core after loss of coolant accident.

The density variation in fluid motion due to temperature gradient form a denser

layer of fluid that exerts a buoyancy force on lighter one in a direction opposite

to gravity. Therefore, the lighter fluid accelerates under the influence of buoy-

ancy force and attains its certain velocity. It is also pertinent to mention that the

steady natural convection boundary layer is formed due to opposing viscous force

and thus the natural convection phenomena is characterized by a dimensionless

combination of some forces that can be named as Grashof number.

2.3.3.2. Forced convection

The mechanism of forced convection is observed as a result, when the surface tem-

perature is different from that of the fluid, heat is transferred as forced convection.

It is observed that the fluid motion for the case of forced convection is due to an

external motive source such as a fan or pump. The phenomena of forced convec-

tion is also very important and has many applications in industry such as radiator

system in vehicles, heating and cooling of parts of the body by blood circulation.

2.3.3.3. Mixed convection

The mechanism of mixed convection is the combination of forced and free con-

vection flows where buoyancy and forced motion effects are very important. It is

very important mode of heat transfer, arise in many transport process in engineer-

ing devices and in nature. The mixed convection flows are controlled by mixed

convection parameter λ which has been defined above.

2.4 Computational techniques

The physical processes in nature are governed by partial differential equations

(PDE’s). For this reason, it is important to understand the physical behaviour of

the model represented by partial differential equations. In addition, knowledge of

20


the mathematical character, properties and solution of the governing equation is

required. For this purpose in this thesis, we have used two methods to find the

numerical solutions of the model problems those are given as:

2.4.1 Finite difference method

In the present thesis, the momentum, magnetic field and energy equations are

under consideration to model the boundary layer flow. The nature of these equa-

tions is non linear and implicit, the analytical solutions of these equations for

the cases of practical interest do not exist. For this purpose, primitive variable

transformation is used to transform the set of equations in convenient form and

then integrate by using finite difference method, using backward difference method

in x-direction and central difference method for y-direction out of which we get

system of tri-diagonal algebraic equations and then solved numerically by using

Gauss elimination technique.

2.4.2 Asymptotic method

The boundary layer flows divided up into two regions, near the surface of the ver-

tical or horizontal plate and away from the surface of the plate. For this purpose,

we used stream function formulation to reduce these equations in convenient form.

The reduced system of equations is expand all the depending functions in terms

of small parameter and get a system of equations by truncating it up to fixed or-

der with their boundary conditions. The solution of these equations are obtained

by Nactsheim-Swigert iteration technique together with six order implicit Runge

Kutta-Butcher initial value solver.

In summing up what has been discussed in this chapter, we shall discuss the

radiative magnetohydrodynamic mixed convection flow past a vertical magnetized

permeable heated plate in Chapter 3 with the help of mathematical model given

in (2.2.2)-(2.2.6). Similarly, we shall investigate the radiative magnetohydrody-

namics natural convection flow past a magnetized vertical permeable plate by

using Eqns. (2.2.2)-(2.2.6) in Chapter 4 when the fluid circulates at the surface

21


of magnetized plate due to surface temperature. Chapter 5 will discuss the radia-

tive fluctuating magnetohydrodynamic mixed convection flow past a magnetized

vertical heated plate. In this chapter, we will investigate the effects of different

parameters such that mixed convection parameter λ, the conduction radiation pa-

rameter Rd, Prandtl number Pr, magnetic Prandtl number Pm, the magnetic force

parameter S and surface temperature θw in terms of amplitude and phase angle.

Moreover, we will discuss the effect of these parameters on the transient shear

stress, rate of heat transfer and current density. Chapter 6 will display the nu-

merical results of radiative fluctuating magnetohydrodynamic natural convection

flow past a magnetized vertical heated plate. In this chapter, we will obtain the

results for radiation parameter Rd, magnetic Prandtl number Pm, Prandtl number

Pr, magnetic force parameter S and surface temperature θw in terms of amplitude

and phase of coefficient of skin friction, rate of heat transfer and current density

at the surface of the plate. We will also investigate the effect of these parameters

on transient coefficient of skin friction, rate of heat transfer and current density.

In all these chapters, we will implement the finite difference method for primitive

variable formulation and asymptotic series solution for stream function to obtain

the numerical solutions.

22

23

Chapter 3

Radiative magnetohydrodynamic mixed convection flow past a magnetized vertical permeable heated plate

Chapter 3: Radiative magnetohydrodynamic mixed convection flow past...

In this chapter, we investigated the thermal radiation effects on hydromagnetic

mixed convection laminar boundary layer flow of a viscous, incompressible and

electrically conducting optically dense grey fluid along a magnetized permeable

surface with a variable magnetic field applied in stream direction at the surface.

The boundary layer equations for the momentum, energy and magnetic field are

reduced to the convenient form for integration using appropriate transformations.

The solutions of the transformed boundary layer equations are then simulated

employing by two methods, namely, (i) finite difference method and the (ii) the

asymptotic series solution for small and large values of local transpiration parame-

ter ξ = (V0x/ν)/Rex1/2 that depends on the surface mass-flux, V0, as well as the

distance x measured from the leading edge of the plate. The pertinent physical

parameters that dominate the flow in terms of local skin-friction Cfx, rate of heat

transfer, Nux and the magnetic intensity Mgx at the surface are the magnetic

field parameter S, conduction-radiation parameter Rd, Prandtl number Pr, the

magnetic Prandtl number Pm, mixed convection parameter λ and the surface

temperature parameter θw.

3.1 Formulation of the mathematical model

We consider the radiation interaction on the laminar two-dimensional magneto-

hydrodynamic mixed convection flow of an electrically conducting, viscous and

incompressible fluid past a uniformly heated vertical porous plate. The x-axis is

taken along the surface and y-axis is normal to it.

A schematic diagram illustrating the flow domain and the coordinate system

is shown in Fig. 3.1. In Fig.1 δM , δH and δT stands for momentum, magnetic field

and thermal boundary layer thicknesses. It is assumed that the surface tempera-

ture Tw of the plate is greater than the ambient fluid temperature T∞.

∂u

∂x+

∂v

∂y= 0 (3.1.1)

u∂u

∂x+ v

∂v

∂y= υ

∂2u

∂y2+

µ

ρ

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)+ gβ(T − T∞) (3.1.2)

24


u=

0,v

=V

0,T

=T

w,

Hx

=H

w(x

),H

y=

0

g

T=

T∞,u

=0,

Hx(

∞)

=0

δΤ

δΗ

δM

O y

x

Fig.3.1 The coordinate system and flow configuration

∂Bx

∂x+

∂By

∂y= 0 (3.1.3)

−γ∂Bx

∂y= (uBy − vBx) (3.1.4)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2− ∂qr

∂y(3.1.5)

where

qr = − 4σ

3αR

∂T 4

∂y

u and v are the components of fluid velocity in x and y-direction respectively, Bx

and By are the x and y-components of magnetic field, qr is the radiative heat flux

in the y direction, α, µ ρ, ν and γ are the thermal diffusion, magnetic permeability,

density, kinematic viscosity and magnetic diffusivity of the medium. The solution

of the above equations should satisfy the following boundary conditions

u(x, 0) = 0, v(x, 0) = ±V0, Bx(x, 0) = B0, T (x, 0) = Tw

u(x,∞) = U∞(x), Bx(x,∞) = 0, T (x,∞) = T∞

(3.1.6)

The nonlinearity of the momentum, hydromagnetic and energy equations make it

difficult to obtain a closed mathematical solution to the problem. However, by

introducing the following non-dimensional dependent and independent variables,

25


we have:

u = U0u, v =ν

LRe

1/2L v, x =

x

L, y =

y

LR

12eLy, Bx = B0Bx,

By =B0

Re1/2L

By, θ =T − T∞4T

(3.1.7)

where 4T is the temperature difference. By using Eqn. (3.1.7) in Eqns. (3.1.1)-

(3.1.6), we have∂u

∂x+

∂v

∂y= 0 (3.1.8)

u∂u

∂x+ v

∂v

∂y=

∂2u

∂y2+ S

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)+ λθ (3.1.9)

∂Bx

∂x+

∂By

∂y= 0 (3.1.10)

− 1

Pm

∂Bx

∂y= (uBy − vBx) (3.1.11)

u∂θ

∂x+ v

∂θ

∂y=

1

Pr

[∂2θ

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θ)3 ∂θ

∂y}]

(3.1.12)

The corresponding boundary condition take the form:

u(x, 0) = 0, v(x, 0) = SL, Bx(x, 0) = 1, θ(x, 0) = 1

u(x,∞) = 1, Bx(x,∞) = 0, T (x,∞) = 0(3.1.13)

In the above conditions SL = (V0L/ν)Re−1/2L is the transpiration parameter. The

parameter arises when we make dimensionless the problem, are given as below:

ReL =U0L

ν, GrL =

gβ4TL3

ν2, Pr =

ν

αλ =

GrL

Re2L

, Pm =υ

γ,

α =K

ρCp

, S =µB2

0

ρU20

, Rd =KαR

4σT 3∞

(3.1.14)

3.2 Methods of solution

To get the set of equations in convenient form for integration, we will introduce

the following one parameter group of transformation for the dependent and inde-

pendent variables:

u = U(ξ, Y ), v = x12 (V + ξ), ϕ = x

12 φ, θ = θ(ξ, Y )

Y = x−12 y, ξ = SLx

12 , Bx =

∂ϕ

∂y, By = −∂ϕ

∂x

(3.2.1)

26


The ξ is the local distribution of the surface mass flux. Here for suction (or

withdrawal) ξ is positive and for injection (or blowing) of fluid ξ is negative and

for solid surface ξ is zero. Where ϕ is the potential function that satisfies the

Eqn. (3.1.11). By using this group of transformations, which satisfies equation of

continuity and by using in Eqns. (3.1.8)-(3.1.12), we have the set of equations:

1

2ξ∂U

∂ξ− 1

2Y

∂U

∂Y+

∂V

∂Y= 0 (3.2.2)

1

2ξU

∂U

∂ξ+

(V + ξ − 1

2Y U

)∂U

∂Y

− S

[−1

2φ

∂2φ

∂Y 2+

1

2ξ

(∂φ

∂Y

∂2φ

∂ξ∂Y− ∂2φ

∂Y 2

∂φ

∂ξ

)]=

∂2U

∂Y 2+ λθ

(3.2.3)

1

Pm

∂2φ

∂Y 2=

1

2Uφ +

1

2ξU

∂φ

∂ξ+

(V + ξ − 1

2Y U

)∂φ

∂Y(3.2.4)

1

Pr

[1 +

4

3Rd

(1 + ∆θ)3

]∂2θ

∂Y 2+

4

Pr∆

1

Rd

(1 + ∆θ)2(∂θ

∂Y)2

=1

2ξU

∂θ

∂ξ+

(V + ξ − 1

2Y U

)− ∂θ

∂Y− Uθ

(3.2.5)

The appropriate boundary conditions satisfied by the above system of equations

are

U(ξ, 0) = V (ξ, 0) = 0, φ′(ξ, 0) = 1, θ(ξ, 0) = 1

V (ξ,∞) = 1, φ′(ξ,∞) = 0, θ(ξ,∞) = 0(3.2.6)

Once we know the solutions of the above equations, we readily can obtain the

values of skin friction, heat transfer and the normal magnetic intensity at the

surface from the following relations in terms of skin-friction, Nusselt number and

magnetic intensity from the following relations:

Re1/2x Cfx = f ′′(ξ, 0) (3.2.7)

Re−1/2x Nux = −θ′(ξ, 0) (3.2.8)

and

Re1/2x Mgx = −g(ξ, 0) (3.2.9)

Now we will discretize the Eqns. (3.2.2)-(3.2.5) with boundary conditions given

in Eqn. (3.2.6), we have a new system of discretised form of equations as follows:

A1Ui+1,j + B1Ui−1,j + C1Ui−1,j = D1 (3.2.10)

27


where

A1 = 1 +1

2(Vi,j − ξi − YjUi,j)4Y (3.2.11)

B1 = −2 +1

2

ξi

4ξUi,j4Y 2 (3.2.12)

C1 = 1− 1

2(Vi,j − ξi − YjUi,j)4Y (3.2.13)

D1 =

[S

24ξ(H1)i,j((H1)i,j − (H1)i,j−1)− 1

2

ξi

4ξUi,jUi,j−1

]4Y 2−

[S

4((H1)i+1,j − (H1)i−1,j) +

S

2((H1)i,j − (H1)i−1,j)(H2)i,j

]4Y

(3.2.14)

where (H1)i,j = (∂φ∂y

)i,j and (H2)i,j = (∂φ∂ξ

)i,j.

Similarly for hydromagnetics equation we have

A2φi+1,j + B2φi,j + C2φi−1,j = D2 (3.2.15)

A2 =1

Pm+

1

2(Vi,j − ξi − YjUi,j)4Y (3.2.16)

B2 = − 2

Pm− 1

2(1 +

ξi

4ξ)(H1)i,j4Y 2 (3.2.17)

C2 =1

Pm− 1

2(Vi,j − ξi − 1

2YjUi,j)4Y (3.2.18)

D2 = −1

2

ξi

4ξUi,jφi,j−14Y 2 (3.2.19)

and the discretised form of energy equation is of the form

A3θi+1,j + B3θi,j + C3θi−1,j = D3 (3.2.20)

where

A3 =1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θi,j)3

]+

1

PrRd

(θw − 1)(1 + (θw − 1)θi,j)24Y

+1

2(Vi,j − ξi − 1

2YjUi,j)4Y

(3.2.21)

B3 = − 2

Pr

[1 +

4

3Rd

(1 + (θw − 1)θi,j)3

]

+2

PrRd

(θw − 1)(1 + (θw − 1)θi,j)24Y

− (1− 1

2

ξi

4ξ)Ui,j4Y 2

(3.2.22)

28


C3 =1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θi,j)3

]

+1

PrRd

(θw − 1)(1 + (θw − 1)θi,j)24Y

− 1

2(Vi,j − ξi − 1

2YjUi,j)4Y

(3.2.23)

D3 = −1

2

ξi

4ξUi,jθi,j−14Y 2 (3.2.24)

The velocity can be calculated directly using equation of continuity (3.2.2) as

shown below:

Vi,j = Vi−1,j − 1

2

(ξ4y

4ξ− Yj

)Ui,j +

1

2ξ4Y

4ξUi,j−1 − 1

2YjUi−1,j (3.2.25)

where i and j denote the grid points along the x and y directions respectively. In

order to find the numerical solution we have discretised the Eqns. (3.2.2)-(3.2.5)

with boundary conditions (3.2.6) by using finite difference method, using backward

difference for x -direction and central difference for y-direction out of which we get

a system of tri-diagonal algebraic equations. These tri-diagonal equations are then

solved by Gaussian elimination technique. The computation is started at ξ = 0.0

and then marches downstream implicitly. Once we know the solution of these

equations, physical quantities of interest such as the coefficient of skin friction,

the coefficient of magnetic intensity and the coefficient of rate of heat transfer at

the surface may be calculated from:

Re12x Cfx = f ′′(ξ, 0), MgxRe

12x = −φ(ξ, 0),

Nux

Re12x

= −(

1 +4

3Rd

)θ′(ξ, 0)

(3.2.26)

3.3 Results and discussion

In present investigation, we have obtained the solutions of the non-similar bound-

ary layer Eqs.(3.2.2)-(3.2.5) with boundary conditions (3.2.6) that governs the flow

of a viscous incompressible and electrically conducting fluid past a magnetized ver-

tical porous plate with surface temperature by using the method discussed in the

preceding section for a wide range of physical parameters S, conduction-radiation

parameter Rd, surface temperature θw, Prandtl number Pr and mixed convection

29


parameter λ, magnetic Prandtl number Pm, against ξ. Now we discuss the effects

of the aforementioned physical parameters of the flow fields as well as on the lo-

cal skin-friction coefficient Re1/2x Cfx, the coefficient of surface magnetic intensity

Re1/2x Mgx and rate of heat transfer Re

−1/2x Nux on the surface of the plate.

Table 3.1 Numerical values of CfxRe12x obtained for Rd= 1.0, 10.0, and θw=1.1

when Pm=0.1, Pr=0.1, λ=1.0, S=0.1 against ξ by two methods.ξ Rd = 1.0 Rd = 10.0

FDM Asymptotic FDM Asymptotic

0.05 1.62225 1.622809sm 1.55037 1.55809sm0.1 1.66669 1.66423sm 1.59222 1.59423sm0.2 1.75771 1.75652sm 1.67797 1.67652sm0.4 1.94733 1.94108sm 1.85681 1.85108sm0.8 2.35008 2.350021sm 2.23701 2.23021sm1.0 2.55973 - 2.43497 -3.0 4.70172 - 4.45669 -4.0 5.71434 - 5.41609 -5.0 6.68702 - 6.34363 -6.0 7.63595 7.63883Lr 7.25476 7.25883Lr7.0 8.57192 8.57900Lr 8.15909 8.15900Lr8.0 9.49946 9.499912Lr 9.06043 9.06912Lr9.0 10.92063 10.99922Lr 9.96012 9.96174Lr10.0 11.33623 11.39390Lr 10.85958 10.85930Lr

3.3.1 Effects of different parameters on skin friction, mag-

netic intensity and rate of heat transfer

In first attempt we have obtained the solution of the non-similar boundary layer

equations governing the mixed convection flow of a viscous incompressible and

electrically conducting fluid along a vertical magnetized porous plate against ξ.

Tables 3.1, 3.2 and 3.3 exhibiting the effects of radiation parameter or Planck

number Rd = 1.0, 10.0 and for the fixed value of parameter λ = 1.0, Pm = 0.1

and Pr=0.1, S=0.1, and surface temperature θw=1.1 on coefficients of skin friction

Re1/2x Cfx, rate of heat transfer Re

−1/2x Nux and magnetic intensity Re

1/2x Mgx at

the surface. From Tables 3.1-3.3, it can easily be seen that an increase in radiation

30


Table 3.2 Numerical values of MgxRe12x obtained for Rd= 1.0, 10.0, and θw=1.1

when Pm=0.1, Pr=0.1, λ=1.0, S=0.1 against ξ by two methods.ξ Rd = 1.0 Rd = 10.0

FDM Asymptotic FDM Asymptotic0.05 1.35080 1.35110sm 1.37691 1.37118sm0.1 1.30772 1.30005sm 1.33024 1.33005sm0.2 1.22683 1.22780sm 1.24886 1.24780sm0.4 1.08388 1.08330sm 1.10133 1.10330sm0.8 0.85853 0.85429sm 0.86924 0.84291sm1.0 0.76950 - 0.77777 -3.0 0.32395 - 0.32414 -4.0 0.23978 - 0.23970 -5.0 0.18862 - 0.18855 -6.0 0.15474 0.15667Lr 0.15470 0.215667Lr7.0 0.13068 0.13286Lr 0.13065 0.13206Lr8.0 0.11269 0.12500Lr 0.11268 0.11500Lr9.0 0.09874 0.10111Lr 0.09873 0.11111Lr10.0 0.08758 0.092741Lr 0.08750 0.081000Lr

Table 3.3 Values of Nux/Re12x against ξ forRd= 1.0, 10.0, and θw=1.1, when

Pm=0.1, Pr=0.1, λ=1.0, S=0.1 against ξ by two methods.ξ Rd = 1.0 Rd = 10.0

FDM Asymptotic FDM Asymptotic0.05 0.29162 0.29725sm 0.34621 0.34725sm0.1 0.29321 0.29901sm 0.34935 0.34824sm0.2 0.29636 0.29954sm 0.35558 0.35654sm0.4 0.30251 0.30559sm 0.36789 0.36559sm0.8 0.31429 0.31369sm 0.39189 0.39369sm1.0 0.31992 - 0.40360 -3.0 0.37116 - 0.51618 -4.0 0.39612 - 0.57401 -5.0 0.42193 - 0.63503 -6.0 0.44887 0.44860Lr 0.69949 0.69860Lr7.0 0.47695 0.47170Lr 0.76723 0.76170Lr8.0 0.50613 0.50480Lr 0.83791 0.83480Lr9.0 0.53636 0.53790Lr 0.91119 0.91790Lr10.0 0.56757 0.563100Lr 0.98672 0.98310Lr

sm stand for small ξ, where Lr for large ξ

31


ξ

Cfx

/Re x1/

2

0.0 1.0 2.0 3.0 4.0 5.0

1

2

3

4

5

1.05.010.020.050.0

Pm = 0.1Pr= 7.0

S = 0.8θw = 1.1

λ = 1.0Rd

ξ

Nu x

/Re x1/

2

0.0 2.0 4.0 6.0 8.0 10.0

100

101

102

1.05.010.020.050.0

Pm = 0.1Pr = 7.0S = 0.8

θw = 1.1λ = 1.0

Rd

ξ

Mgx

Re x1/

2

0.0 2.0 4.0 6.0 8.0 10.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

1.05.010.020.050.0

Pm = 0.1Pr = 7.0S = 0.8

θw = 1.1λ = 1.0 Rd

Fig. 3.2 Numerical solution of (a)skin friction coefficient and (b) coefficient ofrate of heat transfer (c) coefficient of magnetic intensity at the surface against ξfor different values of radiation parameter Rd=1.0, 5.0, 10.0, 20.0, 50.0, Pm=0.1,

Pr=7.0, and S=0.8, θw=1.1, λ=1.0.

parameter Rd leads to decrease in coefficient of local skin friction and increases

in the rate of heat transfer, magnetic intensity at the surface. This phenomenon

can easily be understood from the fact that when Rd increases, the ambient fluid

temperature decreases and Roseland mean absorption coefficient increases which

reduce the skin friction and enhance the rate of heat transfer and magnetic in-

tensity at the surface. In Figs. 3.2(a-c), it is observed that with the increase of

radiation parameter Rd the skin friction decreases and rate of heat transfer and

magnetic intensity at the surface increases. In Figs. 3.3(a-c) it can be seen that

the increase in λ = 0.0, 2.5, 5.0, 7.5, 10 the coefficient of skin friction, heat transfer

increases and magnetic intensity at the surface decreases. It is very interesting fact

that forced convection is dominant mode of flow and heat transfer when buoyancy

parameter λ → 0 but with the increase of λ the buoyancy force acts like pressure

gradient and increase the the fluid motion, hence the coefficients of skin friction,

heat transfer and magnetic intensity increases with the streamwise distance ξ.

Figures 3.4(a-c) are representing the effects of different values of Prandtl num-

ber Pr=0.01, 0.1, 0.71, 7.0, and for fixed values of λ = 1.0, S = 0.4 Pm = 0.1,

Rd=1.0 and θw=1.1 on the coefficients of skin friction, rate of heat transfer and

magnetic intensity at the surface. In these figures, it is observed that with in-

crease of Pr the coefficient of skin friction decreases, coefficient of heat transfer

and magnetic intensity at the surface increases. It is very pertinent to mention

32


ξ

Cfx

/R

e x1/2

0.0 2.0 4.0 6.0 8.0 10.0

5.0

10.0

15.020.0

Pm = 0.5Pr = 0.1

S = 0.1Rd = 10.0

θw=1.1

λ0.02.55.07.510.0

(a) ξ

Nu x

/Re x1/

2

0.0 2.0 4.0 6.0 8.0 10.0

0.4

0.6

0.8

1.0 Pm = 0.5Pr= 0.1

S = 0.1Rd = 10.0

θw = 1.1

λ0.02.55.07.510.0

(b)ξ

Mgx

Re x1/

2

2.0 4.0 6.0 8.0 10.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0 Pm = 0.5Pr= 0.1

S = 0.1Rd = 10.0

θw = 1.1

λ0.02.55.07.510.0

(c)

Fig. 3.3 Numerical solution of (a) skin friction coefficient and (b) coefficient ofrate of heat transfer (c) coefficient of magnetic intensity at the surface against ξfor different values of mixed convection parameter λ=0.0, 2.5 5.0, 7.5,10.0 when

Pm=0.5, Pr=0.1 S=0.1, Rd=10.0 and θw=1.1.

that the increase in the Pr increases the kinematic viscosity (which ratio of dy-

namic viscosity to density of the fluid) of the fluid and decreases the thermal

diffusion which causes the increase in momentum boundary layer thickness and

due to rise in temperature thermal boundary layer becomes thinner. So, these

factors are responsible for the aforementioned phenomena. In Figs. 3.5(a-c) the

effects of different values of magnetic Prandtl number by keeping other parameters

fixed on coefficients of skin friction, heat transfer and magnetic intensity are dis-

played. From these figures, it is shown that the increase in Pm = 1.0, 10.0, 100.0

increase the coefficients of skin friction, heat transfer and decrease the coefficient

of magnetic intensity at the surface. It is also noted that the increase in coeffi-

cients of skin friction, heat transfer very remarkable for large values of magnetic

Prandtl number i.e. for Pm=10.0, 100.0 as compared with magnetic intensity at

the surface. The reason is that with the increase of Pm the magnetic diffusion

γ decreases or product of magnetic permeability, electrical conductivity and kine-

matic viscosity at the surface increases and hence the momentum and thermal

boundary layer thicknesses decrease due to which coefficients of skin friction and

heat transfer increases and magnetic intensity at the surface decreases.

33


ξ

Cfx

/R

e x1/2

0.0 2.0 4.0 6.0 8.0 10.0

5.0

10.0

15.0 Pm = 0.1S = 0.4λ = 1.0Rd = 1.0

θw=1.1

Pr0.010.10.717.0

(a) ξ

Nu x

/Re x1/

2

0.0 2.0 4.0 6.0 8.0 10.010-1

100

101 Pm = 0.1S = 0.4λ = 1.0Rd = 1.0

θw=1.1 Pr0.010.10.717.0

(b) ξ

Mgx

Re x1/

2

0.0 2.0 4.0 6.0 8.0 10.01.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5 Pm = 0.1S = 0.4λ = 1.0

Rd = 1.0θw=1.1

Pr0.010.10.717.0

(c)

Fig. 3.4 Numerical solution of (a) skin friction coefficient and (b) coefficient ofrate of heat transfer (c) coefficient of magnetic intensity at the surface against ξ

for different values of Prandtl number Pr=0.01, 0.1, 0.71,7.0 when Pm=0.1,S=0.4, Rd=1.0, θw=1.1 and λ=1.0.

ξ

Cfx

/R

e x1/2

10-1 100 1010.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0 Pr = 0.1S = 2.0λ = 1.0

Rd = 1.0θw = 1.1

Pm1.010.0100.0

(a) ξ

Nu x

/Re x1/

2

10-1 100 101

0.16

0.18

0.2

0.22

0.24 Pr= 0.1S = 2.0λ = 1.0

Rd = 1.0θw = 1.1

Pm1.010.0100.0

(b)ξ

Mgx

Re x1/

2

10-1 100 1010

0.2

0.4

0.6

0.8

1

1.2

1.4

Pr= 0.1S = 2.0λ = 1.0

Rd = 1.0θw = 1.1

Pm1.010.0100.0

(c)

Fig. 3.5 Numerical solution of (a) skin friction coefficient and (b) coefficient ofrate of heat transfer (c) coefficient of magnetic intensity at the surface against ξ

for different values of magnetic Prandtl number Pm=1.0, 10.0, 100.0 whenPr=0.1, S=2.0, Rd=1.0, θw=1.1 and λ=1.0.

34


3.3.2 Effects of different parameters on velocity, tempera-

ture and magnetic field profiles

Now we will discuss the effects of different physical parameters on the profiles of

the velocity, temperature and the transverse component of magnetic field against

similarity variable η for transpiration parameter ξ=10.0. The effects of parame-

ter λ = 0.0, 2.5, 5.0, 7.5, 10.0, for two values of S=0.0 and 0.8 and for fixed value

of Pm = 1.6, Pr=0.1, ξ=0.5, Rd=10.0 and θw=1.1 on velocity, temperature and

transverse component of magnetic field profiles are shown in Figs. 3.6(a), 3.6(b)

and 3.6(c). The dotted and solid lines in Figs. 3.6(a-c) shown the effects of para-

η

V(η

,ξ)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

(a)

S0.00.8

Pm = 1.6Pr = 0.1ξ = 0.5θw=1.1

Rd = 10.0

λ = 10.0

λ = 7.5

λ = 5.0

λ = 2.5

λ = 0.0

η

θ(η

,ξ)

0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0

(b)

S0.00.8

Pm = 1.6Pr = 0.1ξ = 0.5θw=1.1

Rd = 10.0

λ = 0.0, 2.5, 5.0, 7.5, 10.0

η0.0 1.0 2.0 3.0

0.0

0.2

0.4

0.6

0.8

1.0

(c)

S0.00.8

Pm = 1.6Pr= 0.1ξ = 0.5θw=1.1

Rd = 10.0

φ′(η

,ξ)

λ = 0.0, 2.5, 5.0, 7.5, 10.0

Fig. 3.6 (a) velocity and (b) temperature (c) transverse component of magneticfield profile against η at ξ=0.5 for different values of mixed convection parameter

λ = 0.0, 2.5, 5.0, 7.5, 10.0 when S = 0.0, 0.8 and for Pr=0.1, and Pm=1.6,Rd = 10.0, θw=1.1

meter λ for S = 0 (absence of magnetic field) and S = 0.8 (presence of magnetic

field) respectively. It is concluded that the velocity profile is influenced consid-

erably and increase when the value of λ increases and there is no any significant

changes shows in the absence of magnetic field as shown by dotted lines in Fig.

3.5(a). In Fig 3.6(b) it is shown that the temperature decreases with the increase

of λ and there is no change seen for magnetic field parameter S=0 and S=0.8.

From Fig. 3.6(c), we note that with the increase of parameter λ the effects of

transverse component of magnetic field decreases against η.

Figs. 3.7(a-c) are based on the effects of the S, on the velocity, temperature and

component of transverse magnetic field profiles. These figures clearly show that

35


η

V(η

,ξ)

0.0 2.0 4.0 6.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.00.81.6

S

Pm = 1.0Pr = 0.1λ = 1.0ξ = 0.5Rd=10

θw=1.1

(a)η

θ(η

,ξ)

0.0 2.0 4.0 6.0 8.00.0

0.2

0.4

0.6

0.8

1.0

0.00.81.6

S

Pm = 1.0Pr= 0.1λ = 1.0ξ = 0.5Rd=10

θw=1.1

(b) η0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0

0.00.81.6

S

Pm = 1.0Pr= 0.1λ = 1.0ξ = 0.5Rd=10

θw=1.1

(c)

φ′(η

,ξ)

Fig. 3.7 (a) velocity and (b) temperature (c) transverse component of magneticfield profile against η at ξ=0.5 for different values magnetic field parameter S=

0.0, 0.8, 1.6 when λ =1.0 and for Pr=0.1, and Pm=1.0, Rd = 10.0, θw=1.1

with the increase of S, the velocity profile decreases and the temperature, trans-

verse component of magnetic field profile increases. In Figs. 3.8(a-c) it is noted

that the increase in transpiration parameter increase velocity profile and decrease

the temperature and transverse component of magnetic field profiles. From these

figures it is also concluded that the momentum boundary layer thickness decreases

and thermal boundary layer thickness increases which indicates that transpiration

destabilizes the boundary layer. Finally, in Figs. 3.9(a-c) it is shown that with the

η

V(η

,ξ)

0.0 2.0 4.0 6.0 8.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.11.02.04.08.010.0

ξ

Pm = 2.0Pr= 0.1

S = 0.4λ = 1.0θw=1.1

Rd=10.0

(a)η

θ(η

,ξ)

0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0

0.11.02.04.08.010.0

ξ

Pm = 2.0Pr= 0.1

S = 0.4λ = 1.0θw=1.1

Rd=10.0

(b)η0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

0.11.02.04.08.010.0

ξ

Pm = 2.0Pr = 0.1

S = 0.4λ = 1.0θw=1.1

Rd=10.0

(c)

φ′(η

,ξ)

Fig. 3.8 (a) velocity and (b) temperature (c) transverse component of magneticfield profile against η for different values transpiration parameter ξ=2.0, 1.0, 2.0,4.0, 8.0, 10.0, when S= 0.4, λ =1.0 and for Pr=0.1, Pm=2.0, Rd = 10.0, θw=1.1

increase of Rd and keeping other parameters fixed the velocity and temperature

distribution decreases and transverse component of magnetic field increases.

36


η

V(η

,ξ)

0.0 5.0 10.0 15.0 20.00.0

0.5

1.0

1.5

2.0

2.5

0.10.51.05.010.0

Rd

Pm = 0.1Pr = 0.1

S = 0.4λ = 1.0ξ = 2.0

θw = 1.1

(a) η

θ(η

,ξ)

0.0 5.0 10.0 15.0 20.0

0.0

0.2

0.4

0.6

0.8

1.0

0.10.51.05.010.0

Rd

Pm = 0.1Pr= 0.1

S = 0.4λ = 1.0ξ = 2.0

θw = 1.1

(b)η0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0

0.10.51.05.010.0

Rd

Pm = 0.1Pr = 0.1S = 0.4λ = 1.0ξ = 2.0

θw = 1.1

(c)

(η,ξ

)φ’

Fig. 3.9 (a) velocity and (b) temperature (c) transverse component of magneticfield profile against η for different values radiation parameter Rd=0.1, 0.5, 1.0,

5.0, 10.0 when S= 0.4, λ =1.0 and for ξ=2.0, Pr=0.1, Pm=0.1, θw=1.1

3.4 Asymptotic solutions for small and large lo-

cal transpiration parameter ξ

Now we are heading in finding the solution of the present problem for small and

large values of transpiration parameter ξ. To do this we first reduce the Eqns.

(3.1.1)-(3.1.6) to convenient form by introducing the following transformations:

ψ = x12

[f(η, ξ) + ξ

]

ϕ = x−12 φ(η, ξ), θ = x−1θ(η, ξ)

η = x−12 , ξ = sx

12

(3.4.1)

where η is the similarity variable, ξ be the local transpiration parameter and ψ, φ

are the functions which satisfy the equations of conservation of mass and magnetic

field such that:

u =∂ψ

∂y, v = −∂ψ

∂x, Hx =

∂ϕ

∂y, Hy = −∂ϕ

∂x(3.4.2)

For withdrawal of fluid ξ > 0 whereas for blowing of fluid through the surface of the

plate ξ < 0. Throughout the present computations, we have chosen the value of ξ

positive with reason that in this case we considered the case of suction of the fluid

through the surface. By using Eqns. (3.4.1) and (3.4.2) in Eqns. (3.1.1)-(3.1.6),

37


we will obtain the following dimensionless local non-similarity equations:

f ′′′ +1

2(ff ′′ − Sφφ′′) + ξf ′′ + λθ

=1

2ξ

[f ′

∂f ′

∂ξ− f ′′

∂f

∂ξ− S

(φ′

∂φ′

∂ξ− φ′′

∂φ

∂ξ

)] (3.4.3)

1

Pmφ′′ +

1

2(fφ′ − f ′φ) + ξφ′ =

1

2ξ

[f ′

∂φ

∂ξ− φ′

∂f

∂ξ

](3.4.4)

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θ)3θ′]′

+1

2fθ′ + f ′θ + ξθ′

=1

2ξ

[f ′

∂θ

∂ξ− θ′

∂f

∂ξ

] (3.4.5)

where S =µH2

0

ρU20, Pm = υ

γ, respectively, are known as magnetic field parameter

and magnetic Prandtl number and ∆ = θw − 1 . The corresponding boundary

conditions become

f(0, ξ) = 0, f ′(0, ξ) = 0, φ′(0, ξ) = 1, θ(0, ξ) = 1

f ′(∞, ξ) = 1, φ′(∞, ξ) = 0, θ(∞, ξ) = 0(3.4.6)

It can be seen from Eqns. (3.4.3)-(3.4.5) that for ξ = 0.0, the equations are similar

otherwise these equations are locally non-similar. We can calculate chief physical

parameters

Re1/2x Cfx = f ′′(ξ, 0)

Re1/2x Mgx = −φ(ξ, 0)

Re−1/2x Nux = −

(1 +

4

3Rd

)θ′(ξ, 0)

(3.4.7)

3.4.1 When local transpiration parameter ξ is small

Since near the leading edge, ξ is small (ξ << 1), solutions to the Eqn. (3.4.3)-

(3.4.5) with boundary conditions given in Eqn. (3.4.6) may be obtained by using

the perturbation method. We can expand all the depending functions in powers

of ξ, we consider that

f(ξ, η) =∞∑i

ξifi(η), φ(ξ, η) =∞∑i=0

ξiφi(η), θ(ξ, η) =∞∑i=0

ξiθi(η) (3.4.8)

38


Substituting (3.4.8) into expression (3.4.3)-(3.4.5) and taking the terms only up

to O(ξ2), we will get the system of equations together with boundary conditions

(3.4.6) which is given as under:

f ′′′0 +1

2(f0f

′′0 − Sφ0φ

′′0) + λθ0 = 0 (3.4.9)

1

Pmφ′′0 +

1

2(f0φ

′0 − φ0f

′0) = 0 (3.4.10)

[1 + α (1 + ∆θ0)

3] θ′′0 + 3α∆ (1 + ∆θ0)2 θ′20 +

Pr

2f0θ

′0 + Prf ′0θ0 = 0 (3.4.11)

f0(0, ξ) = 0, f ′0(0, ξ) = 0, φ′0(0, ξ) = 1, θ0(0, ξ) = 1

f ′0(∞, ξ) = 1, φ′0(∞, ξ) = 0, θ0(∞, ξ) = 0(3.4.12)

f ′′′1 +1

2(f0f

′′1 − f ′0f

′1 − S(φ0φ

′′1 − φ′0φ

′1))+ (f ′′0 f1−Sφ′′0φ1)+ f ′′0 +λθ1 = 0 (3.4.13)

1

Pmφ′′1 +

1

2(f0φ

′1 − f ′1φ0) + (f ′0φ1 − f1φ

′0) + φ′0 = 0 (3.4.14)

[1 + α (1 + ∆θ0)

3] θ′′1 + 3α∆ (1 + ∆θ0)2 (θ1θ

′′0 + 2θ′0θ

′1)

+ 6α∆2θ1 (1 + ∆θ0) θ′20

+Pr

2(f0θ

′1 + f ′0θ1) + Pr(f1θ

′0 + θ0f

′1) + Prθ′0 = 0

(3.4.15)

f1(0, ξ) = 0, f ′1(0, ξ) = 0, φ′1(0, ξ) = 0, θ1(0, ξ) = 0

f ′1(∞, ξ) = 0, φ′1(∞, ξ) = 0, θ1(∞, ξ) = 0(3.4.16)

f ′′′2 +1

2

(f0f

′′2 − f ′21 − S(φ0φ

′′2 − φ′21 )

)+ (f1f

′′1 − f ′0f

′2

− S(φ1φ′′1 − φ′0φ

′2)) +

3

2(f ′′0 f2 − Sφ′′0φ2) + f ′′1 + λθ2 = 0

(3.4.17)

1

Pmφ′′2 +

1

2(f0φ

′2 − f ′0φ2) + (f1φ

′1 − f ′1φ1) +

3

2(φ′0f2 − f ′0φ2) + φ′1 = 0 (3.4.18)

[1 + α (1 + ∆θ0)

3] θ′′2 + 3α∆θ′′1θ1 (1 + ∆θ0)2

+ 3α∆θ′′0[θ′′2 + ∆

(2θ2θ0 + θ2

1

)+ ∆2θ0

(θ2θ0 + θ2

1

)]

+ 3α∆[(2θ′2θ

′0 + θ2

1

)(1 + ∆θ0)

2

+ 4∆θ′1θ1θ0 (1 + ∆θ0) + ∆θ20

(2θ2 + ∆(2θ2θ0 + θ2

1))]

+Pr

2(f0θ

′2 + θ1f

′1) + (θ′1f1 + f ′2θ0) +

3Pr

2f2θ

′0 + Prθ′1 = 0

(3.4.19)

39


f2(0, ξ) = 0, f ′2(0, ξ) = 0, φ′2(0, ξ) = 0, θ2(0, ξ) = 0

f ′2(∞, ξ) = 0, φ′2(∞, ξ) = 0, θ2(∞, ξ) = 0(3.4.20)

It is pertinent to mentioned that the set of equations given in (3.4.9) -(3.4.20)

are coupled and non-linear, so the solutions of these equations can be obtained

by the Nachtsheim-Swigert iteration technique together with the sixth order im-

plicit Runge-Kutta-Butcher initial value solver. After knowing the values of the

functions f ′′, φ and θ′ and their derivatives we can calculate the values of the coef-

ficients of skin friction, surface magnetic intensity and heat transfer in the region

near the leading edge against ξ from the following expansion for S=0.1, λ=1.0,

Pm =0.1, Pr=0.1 and radiation parameter Rd=1.0, 10.0, and θw=1.1 respectively.

Re1/2x Cfx = (1.59195 + 1.12282ξ + 1.01047ξ2 + .......)

Re1/2x Mgx = −(1.43230 + 1.02251ξ + 1.34209ξ2 + .....)

Re−1/2x Nux = −(0.21749− 0.00475ξ + 0.09878ξ2.....)

(3.4.21)

The numerical results thus obtained are entered in Table 3.1, 3.2 and 3.3 for

coefficients of skin friction, rate of heat transfer and magnetic intensity at the

surface. We can see that from these tables, the series solution are in excellent

agreement with that of finite difference solutions even for ξ ε[0, 1].

In Table 3.4 the comparison of the solutions obtained by finite difference

method and that of Glauert[36] analytically has been given for the coefficient

of skin friction and magnetic intensity at the surface. It is observed that for

Pm = 1.0, 10.0, radiation parameter Rd = ∞ and the variation in magnetic field

parameter S decrease the skin friction and the skin friction approaches to zero

as S −→ 1.1 and the local magnetic intensity Mgx increases with the increase of

magnetic field parameter S. From table 3.4 it can be seen that the present method

and the analytical results obtained by Glauert [36] are in good agreement. For

magnetic Prandtl number Pm=10.0 the coefficients of skin friction and magnetic

intensity at the surface are also noted in excellent agreement. Further, we see that

for small value of magnetic Prandtl number Pm=0.1 and for magnetic field para-

meter S = π as in the case of Glauert [36] the separation occur at the surface. In

Table 3.5 the value of coefficient of skin friction obtained by other authors Glauert

40


[36] for large magnetic Prandtl number Pm by keeping Rd = ∞ and Davies[2] for

small magnetic Prandtl number by taking magnetic field parameter S=0.1 and

sufficiently small S=0.05 are entered and compare with present results and found

to be in good agreement. Here we notice that the agreement between present

results and results obtained by Glauert[36], Davies[2] are in excellent agreement.

The results presented in Table 3.6 are those obtained from heat transfer in hy-

dromagnetics by Ramamoorthy [6] considering Eckert number equal to zero and

radiation parameter Rd = ∞are compared with the present results. From this

table it can be seen that the numerical results obtained for different values of

magnetic field parameter S and for Pm=0.1, Pr=1.0, λ=0.5 are in agreement

that obtained by Ramamoorthy.

Table 3.4 Values of Re1/2x Cfx and Re

1/2x Mgx obtained by Glauert [2] and present

authors while Rd = ∞, Pm= 1.0 and 10.0 against different values of SS Pm = 1.0 Pm = 10.0

Present Glauert [2] Present Glauert [2]f ′′(0) g(0) f ′′(0) g(0) f ′′(0) g(0) f ′′(0) g(0)

0.0 0.3321 2.1290 0.3321 2.1797 0.3321 0.9525 0.3321 1.00950.1 0.3067 2.1713 0.3025 2.2448 0.3188 0.9631 0.3182 1.02380.2 0.2806 2.2191 0.2729 2.3099 0.3050 0.9744 0.3044 1.03800.4 0.2257 2.3372 0.2138 2.4402 0.2770 0.9995 0.2767 1.06650.6 0.1657 2.5066 0.1547 2.5704 0.2479 1.0285 0.2491 1.09500.8 0.0972 2.8017 0.0955 2.7006 0.2176 1.0629 0.2214 1.1235

3.4.2 When local transpiration parameter ξ is large

Now, attention is given in finding the solution of Eqns. (3.4.3)-(3.4.5) when ξ is

large. The order of magnitude analysis of various terms in these equations shows

that f ′′′ and ξf ′′ are largest terms in Eqn. (3.4.3) and φ′′ and ξφ′ in Eqn. (3.4.4)

and θ′′ and ξθ′ in Eqn. (3.4.5). In the respective equations, both the terms have

to be balanced in magnitude and the only way to do this, is to assume that η is

small and hence its derivative is large. It is essential to find appropriate scaling

for f , φ and θ. On balancing f ′′′ and ξf ′′ in Eqn. (3.4.3) and φ′′ and ξφ′ in Eqn.

41


Table 3.5 Values of Re1/2x Cfx when Rd = ∞, S= 0.1 and 0.05 at ξ=0.0 against

Pm obtain by Glauert[2] and Devies[3] and the present authors.S = 0.0 S = 0.8

Pm Present Glauert [2] Pm Present Davies [3]0.1 0.2888 0.2669 0.1 0.3106 0.31401.0 0.3067 0.3016 0.2 0.3107 0.31572.0 0.3109 0.3078 0.3 0.3153 0.31734.0 0.3145 0.3128 0.5 0.3172 0.31946.0 0.3164 0.3152 0.7 0.3183 0.32048.0 0.3177 0.3167 0.9 0.3191 0.320410.0 0.3186 0.3178 - - -

- - - - - -50 0.3238 0.3237 - - -75 0.3248 0.3247 - - -100 0.3254 0.3254 - - -

Table 3.6 Values of Re−1/2x Nux for different S when Rd = ∞, Pm=0.1, Pr=1.0,

and λ=0.5S Present [7]

0.1 0.64551 0.650480.3 0.61994 0.619580.5 0.59112 0.57693

42


(3.4.4) and θ′′, ξθ′ in Eqn. (3.4.5), it is found that η = O(ξ−1), f = O(ξ−1) and

φ = O(ξ−1). Therefore, following transformations may be introduced.

f(η) = ξ−1f(η), η = ξη, φ(η) = ξ−1φ(η), θ(η) = θ(η) (3.4.22)

By using (3.4.22), the transformed equation will take the form:

f ′′′ + f ′′ + λξ−2θ =1

2ξ−1

[f ′

∂f

∂ξ− f ′′

∂f

∂ξ− S

(φ′

∂φ′

∂ξ− φ′′

∂φ

∂ξ

)](3.4.23)

1

Pmφ′′ + φ′ =

1

2ξ−1

[f ′

∂φ

∂ξ− f ′

∂f

∂ξ

](3.4.24)

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θ)3θ′]′

+ θ′ =1

2ξ−1

[f ′

∂θ

∂ξ− θ′

∂f

∂ξ

](3.4.25)

The regular perturbation of the functions f , φ and θ in power of ξ−2 is given as

under:

f(ξ, η) =1∑

m=0

ξ−2mfm(η), φ(ξ, η) =1∑

m=0

ξ−2mφm(η),

θ(ξ, η) =1∑

m=0

ξ−2mθm(η)

(3.4.26)

By substituting Eqn. (3.4.26) into Eqns. (3.4.23)-(3.4.25), and equating like

powers of ξ and by dropping bars, we have the following set of equations:

O(ξ0):

f ′′′0 + f ′′0 = 0 (3.4.27)

1

Pmφ′′0 + φ′0 = 0 (3.4.28)

[1 + α (1 + ∆θ0)

3] θ′′0 + 3α∆ (1 + ∆θ0)2 θ′20 + Prθ′0 = 0 (3.4.29)

The boundary conditions regarding to the perturbation series expansion are of the

following form:

f0(0) = 0, f ′0(0) = 0, φ′0(0) = 1, θ0(0) = 1

f ′0(∞) = 1, φ′0(∞) = 0, θ0(∞) = 0(3.4.30)

O(ξ−2):

f ′′′1 + f ′′1 + λθ0 = 0 (3.4.31)

1

Pmφ′′1 + φ′1 = 0 (3.4.32)

43


[1 + α (1 + ∆θ0)

3] θ′′1 + 3α∆ (1 + ∆θ0)2 (θ1θ

′′0 + 2θ′0θ

′1)

+ 6α∆2θ1 (1 + ∆θ0) θ′20 + Prθ′1 = 0(3.4.33)

And the boundary conditions regarding to the perturbation series expansion are

of the following form:

f1(0) = 0, f ′1(0) = 0, φ′1(0) = 0, θ1(0) = 0

f ′1(∞) = 0, φ′1(∞) = 0, θ1(∞) = 0(3.4.34)

From the solutions of Eqns. (3.4.27)-(3.4.30) and Eqns. (3.4.31)-(3.4.34), we

obtain

f0(η) = η + e−η − 1 (3.4.35)

f ′′0 (0) = 1 (3.4.36)

φ0(η) = − 1

Pme−ηPm (3.4.37)

φ0(0) = − 1

Pm(3.4.38)

θ′0(η) =

[1 +

4

3Rd

(1 + ∆θ0)3

]Pre−ηPr (3.4.39)

θ′0(0) =

[1 +

4

3Rd

(1 + ∆θ0)3

]Pr (3.4.40)

f ′′1 (η) =λe−ηPr(Pr− 1) + Prλe−η − λe−η

2(Pr− 1)(3.4.41)

f ′′1 (0) =λ(Pr− 1) + Prλ− λ

2(Pr− 1)(3.4.42)

φ1(η) = 0 (3.4.43)

φ1(0) = 0 (3.4.44)

θ′1(η) = 0 (3.4.45)

θ′1(0) = 0 (3.4.46)

Since, now, we know the values of f ′′0 (0), φ0(0), θ′0(0) and f ′′1 (0) φ1(0)and θ′1(0)

from the above solutions we calculate the friction coefficient,R12exCfx, local rate of

44


heat transfer, R− 1

2ex Nux, and the local magnetic intensity, R

12exMgx at the surface

from the following expressions:

Re1/2x Cfx ' ξ +

λ(Pr− 1) + Prλ− λ

2(Pr− 1)ξ−1

Re1/2x Mgx ' ξ

Pm

Re−1/2x Nux '

[1 +

4

3Rd

(1 + ∆θ0)3

]Prξ

(3.4.47)

Numerical value of the local skin friction coefficient, surface magnetic inten-

sity and the local rate of heat transfer are obtained from the Eqn. (3.4.47) for

different values of magnetic field parameter and magnetic Prandtl number and

radiation parameter, surface temperature and Prandtl number respectively in the

down stream region are entered in Tables 3.1, 3.2 and 3.3 respectively. From these

tables it can be seen that for large value of transpiration parameter ξ, the skin

friction Re1/2x Cfx approaches to ξ and the values of coefficient of magnetic inten-

sity Re1/2x Mgx approaches to ξ/Pm and Nusselt number Re

−1/2x Nux approaches

to[1 + 4

3Rd(1 + ∆θ0)

3]Prξ. The comparison of the present results with the nu-

merical results obtained by FDM shows excellent agreement in the down stream

region.

3.5 Conclusion

The physical parameters such as mixed convection parameter λ, transpiration

parameter ξ, magnetic field parameter, magnetic Prandtl number Pr and radiation

parameter Rd, Prandtl number and surface temperature exerts significant influence

on coefficients of skin friction Re1/2x Cfx, heat transfer Re

−1/2x Nux and magnetic

intensity Re1/2x Mgx at the surface.

The coefficient of skin friction decreases, and the coefficient of rate of heat

transfer and magnetic intensity at the surface increase with the increase of Rd.

The coefficients of skin friction, heat transfer increase and magnetic intensity at

the surface decreases with the increase of λ by keeping Rd, θw,Pm, S fixed. The

momentum and thermal boundary layer thicknesses decrease and velocity and

45


temperature profiles increase with the increase of the mixed convection parame-

ter λ. It is also noted that the increase in λ reduce the transverse component of

magnetic field profile. The increase in Prandtl number reduce the coefficient of

skin friction and enhance the coefficient of heat transfer and magnetic intensity at

the surface. The coefficient of skin friction, heat transfer increase and the coeffi-

cient of magnetic intensity decreases with the increase of Pm. The transpiration

parameter ξ plays a significant role in boundary layer, due to increase in tran-

spiration parameter ξ, the momentum and thermal boundary layer thicknesses

decrease and the transverse component of magnetic field profile is reduced. It is

also concluded that an increase of the Rd decreases the local velocity as well as

temperature distribution and enhance the transverse component of magnetic field

at the surface.

46

47

Chapter 4

Radiative magnetohydrodynamic natural convection flow past a magnetized vertical permeable heated plate

Chapter 4: Radiative magnetohydrodynamic natural convection ...

Natural convection phenomena is observed due to fluid motion in which changes

are arising from heating or cooling processing. With this understanding, we will

modify the problem given in Chapter 3 and find the effects of conduction radiation

on hydromagnetic natural convection flow of viscous, incompressible, electrically

conducting and optically dense grey fluid past a magnetized vertical porous plate.

The governing non similar equations are solved by using (i) finite difference method

for entire values of suction parameter ξ and(ii) the asymptotic solution for small

and large values of ξ numerically. The effects of varying the Prandtl number, Pr,

magnetic Prandtl number Pm, magnetic force parameter S, radiation parameter

Rd, and surface temperature θw on coefficients of skin friction, rate of heat transfer

and current density are shown graphically and in tabular form. Finally, an attempt

has been made to examine the effects of above mentioned physical parameters on

velocity profile, temperature distribution and transverse component of magnetic

field.

4.1 Mathematical analysis and governing equa-

tions

Here, we consider a steady two-dimensional magnetohydrodynamic natural con-

vection flow of an electrically conducting, viscous and incompressible fluid past a

uniformly heated vertical porous plate by including radiation effects in the energy

equation.The diagram illustrating the flow domain and the coordinate system is

shown in Fig. 4.1. The x-axis is taken along the surface and y-axis is normal to

it. In Fig.4.1 δM , δT and δH stand for momentum, thermal and magnetic field

boundary layer thicknesses. The boundary layer equations those govern the flow

under consideration are

48


u=

0,v

=V

0=

0,T

=T

w,

Hx

=H

w(x

)

g

T=

T∞,u

=0,

,=

0,H

x(∞

)=

0

δΤ

δΗ

δM

O y

x

Fig. 4.1 The coordinate system and flow configuration

∂u

∂x+

∂v

∂y= 0 (4.1.1)

u∂u

∂x+ v

∂v

∂y= ν

∂2u

∂y2+

µ

ρ

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)+ gβ(T − T∞) (4.1.2)

∂Bx

∂x+

∂By

∂y= 0 (4.1.3)

u∂Bx

∂x+ v

∂Bx

∂y− Bx

∂u

∂x− By

∂u

∂y=

1

γ

∂2Bx

∂y2(4.1.4)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2− ∂qr

∂y(4.1.5)

where

qr = − 4σ

3αR

∂T 4

∂y

u(x, 0) = 0, v(x, 0) = V0, Bx(x, 0) = Bw(x), By(x, 0) = 0, T (x, 0) = Tw

u(x,∞) = 0, Bx(x,∞) = 0, T (x,∞) = 0 (4.1.6)

For convenience, we introduce the following dependent and independent variables

to normalize the boundary layer equations

u =ν

LGr

12Lu, v =

ν

LGr

14Lv, θ =

T − T∞Tw − T∞

Bx =B0

LGr

12LBx, By =

B0

LGr

14LBy, y =

y

LGr

14L , x =

x

L(4.1.7)

49


By substituting Eqn. (4.1.7) into Eqns. (4.1.1)-(4.1.6) the dimensionless boundary

layer equations and boundary conditions are given as follows

∂u

∂x+

∂v

∂y= 0 (4.1.8)

u∂u

∂x+ v

∂u

∂y= θ +

∂2u

∂y2+ S

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)(4.1.9)

∂Bx

∂x+

∂By

∂y= 0 (4.1.10)

u∂Bx

∂x+ v

∂Bx

∂y−Bx

∂u

∂x−By

∂u

∂y=

1

Pm

∂2Bx

∂y2(4.1.11)

u∂θ

∂x+ v

∂θ

∂y=

1

Pr

[∂2θ

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θ)3 ∂θ

∂y}]

(4.1.12)

The boundary conditions to be satisfied by the above equations are

u(x, 0) = 0, v(x, 0) = V0, Bx(x, 0) = 1, By(x, 0) = 0, θ(x, 0) = 1

u(x,∞) = 0, Bx(x,∞) = 0, θ(x,∞) = 0 (4.1.13)

where u and v are dimensionless fluid velocity components in x and y-direction

respectively, Bx and By are the dimensionless x and y-components of magnetic

field, θ is the dimensionless temperature of the fluid in boundary layer.

Pr =ν

α, S =

µB20

ρL2, Pm =

ν

γ, Rd =

KαR

4σT 3∞, GrL =

gβ∆TL3

ν2, θw =

Tw

T∞

where, µ, ν, Pr, S, Pm, Rd, GrL and θw are the physical parameters arises

during dimensioned the problem and have been defined in Chapter 2. In Eqn.

(4.1.6), V0 is surface mass flux, which is assumed to be uniform, when fluid is

being withdrawn through the surface, it is negative and when fluid is being blown

through it is positive. In our present investigation, we shall consider that V0 is for

the case of withdrawal of fluid.


We now turn to get the numerical solutions of the problem, for this purpose, we will

use two methods namely (i) Primitive variable transformation for finite difference

method and (ii) Stream function formulation for asymptotic series solutions near

and away from the leading edge of the plate.

50


4.2.1 Primitive variable formulation

To get the set of equations in convenient form for integration, we define the follow-

ing one parameter of transformations for the dependent and independent variables:

u = x12 U(ξ, Y ), v = x−

14 (V (ξ, Y ) + ξ), Y = x−

14 V (ξ, Y ) = 0

Bx = x12 φ1(ξ, Y ), By = x−

14 φ2(ξ, Y ), θ = θ(ξ, Y ), ξ = V0x

14 (4.2.1)

By using Eqn. (4.2.1) into Eqns. (4.1.8)-(4.1.12) with boundary conditions

(4.1.13) we have1

2U + ξ

∂U

∂ξ− 1

4Y

∂U

∂Y+

∂V

∂Y(4.2.2)

1

2U2 +

1

4ξU

∂U

∂ξ+ (V − 1

4Y U)

∂U

∂Y− ξ

∂U

∂Y= θ +

∂2U

∂Y 2

+ S

[1

2φ2

1 +1

4ξφ1

∂φ1

∂ξ+

(φ2 − 1

4Y φ1

)∂φ1

∂Y

] (4.2.3)

1

2φ1 +

1

4ξ∂φ1

∂ξ− 1

4Y

∂φ1

∂Y+

∂φ2

∂Y= 0 (4.2.4)

1

4ξU

∂φ1

∂ξ+ (V − 1

4Y U)

∂φ1

∂Y− ξ

∂φ1

∂Y

− 1

4ξφ1

∂U

∂ξ− (φ2 − 1

4Y φ1)

∂U

∂Y=

1

Pm

∂2φ1

∂Y 2

(4.2.5)

1

4ξU

∂θ

∂ξ+ (V − 1

4Y U)

∂θ

∂Y− ξ

∂θ

∂Y

=1

Pr

[1 +

4

3Rd

(1 + ∆θ)3

]∂2θ

∂Y 2+

4

Pr∆

1

Rd

(1 + ∆θ)2(∂θ

∂Y)2

(4.2.6)

The appropriate boundary conditions to be satisfied by the above equations are

U(ξ, 0) = V (ξ, 0) = 0, φ1(ξ, 0) = 1, φ2(ξ, 0) = 0, θ(ξ, 0) = 1

U(ξ,∞) = 0, φ1(ξ,∞) = 0, θ(ξ,∞) = 0(4.2.7)

We discretised the Eqns. (4.2.2)-(4.2.7) as have been done in Chapter 3 by us-

ing finite difference method by using central difference along y-axis and backward

difference along x-axis, out of which we got a system of tri-digonal algebraic equa-

tions. These tri-diagonal equations are then solved by Gaussian elimination tech-

nique. The computation is started at X=0, and then marches downstream implic-

itly. Here we have taken 4ξ=0.005 and 4Y =0.01 for ξ and Y grids respectively.

51


Throughout, solutions are obtained for smaller values of Prandtl number Pr, and

magnetic Prandtl number Pm, magnetic force parameter S, radiation parameter

Rd, surface temperature θw which are appropriate for liquid metals that are often

used in nuclear cooling system. Finally, solutions are then obtained for different

values of pertinent physical parameters, namely, the magnetic field parameter, S,

the magnetic Prandtl number, Pm, the Prandtl, Pr, radiation parameter Rd and

surface θw . The values of of Pr are taken here as small that are appropriate for

liquid metal, which is often used as coolant in the nuclear devices. The results

are obtained in coefficients of skin friction, Gr−3/4L x−1/4Cf , rate of heat transfer,

Gr1/4L x1/4Nux and current density Gr

−3/4L x−1/4Jw defined in Eqn. (4.2.8). Effect

of different physical parameters are also obtained in form of velocity, temperature

, and transverse component of magnetic field and shown graphically in Figures

4.6-4.10.

Once we know the solutions of the Eqns. (4.2.2)-(4.2.6), we are in the position

to measure of the physical quantities such as coefficients of skin friction, rate

of heat transfer and current density from the relation given below, which are

important from the application point of view, from the following dimensionless

expressions

Gr−3/4L x−1/4Cf =

(∂u

∂Y

)

Y =0

, Gr1/4L x1/4Nux = −

(1 +

4

3Rd

θ3w

)(∂θ

∂Y

)

Y =0

Gr−3/4L x−1/4Jw =

(∂φ1

∂Y

)

Y =0

(4.2.8)

In the following section, the solution will be obtained for small and large local

transpiration parameter ξ.

4.3 Asymptotic solutions for small and large lo-

cal transpiration parameter ξ

To get the numerical solutions for small and large local transpiration parameter

for the steady state equations, we re-define the flow variables as given below:

Y = x−14 y, u = x

12 f ′(Y ), v = −x−

14

(3

4f(Y )− 1

4Y f ′(Y ) +

1

4ξ∂f

∂ξ+ ξ

)

52


θx = x−1

(1

4ξ∂θ

∂ξ− 1

4Y θ′

), θy = x−

14 θ′

Hx = x12 φ′(Y ), Hy = −x−

14

(3

4φ(Y )− 1

4Y φ′(Y ) +

1

4ξ∂φ

∂ξ

),

ξ = V0x14 ,

(4.3.1)

which reduces the set of equations

f ′′′ +3

4ff ′′ − 1

2f ′2 + θ − S

(3

4φφ′′ − 1

2φ′2

)+ ξf ′′

=1

4ξ

[f ′

∂f ′

∂ξ− f ′′

∂f

∂ξ− S

(φ′

∂φ′

∂ξ− φ′′

∂φ

∂ξ

)] (4.3.2)

1

Pmφ′′′ +

3

4fφ′′ − 3

4f ′′φ + ξφ′′

=1

4ξ

(f ′

∂φ′

∂ξ− φ′

∂f ′

∂ξ+ f ′′

∂φ

∂ξ− φ′′

∂f

∂ξ

) (4.3.3)

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θ)3θ′]′

+3

4fθ′ + ξθ′

=1

4ξ

(f ′

∂θ

∂ξ− θ′

∂f

∂ξ

) (4.3.4)

Boundary conditions to be satisfied by the above equations are

f(ξ, 0) = f ′(ξ, 0) = 0, φ(ξ, 0) = 0, φ′(ξ, 0) = 1, θ(ξ, 0) = 1

f ′(ξ,∞) = 0, φ′(ξ,∞) = 0, θ(ξ,∞) (4.3.5)

4.3.1 When local transpiration parameter ξ is small

Since near the leading edge ξ is small (ξ << 1), we can expand all the depending

functions in power of ξ. Accordingly we consider that

f(ξ, Y ) =∞∑i=0

ξifi(Y ), φ(ξ, Y ) =∞∑i=0

ξiφi(Y ), θ(ξ, Y ) =∞∑i=0

ξiθi(Y ) (4.3.6)

Substituting Eqn. (4.3.6) into Eqns. (4.3.2)-(4.3.4) and taking the term up to

O(ξ) following sets of equations are obtained:

O(ξ0)

f ′′′0 +3

4f0f

′′0 −

1

2f ′20 + θ0 − S

(3

4φ0φ

′′0 −

1

2φ′20

)= 0 (4.3.7)

1

Pmφ′′′0 +

3

4f0φ

′′0 −

3

4f ′′0 φ0 = 0 (4.3.8)

53


[1 + α (1 + ∆θ0)

3] θ′′0 + 3α∆ (1 + ∆θ0)2 θ′20 +

3

4Prf0θ

′0 = 0 (4.3.9)

the boundary conditions are

f0(0) = f ′0(0) = 0, φ0(0) = 0, φ′0(0) = 1, θ0(0) = 1

f ′0(∞) = 0, φ′0(∞) = 0, θ0(∞) = 0 (4.3.10)

O(ξ1)

f ′′′1 +3

4(f0f

′′1 −Sφ0φ

′′1)+ (f ′′0 f1−Sφ′′0φ1)− 5

4(f ′0f

′1−Sφ′0φ

′1)+ θ1 + f ′′0 = 0 (4.3.11)

1

Pmφ′′′1 +

3

4f0φ

′′1 +

5

4f1φ

′′0 −

3

4f ′′0 φ1 − 5

4f ′′0 φ1 − 1

4f ′0φ

′1 +

1

4f ′1φ

′0 + φ′′0 = 0 (4.3.12)

[1 + α (1 + ∆θ0)

3] θ′′1 + 3α∆ (1 + ∆θ0)2 (θ1θ

′′0 + 2θ′0θ

′1)

+ 6α∆2θ1 (1 + ∆θ0) θ′20 + Pr

(3

4f0θ

′1 + f1θ

′0 −

1

4f ′0θ1 + θ′0

)= 0

(4.3.13)

The related order boundary conditions are

f1(0) = f ′1(0) = 0, φ1(0) = 0, φ′1(0) = 0, θ1(0) = 0

f ′1(∞) = 0, φ′1(∞) = 0, θ1(∞) = 0 (4.3.14)

The Eqns. (4.3.7)-(4.3.13) are nonlinear coupled equations, the solutions of these

equations are obtained by Nachtsheim-Swigert iteration technique together with

six order implicit Runge-Kutta-Butcher initial value solver. We can calculate the

values of the coefficients skin friction, rate of heat transfer and current density

at the surface in the region near the leading edge against ξ from the following

expressions

Gr−3/4L x−1/4Cf = f ′′(0) (4.3.15)

Gr−3/4L x−1/4Jw = φ′′(0) (4.3.16)

Gr1/4L x1/4Nu = −

(1 +

4

3Rd

θ3w

)θ′(0) (4.3.17)

The results obtained with the help of the Eqns. (4.3.15)-(4.3.17) are given in

Tables 4.1-4.3 for small values of ξ

54


4.3.2 When local transpiration parameter ξ is large

Now, attention is given in finding the solution of Eqns. (4.3.11)-(4.3.13) along with

boundary conditions (4.3.14) when ξ is large. The order of magnitude analysis of

various terms in these equations shows that the largest terms are f ′′′ and ξf ′′ in

Eqn. (4.3.11), φ′′′ and ξφ′′ in Eqn. (4.3.12) and θ′′, ξθ′ in Eqn. (4.3.13). In the

respective equations, both terms have to be balanced in magnitude and the only

way to do this, is to assume that η is small and hence its derivative is large. It is

essential to find the appropriate scaling for, f , φ, θ and η. On balancing f ′′′ and

ξf ′′ in Eqn. (4.3.11), φ′′′ and ξφ′′ in Eqn. (4.3.12) and θ′′ and ξθ′ in (4.3.12), it

is found that η = O(ξ−1), f = O(ξ−3) and φ = O(ξ−3). Therefore the following

transformations may be introduced

Y = ξ−1η, f(ξ, Y ) = ξ−3F (ξ, η)

φ(ξ, Y ) = ξ−3Φ(ξ, η), θ(ξ, Y ) = Θ(ξ, η) (4.3.18)

by using (4.3.18)into (4.3.11)-(4.3.13), we obtained the following set of equations

F ′′′ + F ′′ + Θ =1

4ξ−3

[F ′∂F ′

∂ξ− F ′′∂F

∂ξ− S

(Φ

∂Φ′

∂ξ− Φ′′∂Φ

∂ξ

)](4.3.19)

1

PmΦ′′′ + Φ′′ =

1

4ξ−3

[F ′∂Φ′

∂ξ− Φ′∂F ′

∂ξ+ F ′′∂Φ

∂ξ− Φ′′∂F

∂ξ

](4.3.20)

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)Θ)3Θ′]′

+ Θ′ =1

4ξ−3

[F ′∂Θ

∂ξ−Θ′∂F

∂ξ

](4.3.21)

Boundary equations to be satisfied by the above equations are

F (ξ, 0) = F ′(ξ, 0) = 0, Φ(ξ, 0) = 0, Φ′(ξ, 0) = 1, Θ(ξ, 0) = 1

F ′(ξ,∞) = 0, Φ′(ξ,∞) = 0, Θ(ξ,∞) = 0 (4.3.22)

Now, we expand the functions F , Φ, Θ in powers of ξ−3

F (ξ, η) =1∑

m=0

ξ−3mFm(η), Φ(ξ, η) =1∑

m=0

ξ−3mΦm(η), Θ(ξ, η) =1∑

m=0

ξ−3mΘm(η)

(4.3.23)

55


By substituting Eqn. (4.3.23) into Eqns. (4.3.19)-(4.3.22) and equating the coef-

ficients of equal powers of ξ from both sides we have

F ′′′0 + F ′′

0 + Θ0 = 0 (4.3.24)

Φ′′′0 + PmΦ′′

0 = 0 (4.3.25)

[1 + α (1 + ∆Θ0)

3] Θ′′0 + 3α∆ (1 + ∆Θ0)

2 Θ′20 + PrΘ′

0 = 0 (4.3.26)

and the boundary conditions are

F0(0) = F ′0(0) = 0, Φ0 = 0, Φ′

0(0) = 1, Θ0(0) = 1

F ′0(∞) = 0, Φ0(∞) = 0, Θ0(∞) = 0 (4.3.27)

from which we see that

F ′′′1 + F ′′

1 + Θ1 = 0 (4.3.28)

Φ′′′1 + PmΦ′′

1 = 0 (4.3.29)[1 + α (1 + ∆Θ0)

3] Θ′′1 + 3α∆ (1 + ∆Θ0)

2 (Θ1Θ′′0 + 2Θ′

0Θ′1)

+ 6α∆2Θ1 (1 + ∆Θ0) Θ′20 + PrΘ′

1 = 0(4.3.30)

F1(0) = F ′1(0) = 0, Φ1 = 0, Φ′

1(0) = 0, Θ1(0) = 0

F ′1(∞) = 0, Φ1(∞) = 0, Θ1(∞) = 0 (4.3.31)

The solution obtained by these equations enables us to calculate the solution of

different parameters for large values of ξ from the following expressions

Gr−3/4L x−1/4Cf = F ′′(0) (4.3.32)

Gr−3/4L x−1/4Jw = Φ′′(0) (4.3.33)

Gr1/4L x1/4Nu = −

(1 +

4

3Rd

θ3w

)Θ′(0) (4.3.34)

The results obtained by relations (4.3.32)-(4.3.34) are given in Tables 4.1-4.3 for

large values of ξ and compared with the solution that obtained by finite difference

method and found to be in good agreement.

56


Table 4.1 Numerical values of coefficient of skin friction Gr−3/4L x−1/4Cfx

obtained for θw= 0.5, 1.5,2.5 when Pm=0.1, Rd=20.0, S=0.2, Pr=0.1,against ξby two methods.

ξ θw = 0.5 θw = 1.5 θw = 2.5FDM Asymptotic FDM Asymptotic FDM Asymptotic

0.0 1.92958 1.92021† 1.93740 1.92833† 1.94480 1.97100†0.1 2.01764 1.94259† 2.02628 1.95215† 2.03428 2.24932†0.5 2.45830 2.05452† 2.47116 2.07127† 2.48307 2.51093†1.0 2.97060 2.94431† 2.99006 2.92201† 3.00807 3.08375†2.0 3.66365 - 3.70120 - 3.73592 -4.0 3.22690 - 3.31710 - 3.40079 -8.0 1.29661 1.26562†† 1.35990 1.31601†† 1.42297 1.41265††10.0 1.01153 1.01000 †† 1.06063 1.01000†† 1.10971 1.01000††

Table 4.2 Numerical values of coefficient of heat transfer Gr1/4L x1/4Nux obtained

for θw= 0.5, 1.5,2.5 when Pm=0.1, Rd=20.0, S=0.2, Pr=0.1,against ξ by twomethods.

ξ θw = 0.5 θw = 1.5 θw = 2.5FDM Asymptotic FDM Asymptotic FDM Asymptotic

0.0 0.15153 0.14548† 0.14948 0.14565† 0.14758 0.14228†0.1 0.15506 0.15374† 0.15284 0.15409† 0.15079 0.15088†0.5 0.17247 0.16505† 0.16939 0.16430† 0.16657 0.16391†1.0 0.19413 0.19419† 0.18992 0.18655† 0.18608 0.18520†2.0 0.24000 - 0.23321 - 0.22705 -4.0 0.36708 - 0.35233 - 0.33906 -8.0 0.77546 0.70824†† 0.73907 0.78187†† 0.72594 0.75631††10.0 0.99261 1.00000†† 0.98721 0.95672†† 0.98584 0.94334††

Table 4.3. Numerical values of Gr−3/4L x−1/4Jw obtained for θw= 0.5, 1.5,2.5 when

Pm=0.1, Rd=20.0, S=0.2, Pr=0.1,against ξ by two methods.ξ θw = 0.5 θw = 1.5 θw = 2.5

FDM Asymptotic FDM Asymptotic FDM Asymptotic0.0 0.01203 0.01569† 0.01119 0.01750† 0.01040 0.01206†0.1 0.01548 0.01707† 0.01461 0.01708† 0.01380 0.01076†0.5 0.03925 0.03400† 0.03827 0.03299† 0.03735 0.03194†1.0 0.08008 0.08016† 0.07903 0.07539† 0.07805 0.07827†2.0 0.17898 - 0.17791 - 0.17692 -4.0 0.39006 - 0.38919 - 0.38836 -8.0 0.79767 0.80000†† 0.79756 0.80000†† 0.79733 0.80000††10.0 0.99829 1.00000†† 0.99821 1.0000†† 0.99812 1.00000 ††

Here and after here † and †† stands for small and large values of ξ

57



Eqns. (4.1.8)-(4.1.12) along with boundary conditions (4.1.13) have been solved

numerically for all values of the transpiration parameter ξ by using finite dif-

ference method. Similarly Eqns. (4.3.10)-(4.3.12) and the boundary conditions

(4.3.13) sufficiently near to the plate and away from the plate have been solved

by using asymptotic series solutions. Later, to test the accuracy of the results

obtained by finite difference method is compared with the results obtained by as-

ymptotic series solution and found to be in excellent agreement. We shall now give

a brief discussion on the effects of Prandtl number Pr, magnetic force parameter

S, magnetic parameter Pm, radiation parameter Rd and surface temperature θw

on coefficients of skin friction Gr−3/4L x−1/4Cf , rate of heat transfer Gr

1/4L x1/4Nu

and current density Gr−3/4L x−1/4Jw in Section 4.1. The detail of velocity profile,

temperature distribution and transverse component of the magnetic field for vary-

ing the parameters Pr, Pm, S, Rd and θw for different values of transpiration

parameter ξ is given in section 4.2.

4.4.1 The effect of physical parameters on skin friction,

current density and rate of heat transfer

Figs. 4.2(a-c) illustrates the influence of different values of radiation parameter

Rd on coefficients of skin friction, rate of heat transfer and current density at

the surface. From these figures, it is shown that the coefficient of skin friction

decreases and coefficient of rate of heat transfer increases at very gross margin

but coefficient of current density increases very slightly up to ξ = 100. The

relationship of Rd = KαR

4σ T∞3 means that with the increase of parameter Rd, the

product of thermal diffusivity and roseland absorption coefficient increases and

ambient fluid temperature decreases which slow down the motion of the fluid. So

the coefficient of skin friction decreases in downstream regime very grossly and rate

of heat transfer increases due to natural convection. The variation in parameter

Rd will therefore generally have no inhibiting effect on the development of current

58


ξ

Gr L-3

/4x-1

/4C

f

0.0 2.0 4.0 6.0 8.0 10.0

2.0

3.0

4.0

5.0

6.0

1.02.55.010.0

Rd

S= 0.1, Pm= 0.1, Pr = 0.1, θw = 1.1

(a) ξ

Gr L1/

4 x1/4 N

u x

0.0 2.0 4.0 6.0 8.0 10.00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1.02.55.010.0

Rd

S = 0.1, Pm = 0.1, Pr = 0.1, θw = 1.1

(b) ξ

Gr L-3

/4x-1

/4J w

10-1 100 101

0.0

0.2

0.4

0.6

0.8

1.0

1.02.55.010.0

Rd

S= 0.1,Pm= 0.1, Pr = 0.1,θw = 1.1

(c)

Fig. 4.2 The behavior of coefficients of (a) skin friction Gr−3/4L x−1/4Cf (b) rate

of heat transfer Gr1/4L x1/4Nux and (c) current density Gr

−3/4L x−1/4Jw at the

surface against ξ for different values of radiation parameter Rd=1.0, 2.5, 5.0, 10.0when Pm=0.1, Pr=0.1, and S=0.1, θw=1.1.

ξ

Gr L-3

/4x-1

/4C

f

0.0 2.0 4.0 6.0 8.0 10.0

3.0

4.0

5.0

6.0

0.00.30.60.9

S

Pr= 0.1, Pm= 0.1, Rd = 1.0, θw = 1.1

(a) ξ

Gr L1/

4 x1/4 N

u x

10-1 100 1010.1

0.12

0.14

0.16

0.18

0.00.30.60.9

S = 0.1,Pm = 0.1, Pr = 0.1,θw = 1.1

S

ξ

Gr L-3

/4x-1

/4J w

10-1 100 101

0.0

0.2

0.4

0.6

0.8

1.0

0.00.30.60.9

S

Pr= 0.1,Pm= 0.1, Rd = 1.0,θw = 1.1

(c)

Fig. 4.3 The behavior of coefficients of (a) skin friction Gr−3/4L x−1/4Cfb) rate of

heat transfer Gr1/4L x1/4Nux and (c) current density Gr

−3/4L x−1/4Jw at the surface

against ξ for different values of magnetic force parameter S = 0.0, 0.3, 0.6, 0.9when Pm=0.1, Rd=1.0, and Pr=0.1, θw=1.1.

density. The increase in S increase the coefficient of skin friction actively in the

middle range of the surface of the plate but the coefficients rate of heat transfer

and current density increases at very low margin near the surface of the plate and

no difference shown after ξ = 1.0 in Figs. 4.3(a-c).

The reason is that with the increase of magnetic force parameter the magnetic

energy increases which extract the kinetic energy of the fluid, thus the coefficients

of skin friction, rate of heat transfer and current density increases.

Figs. 4.4(a-c) illustrates the response of the different values of the Pm on

coefficients of skin friction, rate of heat transfer and current density.

It can be seen that the coefficient of skin friction decreases slightly in the

59


ξ

Gr L-3

/4x-1

/4C

f

0.0 2.0 4.0 6.0 8.0 10.0

2.0

3.0

4.0

5.0

6.0

0.0010.010.050.1

Pm

S= 0.1, Pr = 0.1, Rd = 1.0, θw = 1.1

(a) ξ

Gr L1/

4 x1/4 N

u x

10-1 100 1010.1

0.12

0.14

0.16

0.18

0.0010.010.050.1

S = 0.1, Rd = 1.0, Pr = 0.1,θw = 1.1

Pm

(b)ξ

Gr L-3

/4x-1

/4J w

10-1 100 101

0.0

0.2

0.4

0.6

0.8

1.0

0.0010.010.050.1

Pm

S= 0.1, Pr = 0.1, Rd = 1.0,θw = 1.1

(c)

Fig. 4.4 The behavior of coefficients of (a) skin friction Gr−3/4L x−1/4Cf (b) rate



surface against ξ for different values of magnetic Prandtl number Pm=0.001,0.01, 0.05, 0.1 when Rd=1.0, Pr=0.1, and S=0.1, θw=1.1.

downstream regime and the coefficient of heat transfer remains unchanged. We

can also see that the coefficient of current density shows its maximum response

for small value of Pm=0.001 and decreases very actively and is exactly zero for

pm=0.1. This happens because with the increase of Pm the induced current with

in the boundary layer tends to spread away from the surface and this results in

thickening the momentum and magnetic field boundary layer thickness, but in the

case of heat transfer this factor is not dominant in the flow domain. With this

reason coefficients of skin friction and current density decreases but the coefficient

of rate of heat transfer remains unchanged. Finally in Figs. 4.5(a-c), we have

shown the effects of variation of Pr on the physical quantities such as coefficients

of skin friction, rate of heat transfer and current density at the surface.

We have examined that the coefficient of skin friction decreases moderately

near the surface but grossly decreases in down stream regime, similarly the rate

of heat transfer increases very slowly in upstream regime but this margin extend

in downstream regime and small increase in the current density is also seen in

upstream regime. The reason is that the increase in the value of Pr correspond

to rise in kinematic viscosity of the fluid and reduce the thermal diffusion. It

is very interesting to note that the rise in kinematic viscosity leads to thick the

momentum boundary layer thickness and reduction in thermal diffusion give a

thin thermal boundary layer thickness which is responsible for the aforementioned

60


ξ

Gr L-3

/4x-1

/4C

f

2.0 4.0 6.0 8.0 10.00.0

1.0

2.0

3.0

4.0

5.0

6.0

0.010.050.080.1

Pr

S= 0.1, Pm= 0.1, Rd = 10.0, θw = 1.1

(a) ξ

Gr L1/

4 x1/4 N

u x

2.0 4.0 6.0 8.0 10.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.010.050.080.1

Pr

S = 0.1, Pm = 0.1, Rd = 10.0, θw = 1.1

(b)ξ

Gr L-3

/4x-1

/4J w

10-1 100 101

0.0

0.2

0.4

0.6

0.8

1.0

0.010.050.080.1

Pr

S= 0.1,Pm= 0.1, Rd = 10.0,θw = 1.1

(c)

Fig. 4.5 The behavior of coefficient of (a) skin friction Gr−3/4L x−1/4Cf (b) rate



surface against ξ for different values of Prandtl number Pr=0.01, 0.05, 0.08, 0.1when Rd=10.0, Pm=0.1, and S=0.1, θw=1.1.

phenomena.

In Tables 4.1-4.3 the comparison of the numerical solutions obtained by finite

difference method with asymptotic series method in upstream and downstream

regime ξ is given to test the numerical accuracy of the results obtained by both

methods. It is found that as θw increases the coefficients of skin friction, and rate of

heat transfer increases in downstream regime but small changes in numerical values

of coefficient of current density are seen throughout the entire regime. From this

physical phenomena, we can observe that with the increase of surface temperature

the ambient fluid temperature in the domain of fluid flow decreases and by the

Fourier law of heat transfer, the flow direction of heat transfer is towards down

stream regime where we can see the changes in coefficients of skin friction and

rate of heat transfer with good margin and the coefficient of current density is

dominated. From these tables we can also claim that the results obtained by both

methods are in excellent agreement.

4.4.2 The effects of physical parameters on velocity, tem-

perature and transverse component of magnetic field

The velocity, temperature and transverse component of magnetic field distribu-

tions obtained by the finite difference method for various values of transpiration

parameter ξ are displayed in Figs. 4.6-4.10. The aim of these figures is to display

61


how the profile vary in ξ. The transpiration parameter ξ in present investigation

is taken as positive for suction. It is shown that the values of velocity, temper-

ature and transverse component of magnetic field decreases in magnitude as ξ

increases in Figs. 4.6-4.10. This phenomena establish the very strong reason that

the suction slow down the motion of the fluid in the down stream regime and

the values of the aforementioned physical quantities decreases. Thus the numer-

ical results in Fig. 4.6(a-c) indicates that the momentum, thermal and magnetic

field boundary layers thicknesses decreases as ξ=1.0,3.0,5.0,8.0,10.0 increases for

two different values of radiation parameter Rd=1.0,10.0 and for constant values of

Pr=0.1, Pm=0.1, S=0.1 and θw=1.1.

Y0.0 2.0 4.0 6.0 8.0

0.0

0.5

1.0

1.5

2.0 Rd

10.01.0

ξ1.03.05.08.010.0

(a)

S = 0.1, Pr = 0.1, Pm = 0.1, θw = 1.1

f/ (0

)

Y

θ’(0

)

0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0 Rd

10.0

1.0

ξ1.03.05.08.010.0

(b)

S = 0.1, Pr = 0.1, Pm = 0.1, θw = 1.1

Y0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0 Rd

10.0

1.0

ξ1.03.05.08.010.0

(c)

S = 0.1, Pr = 0.1, Pm = 0.1, θw = 1.1

φ 1/ (0)

Fig. 4.6 (a) velocity profile (b) temperature distribution and (c) transversecomponent of magnetic field for various values of ξ = 1.0, 3.0, 5.0, 8.0, 10.0

against Y for different values of radiation parameter Rd=1.0, 10.0 when Pr=0.1,S=0.1, Pr=0.1, θw=1.1.

It can be seen that the increase of Rd in the fluid, the values of velocity,

temperature and transverse component of magnetic field decreases, which leads to

a decrease in momentum, thermal and magnetic field boundary layer thicknesses.

The variation in the parameter S=0.0,0.4 for the case of suction increase the

values momentum and magnetic field profile and no change is seen in temperature

distribution, which is expected because the direction of the magnetic field is in

favor of the flow which can be seen in Figs 4.7(a-c).

In Figs. 4.8(a-c), we can see the effect of Pr=0.01,0.1 by keeping other pa-

rameters fixed, it is noted that the values of the velocity decreases slightly but

temperature distribution decreases and separated into regions and no change is

62


Y0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.5

1.0

1.5

2.0S

0.40.0

ξ1.03.05.08.010.0

(a)

Rd = 1.0, Pr = 0.1, Pm = 0.1, θw = 1.1

f/ (0

)

Y0.0 2.0 4.0 6.0 8.0 10.00.0

0.2

0.4

0.6

0.8

1.0S

0.40.0

ξ1.03.05.08.010.0

(b)

Rd = 1.0, Pr = 0.1, Pm = 0.1, θw = 1.1

θ(0)

Y0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0 S0.40.0

ξ1.03.05.08.010.0

(c)

Rd = 1.0, Pr = 0.1, Pm = 0.1, θw = 1.1

φ 1/ (0)

Fig. 4.7 (a) velocity profile (b) temperature distribution and (c) transversecomponent of magnetic field for various values of ξ = 1.0, 3.0, 5.0, 8.0, 10.0against Y for different values of magnetic force parameter S=0.0, 0.4 when

Pr=0.1, Rd=1.0, Pr=0.1, θw=1.1.

Y0.0 2.0 4.0 6.0 8.0

0.0

0.5

1.0

1.5

2.0 Pr0.10.01

ξ1.03.05.08.010.0

(a)

Rd = 1.0, S = 0.1, Pm = 0.1, θw = 1.1

f/ (0

)

Y2.0 4.0 6.0 8.0 10.00.0

0.2

0.4

0.6

0.8

1.0 Pr0.010.1

ξ1.03.05.08.010.0

(b)

Rd = 1.0, S = 0.1, Pm = 0.1, θw = 1.1

θ(0)

Y2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0 Pr0.01

0.1

ξ1.03.05.08.010.0

(c)

Rd = 1.0, S = 0.1, Pm = 0.1, θw = 1.1

φ 1/ (0)

Fig. 4.8 (a) velocity profile (b) temperature distribution and (c) transversecomponent of magnetic field for various values of ξ = 1.0, 3.0, 5.0, 8.0, 10.0against Y for different values of Prandtl number Pr=0.01, 0.1 when Rd=1.0,

S=0.1, Pr=0.1, Pm=0.1, θw=1.1.

seen in transverse component of magnetic field, which is understandable because

the role of Pr in magnetic field equation is not so prominent.

The effects of varying the Pm=0.01,0.1 for Pr=0.1, S=0.1, Rd=1.0 and θw=1.1

on the velocity, temperature and transverse component of magnetic field are shown

in Figs. 4.9(a-c).

It is clear from these figures that with the increase of magnetic Prandtl number

Pm and the suction is also present there, the value of the velocity profile increases

slightly no change seen in temperature distribution and the transverse component

of magnetic field decreases drastically and separated into two part in the flow

domain. Whereas from Figs. 4.10(a-c), we can see that with the increase of

63


Y0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.5

1.0

1.5

2.0 Pm0.10.01

ξ1.03.05.08.010.0

(a)

Rd = 1.0, S = 0.1, Pr = 0.1, θw = 1.1

f/ (0

)

Y0.0 2.0 4.0 6.0 8.0 10.00.0

0.2

0.4

0.6

0.8

1.0 Pm0.010.1

ξ1.03.05.08.010.0

(b)

Rd = 1.0, S = 0.1, Pr = 0.1, θw = 1.1

θ(0)

Y2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0 Pm0.01

0.1

ξ1.03.05.08.010.0

(c)

Rd = 1.0, S = 0.1, Pr = 0.1, θw = 1.1

φ 1/ (0)


against Y for different values of magnetic Prandtl number Pm=0.01, 0.1 whenS=0.1, Rd=1.0, Pr=0.1, θw=1.1.

Y0.0 2.0 4.0 6.0 8.0

0.0

0.5

1.0

1.5

2.0 θω

1.10.5

ξ1.03.05.08.010.0

(a)

Rd = 1.0, S = 0.1, Pr = 0.1, Pm = 0.1

f/ (0

)

Y

θ’(0

)

0.0 2.0 4.0 6.0 8.0 10.00.0

0.2

0.4

0.6

0.8

1.0 θω

0.51.1

ξ1.03.05.08.010.0

(b)

Rd = 1.0, S = 0.1, Pr = 0.1, Pm = 0.1

Y0.0 2.0 4.0 6.0 8.0 10.0

0.0

0.2

0.4

0.6

0.8

1.0 θω0.5

1.1

ξ1.03.05.08.010.0

(c)

Rd = 1.0, S = 0.1, Pr = 0.1, Pm = 0.1

φ 1/ (0)


against Y for different values of surface temperature θw=0.5, 1.1 when S=0.1,Rd=1.0, Pr=0.1, Pm=0.1.

surface temperature the values of velocity and temperature distribution increases

thus momentum and thermal boundary layer thicknesses increases. The transverse

component of magnetic field increases for ξ = 1.0 and change is seen for other

values of transpiration parameter ξ.

4.5 Conclusion

In summing up what has been discussed above, we are remarking the effects of

different physical parameters such as radiation parameter, magnetic force parame-

ter, Prandtl number, magnetic Prandtl number and the surface temperature on

64


coefficients of skin friction Gr−3/4L x−1/4Cf , rate of heat transfer Gr

1/4L x1/4Nu and

current density Gr−3/4L x−1/4Jw which are discussed as below.

The coefficient of skin friction decreases and the coefficient of rate of heat

transfer increases in the downstream regime but the coefficient of current density

increases slightly upto the value of ξ=1.0 with the increase of Rd. The increase

in magnetic force parameter increase the coefficient of skin friction actively in the

middle range of the flow domain but the coefficient of the rate of heat transfer

and current density increases at very low difference near the surface of the plate.

With varying the magnetic Pradtl number, the coefficient of skin friction decreases

slightly in the downstream regime and the coefficient of rate of heat transfer re-

mains unchanged but the coefficient of current density decreases very actively. It

is observed that the coefficient of skin friction decreases but the coefficient of rate

of heat transfer and current density increases with the increase of Pr. It is also

observed that the coefficient of skin friction and rate of heat transfer increases and

small changes in numerical values of coefficient of current density are noticed with

the increase of surface temperature. The velocity, temperature and transverse

component of magnetic field distributions decreases in magnitude as ξ increases.

The numerical results indicate that the momentum, thermal and magnetic field

boundary layer thicknesses decrease for the case of suction for two different values

of Rd. The variation in the magnetic force parameter for the case of suction leads

to decrease the value of momentum and magnetic field profile but no change is

seen in temperature distribution. It is noted that the values of velocity profile

decreases slightly but the temperature distribution decreases and separated into

two regions in the flow domain and there is no change seen in transverse compo-

nent of magnetic field with the change of Pr. It is also observed that for varying

the magnetic Prandtl number, the values of velocity profile increases slightly and

no change is seen in temperature distribution but transverse component of mag-

netic field decreases and separated into two regions in flow domain. It is also

concluded that with the increase of surface temperature the values of velocity and

temperature distribution increase, thus momentum and thermal boundary layer

thicknesses increase. The transverse component of magnetic field increases very

65


slowly only for ξ = 1.0.

66

67

Chapter 5

Radiative fluctuating magnetohydrodynamic mixed convection flow past a magnetized vertical heated plate

Chapter 5: Radiative fluctuating magnetohydrodynamic mixed ...

In this chapter, we will further extend the problem given in Chapter 3 and mod-

ify the mathematical model in unsteady form to analyze the effect of conduction

radiation on the fluctuating hydromagnetic mixed convection flow past a magne-

tized vertical plate, when the magnetic field, surface temperature and free stream

velocity oscillates in magnitudes simultaneously. The effects of variation in the

mixed convection parameter λ, the conduction-radiation parameter Rd, Prandtl

number Pr, the magnetic Prandtl number Pm, the magnetic force parameter S

and the surface temperature θw, have been discussed in terms of amplitudes and

phases of shear stress, rate of heat transfer and current density at the surface. The

effects of these parameters on the the transient shear stress, rate of heat transfer

and current density have also been discussed in detail. For the purpose of the

numerical solutions we have applied two methods(i) finite difference method for

primitive variable formulation and (ii) the asymptotic series solution for stream

function formulation. The results obtained by both the methods are found to be

in good agreement.

5.1 Basic equations and the flow model

The physical system investigated here is shown in Fig. 5.1. As we have done

in the earlier chapter here we also considere a two dimensional fluctuating mixed

convection flow of an electrically conducting, viscous, incompressible fluid with the

effect of radiation past a magnetized vertical heated plate is investigated. Here, we

assume that the magnetic field, surface temperature and the free stream velocity

oscillates in magnitude about a constant non-zero mean.

In this physical system we have taken x-axis along the surface and y-axis is

normal to it. Here in Fig. 5.1, δM , δT and δH represent the momentum, thermal

and magnetic field boundary layer thicknesses. The dimensionless equations for

the unsteady boundary layer flow may be expressed as follows :

∂u

∂x+

∂v

∂y= 0 (5.1.1)

∂u

∂τ+ u

∂u

∂x+ v

∂v

∂y=

dU

dτ+

∂2u

∂y2+ S

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)+ λθ (5.1.2)

68


u=

0,T

=T

w(x

),B

x=

Bw(x

),B

y=0

T=

T∞,u

=U

(t),

Bx=

By

=0

δΗ

δM

Oy

x

U(t)

Hx

Hy

u

v

δΤ

Fig.5.1 The coordinate system and flow configuration

∂Bx

∂x+

∂By

∂y= 0 (5.1.3)

∂Bx

∂τ+ u

∂Bx

∂x+ v

∂Bx

∂y− Bx

∂u

∂x− By

∂u

∂y=

1

Pm

∂2Bx

∂y2(5.1.4)

∂θ

∂τ+ u

∂θ

∂x+ v

∂θ

∂y=

1

Pr

[∂2θ

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θ)3 ∂θ

∂y}]

(5.1.5)

The dimensionless boundary conditions for the problem now become:

u(x, 0) = 0, v(x, 0) = 0, Bx(x, 0) = B0(τ), By(x, 0) = 0, θ(x, 0) = θ0(τ)

u(x,∞) = U(τ), Bx(x,∞) = 0, θ(x,∞) = 0(5.1.6)

Once we know the functions u, Bx and θ and their derivatives we can obtain the

values of dimensionless shear stress, τs, current density Jm and the rate of heat

transfer qt from the following relations:

τs =

(τwL

µ0Re12L

)=

(∂u

∂y

)

y=0

, Jm =

(JwL

B0Re12L

)=

(∂Bx

∂y

)

y=0

qt =

(1 +

4

3Rd

θ3w

)qw =

(∂θ

∂y

)

y=0

(5.1.7)

Here U(τ), is the component of velocity outside the boundary layer which is given

as below:

U(τ) = 1 + εeiτ

69


In above expression, it is assumed that ε is small amplitude oscillation of free

stream velocity, surface magnetic intensity and surface temperature.

The solution of the above system of Eqns. (5.1.1)-(5.1.6) will be obtained

in terms of complex functions, only the real parts of which will have physical

significance. We write u, v, Bx, By and θ as the sum of steady and oscillating

components as proposed by Chawala [38]

u =[us(x, y) + εeiτut(x, y)

]

v =[vs(x, y) + εeiτvt(x, y)

]

Bx =[Bxs(x, y) + εeiτBxt(x, y)

]

By =[Bys(x, y) + εeiτByt(x, y)

]

θ =[θs(x, y) + εeiτθt(x, y)

]

(5.1.8)

where, u0(x, y), vs(x, y), Bxs(x, y), Bys(x, y), θs(x, y) and ut(x, y), vt(x, y), Bxt(x, y),

Byt(x, y), θ1(x, y) are, respectively, the steady and fluctuating parts of the flow

variables.

By substituting Eqn. (5.1.8) into Eqns. (5.1.1)-(5.1.5) with boundary condi-

tions given in Eqn. (5.1.6), we have the steady and unsteady system of equations

given as below∂us

∂x+

∂vs

∂y= 0 (5.1.9)

us∂us

∂x+ vs

∂vs

∂y=

∂2us

∂y2+ S

(Bxs

∂Bxs

∂x+ Bys

∂Bxs

∂y

)+ λθs (5.1.10)

∂Bxs

∂x+

∂Bys

∂y= 0 (5.1.11)

us∂Bxs

∂x+ vs

∂Bxs

∂y−Bxs

∂us

∂x−Bys

∂us

∂y=

1

Pm

∂2Bxs

∂y2(5.1.12)

us∂θs

∂x+ vs

∂θs

∂y=

1

Pr

[∂2θs

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θs)

3∂θs

∂y}]

(5.1.13)

The corresponding boundary conditions are:

us(x, 0) = 0, vs(x, 0) = 0, Bxs(x, 0) = 1, Bys(x, 0) = 0, θs(x, 0) = 1

u0(x,∞) −→ 1, Bxs(x,∞) −→ 0, θs(x,∞) −→ 0(5.1.14)

70


We obtain the following coupled equations for the solution of fluctuating flow

by collecting terms of the first power of ε.

∂ut

∂x+

∂vt

∂y= 0 (5.1.15)

iut + us∂ut

∂x+ ut

∂us

∂x+ vs

∂ut

∂y+ vt

∂us

∂y= i +

∂2us

∂y2

+ S

(Bxs

∂Bxt

∂x+ Bxt

∂Bxs

∂x+ Bys

∂Bxt

∂y+ Byt

∂Bxs

∂y

)+ λθt

(5.1.16)

∂Bxt

∂x+

∂Byt

∂y= 0 (5.1.17)

iBxt + us∂Bxt

∂x+ ut

∂Bxs

∂y+ vs

∂Bxt

∂y+ vt

∂Bxs

∂y

−Bxs

∂ut

∂x−Bxt

∂us

∂x−Bys

∂ut

∂x−Byt

∂us

∂y=

1

Pm

∂2Bxt

∂y2

(5.1.18)

iθt + us∂θt

∂x+ ut

∂θs

∂x+ vs

∂θt

∂y+ vt

∂θs

∂y=

1

Pr

[∂2θt

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θs)

3∂θt

∂y+ 3(1 + (θw − 1)θs)

2∆θ1∂θs

∂y}]

(5.1.19)

the corresponding boundary conditions are:

ut(x, 0) = 0, vt(x, 0) = 0, Hxt(x, 0) = 1, Hyt(x, 0) = 0, θt(x, 0) = 1

ut(x,∞) −→ 1, Bxt(x,∞) −→ 0, θt(x,∞) −→ 0(5.1.20)

By using expressions (5.1.15)-(5.1.19) with boundary conditions (5.1.20), we

can find the solution of fluctuation flow for momentum, magnetic filed and energy

equations.


We now obtain the numerical solution of Eqns. (5.1.9)-(5.1.19) by using straight

forward finite difference method for all values of ξ and asymptotic series solution

of these equations for small and large values of ξ. To deal with this, we will use

two techniques (i) Primitive variable transformation and (ii) the stream function

formulation, which are given below.

71



In order to find the solution of the steady part of the problem posed in Eqns.

(5.1.9) to (5.1.14), we first give attention to reduce the equations to the convenient

form by the use of the primitive variable formulation for the steady part of the

flow field which is given as:

us = Us(ξ, Y ), vs = ξ−12 Vs(ξ, Y ), θs = θs(ξ, Y )

Bxs = Bxs(ξ, Y ), Bys = ξ−12 Bys(ξ, Y ), Y = ξ−

12 y, ξ = x (5.2.1)

Now following equations are derived by applying the transformations given in Eqn.

(5.2.1) into Eqns. (5.1.9) to (5.1.14):

ξ∂Us

∂ξ− Y

2

∂Us

∂Y+

∂Vs

∂Y= 0 (5.2.2)

ξUs∂Us

∂ξ+

(Vs − 1

2Y Us

)∂Us

∂Y=

∂2Us

∂Y 2

+ S

[(Bys −

Y

2Bxs

)∂Bxs

∂Y+ ξBxs

∂Bxs

∂ξ

]+ λθs

(5.2.3)

ξ∂Bxs

∂ξ− Y

2

∂Bxs

∂Y+

∂Bys

∂Y= 0 (5.2.4)

ξUs∂Bxs

∂ξ+

(Vs − 1

2Y Us

)∂Bxs

∂Y

−(

Vs − 1

2Bxs

)∂Us

∂Y− ξBxs

∂Us

∂ξ=

1

Pm

∂2Bxs

∂Y 2

(5.2.5)

ξUs∂θs

∂ξ+

(Vs − 1

2Y Us

)∂θs

∂Y=

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θs)3

]∂2θs

∂Y 2

+4

Pr∆

1

Rd

(1 + ∆θs)2(

∂θs

∂Y)2

(5.2.6)

The boundary conditions to be satisfied by the above system of equations are

Us(ξ, 0) = Vs(ξ, 0) = 0, Bys(ξ, 0) = 0, Bxs(ξ, 0) = 1, θs(ξ, 0) = 1

Us(ξ,∞) = 1, Bxs(ξ,∞) = 0, θs(ξ,∞) = 0(5.2.7)

and the primitive variable formulations for the fluctuating part of the flow is

given as follows.

ut = Ut(ξ, Y ), vt = ξ−12 Vt(ξ, Y ), θt = θ1(ξ, Y )

72


Bxt = Bxt(ξ, Y ), Byt = ξ−12 Byt(x, Y ), Y = ξ−

12 y, ξ = x (5.2.8)

Now using the formulations given in Eqn. (5.2.8) into into the Eqns. (5.1.16)-

(5.1.20), we have the following system of reduced equations.

ξ∂Ut

∂ξ− Y

2

∂Ut

∂Y+

∂Vt

∂Y= 0 (5.2.9)

iξUt + ξUs∂Ut

∂ξ+

(Vs − Y

2Us

)∂Ut

∂Y

+

(Vt − Y

2Ut

)∂Us

∂Y= iξ +

∂2Ut

∂Y 2

+ S[xBxt

∂Bxt

∂ξ+

(Bys −

Y

2Bxs

)∂Bxt

∂Y+

(Byt −

Y

2Bxt)

∂Bxs

∂Y

]+ λθt

(5.2.10)

ξ∂Bxt

∂ξ− Y

2

∂Bxt

∂Y+

∂Byt

∂Y= 0 (5.2.11)

iξBxt + ξUs∂Bxt

∂ξ+

(Vs − Y

2Us

)∂Bxt

∂Y+

(Vt − Y

2Ut

)∂Bxs

∂Y

− ξBxs

∂Ut

∂ξ+

(Bys −

Y

2Bxs

)∂Ut

∂Y

+

(Byt −

Y

2Bxt

)∂Us

∂Y=

1

Pm

∂2Bxt

∂Y 2

(5.2.12)

iξθt + ξUs∂θt

∂ξ+

(Vs − Y

2Us

)∂θt

∂Y

+

(Vt − Y

2Ut

)∂θs

∂Y=

1

Pr

[1 +

4

3Rd

(1 + ∆θt)3

]∂2θt

∂Y 2

+4∆

PrRd

[2∆(1 + ∆θs)(

∂θs

∂Y)2 + (1 + ∆θs)

2(∂2θs

∂Y 2)

]θt

(5.2.13)

The corresponding boundary conditions are as follows:

Ut(ξ, 0) = Vt(ξ, 0) = 0, Byt(ξ, 0) = 0 Bxt(ξ, 0) = 1, θt(ξ, 0) = 1

Ut(ξ,∞) = 1, Bxt(ξ,∞) = 0, θt(ξ,∞) = 0 (5.2.14)

Once we obtained the functions Us(ξ, Y ), Vs(ξ, Y ), Bxs(ξ, Y ), Bys(ξ, Y ), θs(ξ, Y ),

now we can obtain the values of the flow variables of the fluctuating part of the

problem given in Eqns. (5.2.9)-(5.2.13) by finite difference method. Here, we rep-

resent the available solutions in terms of amplitudes and phases of shear stress,

73


rate of heat transfer and current density, we can write it in the following relations

As =√

U2i + U2

r , At =√

θ2i + θ2

r , Am =√

H2i + H2

r

φs = arctan

(Ui

Ur

), φt = arctan

(θi

θr

), φm = arctan

(Hi

Hr

)

where (Ur,Ui), (θr,θi) and (Hr,Hi) are corresponding real and imaginary parts of

the shear stress, rate of heat transfer and current density at the surface.

5.2.2 Asymptotic solutions for small and large parameter

ξ

As we have done in the earlier case, here we also proceed to find the asymptotic

solutions for small and large values of the parameter ξ of the steady and fluctuating

parts of the present problem governed by the Eqns. (5.1.1) - (5.1.6). To do this

we introduce the the following similarity transformation as given below:

u = F ′1(η, ξ), v = x−1/2

(1

2F1(η, ξ) + ξ

∂F1

∂ξ

)

Bx = x−1/2

(1

2G1(η, ξ) + ξ

∂G1

∂ξ

), By = G′

1(η, ξ) (5.2.15)

θ = θ1(ξ, η), η = x−1/2y, ξ = x

Now substituting Eqn. (5.2.15) into Eqns. (5.1.1) - (5.1.6) we have the following

non-similarity equations for the fluctuating part of the problem:

F ′′′1 +

1

2(F0F

′′1 + F ′′

0 F1 − S(G0G′′1 + G′′

0G1)) + λθ1

= ξ

[iF ′

1 + F ′0

∂F ′1

∂ξ− F ′′

0

∂F1

∂ξ− S

(G′

0

∂G′1

∂ξ−G′′

0

∂G1

∂ξ

)− i

] (5.2.16)

1

PmG′′′

1 +1

2(F0G

′′1 + F1G

′′0 − F ′′

0 G1 −G0F′′1 )

= ξ

[iG′

1 + F ′0

∂G′1

∂ξ−G′

0

∂F ′1

∂ξ+ F ′′

0

∂G1

∂ξ−G′′

0

∂F1

∂ξ

] (5.2.17)

1

Pr

[1 +

4

3Rd

(1 + ∆θ1)3 θ′

]′+

1

2(F0θ

′1 + F1θ

′0)

= ξ

[iθ1 + F ′

0

∂θ1

∂ξ− θ′0

∂F1

∂ξ

] (5.2.18)

74


where primes denotes differentiation with respect to η and the boundary conditions

are

F1(0) = F ′1(0) = 0, G1(0) = 0, G′

1(0) = 1, θ1(0) = 1

F ′1(∞) = 1, G′

1(∞) = 0, θ1(∞) = 0 (5.2.19)

Here F0(η), G0(η) and θ0(η) in Eqns. (5.2.16) - (5.2.18) are the functions obtained

from the following similarity equation that represent the steady mean parts of the

problem:

F ′′′0 +

1

2(F0F

′′0 − SG0G

′′0) + λθ0 = 0 (5.2.20)

1

PmG′′′

0 +1

2(F0G

′′0 − F ′′

0 G0) = 0 (5.2.21)

1

Pr

[1 +

4

3Rd

(1 + ∆θ0)3 θ′

]′+

1

2F0θ

′0 = 0 (5.2.22)

F0(0) = F ′0(0) = 0, G0(0) = 0, G′

0(0) = 1, θ0(0) = 1

F ′0(∞) = 1, G′

0(∞) = 0, θ0(∞) = 0 (5.2.23)

It needs to be mentioned that Eqns. (5.2.20)-(5.2.22) are obtained by the use of

the following transformations:

u0 = F ′0(η), v0 =

1

2x−1/2F0(η)

Bx0 =1

2x−1/2G0(η), By0 = G′

0(η)

θ0 = θ0(η), η = x−1/2y (5.2.24)

5.2.2.1. When parameter ξ is small

The functions F1, G1 and θ1 are expanded in power series in ξ, that is, we take

F1(ξ, η) =n∑

i=0

(iξ)nfn(ξ, η), G1(ξ, η) =n∑

i=0

(iξ)nφn(ξ, η),

θ1(ξ, η) =n∑

i=0

(iξ)nϕn(ξ, η)

(5.2.25)

Using Eqns. (5.2.25), into above Eqns. (5.2.16)-(5.2.18), with boundary conditions

given in Eqn. (5.2.19) and equating like powers of iξ, we have

O(iξ)0:

f ′′′0 +1

2(F0f

′′0 + F ′′

0 f0 − S(G0φ′′0 + G′′

0φ0)) + λϕ0 = 0 (5.2.26)

75


1

Pmφ′′′0 +

1

2(F0φ

′′0 + G′′

0f0 − F ′′0 φ0 −G0f

′′0 ) = 0 (5.2.27)

[1 + α (1 + ∆ϕ0)

3] ϕ′′0 + 3α∆ (1 + ∆ϕ0)2 ϕ′20

+Pr

2(F0ϕ

′0 + θ′0f0) = 0

(5.2.28)

f0(0) = f ′0(0) = 0, φ0(0) = 0, φ′0(0) = 1, ϕ0(0) = 1

f ′0(∞) = 1, φ′0(∞) = 0, ϕ0(∞) = 0(5.2.29)

O(iξ)1:

f ′′′1 +1

2(F0f

′′1 − SG0φ

′′1) +

3

2(F ′′

0 f1 − SG′′0φ1) + λϕ1

= f ′0 − 1 + (f ′0f′1 − SG′

0φ′1)

(5.2.30)

1

Pmφ′′′1 +

1

2(F0φ

′′1 −G0f

′′1 ) +

3

2(G′′

0f1 − F ′′0 φ1) = F ′

0φ′1 −G′

0f′1 (5.2.31)

[1 + α (1 + ∆ϕ0)

3] ϕ′′1 + 3α∆ (1 + ∆ϕ0)2 (ϕ1ϕ

′0 + 2ϕ0ϕ

′1)

+ 6α∆2ϕ1 (1 + ∆ϕ0) ϕ′20 +Pr

2F0ϕ

′1 +

3Pr

2θ′0f1

= Prϕ0 + PrF ′0ϕ1

(5.2.32)

f1(0) = f ′1(0) = 0, φ1(0) = 0, φ′1(0) = 0, ϕ0(0) = 0

f ′1(∞) = 0, φ′1(∞) = 0, ϕ1(∞) = 0(5.2.33)

O(iξ)2:

f ′′′2 +1

2(F0f

′′2 − SG0φ

′′2) +

3

2(F ′′

0 f2 − SG′′0φ2) + λϕ2

= f ′1 + 2 (F ′0f′2 − SG′

0φ′2)

(5.2.34)

1

Pmφ′′′2 +

1

2(F0φ

′′2 −G0f

′′2 ) +

3

2(G′′

0f2 − F ′′0 φ2)

= φ′1 + 2 (F ′0φ′2 −G′

0f′2)

(5.2.35)

[1 + α (1 + ∆ϕ0)

3] ϕ′′2 + 3α∆ϕ′′1ϕ1 (1 + ∆ϕ0)2

+ 3α∆ϕ′′0[ϕ′′2 + ∆

(2ϕ2ϕ0 + ϕ2

1

)+ ∆2ϕ0(ϕ2ϕ0 + ϕ2

1)]

+ 3α∆[(2ϕ′2ϕ

′0 + ϕ2

1

)(1 + ∆ϕ0)

2

+ 4∆ϕ′1ϕ1ϕ0 (1 + ∆ϕ0) + ∆ϕ20

(2ϕ2 + ∆(2ϕ2ϕ0 + ϕ2

1))]

+Pr

2F0ϕ

′2 +

3Pr

2θ′0f2 = Prϕ1 + 2PrF ′

0ϕ2

(5.2.36)

76


f2(0) = f ′2(0) = 0, φ2(0) = 0, φ′2(0) = 0, ϕ2(0) = 0

f ′2(∞) = 0, φ′2(∞) = 0, ϕ2(∞) = 0(5.2.37)

The Eqns. (5.2.26)-(5.2.37) are solved using Runge-Kutta-Butcher initial value

solver together with iteration scheme of Nachtsheim and Swigert [76]. The solution

of these equations enable us for calculating different physical quantities near the

leading edge, such as shear stress, rate of heat transfer and current density at the

surface with the help of following relations.

F ′′1 (0) = (f ′′0 − ξ2f ′′2 + · · · ) + i(f ′′1 − ξ3f ′′3 + · · · ) = F ′′

1r(0) + iF ′′1i(0)

G′′1(0) = (ϕ′′0 − ξ2ϕ′′2 + · · · ) + i(ϕ′′1 − ξ3ϕ′′3 + · · · ) = G′′

1r(0) + iG′′1i(0)

θ′1(0) = (φ′0 − ξ2φ′2 + · · · ) + i(φ′1 − ξ3φ′3 + · · · ) = θ′1r(0) + iθ′1i(0) (5.2.38)

Here, (F ′′1r, F

′′1i), (G′′

1r, G′′1i) and (θ′1r, θ1i) are the real and imaginary parts of the

shear stress, current density and rate of heat transfer. With the help of real and

imaginary parts of shear stress, current density and rate of heat transfer we can

calculate the amplitude and phase of shear stress, current density and rate of heat

transfer as follows.

As =√

F 2′′1i + F 2′′

1r , At =√

θ21i′ + θ2

1′r, Am =√

G21i′′ + G2

1r′′

φs = arctan

(F ′′

1i

F ′′1r

), φt = arctan

(θ′1i

θ′1r

), φm = arctan

(G′′

1i

G′′1r

) (5.2.39)

It is noted that As, At and Am are the amplitudes of the coefficients of skin

friction, rate of heat transfer and current density, and φs, φt and φm are the phase

angles of aforementioned physical quantities.

5.2.2.2. When parameter ξ is large

The solution for large ξ can be find by introducing the parameter α = ξ−12 in

Eqns. (5.2.16)-(5.2.18), we have the following system of equations:

α2F ′′′1 + i(1− F ′

1) +α2

2(F0F

′′1 + F ′′

0 F1 − S(G0G′′1 + G′′

0G1)) + λθ1

+1

2α3

[F ′

0

∂F ′1

∂α− F ′′

0

∂F1

∂α− S

(G′

0

∂G′1

∂α−G′′

0

∂G1

∂α

)]= 0

(5.2.40)

77


α2

PmG′′′

1 − iG′1 +

α2

2(F0G

′′1 + F1G

′′0 − F ′′

0 G1 −G0F′′1 )

+α3

2

[F ′

0

∂G′1

∂α−G′

0

∂F ′1

∂α+ F ′′

0

∂G1

∂α−G′′

0

∂F1

∂α

]= 0

(5.2.41)

α2

Pr

[1 + α (1 + ∆θ1)

3 θ′]′ − iθ1 +

α2

2(F0θ

′1 + F1θ

′0)

+1

2α3

[F ′

0

∂θ1

∂α− θ′0

∂F1

∂α

]= 0

(5.2.42)

and the boundary conditions are given as:

F1(0) = F ′1(0) = 0, G1(0) = 0, G′

1(0) = 1, θ1(0) = 1

F ′1(∞) = 1, G′

1(∞) = 0, θ1(∞) = 0(5.2.43)

a. Outer solutions

In view of the parameter α as introduced above in Eqns. (5.2.40)-(5.2.42), we

suggest an expansion of the form:

F1(α, η) ∼∑n=0

αnFn(η), G1(α, η) ∼∑n=0

αnGn(η),

θ1(α, η) ∼∑n=0

αnθn(η)(5.2.44)

for ξ−12 → 0, and when η = O(1) substituting Eqn. (5.2.44) into Eqns. (5.2.40)-

(5.2.43), and equating like powers of ξ, we have the system of equations for outer

solution of the form:

F o1 (α, η) = η + C0 + αC1 +

α2

2i

[(ηF ′

0 + F ′0C0 − F0 − SG′

0D0) +λ

4s2(ηθ0 + C0θ0)

]

+ α3C2 + .......

Go1(α, η) = D0 + αD1 +

α2

2i(ηG′

0 + G′0c0 −G0 −D0F

′0) + α3D2 + ............

θo1(α, η) = α2(

1

i

[1 + α

(1 + ∆θ0

)3]θ′′0

+ 3α∆(1 + ∆θ0

)2θ′20 +

1

2iθ′0(η + C0)) + ............

(5.2.45)

The unknown constants C0, C1, C2 and D0, D1, D2 will be determined by matching

procedure to be discussed later.

b. Inner solutions

78


To find the inner solution, we will define the independent and dependent vari-

ables:

α = ξ−12 , η = ξ

12 η F1(ξ, η) = ξ−

12 f(α, η),

G1(ξ, η) = ξ−12 φ(α, η) θ1(ξ, η) = θ(α, η)

(5.2.46)

By substituting Eqn. (5.2.46) in Eqns. (5.2.40)-(5.2.43), we get the solution of

the steady part

F0(η) =1

2c0η

2

G0(η) = η +1

2d0η

2

θ0(η) = 1 + e0η

(5.2.47)

We have the equation of the following form

f ′′′ − i(f ′ − 1) + α2λθ − α3

4

(c0η

2f ′′ − 2c0ηf ′ + S(2d0ηφ′ − d0η2φ′′ + 2φ′α

)

+α4

2(c0ηf ′α − c0fα − Sd0ηφ′α + Sd0φα) + O(α5) = 0

(5.2.48)

1

Pmφ′′′ − iφ′ − α3

4

(c0η

2φ′′ + 2c0ηφ′ − d0η2f ′′ − 2d0ηf ′ + 2f ′α

)

α4

2(c0ηφ′α − d0ηf ′α + c0φα − d0fα) + O(α5) = 0

(5.2.49)

1

Pr

[1 + α (1 + ∆θ)3] θ′′ − iθ +

α2

2e0f − α3

4

(c0η

2θ′ − 2e0ηf ′)

+α4

2(c0ηθα − e0fα) + O(α5) = 0

(5.2.50)

By equating like power of α, we can obtain the set of of differential equations,

the solution of that obtain differential equations give the inner solution in the

following form:

f(α, η) = η +1

s

(e−sη − 1

)+ α2 λ

s3Pr(1− Pr)e−s

√Prη + · · ·

φ(α, η) =1

s√

Pm

(1− e−s

√Pmη

)− α3

2((c0

√Pm + d0)s

+9

2(c0 + d0)

√Pm + 18Pm2)

e−s√

Pmη

Pms2+ · · ·

79


θ(α, η) = −(

3α4(1 +4θ0)

2(1 + α(1 +4θ0)3)+ s

√9α242(1 +4θ0)2)− 4(1 + α(1 +40)3)Pr

2(1 + α(1 +40)3)

)

Exp−(

3α4(1 +4θ0)

2(1 + α(1 +4θ0)3)− s

√9α242(1 +4θ0)2)− 4(1 + α(1 +40)3)Pr

2(1 + α(1 +40)3)

)η

+ s

√Prs

2(1 + α(1 +4θ0)3)e−

√Prsη + · · ·

(5.2.51)

Since the inner and outer expansions of the fluctuating in magnetic field having

over lapping domain of validity, we should match them. Although each term in the

inner expansion is completely determined, unknown constants are introduced in

the outer expansion at each order. They are determined by the following matching

principle:

limη→∞

F i1(α, η) = lim

η→0F o

1 (α, η), limη→∞

θi1(α, η) = lim

η→0θo1(α, η),

limη→∞

Gi1(α, η) = lim

η→0Go

1(α, η)(5.2.52)

Carrying on the limit on the left and using Eqn. (5.2.45) into Eqn. (5.2.51),

neglecting exponentially small terms,

F i1(α, η) ∼ −η +

1

s

(e−sη − 1

)+ α2 λ

s3Pr(1− Pr)e−s

√Prη + · · ·

Gi1(α, η) ∼ 1

s√

Pm

(1− e−s

√Pmη

)−α3

2

((c0

√Pm + d0)s +

9

2(c0 + d0)

√Pm + 18Pm2

)

e−s√

Pmη

Pms2+ · · ·

θi1(α, η) ∼ −

(3α4(1 +4θ0)

2(1 + α(1 +4θ0)3)+ s

√9α242(1 +4θ0)2)− 4(1 + α(1 +40)3)Pr

2(1 + α(1 +40)3)

)

Exp−(

3α4(1 +4θ0)

2(1 + α(1 +4θ0)3)− s

√9α242(1 +4θ0)2)− 4(1 + α(1 +40)3)Pr

2(1 + α(1 +40)3)

)η

+ s

√Prs

2(1 + α(1 +4θ0)3)e−

√Prsη + · · ·

(5.2.53)

The superscripts i and o denote the inner and outer expansion respectively. With

the help of matching procedure for each term in Eqn. (5.2.45) there should be

80


a corresponding term in Eqn. (5.2.53). This can be achieved by choosing the

unknown constants in the following way

c0 = 0, c1 = −s−1, D0 = 0, D1 =s−1

√Pm

, D3 = 0

and where α = ξ−12

c. Composite solution for ξ →∞Following Ackerberg and Philips [49] the inner and outer expansions may be

written as By using (5.2.45) and (5.2.51) as

F1 =ξ−

12

s

(e−sξ

12 η − 1

)+ ξ−

32

λ

s3Pr(1− Pr)e−s

√Prξ

12 η

− 5

4s2iξ−2

(c0 + Sd0

√Pm

) e−sξ12 η

s2+

ξ−1

2i[(ηF ′

0 − F0 − SG′0)] + · · ·

G1 =ξ−

12

s√

Pm

(1− e−s

√Pmξ

12 η

)− ξ−2

2Pms2((c0

√Pm + d0)s

+9

2(c0 + d0)

√Pm + 18Pm2)e−s

√Pmη +

ξ−1

2i(ηG′

0 −G0) + · · ·

θ1 = −(

3α4(1 +4θ0)

2(1 + α(1 +4θ0)3)+ s

√9α242(1 +4θ0)2)− 4(1 + α(1 +40)3)Pr

2(1 + α(1 +40)3)

)

Exp−(

3α4(1 +4θ0)

2(1 + α(1 +4θ0)3)− s

√9α242(1 +4θ0)2)− 4(1 + α(1 +40)3)Pr

2(1 + α(1 +40)3)

)η

+ s

√Prs

2(1 + α(1 +4θ0)3)e−

√Prsη + · · ·

(5.2.54)

The expression for shear stress, current density and rate of heat transfer are given

as:

F ′′1 (0) ∼ sξ

12 + ξ−

12

λ

s(1− Pr)− 5

4iξ−1

(c0 + Sd0

√Pm

)+ · · ·

G′′1(0) ∼ −sPmξ

12 − ξ−1

2

((c0

√Pm + d0)s +

9

2(c0 + d0)

√Pm + 18Pm2

)+ · · ·

θ′1(0) ∼ s

(3α4(1 +4θ0)

2(1 + α(1 +4θ0)3)+ s

√9α242(1 +4θ0)2)− 4(1 + α(1 +40)3)Pr

2(1 + α(1 +40)3)

)2

− s2√

Pr

√Prs

2(1 + α(1 +4θ0)3)+ · · ·

(5.2.55)

81


Here s = i+12

c0 and d0 are known values from steady solution. By separating the

real and imaginary part from Eqn. (5.2.55) we can find the numerical values am-

plitudes and phases angle of coefficients of skin friction, heat transfer and current

density for large ξ by using the following relations just to follow Hossain and Banu

[52].

As =√

F ′′21i + F ′′2

1r Am =√

G′′21i + G′′2

1r At =√

θ′21i + θ′21r

φs = arctan

(F ′′

1i

F ′′1r

)φm = arctan

(G′′

1i

G′′1r

)qt = arctan

(θ′1i

θ′1r

)

By using this relation we can obtain the solution for large ξ that is given in Figs.

5.7(a-b) and 8(a-b) graphically.


Here, the effect of conduction radiation on fluctuating hydromagnetic flow past a

magnetized vertical heated plate has been investigated numerically. For numerical

solution of the dimensionless equation that govern the flow, we have employed the

finite difference approach with Gaussian elimination technique for entire values

of ξ and asymptotic series solution for small and large value of ξ. The effects of

physical quantities such as conduction radiation parameter Rd, mixed convection

parameter λ, Prandtl number Pr, magnetic Prandtl number Pm, magnetic force

parameter S and surface temperature θw on amplitude and phase of shear stress

, rate of heat transfer and current density at the surface are discussed below.

The effects of these parameters on transient shear stress, current density and heat

transfer are also given in detail.

5.3.1 Effects of physical parameters upon amplitude and

phase of rate of heat transfer, shear stress and cur-

rent density

Figs. 5.2(a-b), 5.3(a-b) and 5.4(a-c) depict the effect of Rd against ξ on the am-

plitude and phase angle of rate of heat transfer, shear stress and current density

82


respectively, keeping other parameters of the flow field to be constant. It is in-

teresting to observe that with the increase of Rd = 0.1, 1.0, 5.0, 50.0 and for

Pr=0.71, S=0.8, Pm=1.0, λ=1.0 and θw=1.1 the amplitude and phase angle of

heat transfer increases, amplitude of shear stress and current density decreases

where as the phase angle of shear stress and current density increases.

ξ

At

0.0 2.0 4.0 6.0 8.0 10.0

0.5

1

1.5

2

2.5

3Pr = 0.71Pm = 1.0

S = 0.8λ = 1.0θw= 1.1

(a)

Rd = 0.1

Rd = 1.0

Rd = 5.0

Rd = 50.0

ξφ t

0.0 2.0 4.0 6.0 8.0 10.00

10

20

30

40

50

0.11.05.050.0

Pr = 0.71Pm = 1.0

S = 0.8λ = 1.0θw= 1.1

(b)

Rd

Fig. 5.2 Numerical solution of amplitude and phase angle of heat transfer fordifferent values of Rd=0.1, 1.0, 5.0, 50 while Pr = 0.71, Pm= 1.0, λ = 1.0, θw =

1.1 and S=0.8

ξ

As

10-1 100 101

1.0

1.5

2.0

2.5

3.0

0.11.05.050.0

Pr = 0.71Pm = 1.0

S = 0.8λ = 1.0θw= 1.1

(a)

Rd

ξ

φ s

2.0 4.0 6.0 8.0 10.0

10.0

20.0

30.0

40.0

50.0

0.11.05.050.0

Pr = 0.71Pm = 1.0

S = 0.8λ = 1.0θw= 1.1

(b)

Rd

Fig. 5.3 Numerical solution of amplitude and phase angle of shear stress fordifferent values of Rd=0.1, 1.0, 5.0, 50 while Pr = 0.71, Pm= 1.0, λ = 1.0, θw =

1.1 and S=0.8

We mark this trend with the understanding that when Rd increases, the ambi-

ent fluid temperature decreases and Roseland mean absorption coefficient increases

which enhance the rate of heat transfer and reduce the fluid motion which thicken

the momentum boundary layer thickness and thin thermal boundary layer thick-

ness. Figs. 5.5(a-b), it is seen that the amplitude of current density decreases

and the phase of current density increases. We describe this phenomena as by

83


ξ

Am

10-1 100 1010.5

1.0

1.5

2.0

2.5

3.0

0.11.05.050.0

Pr = 0.71Pm = 1.0

S = 0.8λ = 1.0θw= 1.1

(a)

Rd

ξ

φ m

0.0 2.0 4.0 6.0 8.0 10.00.0

10.0

20.0

30.0

40.0

50.0

0.11.05.050.0

Pr = 0.71Pm = 1.0

S = 0.8λ = 1.0θw= 1.1

(b)

Rd

Fig. 5.4 Numerical solution of amplitude and phase angle of current density fordifferent values of Rd=0.1, 1.0, 5.0, 50 while Pr = 0.71, Pm= 1.0, λ = 1.0, θw =

1.1 and S=0.8

ξ

Am

10-1 100 101

0.0

1.0

2.0

3.0

FDMSERASS

Pm

Pr = 0.1S = 0.02λ = 0.5Rd = 1.0θ w = 1.0

(a)

Pm

0.01

0.1

0.5

ξ

φ m

5 10 150.0

10.0

20.0

30.0

40.0

50.0

0.010.10.5Pm

Pr = 0.1S = 0.02λ = 0.5

Rd = 1.0θ w = 1.0

(b)

FDMSERASS

Fig. 5.5 Comparison of numerical solutions with finite difference method (forentire ξ ) and asymptotic solutions (for small and large ξ) for amplitude andphase of current density for different values of Pm= 0.01, 0.1, 0.5 while Rd =

1.0, Pr= 0.1, λ = 0.5, θw = 1.0 and S=0.02

the increase of Pm the induced current within the boundary layer tends to spread

away from the surface and this results in thickening of the boundary layer, thus

the amplitude of current density decreases and phase increases.

It is also noted that the comparison of both methods i.e finite difference method

and asymptotic solution for entire ξ and large, and small ξ is in good agreement.

Its is also noted that with the increase of magnetic force parameter the amplitude

and phase angle of shear stress decreases in figures 5.6(a-b). We mark this trend

as when the magnetic force parameter increases there are wave like disturbance

generate within the boundary layer.These disturbance are, in fact, hydromagnetic

waves which become more and more concentrated as the strength of the magnetic

force parameter S is increased so amplitude and phase of shear stress decrease.

The comparison of finite difference method with that of asymptotic solutions is

84


ξ

As

10-1 100 101

1.0

2.0

3.0

4.0 Pr = 0.015Pm = 0.1

λ = 1.0Rd= 10.0θw = 0.5

(a)

S0.000.150.30

ξ

φ s

10-1 100 101

0.0

10.0

20.0

30.0

40.0

50.0 Pr = 0.015Pm = 0.1

λ = 1.0Rd= 10.0θw = 0.5

S0.000.150.30

(b)

Fig. 5.6 Comparison of numerical solutions with finite difference method (forentire ξ ) and asymptotic solutions (for small and large ξ) for amplitude and

phase of shear stress for different values of S= 0.0, 0.15, 0.30 while Rd = 10.0,Pr= 0.015, λ = 1.0, θw = 0.5 and Pm= 0.1

ξ

At

0.0 2.0 4.0 6.0 8.0 10.0

0.0

2.0

4.0

6.0

8.0

0.10.30.717.0

S = 0.4Pm = 1.0Rd = 10.0

λ = 1.0θw= 1.1

(a)

Pr

ξ

φ t

0.0 2.0 4.0 6.0 8.0 10.00.0

10.0

20.0

30.0

40.0

50.0

0.10.30.717.0

S = 0.4Pm = 1.0Rd = 10.0

λ = 1.0θw= 1.1

(b)

Pr

Fig. 5.7 Numerical solution of amplitude and phase angle of rate of heattransfer for different values of Pr=0.1, 0.3, 0.71, 7.0 while Rd = 10.0, Pm= 1.0,

λ = 1.0, θw = 1.1 and S=0.4

also shown with good agreement in these figures.

The amplitude and phase of heat transfer increase, the amplitude of shear

stress and current density decrease but phase angle of shear stress and current

density increase as Pr= 0.1, 0.3, 0.71, 7.0 increases for Rd= 10.0, Pm= 1.0, S=

0.4, λ= 1.0, and θw= 1.1 in Figs. 5.7(a-b), 5.8(a-b) and 5.9(a-b) respectively. The

reason is that with the increase in the Prandtl number Pr the kinematic viscosity

of the fluid increase and thermal diffusion decreases that rise the temperature and

thermal boundary layer becomes thinner and momentum boundary layer becomes

thicker which results the aforementioned phenomena. The numerical solutions for

amplitude and phase of shear stress and current density are presented in Figs.

5.10(a-b), 5.11(a-b) and 5.12(a-b) respectively. In these figures it is shown that

85


ξ

As

10-1 100 1010.5

1.0

1.5

2.0

2.5

3.0

3.5

0.10.30.717.0

S = 0.4Pm = 1.0Rd = 10.0

λ = 1.0θw= 1.1

(a)

Pr

ξ

φ s

2.0 4.0 6.0 8.0 10.0

10.0

20.0

30.0

40.0

50.0

0.10.30.717.0

S = 0.4Pm = 1.0Rd = 10.0

λ = 1.0θw= 1.1

(b)

Pr

Fig. 5.8 Numerical solution of amplitude and phase angle of shear stress fordifferent values of Pr=0.1, 0.3, 0.71, 7.0 while Rd = 10.0, Pm= 1.0, λ = 1.0, θw

= 1.1 and S=0.4

ξ

Am

10-1 100 1010.5

1.0

1.5

2.0

2.5

3.0

0.10.30.717.0

S = 0.4Pm = 1.0Rd = 10.0

λ = 1.0θw= 1.1

(a)

Pr

ξ

φ m

10-1 100 1010.0

10.0

20.0

30.0

40.0

50.0

0.10.30.717.0

S = 0.4Pm = 1.0Rd = 10.0

λ = 1.0θw= 1.1

(b)

Pr

Fig. 5.9 Numerical solution of amplitude and phase angle of current density fordifferent values of Pr=0.1, 0.3, 0.71, 7.0 while Rd = 10.0, Pm= 1.0, λ = 1.0, θw

= 1.1 and S=0.4

for different values of θw when the values of other parameters are constant the

amplitude and phase angle of rate of heat transfer decreases, the amplitudes of

shear stress and current density increases and phases of shear stress and current

density decreases. The reason is that the increase in surface temperature by well

know Fourier law of heat transfer increase the rate of heat transfer towards ambient

fluid and also enhance the fluid motion at the surface which support the physical

reasoning of the fluid flow phenomena in the flow domain for different values of

θw.

86


ξ

At

2.0 4.0 6.0 8.0 10.00.0

0.2

0.4

0.6

0.8

1.0

0.00.30.61.1

S = 0.6Pm = 2.0Pr = 0.1Rd = 0.1

λ = 1.0

(a)

θw

ξ

φ t

0.0 2.0 4.0 6.0 8.0 10.00.0

10.0

20.0

30.0

40.0

50.0

0.00.30.61.1

S = 0.6Pm = 2.0Pr = 0.1

Rd = 0.1λ = 1.0

(b)

θw

Fig. 5.10 Numerical solution of amplitude and phase angle of heat transfer fordifferent values of θw=0.0, 0.3, 0.6, 1.1 while Rd = 0.1, Pm= 2.0, λ = 1.0, Pr =

0.1 and S=0.6

ξ

As

10-1 100 1011.0

1.5

2.0

2.5

3.0

3.5

0.00.30.61.1

S = 0.6Pm = 2.0Pr = 0.1Rd = 0.1

λ = 1.0

(a)

θw

ξ

φ s

10-1 100 101

0.0

10.0

20.0

30.0

40.0

50.0

0.00.30.61.1

S = 0.6Pm = 2.0Pr = 0.1Rd = 0.1

λ = 1.0

(b)

θw

Fig. 5.11 Numerical solution of amplitude and phase angle of shear stress fordifferent values of θw=0.0, 0.3, 0.6, 1.1 while Rd = 0.1, Pm= 2.0, λ = 1.0, Pr =

0.1 and S=0.6

5.3.2 Effects of physical parameters upon transient rate of

heat transfer, shear stress and current density

In this section, the effects of physical parameters are described, such as conduction

radiation parameter Rd, mixed convection parameter λ, magnetic force parameter

S, magnetic Prandtl number Pm, dimensionless coordinate ξ, surface temperature

θw on the transient rate of heat transfer, shear stress and current density against

τ with the help of the following relations

qt = [q0 + εAt cos(τ + φt)]

τs = [τ0 + εAs cos(τ + φs)]

Jm = [J0 + εAm cos(τ + φm)] (5.3.1)

87


ξ

As

10-1 100 1011.0

1.5

2.0

2.5

0.00.30.61.1

S = 0.6Pm = 2.0Pr = 0.1Rd = 0.1

λ = 1.0

(a)

θw

ξ

φ m

10-1 100 101

0.0

10.0

20.0

30.0

40.0

50.0

0.00.30.61.1

S = 0.6Pm = 2.0Pr = 0.1Rd = 0.1

λ = 1.0

(b)

θw

Fig. 5.12 Numerical solution of amplitude and phase angle of current densityfor different values of θw=0.0, 0.3, 0.6, 1.1 while Rd = 0.1, Pm= 2.0, λ = 1.0, Pr

= 0.1 and S=0.6

τ

q t

0.0 10.0 20.0 30.0 40.0 50.0

0.20

0.22

0.24

0.26

0.28

0.30

0.32

Pm=0.5, θw=1.1 , Pr = 0.71, Rd = 1.0λ = 0.1, ε = 0.05, ξ = 10.0

S0.0

0.5

1.0

(a)τ

τ s

0.0 10.0 20.0 30.0 40.0 50.0

0.10

0.20

0.30

0.40

0.50

0.60

0.70

Pm=0.5, θw=1.1 , Pr = 0.71, Rd = 1.0λ = 0.1, ε = 0.05, ξ = 10.0

S

1.0

0.5

0.0

(b) τ

J m

0.0 10.0 20.0 30.0 40.0 50.0

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

Pm=0.5,θw=1.1 , Pr = 0.71, Rd = 1.0λ = 0.1,ε = 0.05,ξ = 10.0

S

1.0

0.5

0.0

(c)

Fig. 5.13 Solutions for transient (a) heat transfer (c) shear stress and (b)current density against τ for different values of S = 0.0, 0.5, 1.0 and for Pr=

0.71, Rd = 1.0, θw = 1.1 and Pm= 0.5, λ = 0.1, ξ = 10.0 and ε = 0.05

where q0, τ0 and J0 are the rate of heat transfer, shear stress and current density

comes from steady part, and similarly (At, As, Am) and (φt, φs, φm) are amplitudes

and phases of rate of heat transfer, shear stress and current density comes from

fluctuating part and ε is small amplitude oscillation.

Figs. 5.13(a-c) depict that the transient rate of heat transfer and shear stress

decreases as magnetic force parameter S increases and there is no significant effect

can be seen for the case of current density due to very poor role of parameter S

in equation in Eqn. (4). In Figs. 5.14(a-c), it can be seen that the transient rate

of heat transfer, shear stress and current density increases prominently with the

increase of Pm while values of other parameters are constant. The transient rate

of heat transfer increases and shear stress decreases with the increases of Pr but

there is no changes seen in transient current density due to poor contribution of

88


τ

q t

0.0 10.0 20.0 30.0 40.0 50.0

0.20

0.22

0.24

0.26

0.28

0.30

0.32

S=0.5, θw=1.1 , Pr = 0.71, Rd = 1.0λ = 0.1, ε = 0.05, ξ=10.0

1.0

0.5

0.1

(a)

Pm

τ

τ s

0.0 10.0 20.0 30.0 40.0 50.0

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

S=0.5,θw=1.1 , Pr = 0.71, Rd = 1.0λ = 0.1,ε = 0.05,ξ=10.0

1.0

0.5

0.1

(b)

Pm

τ

J m

0.0 10.0 20.0 30.0 40.0 50.0

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

S=0.5,θw=1.1 , Pr = 0.71, Rd = 1.0λ = 0.1,ε = 0.05

1.0

0.5

0.1

(c)

Pm

Fig. 5.14 Solutions for transient (a) heat transfer (b) shear stress and (c)current density against τ for different values of Pm = 0.1, 0.5, 1.0 and for Pr=

0.71, Rd = 1.0, θw = 1.1 and S= 0.5, λ = 0.1, ξ = 10.0 and ε = 0.05

τ

q t

0.0 10.0 20.0 30.0 40.0 50.0

-0.20

0.00

0.20

0.40

0.60

S=0.5,θw=1.1 ,Pm = 0.1, Rd = 0.1λ = 0.1,ε = 0.05,ξ=10.0

7.0

0.71

0.1

(a)

Pr

τ

τ s

0.0 10.0 20.0 30.0 40.0 50.00.25

0.30

0.35

0.40

0.45

0.50

0.55

S=0.5, θw=1.1 , Pm = 0.1, Rd = 0.1λ = 0.1, ε = 0.05, ξ=10.0

7.0

0.71

0.1

(b)

Pr

τ

J m

0.0 10.0 20.0 30.0 40.0 50.0-0.28

-0.26

-0.24

-0.22

-0.20

-0.18

-0.16

-0.14

S=0.5,θw=1.1 , Pr = 0.71, Rd = 1.0λ = 0.1,ε = 0.05,ξ=10.0

(c)

Fig. 5.15 Solutions for transient (a) heat transfer (b) shear stress and (c)current density against τ for different values of Pr = 0.1, 0.71, 7.0 and for S=

0.5, Rd = 1.0, θw = 1.1 and Pm= 0.5, λ = 0.1, ξ = 10.0 and ε = 0.05

Pr in Eqn (5), which can be seen in Figs. 5.15(a)-5.15(c). From Figs. 5.16(a-c),

it can seen that the transient rate of heat transfer increases and transient shear

stress decrease and current density increases very slightly. In Figs. 5.17(a-c) It can

be seen that the transient rate of heat transfer, shear stress and current density

increase with the increase of λ for the fixed values of other parameters.

In Figs. 5.18(a-c), due to the variation in surface temperature for fixed values

of Pm, S, λ, Pr and Rd, small amplitude ε and nondimensional parameter ξ the

transient rate of heat transfer qt decreases and transient shear stress τs increases

and there is no change in transient current density seen. Finally, in Figs. 5.19(a-

c), the increase in parameter ξ decrease the transient heat transfer and increase

transient shear stress and current density.

89


τ

q t

0.0 10.0 20.0 30.0 40.0 50.00.00

0.05

0.10

0.15

0.20

0.25

0.30

S=1.0, θw=1.1 , Pm = 0.1, Pr = 0.71λ = 0.1, ε = 0.05, ξ=10.0

(a)

Rd = 10.0

Rd = 5.0

Rd = 0.1

τ

τ s

0.0 10.0 20.0 30.0 40.0 50.0

0.05

0.10

0.15

0.20

0.25

S=1.0, θw=1.1 , Pm = 0.1, Pr = 0.71λ = 0.1, ε = 0.05, ξ=10.0

(b)

Rd = 10.0

Rd = 5.0

Rd = 0.1

τ

J m

0.0 10.0 20.0 30.0 40.0 50.0-0.28

-0.26

-0.24

-0.22

-0.20

-0.18

-0.16

-0.14

S=1.0,θw=1.1 ,Pm = 0.1, Pr = 0.71λ = 0.1,ε = 0.05,ξ=10.0

(c)

Rd = 10.0

Rd = 5.0

Rd = 0.1

Fig. 5.16 Solutions for transient (a) heat transfer (b) shear stress and (c)current density against τ for different values of Rd = 0.1, 5.0, 10.0 and for Pr=

0.71, S = 1.0, θw = 1.1 and Pm= 0.1, λ = 0.1, ξ = 10.0 and ε = 0.05

τ

q t

0.0 10.0 20.0 30.0 40.0 50.00.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

S=1.0, θw=1.1 , Pm = 0.1, Pr = 0.71Rd = 1.0, ε = 0.05, ξ=10.0

(a)

λ = 5.0

λ = 1.0

λ = 0.1

τ

τ s

0.0 10.0 20.0 30.0 40.0 50.0-1.00

0.00

1.00

2.00

3.00

4.00

5.00

S=1.0,θw=1.1 ,Pm = 0.1, Pr = 0.71Rd = 1.0,ε = 0.05,ξ=10.0

(b)

λ = 5.0

λ = 1.0

λ = 0.1

τ

J m

0.0 10.0 20.0 30.0 40.0 50.0

-0.25

-0.20

-0.15

-0.10

S=1.0,θw=1.1 ,Pm = 0.1, Pr = 0.71Rd = 1.0,ε = 0.05,ξ=10.0

(c)

λ = 5.0

λ = 1.0

λ = 0.1

Fig. 5.17 Solutions for transient (a) heat transfer (b) shear stress and (c)current density against τ for different values of λ = 0.1, 1.0, 5.0 and for Pr=

0.71, Rd = 1.0, θw = 1.1 and Pm= 0.1, S = 1.0, ξ = 10.0 and ε = 0.05

5.4 Conclusion

Now we summarize the results of physical interest on the amplitude and phase of

shear stress, rate of heat transfer and current density in flow field at the surface.

It is to be noted that with the increase of conduction radiation parameter

Rd, the amplitude and phase of heat transfer increases and amplitude of shear

stress and current density decreases and phase angle of shear stress and current

density increases. It is also to be noted that the transient rate of heat transfer

and shear stress increases as the radiation conduction parameter increases. It

is concluded that the amplitude and phase of rate of heat transfer increases very

actively with increase of Prandtl number , the amplitude of shear stress and current

density decrease and phase angle of both physical quantities increases. Similarly,

90


τ

q t

0.0 10.0 20.0 30.0 40.0 50.0

0.11

0.12

0.13

0.14

0.15

0.16

0.3

0.5

θw

0.1

Pm=0.4, S = 0.1, Pr = 0.1, Rd = 0.1λ = 0.5, ε = 0.05, ξ=10.0

(a)τ

τ s

0.0 10.0 20.0 30.0 40.0 50.01.26

1.28

1.30

1.32

1.34

1.36

1.38

1.40

0.3

0.5

θw

0.1

Pm=0.4, S = 0.1, Pr = 0.1, Rd = 0.1λ = 0.5, ε = 0.05, ξ=10.0

(b) τ

J m

0.0 10.0 20.0 30.0 40.0 50.0-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.10.30.5

Pm=0.4,S = 0.1, Pr = 0.1, Rd = 0.1λ = 0.5,ε = 0.05,ξ=10.0

θw

(c)

Fig. 5.18 Solutions for transient (a) heat transfer (b) shear stress and (c)current density against τ for different values of θw = 0.1, 0.3, 0.5 and for Pr=

0.1, Rd = 0.1, λ = 0.5 and Pm= 0.4, S = 0.1, ξ = 10.0 and ε = 0.05

τ

q t

0.0 10.0 20.0 30.0 40.0 50.00.186

0.187

0.188

0.189

0.190

0.191

0.192

0.193 ξ

10.0

2.5

1.0

Pm=0.1, S = 0.4, Pr = 0.1, Rd = 10.0λ = 1.0, ε = 0.05, θw=1.1,

(a) τ

τ s

0.0 10.0 20.0 30.0 40.0 50.0

1.750

1.800

1.850

1.900

1.950 ξ10.0

2.5

1.0

Pm=0.1, S = 0.4, Pr = 0.1, Rd = 10.0λ = 1.0, ε = 0.05, θw=1.1

(b) τ

J m

0.0 10.0 20.0 30.0 40.0 50.0

-0.230

-0.220

-0.210

-0.200

-0.190

-0.180

-0.170

-0.160 ξ10.0

2.5

1.0

Pm=0.1,S = 0.4, Pr = 0.1, Rd = 10.0λ = 1.0,ε = 0.05,θw=1.1

(c)

Fig. 5.19 Solutions for transient (a) heat transfer (b) shear stress and (c)current density against τ for different values of ξ = 1.0, 2.5, 10.0 and for Pr=

0.1, Rd = 10.0, λ = 1.0 and Pm= 0.1, S = 0.4, θw = 1.1 and ε = 0.05

the transient rate of heat transfer increases and shear stress decreases with the

increases of Prandtl number but no changes is seen in transient current density

It is observed that the amplitude and phase of the shear stress decrease with the

increase of magnetic force parameter. The transient rate of heat transfer increases

but the transient shear stress decreases and no change is seen in transient current

density as magnetic force parameter increases. It is also noted that transient

rate of heat transfer, shear stress and current density increase, when the mixed

convection parameter increases for remaining the other parameter constant. With

the variation in surface temperature the amplitude and phase angle of rate of heat

transfer decrease, the amplitudes of shear stress and current density increase and

phases of shear stress and current density decrease. The transient rate of heat

91


transfer decreases and transient shear stress increases but no change is seen in

the transient current density. The amplitude of current density decreases but the

phase of current density increases as the magnetic Prandtl number increases. The

transient shear stress rate of heat transfer and current density are increased as Pm

increases. The asymptotic solutions for small and large values of dimensionless

streamwise coordinate, ξ, are found to be in good agreement that obtained by

finite difference method for entire value of ξ.

92

93

Chapter 6

Radiative fluctuating magnetohydrodynamic natural convection flow past a magnetized vertical heated plate

Chapter 6: Radiative fluctuating magnetohydrodynamic natural...

The work presented in the previous Chapter 4 is extended here in Chapter 6 to

investigate the effect of thermal radiation on fluctuating hydro-magnetic natural

convection flow of viscous, incompressible, electrically conducting and optically

dense grey fluid past a magnetized vertical plate; when the magnetic field and

surface temperature oscillate in magnitude about a constant non zero mean. The

numerical solutions have been obtained for different values of radiation parameter

Rd, magnetic Prandtl number Pm, magnetic force parameter S, Prandtl number

Pr and surface temperature θw in terms of amplitude and phase of coefficients

of skin friction, rate of heat transfer and current density at the surface of the

plate. Moreover, the effects of these parameters on transient coefficients of skin

friction, rate of heat transfer and current density have been discussed. The finite

difference method for primitive variable formulation and asymptotic series solution

for stream function formulation have been used to obtain the numerical solution

of the boundary layer flow field.

6.1 Mathematical analysis and governing equa-

tions

We consider a fluctuating two-dimensional magnetohydrodynamic natural convec-

tion flow of an electrically conducting, viscous, incompressible and optically dense

grey fluid past a uniformly heated and magnetized vertical plate in the presence

of radiative heat flux in the energy equation.The flow configuration and the coor-

dinate system is shown in Fig. 6.1.

We have taken x-axis along the surface and y-axis is normal to it. In Fig.6.1

δM , δT and δH stand for momentum, thermal and magnetic field boundary layer

thicknesses. The governing equations of the problem in the form of boundary layer

for unsteady flow can be written as:

∂u

∂x+

∂v

∂y= 0 (6.1.1)

94


u=v

=0,

T=

Tw(x

),B

x=

Bw(x

),B

y=0

u=0

,T=T

∞,

Bx(

∞)=

0

δΗ

δM

Oy

x

Hx

Hy

u

v

δT

Fig. 6.1 The coordinate system and flow configuration

∂u

∂t+ u

∂u

∂x+ v

∂v

∂y= ν

∂2u

∂y2+

µ

ρ

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)+ gβ(T − T∞) (6.1.2)

∂Bx

∂x+

∂By

∂y= 0 (6.1.3)

∂Bx

∂t+ u

∂Bx

∂x+ v

∂Bx

∂y− Bx

∂u

∂x− By

∂u

∂y=

1

γ

∂2Bx

∂y2(6.1.4)

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2− ∂qr

∂y(6.1.5)

where

qr = − 4σ

3αR

∂T 4

∂y

The boundary conditions which satisfy by the above equations are:

u(x, 0) = 0, v(x, 0) = V0, Bx(x, 0) = Bw(x), By(x, 0) = 0, T (x, 0) = Tw

u(x,∞) = 0, Bx(x,∞) = 0, T (x,∞) = 0 (6.1.6)

For convenience, we introduce the following dependent and independent variables

to normalize the boundary layer equations

t =Gr

− 12

L τ

L−2, u =

ν

LGr

12Lu, v =

ν

LGr

14Lv, θ =

T − T∞Tw − T∞

95


Bx =B0

LU0

Gr12LBx, By =

B0

LU0

Gr14LBy, y =

y

L−1Gr

− 14

L , x =x

L(6.1.7)

By substituting Eqn. (6.1.7) into Eqns. (6.1.1)-(6.1.6) the dimensionless boundary

layer equations and boundary conditions are given as follows

∂u

∂x+

∂v

∂y= 0 (6.1.8)

∂u

∂τ+ u

∂u

∂x+ v

∂u

∂y= θ +

∂2u

∂y2+ S

(Bx

∂Bx

∂x+ By

∂Bx

∂y

)(6.1.9)

∂Bx

∂x+

∂By

∂y= 0 (6.1.10)

∂Bx

∂τ+ u

∂Bx

∂x+ v

∂Bx

∂y−Bx

∂u

∂x−By

∂u

∂y=

1

Pm

∂2Bx

∂y2(6.1.11)

∂θ

∂τ+ u

∂θ

∂x+ v

∂θ

∂y=

1

Pr

[∂2θ

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θ)3 ∂θ

∂y}]

(6.1.12)

The dimensionless boundary conditions are:

u(x, 0) = 0, v(x, 0) = 0, Bx(x, 0) = B(τ), By(x, 0) = 0, θ(x, 0) = θ(τ)

u(x,∞) = 0, Bx(x,∞) = 0, θ(x,∞) = 0 (6.1.13)

where B(τ) and θ(τ) are the components of velocity and temperature which can

be defined as follows

B(τ) = 1 + εeiτ , θ(τ) = 1 + εeiτ (6.1.14)

In Eqn. (6.1.14) ε is assumed to be small amplitude of oscillation of free stream

magnetic intensity and surface temperature. In keeping view of Eqn. (6.1.14)

the solution of Eqns. (6.1.8)-(6.1.12) with boundary conditions (6.1.13) will be

obtained as a complex functions by considering the real part which have physical

significance in nature. We will write u, v, Bx, By and θ as the sum of steady and

oscillating components as proposed by Chwala [37].

u =[u0(x, y) + εeiτu1(x, y)

]

v =[v0(x, y) + εeiτv1(x, y)

]

Bx =[Bx0(x, y) + εeiτBx1(x, y)

]

By =[By0(x, y) + εeiτBy1(x, y)

]

θ =[θ0(x, y) + εeiτθ1(x, y)

]

(6.1.15)

96


Here u0(x, y), v0(x, y), Bx0(x, y), By0(x, y), θ0(x, y) and u1(x, y), v1(x, y), Bx1(x, y),

By1(x, y), θ1(x, y) are the steady and fluctuating parts of the flow variables.

By using Eqn. (6.1.15) into Eqns. (6.1.8)-(6.1.12) and by collecting terms of

the first power of ε, we have, steady and unsteady system of equations sd given

below

∂u0

∂x+

∂v0

∂y= 0 (6.1.16)

u0∂u0

∂x+ v0

∂v0

∂y=

∂2u0

∂y2+ S

(Bx0

∂Bx0

∂x+ By0

∂Bx0

∂y

)+ θ0 (6.1.17)

∂Bx0

∂x+

∂By0

∂y= 0 (6.1.18)

u0∂Bx0

∂x+ v0

∂Bx0

∂y−Bx0

∂u0

∂x−By0

∂u0

∂y=

1

Pm

∂2Hx0

∂y2(6.1.19)

u0∂θ0

∂x+ v0

∂θ0

∂y=

1

Pr

[∂2θ0

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θ0)

3∂θ0

∂y}]

(6.1.20)

and the corresponding boundary conditions are:

u0(x, 0) = 0, v0(x, 0) = 0, Bx0(x, 0) = 1, By0(x, 0) = 0, θ0(x, 0) = 1

u0(x,∞) −→ 0, Bx0(x,∞) −→ 0, θ0(x,∞) −→ 0(6.1.21)

and

∂u1

∂x+

∂v1

∂y= 0 (6.1.22)

iu1 + u0∂u1

∂x+ u1

∂u0

∂x+ v0

∂u1

∂y+ v1

∂u0

∂y=

∂2u1

∂y2

+ S

(Bx0

∂Bx1

∂x+ Bx1

∂Bx0

∂x+ By0

∂Bx1

∂y+ By1

∂Bx0

∂y

)+ θ1

(6.1.23)

∂Bx1

∂x+

∂By1

∂y= 0 (6.1.24)

iBx1 + u0∂Bx1

∂x+ u1

∂Bx0

∂y+ v0

∂Bx1

∂y+ v1

∂Bx0

∂y

−Bx0

∂u1

∂x−Bx1

∂u0

∂x−By0

∂u1

∂x−By1

∂u0

∂y=

1

Pm

∂2Bx1

∂y2

(6.1.25)

97


iθ1 + u0∂θ1

∂x+ u1

∂θ0

∂x+ v0

∂θ1

∂y+ v1

∂θ0

∂y

=1

Pr[∂2θ1

∂y2+

4

3Rd

∂

∂y{(1 + (θw − 1)θ0)

3∂θ1

∂y

+ 3(1 + (θw − 1)θ0)2∆θ1

∂θ0

∂y}]

(6.1.26)

the corresponding boundary conditions are:

u1(x, 0) = 0, v1(x, 0) = 0, Bx1(x, 0) = 1,

By1(x, 0) = 0, θ1(x, 0) = 1

u1(x,∞) −→ 0, Bx1(x,∞) −→ 0, θ1(x,∞) −→ 0

(6.1.27)

We can find the solution of steady part functions u0, v0, Bx0 , By0 and θ0

by using Eqns. (6.1.16)-(6.1.21). By using these solutions in Eqns. (6.1.22)-

(6.1.27), we can quit from the situation of non linearity and can find the solution

of fluctuating flow for momentum, energy and magnetic field equations.

6.2 Solution methodology

We now turn to get the numerical solutions of the problem, for this purpose we will

use two methods namely (i) Primitive variable transformation for finite difference

method and (ii) Stream function formulation for asymptotic series solutions near

and away from the leading edge of the plate.


To get the set of equations in convenient form for integration, we define the follow-

ing one parameter of transformations for the dependent and independent variables:

6.2.1.1. Transformations for steady case

Here we introduced transformation for steady case

u0 = x12 U0(X,Y ), v0 = x−

14 V0(X,Y ), Y =

1

2x−

14 y

98


Bx0 = x12 φ1s(X, Y ), By0 = x−

14 φ2s(X,Y ), θ0 = θ0(X, Y ) (6.2.1)


(21) we have1

2U0 + X

∂U0

∂X− 1

4Y

∂U0

∂Y+

∂V0

∂Y= 0 (6.2.2)

1

2U2

0 + XU0∂U0

∂X+ (V0 − 1

4Y U0)

∂U0

∂Y= θ0 +

∂2U0

∂Y 2

+ S

[1

2φ2

1s + Xφ1s∂φ1s

∂X+

(φ2s − 1

4Y φ1s

)∂φ1s

∂Y

] (6.2.3)

1

2φ1s + X

∂φ1s

∂X− 1

4Y

∂φ1s

∂Y+

∂φ2s

∂Y= 0 (6.2.4)

XU0∂φ1s

∂X+ (V0 − 1

4Y U0)

∂φ1s

∂Y

−Xφ1s∂U0

∂X− (φ2s − 1

4Y φ1s)

∂U0

∂Y=

1

Pm

∂2φ1s

∂Y 2

(6.2.5)

XU0∂θ0

∂X+ (V0 − 1

4Y U0)

∂θ0

∂Y=

1

Pr

[1 +

4

3Rd

(1 + ∆θ0)3

]∂2θ0

∂Y 2

+4

Pr∆

1

Rd

(1 + ∆θ0)2(

∂θ0

∂Y)2

(6.2.6)

where ∆ = θw − 1, The appropriate boundary conditions to be satisfied above

equations are

U0(X, 0) = V0(X, 0) = 0, φ1s(X, 0) = 1, φ2s(X, 0) = 0, θ0(X, 0) = 1

U0(X,∞) = 0, φ1s(X,∞) = 0, θ0(X,∞) = 0(6.2.7)

and similarly we have transformation for unsteady case

6.2.1.2. Transformations for unsteady case

Here we introduce transformation for unsteady case

u1 = x12 U1(X,Y ), v1 = x−

14 V1(X,Y ), Y =

1

2x−

14 y

Hx1 = x12 φ1us(X, Y ), Hy1 = x−

14 φ2u(X, Y ), θ = θ(X, Y ) (6.2.8)


(6.1.27) we have the following system of equations:

99


1

2U1 + X

∂U1

∂X− 1

4Y

∂U1

∂Y+

∂V1

∂Y= 0 (6.2.9)

1

2U2

1 + XU0∂U1

∂X+ (V0 − 1

4Y U0)

∂U1

∂Y

+ XU1∂U0

∂X+ (V1 − 1

4Y U1)

∂U0

∂Y+ iU1 = θ1 +

∂2U1

∂Y 2

+ S[φ1sφ1u + Xφ1s∂φ1u

∂X+ (φ2s − 1

4Y φ1s)

∂φ1u

∂Y

+ Xφ1u∂φ1s

∂X+ (φ2u − 1

4Y φ1u)

∂φ1s

∂Y]

(6.2.10)

1

2φ1u + X

∂φ1u

∂X− 1

4Y

∂φ1u

∂Y+

∂φ2u

∂Y= 0 (6.2.11)

XU0∂φ1u

∂X+ (V0 − 1

4Y U0)

∂φ1u

∂Y+ XU1

∂φ1s

∂X+ (V1 − 1

4Y U1)

∂φ1s

∂Y

− [Xφ1s∂U1

∂X+ (φ2s − 1

4Y φ1s)

∂U1

∂Y+ Xφ1u

∂U0

∂X

+ (φ2u − 1

4Y φ1u)

∂U0

∂Y] + iφ1u =

1

Pm

∂2φ2u

∂Y 2

(6.2.12)

XU1∂θ0

∂X+ U0

∂θ1

∂X+ (V0 − 1

4Y U0)

∂θ1

∂Y+ (V1 − 1

4Y U1)

∂θ0

∂Y+ iθ1

=1

Pr

[1 +

4

3Rd

(1 + ∆θ1)3

]∂2θ1

∂Y 2

+4∆

PrRd

[2∆(1 + ∆θ0)(

∂θ0

∂Y)2 + (1 + ∆θ0)

2(∂2θ0

∂Y 2)

]θ1

(6.2.13)

The appropriate boundary conditions which satisfy by the above equations are

U1(X, 0) = V1(X, 0) = 0, φ1u(X, 0) = 1, φ2u(X, 0) = 0, θ1(X, 0) = 1

U1(X,∞) = 0, φ1u(X,∞) = 0, θ1(X,∞) = 0(6.2.14)

The above system of transformed equations given in Eqns. (6.2.2)-(6.2.6) and

Eqns. (6.2.9)-(6.2.13) along with their boundary conditions are descritized by

using finite difference method, backward difference for x-direction and central

difference for y-direction. By this way, we get the system of tri-diagonal equa-

tions. This set of tri-diagonal equations is solved by using Gaussian elimination

100


technique. The computation is started at X=0, and then marches downstream

implicitly.

Once we obtained the solutions of the functions U0(X, Y ), V0(X,Y ), φ1s(X, Y ),

φ2s(X, Y ) and θ0(X,Y ), then we can easily find the solutions of flow variables of

the fluctuating part of the problem given in equations (6.2.10)-(6.2.13). After

several runs of convergence we need to find the values of the coefficients of skin

friction Gr−3/4L x−1/4Cf , rate of heat transfer Gr

1/4L x1/4Nux and current density

Gr−3/4L x−1/4Jw at the surface for smaller values of Pr and Pm, S, Rd and θw that

can be defined as follows:

τ = Gr−3/4L x−1/4Cf =

(∂u

∂Y

)

Y =0

, Q = Gr1/4L x1/4,

Nux = −(

1 +4

3Rd

θ3w

)(∂θ

∂Y

)

Y =0

, J = Gr−3/4L x−1/4Jw =

(∂φ1

∂Y

)

Y =0

(6.2.15)

To represent the available solution in terms of amplitude and phase of coefficient

of skin friction, rate of heat transfer and current density, we proceed as follows:

As =√

τ 2r + τ 2

i , Am =√

J2r + J2

i , At =√

Q2r + Q2

i

φs = tan−1

(τi

τr

), φm = tan−1

(Ji

Jr

), φt = tan−1

(Qi

Qr

)(6.2.16)

Here, (τr, τi), (Jr, Ji) and (Qr, Qi) are the real and imaginary parts of the

coefficients of skin friction, rate of heat transfer and current density. The numerical

solutions obtained by using formulations given in Eqn. (6.2.16) are shown in Figs.

6.2-6.10 graphically for the different values of different parameters in terms of

amplitude and phase angle against ξ

6.2.2 Asymptotic solution for small and large parameter ξ

As we have done in earlier chapters, here we also propose to find the numerical

solutions for small and large values of non-dimensional parameter ξ.

101


6.2.2.1 When parameter ξ is small

For small values of ξ, we have two cases, one for steady and other for unsteady

flow.

a. The case for steady flow

For steady flow we have the following group of transformations

ψ = x3/4f0(Y ), g = x3/4φ0(Y ),

Y = x−1/4y, θ0 = θ0(Y ), ξ = x(6.2.17)

The components of velocity and magnetic field are defined as:

u0 =∂ψ

∂y, v0 = −∂ψ

∂x

Hx0 =∂g

∂y, Hy0 = −∂g

∂x

u0 = x12 f ′0(Y ), v0 = −x−

14

(3

4f0(Y )− 1

4Y f ′0(Y )

)

∂θ0

∂x= − Y

4xθ′0,

∂θ0

∂y= x−

14 θ′0

Hx0 = x12 φ′0(Y ), Hy0 = −x−

14

(3

4φ0(Y )− 1

4Y φ′0(Y )

)(6.2.18)

By using Eqn. (6.2.18) into Eqns. (6.1.16)-(6.1.21), we have the reduced set

of equations:

f ′′′0 +3

4f0f

′′0 −

1

2f ′20 + θ0 − S

(3

4φ0φ

′′0 −

1

2φ′20

)= 0 (6.2.19)

1

Pmφ′′′0 +

3

4f0φ

′′0 −

3

4f ′′0 φ0 = 0 (6.2.20)

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θ0)3θ′0

]′+

3

4f0θ

′0 = 0 (6.2.21)

Boundary conditions to be satisfied by the above equations are

f0(0) = f ′0(0) = 0, φ0(0) = 0, φ′0(0) = 1, θ0(0) = 1

f ′0(∞) = 0, φ′0(∞) = 0, θ0(∞) = 0 (6.2.22)

102


b. The case for unsteady flow

The case when the flow is unsteady, for this we have the following group of trans-

formations:

ψ = x3/4F (ξ, Y ), G = x3/4Φ(ξ, Y ),

Y = x−1/4y, θ = θ(ξ, Y ), x = ξ(6.2.23)

The components of velocity and magnetic field are defined as:

u1 =∂ψ

∂y, v1 = −∂ψ

∂x

Hx1 =∂G

∂y, Hy1 = −∂G

∂x

From this we have

u1 = x1/2F ′, v1 = −x−1/4

(3

4F − 1

4Y F ′ +

1

2ξ∂F

∂ξ

)

Hx1 = x1/2Φ′, Hy1 = −x−1/4

(3

4Φ− 1

4Y Φ′ +

1

2ξ∂Φ

∂ξ

)

θ1 = θ1(ξ, Y )

(6.2.24)

By using Eqn. (6.2.24) into Eqns. (6.1.26)-(6.1.20), we have the following system

of equations:

F ′′′ +3

4(f0F

′′ + f ′′0 F )− f ′0F′ + θ1 − S

(3

4φ0Φ

′′ +3

4φ′′0Φ− φ′0Φ

′)− iF ′

=1

2ξ

[f ′0

∂F ′

∂ξ− f ′′0

∂F

∂ξ− S

(φ′0

∂Φ′

∂ξ− φ′′0

∂Φ

∂ξ

)]

1

PmΦ′′′ +

3

4(f0Φ

′′ + Fφ′′0)−3

4(f ′′0 Φ + F ′′φ0)− iΦ′

=1

2ξ

[(f ′0

∂Φ′

∂ξ− φ′0

∂F ′

∂ξ

)−

(φ′′0

∂F

∂ξ− f ′′0

∂Φ

∂ξ

)]

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)θ0)3θ′1

]′+

3

4(f0θ

′1 + Fθ′0)

− iθ1 =1

2ξ

[f ′0

∂θ′1∂ξ

− θ′0∂F

∂ξ

] (6.2.25)

103


The corresponding boundary conditions are

F (ξ, 0) = F ′(ξ, 0) = 0, Φ(ξ, 0) = 0, Φ′(ξ, 0) = 1, θ1(ξ, 0) = 1

F ′(ξ,∞) = 0, Φ′(ξ,∞) = 0, θ1(ξ,∞) = 0 (6.2.26)

Now we will expand all the depending functions in power of iξ and taking the

times upto O(iξ), from this we have the following set of equations:

F ′′′0 +

3

4(f0F

′′0 + f ′′0 F0)− f ′0F

′0 + θ10 + S

(3

4(φ0Φ

′′0 + φ′′0Φ0)φ

′0Φ

′0

)= 0 (6.2.27)

1

PmΦ′′′

0 +3

4f0φ

′′0 +

3

4F0φ

′′0 −

3

4f ′′0 Φ0 − 3

4F ′′

0 φ0 = 0 (6.2.28)

[1 + α1 (1 + ∆θ0)

3] θ′′10 + 3α1∆ (1 + ∆θ0)2 θ′210 +

3

4Pr(f0θ

′10 + F0θ

′0) = 0 (6.2.29)

Here, α1 = 43Rd

and 4 = θw − 1

The corresponding boundary conditions are

F0(0) = F ′0(0) = 0, Φ0(0) = 0, Φ′

0(0) = 1, θ10(0) = 1

F ′0(∞) = 0, Φ′

0(∞) = 0, θ10(∞) = 0 (6.2.30)

O(ξ1)

F ′′′1 +

3

4(f0F

′′1 +f0F

′′1 )−f ′0F

′1+θ11−F ′

0+S

(3

4φ0Φ

′′1 +

3

4φ′′0Φ1 − φ′0Φ

′1

)= 0 (6.2.31)

1

PmΦ′′′

1 +3

4(f0Φ

′′1 − φ0F1′′)+

5

4(φ′′0F1 − f ′′0 Φ1)− 1

2(f ′0Φ

′1−φ′0F

′1)−Φ′

0 = 0 (6.2.32)

[1 + α (1 + ∆θ0)

3] θ′′11 + 3α1∆ (1 + ∆θ0)2 (θ11θ

′′0 + 2θ′0θ

′11)

+ 6α1∆2θ11 (1 + ∆θ0) θ′20 + Pr

3

4f0θ

′11 +

5

4F1θ

′0 −

1

2f ′0θ11 − θ0 = 0

(6.2.33)

The related order boundary conditions are

F1(0) = F ′1(0) = 0, Φ1(0) = 0, Φ′

1(0) = 0, θ11(0) = 0

F ′1(∞) = 0, Φ′

1(∞) = 0, θ11(∞) = 0 (6.2.34)

The Eqns. (6.2.27)-(6.2.29) are very helpful to find the solutions of steady part

variables f0, φ0 and θ0. The Eqns. (6.2.31)-(6.2.33) become easy when we use the

solution of steady part from Eqns. (6.2.27)-(6.2.29) and provide a platform to find

104


the solution of fluctuating part of the problem. The solutions of these equations are

obtained by Nachtsheim-Swigert [76] iteration technique together with six order

implicit Runge-Kutta-Butcher initial value solver. We can calculate the values of

the coefficients skin friction, rate of heat transfer and current density in terms

of amplitude and phase angle at the surface in the region near the leading edge

against ξ from the following expressions

τ = Gr−3/4L x−1/4Cf = f ′′(0) (6.2.35)

J = Gr−3/4L x−1/4Jw = φ′′(0) (6.2.36)

Q = Gr1/4L x1/4Nu = −

(1 +

4

3Rd

θ3w

)θ′(0) (6.2.37)

As =√

τ 2r + τ 2

i , Am =√

J2r + J2

i , At =√

Q2r + Q2

i

φs = tan−1

(τi

τr

), φm = tan−1

(Ji

Jr

), φt = tan−1

(Qi

Qr

)(6.2.38)

The quantities (τr, τi), (Jr, Ji) and (Qr, Qi) are the real and imaginary part of

the coefficients of skin friction, current density and the rate of heat transfer.

The results obtained with the help of the equations (6.2.38) are given in Tables

6.1-6.3 for small values of ξ.

Table 6.1 Numerical values of amplitude and phase angle of heat transferobtained for S= 0.0, 0.1 when Pm=0.1, Rd=50.0, θw=2.0, Pr=0.1,against ξ by

two methods.S=0.0 S=0.1

ξ FDM Asymptotic FDM AsymptoticAt φt At φt At φt At φt

0.0 0.4141 0.0000 0.4048† 0.0000† 0.4045 0.0000 0.4065† 0.0000 †0.2 0.4057 2.2562 0.4092† 2.4936† 0.4048 2.2728 0.4109† 2.5115†0.4 0.4068 4.5102 0.4221† 4.6780† 0.4060 4.5420 0.4241† 4.7039†0.6 0.4087 6.7549 0.4430† 6.3065† 0.4080 6.8997 0.4454† 6.3240†0.8 0.4115 8.9827 0.4714† 8.2285† 0.4109 9.0377 0.4742† 9.0668†1.0 0.4152 11.1802 0.4825† 11.6054† 0.4146 11.2419 0.4859† 21.5600†2.0 0.4463 21.2891 - - 0.4461 21.3537 - -4.0 0.5544 35.0198 - - 0.5544 35.0555 - -6.0 0.6870 41.7353 0.6360‡ 44.3310‡ 0.6870 41.7557 0.6360‡ 44.3310‡8.0 0.8200 44.6930 0.7344‡ 44.7323‡ 0.8201 44.7066 0.7344‡ 44.7321‡10.0 0.9428 45.0000 0.8202‡ 44.9999‡ 0.9429 45.0000 0.8202‡ 44.9999‡

105


Table 6.2 Numerical values of amplitude and phase angle of coefficient of skinfriction obtained for S= 0.0, 0.1 when Pm=0.1, Rd=50.0, θw=2.0,

Pr=0.1,against ξ by two methods.S=0.0 S=0.1


0.0 0.8303 0.0000 0.8240† 0.0000† 0.7387 0.0000 0.7811† 0.0000†0.2 0.8259 4.8921 0.8538† 4.1152† 0.7348 4.8618 0.7538† 4.3383†0.4 0.8125 9.6693 0.8873† 9.5277† 0.7229 9.6104 0.7309† 9.7034†0.6 0.7918 14.5824 0.8490† 14.5099† 0.7046 14.0623 0.7191† 14.9031†0.8 0.7655 18.2740 0.7849† 18.2187† 0.6813 18.1588 0.7099† 18.5839†1.0 0.7363 21.9605 0.7589† 21.9199† 0.6555 21.8104 0.6893† 21.5093†2.0 0.5864 34.3274 - - 0.5223 34.0844 - -4.0 0.4090 42.5114 - - 0.3668 42.3844 - -6.0 0.3258 44.6935 0.3345‡ 45.0000‡ 0.2917 44.6151 0.3165‡ 45.0000‡8.0 0.2773 44.7741 0.2944‡ 45.0000‡ 0.2479 44.8137 0.2843‡ 45.0000‡10.0 0.2452 45.0000 0.2658‡ 45.0000‡ 0.2192 45.0000 0.2461‡ 45.0000‡

Table 6.3 Numerical values of amplitude and phase angle of coefficient of currentdensity obtained for S= 0.0, 0.1 when Pm=0.1, Rd=50.0, θw=2.0,

Pr=0.1,against ξ by two methods.S=0.0 S=0.1


0.0 0.4289 0.0000 0.3977† 0.0000† 0.4193 0.0000 0.4017† 0.0000†0.2 0.4205 2.2042 0.4076† 2.6726† 0.4197 2.2185 0.4068† 2.6944†0.4 0.4216 4.4053 0.4172† 4.9846† 0.4209 4.4329 0.4220† 4.0047†0.6 0.4236 6.5960 0.4410† 6.9519† 0.4229 6.6350 0.4466† 6.6337†0.8 0.4264 8.7687 0.4733† 8.5056† 0.4258 8.8168 0.4799† 8.4105†1.0 0.4301 10.9106 0.5091† 10.6271† 0.4295 10.9649 0.5067† 10.4374†2.0 0.4613 20.7669 - - 0.4610 20.8261 - -4.0 0.5691 34.2562 - - 0.5690 34.2922 - -6.0 0.7041 41.0385 0.7754‡ 44.3000‡ 0.7014 40.9730 0.7754‡ 44.3000‡8.0 0.8342 43.9551 0.8953‡ 44.3000‡ 0.8342 43.9714 0.8953‡ 44.3000‡10.0 0.9568 45.0000 1.0000‡ 44.3000‡ 0.9568 45.0000 1.0000‡ 44.3000‡

Here, † and ‡ are stands for small and large values of ξ respectively.

106


6.2.2.2 When parameter ξ is large

To find the solution for (ξ ≥ 1) in downstream regime, the magnitude analysis of

various largest terms F ′′′ and ξF ′′, Φ′′′ and ξΦ′′ and θ′′1 , ξθ′1 in Eqn. (6.2.25) is

required. For this purpose it is necessary to find the appropriate scaling for, F ,

Φ, θ1 and η. On balancing F ′′′ and ξF ′′ , φ′′′ and ξφ′′, θ′′ and ξθ′ in Eqn. (6.2.25),

it is found that η = O(ξ−1/2), F = O(ξ3/2) and Φ = O(ξ3/2). Here, we introduced

the following transformations:

η = ξ−12 Y, F = ξ

32 F (ξ, η)

Φ = ξ32 Φ(ξ, η), θ = Θ(ξ, η) (6.2.39)

by using Eqn. (6.2.39)into Eqn. (6.2.25), we obtained the following set of equa-

tions:

F ′′′ − iF ′ + Θ +3

4ξ−

12 f0F

′′ − 1

2ξ−1f ′F ′

− S

(3

4ξ−

12 φ0Φ

′′ +3

4ξ−

32 φ′′0Φ− ξ−1φ′0Φ

′)

=1

2

[f ′0

(∂F ′

∂ξ+

η

2ξF ′′

)− ξ−

12 f0

(∂F

∂ξ+

η

2ξF ′

)]

− 1

2S

[φ′0

(∂Φ′

∂ξ

)+

1

2ξ−1φ0

(∂Φ′

∂ξ+

η

2ξΦ′

)]

(6.2.40)

1

PmΦ′′′ − iΦ′ +

3

4ξ−

12

(f0Φ

′′ + φ′′0F)

+1

2ξ−1

(f ′0Φ

′ − φ′0F′)

=1

2f ′0

(∂Φ

∂ξ+

η

2ξΦ′

)− 1

2φ′0

(∂F

∂ξF ′′

)

+1

2ξ−

12

[f ′′0

(∂Φ

∂ξ+

η

2ξΦ′

)− φ′′0

(∂F

∂ξ+

η

2ξF ′

)](6.2.41)

1

Pr

[1 +

4

3Rd

(1 + (θw − 1)Θ)3Θ′]′− iΘ +

3

4ξ−

12 f0Θ

′ − 1

2ξ−1f ′0Θ

=1

2

[f ′

(∂Θ

∂ξ+

η

2ξΘ′

)− ξ−

12 f0

(∂F

∂ξ+

η

2ξF ′

)] (6.2.42)

Boundary equations to be satisfied by the above equations are

F (ξ, 0) = F ′(ξ, 0) = 0, Φ(ξ, 0) = 0, Φ′(ξ, 0) = 1, Θ(ξ, 0) = 1

107


F ′(ξ,∞) = 0, Φ′(ξ,∞) = 0, Θ(ξ,∞) = 0 (6.2.43)

In the region near the surface the function f0, φ0 and θ0, can be represented with

good accuracy by

f0 = a2η2 + a3η

3 + a4η4 + .....

φ0 = η + c1η2 + c2η

3 + c3η4 + .....

θ0 = 1 + b1η + b2η2 + b3η

3 + .....

(6.2.44)

where according to the equations() to ()

a2 =1

2f ′′0 (0), a3 =

1

6, a4 =

Sφ′′0(0)− θ′0(0)

24

c1 =1

2φ′′0(0)

b1 = θ′0(0), b2 = −3α14Rd(1 +4)2θ′201 + α1(1 +4)3

(6.2.45)

By using Eqn. (6.2.45) into Eqn. (6.2.44), we can find the solution of variables

f0, φ0 and θ0 of the steady part of the problem. Now we are at position to expand

the functions F , Φ, Θ in powers of ξ−12 as follows:

F (ξ, η) =∞∑i=1

ξ−1/2Fi(η), Φ(ξ, η) =∞∑i=1

ξ−1/2gi(η), Θ(ξ, η) =∞∑i=1

ξ−1/2Θi(η)

(6.2.46)

Substituting in above and equating the coefficients of equal powers of ξ from both

sides we have

F ′′′0 − iF ′

0 + Θ0 = 0 (6.2.47)

g′′′0 − iPmg′0 = 0 (6.2.48)

[1 + α1 (1 + ∆Θ0)

3] Θ′′0 + 3α∆ (1 + ∆Θ0)

2 Θ′20 − iPrΘ0 = 0 (6.2.49)

and the boundary conditions are

F0(0) = F ′0(0) = 0, Φ0 = 0, Φ′

0(0) = 1, Θ0(0) = 1

F ′0(∞) = 0, Φ0(∞) = 0, Θ0(∞) = 0 (6.2.50)

Solutions for F0, g0 and Θ0 are given as

F ′0(η) =

1

1− Pr

(e−

√iPrη − e−

√iPrη

)

g′0(η) = e−iPmη

Θ0(η) =e−iPrη

[1 + α1 (1 + ∆Θ0)

3](6.2.51)

108


from which we see that for large ξ

F ′′0 (0) =

1− i

1 +√

Pr√

2ξ

g′0(0) = −(1 + i)

√ξPm

2

Θ′0(0) = −

(1 + i)√

ξPr2[

1 + α1 (1 + ∆Θ0)3]

(6.2.52)

In Eqn. (6.2.52), we will separate real and imaginary parts, we can find the

solution of fluctuating part of the problem for coefficients of rate of heat transfer,

skin friction and current density in terms of amplitude and phase angle for large

values of ξ. Hence defining

τ = Gr−3/4L x−1/4Cf (6.2.53)

J = Gr−3/4L x−1/4Jw (6.2.54)

Q = Gr1/4L x1/4Nu = −

(1 +

4

3Rd

θ3w

)Θ′(0) (6.2.55)

As =√

τ 2r + τ 2

i , Am =√

J2r + J2

i , At =√

Q2r + Q2

i

φs = tan−1

(τi

τr

), φm = tan−1

(Ji

Jr

), φt = tan−1

(Qi

Qr

)(6.2.56)

The quantities (τr, τi), (Jr, Ji) and (Qr, Qi) are the real and imaginary part of the

coefficients of skin friction, current density and the rate of heat transfer.

The results obtained by Eqn. (6.2.56) are given in Tables 6.1-6.3 for large

values of ξ and compared with the solution that obtained by finite difference

method and found to be in reasonable agreement.


In this section we shall briefly explain the physical behavior of different physical

parameters on coefficients of skin friction, rate of heat transfer and current density

in terms of amplitude and phase angle. Moreover the effects of these parameters

are also exhibit in terms of transient coefficients of skin friction, rate of heat

transfer and current density.

109


6.3.1 Effects of physical parameters upon amplitude and

phase of rate of heat transfer, coefficient of skin fric-

tion and current density

Figures 6.2(a-b), 6.3(a-c) and 6.4(a-b)) shows the effect of different values of radi-

ation parameter Rd = 1.0, 2.5, 5.0, 10.0 when Pm = 0.5,S = 0.3, Pr = 0.71, and

ratio of wall temperature to ambient fluid temperature is chosen θw= 0.5. From

this analysis it is concluded that with the increase of Rd the amplitude and phase

angle of heat transfer increases where amplitude and phase angle of coefficient

of skin friction decreases and there is no prominent change seen for the case of

coefficient of current density.

X

Am

plitu

deof

heat

tran

sfer

(At)

0.0 2.0 4.0 6.0 8.0 10.0

0.5

1.0

1.5

2.0

2.5

1.02.55.010.0

Rd

Pm = 0.5S = 0.3Pr = 0.71

θw = 0.5

X

Pha

sean

gle

ofhe

attr

ansf

er(

φt)

10-2 10-1 100 101

0.0

10.0

20.0

30.0

40.0

50.0

1.02.55.010.0

Rd

Pm = 0.5S = 0.3Pr = 0.71

θw = 0.5

Fig. 6.2 Numerical solution of amplitude and phase angle of heat transfer fordifferent values of Rd=1.0, 2.5, 5.0, 10.0 while Pr = 0.71, Pm= 0.5, θw = 0.5 and

S=0.3

X

Am

plitu

deof

coef

fici

ents

kin

fric

tion

(As)

0.0 2.0 4.0 6.0 8.0 10.00.1

0.2

0.3

0.4

0.5

0.6

1.02.55.010.0

Rd

Pm = 0.5S = 0.3

Pr = 0.71θw = 0.5

X

Pha

sean

gle

ofco

effic

ient

skin

fric

tion

(φ

s)

0.0 2.0 4.0 6.0 8.0 10.0-50.0

-45.0

-40.0

-35.0

-30.0

-25.0

-20.0

1.02.55.010.0

Rd

Pm = 0.5S = 0.3Pr = 0.71θw = 0.5

Fig. 6.3 Numerical solution of amplitude and phase angle of coefficient of skinfriction for different values of Rd=1.0, 2.5, 5.0, 10.0 while Pr = 0.71, Pm= 0.5,

θw = 0.5 and S=0.3

110


X

Am

plitu

deof

curr

entd

ensi

ty(Αm

)

10-2 10-1 100 1010.0

0.5

1.0

1.5

2.0

2.5

1.02.55.010.0

Rd

Pm = 0.5S = 0.3Pr = 0.71θw = 0.5

X

Pha

sean

gle

ofcu

rren

tden

sity

(φ m

)

10-2 10-1 100 101

0.0

10.0

20.0

30.0

40.0

50.0

1.02.55.010.0

Rd

Pm = 0.5S = 0.3Pr = 0.71θw = 0.5

Fig. 6.4 Numerical solution of amplitude and phase angle of current density fordifferent values of Rd=1.0, 2.5, 5.0, 10.0 while Pr = 0.71, Pm= 0.5, θw = 0.5 and

S=0.3

X

Am

plitu

deof

heat

tran

sfer

(A t)

10-2 10-1 100 101

0.5

1.0

1.5

2.0

2.5

0.10.30.50.7

Pm

Rd = 10.0S = 0.2Pr = 0.71

θw = 0.5

X

Pha

sean

gle

ofhe

attr

ansf

er(

φt)

10-2 10-1 100 101

0.0

10.0

20.0

30.0

40.0

50.0

0.10.30.50.7

Pm

Rd = 10.0S = 0.2Pr = 0.71

θw = 0.5

Fig. 6.5 Numerical solution of amplitude and phase angle of heat transfer fordifferent values of Pm=0.1, 0.3, 0.5, 0.7 while Pr = 0.71, Rd= 10.0, θw = 0.5 and

S=0.2

To explain this phenomena physically, we mark this trend with the understand-

ing that when Rd increases, the ambient fluid temperature decreases and according

to the Fourier law of heat transfer the flow of heat is towards the ambient fluid

and at the surface the fluid motion is slowdown and it is also pertinent to mention

that the role of Rd for the case of current density is very poor thus the amplitude

and phase of heat transfer is dominant over other physical quantities. From Figs.

6.5(a-b)-6.7(a-b), we can see the effects of different values of Pm.

From these figures it is clear that with the increases of Pm there is very slight

change is noted in amplitude and phase angle of heat transfer but amplitude and

phase angle of skin friction decreases where amplitude and phase angle of current

density increases prominently that can be seen from Figs. 6.7(a-b).

The reason is that with the increase of Pm the induced current with in the

111


X

Am

plitu

deof

coef

fici

ento

fsk

infr

ictio

n(A

s)

2.0 4.0 6.0 8.0 10.0

0.2

0.3

0.4

0.5

0.6

0.7

0.10.30.50.7

Pm

Rd = 10.0S = 0.2Pr = 0.71

θw = 0.5

X

Pha

sean

gle

ofco

effic

ient

ofsk

infr

ictio

n(

φs)

2.0 4.0 6.0 8.0 10.0-50.0

-45.0

-40.0

-35.0

-30.0

-25.0

-20.0

0.10.30.50.7

Pm

Rd = 10.0S = 0.2Pr = 0.71

θw = 0.5

Fig. 6.6 Numerical solution of amplitude and phase angle of coefficient of skinfriction for different values of Pm=0.1, 0.3, 0.5, 0.7 while Pr = 0.71, Rd= 10.0,

θw = 0.5 and S=0.2

X

Am

plitu

deof

coef

fici

ento

fcu

rren

tden

sity

(Am

)

0.0 2.0 4.0 6.0 8.0 10.0

0.5

1.0

1.5

2.0

2.5

3.00.10.30.50.7

Pm

Rd = 10.0S = 0.2Pr = 0.71

θw = 0.5

X

Pha

sean

gle

ofco

effi

cien

tof

curr

entd

ensi

ty(φ

m)

0.0 2.0 4.0 6.0 8.0 10.00.0

10.0

20.0

30.0

40.0

50.0

0.10.30.50.7

Pm

Rd = 10.0S = 0.2Pr = 0.71

θw = 0.5

Fig. 6.7 Numerical solution of amplitude and phase angle of coefficient ofcurrent density for different values of Pm=0.1, 0.3, 0.5, 0.7 while Pr = 0.71, Rd=

10.0, θw = 0.5 and S=0.2

boundary layer tends to spread away from the surface and this results in thickening

of the boundary layer, thus the amplitude and phase of current density increases

for the case of natural convection. From Figs. 6.8(a-b), 6.9(a-b) and 6.10(a-c), it

is is observed that with the increase of the ratio of wall temperature to ambient

fluid temperature the amplitude and phase of rate of heat transfer decreases where

the amplitude and phase of coefficient of skin friction increases and there is active

change seen for the case of the amplitude and phase of current density.

The amplitude and phase angle of coefficients of rate of heat transfer, skin

friction and current density for different values of S is given in Tables 6.1-6.3.

From these tables, it is found that the amplitude and phase angle of heat transfer

decreases and similarly the amplitude and phase angle of the coefficient of skin

friction is also decreases. It is also evident from table 3. that the amplitude

112


X

Am

plitu

deof

heat

tran

sfer

(At)

0.0 2.0 4.0 6.0 8.0 10.0

0.5

1.0

1.5

2.0

2.5

3.00.00.51.52.0

θω

Rd = 1.0S = 0.2Pr = 0.71

Pm = 0.5

X

Pha

seof

heat

tran

sfer

(A t)

10-2 10-1 100 101

0.0

10.0

20.0

30.0

40.0

50.0

0.00.51.52.0

θω

Rd = 1.0S = 0.2Pr = 0.71

Pm = 0.5

Fig. 6.8 Numerical solution of amplitude and phase angle of heat transfer fordifferent values of θw=0.0, 0.5, 1.5, 2.0 while Pr = 0.71, Pm= 0.5, Rd = 1.0 and

S=0.2

X

Am

plitu

deof

coef

fici

ento

fsk

infr

ictio

n(A

s)

0.0 2.0 4.0 6.0 8.0 10.00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00.51.52.0

θω

Rd = 1.0S = 0.2Pr = 0.71

Pm = 0.5

X

Pha

seof

coef

ficie

ntof

skin

fric

tion

(φs)

0.0 2.0 4.0 6.0 8.0 10.0-50.0

-45.0

-40.0

-35.0

-30.0

-25.0

-20.0

0.00.51.52.0

θω

Rd = 1.0S = 0.2Pr = 0.71

Pm = 0.5

Fig. 6.9 Numerical solution of amplitude and phase angle of coefficient of skinfriction for different values of θw=0.0, 0.5, 1.5, 2.0 while Pr = 0.71, Pm= 0.5, Rd

= 1.0 and S=0.2

and phase angle of coefficient of current density increases. This situation happen

because the imposition of magnetic field parameter decelerates the motion of the

fluid that thicken the boundary layer thickness and generate the magnetic current,

for this reason the amplitude and the phase angle of coefficients of rate of heat

transfer and skin friction decreases and the amplitude and phase angle of current

density increases.

6.3.2 Effects of physical parameters upon transient rate of

heat transfer, shear stress and current density

In the present section, we are going to explain the physical profiles of transient

rate of heat transfer, skin friction and current density at the surface of vertical

113


X

Am

plitu

deof

coef

ficie

ntof

curr

entd

ensi

ty(A m

)

10-2 10-1 100 1010.0

0.5

1.0

1.5

2.0

2.5

0.00.51.52.0

θω

Rd = 1.0S = 0.2Pr = 0.71

Pm = 0.5

X

Pha

seof

coef

ficie

ntof

curr

entd

ensi

ty(

φm

)

10-2 10-1 100 1010.0

10.0

20.0

30.0

40.0

50.0

0.00.51.52.0

θω

Rd = 1.0S = 0.2Pr = 0.71

Pm = 0.5

Fig. 6.10 Numerical solution of amplitude and phase angle of coefficient ofcurrent density for different values of θw=0.0, 0.5, 1.5, 2.0 while Pr = 0.71, Pm=

0.5, Rd = 1.0 and S=0.2.

plate, for this purpose we define the following relations:

τt = [τt0 + εAt cos(τ + φt)]

τs = [τs0 + εAs cos(τ + φs)]

τm = [τm0 + εAm cos(τ + φm)] (6.3.1)

It is necessary to mention that τt0, τs0 and τm0 are rate of heat transfer, skin fric-

tion and current density that comes from steady part, and similarly (At, As, Am)

and (φt, φs, φm) are amplitudes and phases angle of rate of heat transfer, skin

friction and current density comes from fluctuating part and ε is small amplitude

oscillation.

τ

Tra

nsie

ntra

teof

heat

tran

sfer

(τt)

0.0 10.0 20.0 30.0 40.0 50.00.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46 Rd

Pm = 0.8, S = 0.2, Pr = 0.71, θw= 1.1,ε = 0.05

1.0

5.0

10.0

(a) τTra

nsie

ntco

effi

cien

tof

skin

fric

tion

(τs)

0.0 10.0 20.0 30.0 40.0 50.00.8

1

1.2

1.4

1.6

1.8

2Rd

Pm = 0.8, S = 0.2, Pr = 0.71, θw= 1.1,ε = 0.05

1.0

5.0

10.0

(b)

Fig. 6.11 Numerical solution of transient (a) rate of heat transfer (b) coefficientof skin friction for different values of Rd=1.0, 5.0, 10.0 while Pr = 0.71, Pm=

0.8, θw = 1.1 and S=0.2, ξ = 10.0 and ε = 0.05

114


τ

Tra

nsie

ntra

teof

heat

tran

sfer

(τt)

0.0 10.0 20.0 30.0 40.0 50.00.250.260.270.280.290.3

0.310.320.330.340.350.360.37

S

Pm = 0.8, Rd = 1.0, Pr = 0.71, θw= 1.1,ε = 0.05

0.8

0.4

0.0

(a) τ

Tra

nsie

ntco

effi

cien

tof

skin

fric

tion

(τs)

0.0 10.0 20.0 30.0 40.0 50.01.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65S

Pm = 0.8, Rd = 1.0, Pr = 0.71, θw= 1.1,ε = 0.05

0.0

0.4

0.8

(b)

Fig. 6.12 Numerical solution of transient (a) rate of heat transfer (b) coefficientof skin friction for different values of S=0.0, 0.4, 0.8 while Pr = 0.71, Pm= 0.8,

θw = 1.1 and Rd=1.0, ξ = 10.0 and ε = 0.05

The effect of radiation parameter on coefficients of transient rate of heat trans-

fer and skin friction are shown in Figs. 6.11(a-b) with other parameter fixed. It

is observed that, the transient rate of heat transfer increases and transient coef-

ficient of skin friction reduces with the increase of parameter Rd. Figs. 6.12(a-b)

illustrated that the coefficients of transient rate of heat transfer and skin friction

reduces very prominently against dimensionless time τ with the increase of mag-

netic force parameter S. The effect of Pm on the coefficients of skin friction and

current density have been exhibited in Figs. 6.13(a-b). It is observed that the

increase in parameter Pm increases the transient coefficient of skin friction and

reduces the transient coefficient of current density. Figs. 6.(a-b) displays the effect

of dimensionless parameter ξ on coefficients of transient rate of heat transfer and

skin friction. From these figures, it is noted that with the increase of ξ in down

stream the transient rate of heat transfer and skin friction reduces in the case of

natural convection.

6.4 Conclusion

In this study, emphasis was given on the effect of different physical parameters

on chief physical quantities those are very important in the field of mechanical

engineering such as coefficients of rate of heat transfer, skin friction and current

density. From the brief study of figures and tables, our findings are given as:

115


τ

Tra

nsie

ntce

ffic

ient

ofsk

infr

ictio

n(τ

s)

0.0 10.0 20.0 30.0 40.0 50.00.5

1

1.5

2

2.5

Pm

S = 0.8, Rd = 1.0, Pr = 0.71, θw= 1.1,ε = 0.05

1.0

0.5

0.1

(a) τ

Tra

nsie

ntce

ffici

ento

fcur

rent

dens

ity(

τ m)

0.0 10.0 20.0 30.0 40.0 50.0

-4

-3

-2

-1

0

1

2

Pm

S = 0.8, Rd = 1.0, Pr = 0.71,θw= 1.1,ε = 0.05

1.0

0.5

0.0

(b)

Fig. 6.13 Numerical solution of transient (a) rate of heat transfer (b) coefficientof current density for different values of Pm=0.1, 0.5, 1.0 while Pr = 0.71, S=

0.8, θw = 1.1 and Rd=1.0, ξ = 10.0 and ε = 0.05

τ

Tra

nsie

ntra

teof

heat

tran

sfer

(τt)

0.0 10.0 20.0 30.0 40.0 50.0

0.322

0.324

0.326

0.328

0.33 ξ

S = 0.4, Rd = 1.0, Pr = 0.71, θw= 1.1,Pm = 0.8, ε = 0.05

10.0

2.5

0.1

(a)τ

Tra

nsie

ntce

ffic

ient

ofsk

infr

ictio

n(τ

s)

0.0 10.0 20.0 30.0 40.0 50.0

1.48

1.5

1.52

1.54

ξ

S = 0.4, Rd = 1.0, Pr = 0.71, θw= 1.1,Pm = 0.8, ε = 0.05

10.0

2.5

0.1

(b)

Fig. 6.14 Numerical solution of transient (a) rate of heat transfer (b) coefficientof skin friction for different values of ξ=1.0, 2.5, 10.0 while Pr = 0.71, S= 0.8, θw

= 1.1 and Rd=1.0, Pm = 0.8 and ε = 0.05

It is observed that with the increase of conduction radiation parameter, the

amplitude and phase angle of heat transfer increases but coefficient of skin friction

decreases. It is also noted that the transient rate of heat transfer increases and skin

friction in terms of amplitude and phase angle reduces as the radiation conduction

parameter increases. It is concluded that the amplitude and phase angle of rate

of heat transfer have no significance change with the increase of magnetic Prandtl

number. There is very active increase for the case of amplitude and phase angle

of coefficients of skin friction and current density is noted with the increase Pm.

The transient coefficient of skin friction increases and coefficient of current density

decreases with the increase of Pm. It is also observed that the amplitude and phase

of heat transfer is decreased with the increase of parameter θw and amplitude and

116


phase angle of coefficient of skin friction are increasing with the increase of ratio

of the surface temperature to the ambient fluid temperature. The coefficients

of the rate of heat transfer and skin friction in terms of amplitude and phase

angle decrease and current density increases with the increase of magnetic force

parameter S. The asymptotic solutions for small and large values of dimensionless

streamwise coordinate, ξ for different values of magnetic force parameter S when

other parameters are fixed that to be found in reasonable agreement with those

which are obtained by finite difference method for entire value of ξ.

117

118

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