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
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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
Fluctuating Hydromagnetic Flow of Viscous
Incompressible Fluid past a Magnetized Heated
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
CIIT/FA08-PMT-007/ISB
Spring, 2012
iii
Fluctuating Hydromagnetic Flow of Viscous
Incompressible Fluid past a Magnetized Heated
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
Fluctuating Hydromagnetic Flow of Viscous
Incompressible Fluid past a Magnetized Heated
Surface
By
Muhammad Ashraf
CIIT/FA08-PMT-007/ISB
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 Methods of solution 50
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 Results and discussion 58
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 Methods of solution 71
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
5.3 Results and discussion 82
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
Figure 3.3 Numerical solution of (a) coefficient of skin friction (b) rate of heat
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
Figure 3.4 Numerical solution of (a) coefficient of skin friction (b) rate of heat
Figure 3.5 Numerical solution of (a) coefficient of skin friction (b) rate of heat
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
Figure 4.3 The behavior of coefficients of (a) skin friction (b) rate of heat
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
Figure 4.5 The behavior of coefficients of (a) skin friction (b) rate of heat
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
Figure 4.8 Velocity (b) temperature distribution (c) transverse component of
magnetic field for different values of Prandtl number Pr 63
Figure 4.9 Velocity (b) temperature distribution (c) transverse component of
magnetic field for different values of magnetic Prandtl number Pm 64
Figure 4.10 Velocity (b) temperature distribution (c) transverse component of
magnetic field for different values of transpiration parameterξ 64
Figure 5.1 The coordinate system and flow configuration 69
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
different values of radiation parameter dR 83
Figure 5.4 Numerical solution of amplitude of phase angle of current density for
different values of radiation parameter dR 84
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
Figure 5.15 Solution for transient (a) heat transfer (b) shear stress (c) current
density for different values of magnetic Prandtl number mP
89
Figure 5.16 Solution for transient (a) heat transfer (b) shear stress (c) current
density for different values of Prandtl number Pr
90
Figure 5.17 Solution for transient (a) heat transfer (b) shear stress (c) current
density for different values of radiation parameter dR
90
Figure 5.18 Solution for transient (a) heat transfer (b) shear stress (c) current
density for different values of mixed convection parameter λ
91
Figure 5.19 Solution for transient (a) heat transfer (b) shear stress (c) current
density for different values of surface temperature wθ
91
Figure 6.1 The coordinate system and flow configuration 95
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
Figure 6.5 Numerical solution of amplitude of phase angle of rate of heat transfer
for different values of magnetic Prandtl number mP
111
Figure 6.6 Numerical solution of amplitude of phase angle of coefficient of skin
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
Figure 6.8 Numerical solution of amplitude of phase angle of rate of heat transfer
Figure 6.9 Numerical solution of amplitude of phase angle of coefficient of skin
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
Figure 6.12 Numerical solution of transient (a) rate of heat transfer (b)
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
Figure 6.14 Numerical solution of transient (a) rate of heat transfer (b)
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
temperature wθ by two methods
57
Table 4.3 Numerical values of coefficient of current density obtained for surface
temperature wθ by two methods
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
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
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.
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.
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
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.
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),
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
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
Fig. 4.9 (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 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)
Fig. 4.10 (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 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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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,
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
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
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
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=
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:
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=
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