Department of Mathematics Quaid-i-Azam University

On stretched flows of rate type fluids

By

Sabir Ali Shehzad

Department of Mathematics Quaid-i-Azam University

Islamabad, Pakistan 2014


By

Sabir Ali Shehzad

Supervised By

Prof. Dr. Tasawar Hayat




By

Sabir Ali Shehzad

A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT OF THE REQUIREMENT

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

Supervised By

Prof. Dr. Tasawar Hayat



Contents

1 Basics of fluid mechanics 5

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Fundamental laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Law of conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 Law of conservation of linear momentum . . . . . . . . . . . . . . . . . . . 10

1.3.3 Equation of heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Boundary layer equations of rate type fluids . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 Maxwell fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Oldroyd-B fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.3 Jeffrey fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Homotopy analysis method (HAM) . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Steady flow of Maxwell fluid with convective boundary conditions 18

2.1 Governing problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Homotopy analysis solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Convergence of the homotopy solutions . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Graphical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Flow of Maxwell fluid subject to power law heat flux and heat source 31

3.1 Problems development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


1


3.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 On radiative flow of Maxwell fluid with variable thermal conductivity 48

4.1 Governing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Solutions employing HAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 On three-dimensional flow of Maxwell fluid over a stretching surface with

convective boundary conditions 61


5.2 Series solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Convergence analysis and discussion of results . . . . . . . . . . . . . . . . . . . . 66


6 MHD three-dimensional flow of Maxwell fluid with variable thermal conduc-

tivity and heat source/sink 78

6.1 Mathematical formulation of the problems . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3 Convergence analysis and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Hydromagnetic steady flow of Maxwell fluid over a bidirectional stretching

surface with prescribed surface temperature and prescribed surface heat flux 93

7.1 Flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94


7.3 Convergence of series solutions and discussion . . . . . . . . . . . . . . . . . . . . 97


2

8 Three-dimensional flow of an Oldroyd-B fluid over a surface with convective

boundary conditions 110

8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110


8.3 Convergence analysis and discussion of results . . . . . . . . . . . . . . . . . . . . 115

8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

9 Radiative flow of Jeffrey fluid in a porous medium with power law heat flux

and heat source 129





9.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

10 Radiative flow of Jeffrey with variable thermal conductivity in porous medium142

10.1 Mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


10.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


11 Influence of thermophoresis and Joule heating on the radiative flow of Jeffrey

fluid with mixed convection 153

11.1 Flow formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153


11.3 Convergence analysis and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 158

11.4 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

12 Three-dimensional flow of Jeffrey fluid with convective surface boundary con-

ditions 173

12.1 Statement of the problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173


3




13 Three-dimensional flow of Jeffrey fluid over a bidirectional stretching surface

with heat source/sink 187

13.1 Heat transfer analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187





4

Chapter 1

Basics of fluid mechanics

1.1 Introduction

This chapter consists of literature survey for rate type fluids. Review of previous related stud-

ies for heat transfer analysis with thermal radiation, heat generation/absorption and variable

thermal conductivity is made. Constitutive equations of Maxwell, Oldroyd-B and Jeffrey fluids

are include. The boundary layer equations for two and three-dimensional flows of rate type

fluids are also given.

1.2 Background

Navier-Stokes equations are inadequate to characterize the rheological properties of complex

fluids involve in industrial processes. Examples of such fluids include polymer solutions, paints,

certain oils, asphalt, mud etc. Also these materials are diverse in the characteristics. Hence

different constitutive equations were developed to predict the rheological characteristics of such

materials. Further there are many rheological complex fluid models which do not show the

characteristics of relaxation and retardation times. The models presented in the literature are

mainly classified into three categories namely the differential, rate and integral types. The

differential and rate type fluid models are utilized to predict the response of the materials

which have slight memory like dilute polymeric solutions. On the other hand, the integral

type fluid models are used to describe the characteristics of the fluids which have considerable

5

memory such as polymeric melts. There are many non-Newtonian fluid models like second,

third and fourth grades fluid but these fluid models are unable to predict the properties of

relaxation/retardation times. To predict such characteristics, Maxwell, Oldroyd-B and Jeffrey

fluid models [1] were developed. These fluid models are very popular amongst the researchers.

These models are known as rate type non-Newtonian fluids. Such fluids occur mainly in most

polymeric and biological liquids.

Rajagopal and Srinivasa [2] developed a thermodynamic approach for modelling a class of

rate fluids. Rajagopal [3] presented exact solutions for unidirectional flow of an Oldroyd-B

fluid between two infinite parallel plates. Tan and Xu [4] analyzed the unidirectional flow

of viscoelastic fluid with fractional Maxwell model. The flow is generated due to suddenly

moved surface. The plates are rotating about non-coincident axes. The flow of Maxwell fluid

in a channel with suction was presented by Choi et al. [5]. Zierep and Fetecau [6] discussed

the Rayleigh-Sokes problem for unidirectional flow of Maxwell fluid with initial or boundary

conditions. They also investigated the Stokes second problem in this study. Fetecau et al. [7]

carried out a study to discuss the unsteady unidirectional flow of an Oldroyd-B fluid over a

plate. The flow is generated due to constantly accelerating plate. They developed the solution

corresponding to Maxwell, second grade and Oldroyd-B fluid fluids. Bergstrom [8] investigated

the hydrodynamic stability of Jeffrey fluid for small disturbance. The flow here is passed

through a circular cylinder. Peristaltic flow of Jeffrey fluid in a tube with magnetic field was

studied by Hayat and Nasir [9]. Peristaltic flow of an electrically conducting Jeffrey fluid in an

asymmetric channel was studied by Kothandapani and Srinivas [10]. They discussed the flow

phenomenon in the wave frame of reference which is moving with the velocity of the wave. Here

we mentioned some more studies [11-20] relevant to the unidirectional flows of rate type fluids.

The boundary layer flow generated by a stretching sheet is subject of abundant studies due

to its interesting and practical applications in the industrial and technological applications.

Examples of such applications are the boundary layer along the material handling conveyers

and along a liquid film in condensation, cooling of an infinite metallic plate in a cooling bath,

spinning of fibers, continuous casting, glass blowing, aerodynamic extrusion of plastic sheets,

continuous stretching of plastic films, etc. The extrude from a die is generally drawn and

simultaneously stretched into a sheet which is then solidified through gradual cooling by direct

6

contact. In such processes the characteristics of final product greatly depend on the rate of

cooling which is fixed by the structure of the boundary layer near the moving strip. Sakiadis

[21] introduced the concept of boundary layer over a moving solid surface. After Sakiadis,

Crane [22] investigated the boundary layer flow of viscous fluid over a stretching surface and

presented the closed form solutions. Mcleod and Rajagopal [23] explored the uniqueness of the

flow of Navier-Stokes fluid over a stretching sheet. The rotating flow generated by the stretching

surface was analyzed by Wang [24]. Wang [25] also discussed the three-dimensional boundary

layer flow of viscous fluid over a continuously stretching sheet. Unsteady flow of rotating viscous

fluid over a stretching sheet was studied by Rajeswari and Nath [26]. Ariel [27,28] discussed

the axisymmetric flow of viscous and second grade fluids generated by the stretching surface.

He provided the perturbation solution. Three-dimensional boundary layer flow of Newtonian

fluid by a stretching surface was examined by Ariel [29]. He computed exact and analytical

solutions for the nonlinear problem. Mahapatra et al. [30] discussed the boundary layer flow

of an incompressible viscoelastic fluid past a permeable stretching surface near an oblique

stagnation point. Liao [31] provided a new branch of solutions of boundary layer flow of viscous

fluid over a linearly stretching sheet. Three-dimensional flow of viscoelastic fluid induced by

stretching sheet was analytically addressed by Hayat et al. [32]. Ayub et al. [33] presented

the homotopic solutions of an incompressible stagnation point flow of second grade fluid past

a stretching surface. Abbas et al. [34] considered the hydromagnetic flow of viscoelastic fluid

over a stretching surface. Here the flow is induced due to the oscillation of stretching sheet.

Mahapatra et al. [35] presented an analysis to examine the magnetohydrodynamic (MHD)

stagnation point flow of power-law fluid past a stretching surface. MHD boundary layer flow

of micropolar fluid past a nonlinear stretching surface was studied by Hayat et al. [36]. They

developed the series solution for this analysis. Aïboud and Saouli [37] presented an analysis

to study the entropy generation effects in magnetohydrodynamic flow of non-Newtonian fluid

towards a stretching sheet. Influence of variable viscosity in an unsteady flow of viscous fluid

generated by stretching sheet was examined by Dandapat and S. Chakraborty [38]. Akyildiz et

al. [39] discussed the existence of solutions for third order nonlinear boundary value problems

over stretching surfaces. Ahmad and Asghar [40] addressed the effect of transverse magnetic

field in second grade fluid flow over a stretching surface.

7

Heat transfer in the flow induced by a stretching sheet is important in the industrial and

metallurgical processes like manufacture of plastic and rubber sheets, continuous cooling of

fiber spinning, annealing and thinning of copper wires and many others. In addition heat

transfer with thermal radiation has important applications in engineering and physics. Thermal

radiation effects are prominent when the process occur at high temperature. Such effects are

particularly involved in nuclear industry, missiles, satellites, propulsion devices for air-craft,

semiconductor wafers etc. Chamkha [41] presented a study to examine the effect of thermal

radiation in a fluid particle flow past a stretching surface. He presented the numerical solutions

for the considered flow problems. Cortell [42] discussed the flow of viscous fluid over a nonlinear

stretching surface with viscous dissipation and thermal radiation effects. Series solution for

the boundary layer radiative flow of viscous fluid by an exponentially stretching sheet was

given by Sajid and Hayat [43]. Hayat et al. [44] addressed the boundary layer radiative

flow of non-Newtonian second grade fluid with heat transfer past a linear stretching sheet.

Numerical solutions for the steady boundary layer flow of viscous fluid over a moving surface

with radiation effects were computed by Mukhopadhyay et al. [45]. Pal and Talukdar [46]

numerically investigated the effects of Joule heating and chemical reaction in MHD mixed

convection flow of viscous fluid over a permeable surface with porous medium and thermal

radiation. Radiative flow of micropolar fluid over a surface with heat and mass transfer effects

was analytically addressed by Hayat et al. [47]. Unsteady buoyancy-driven flow subject to

thermal and mass diffusion, heat and mass transfer, chemical reaction and Soret effects over

a surface was analytically examined by Pal and Talukdar [48]. Hayat et al. [49] provided

the series solution for the mixed convection flow of viscous fluid with thermal radiation and

variable free stream over an unsteady stretching surface. Motsumi and Makinde [50] studied the

boundary layer flow of nanofluid over a vertical flat surface in the presence of thermal radiation

and viscous dissipation.

Heat generation or absorption effects are quite prominent in the operations which involve

heat removal from nuclear fuel debris, underground disposal of radioactive waste material, dis-

associating fluids in packed-bed reactors, storage of food stuffs and many others. It is commonly

known fact that heat generation/absorption play a vital role in controlling the heat transfer

rate during the manufacturing processes. Magyari and Chamkha [51] provided the analytical

8

solutions for the effects of heat generation/absorption and first order chemical reaction in a

micropolar fluid flow over a uniformly permeable surface. Effects of heat source/sink and Hall

current in the flow of viscous fluid with heat and mass transfer over a continuously moving

surface with chemical reaction were considered by Saleem and El-Aziz [52]. Analytic solution

of MHD flow of two types of viscoelastic fluids over a stretching sheet with viscous dissipation

and internal heat generation was constructed by Chen [53]. In another study Chen [54] ad-

dressed the mixed convection power law fluid flow over a stretching surface in the presence of

magnetic field and internal heat generation/absorption. Natural convection flow with temper-

ature dependent viscosity over an inclined flat plate with heat source was studied by Siddiqa

et al. [55]. Van Gorder and Vajravelu [56] presented an analysis to examine the convective

heat transfer in an electrically conducting fluid over a stretching surface with suction/injection

and heat source/sink. Rana and Bhargava [57] obtained the numerical solutions for the flow

of nanofluid with heat generation/absorption. Series solutions for the stagnation point flow

of nanofluid with heat source/sink were constructed by Alsaedi et al. [58]. Noor et al. [59]

presented the numerical solutions for heat and mass transfer in MHD flow of viscous fluid over

an inclined surface with thermophoresis, Joule heating and heat source/sink. Soret and heat

generation effects in unsteady flow of an electrically conducting fluid over a permeable surface

were investigated by Turkyilmazoglu and Pop [60].

It is noted that all the above mentioned studies dealt with the constant thermal conductivity

but it is now proven that the thermal conductivity of the fluid varies linearly with temperature

from 00 to 4000 [61]. Heat transfer analysis in the boundary layer flow of viscous fluid over

a linear stretching surface with temperature dependent thermal conductivity was investigated

by Chiam [62]. Chiam [63] also examined the effect of temperature dependent thermal con-

ductivity in stagnation point flow of viscous fluid toward a stretched sheet. The influences

of temperature dependent viscosity and variable thermal conductivity on unsteady flow with

suction and injection over a vertical plate were discussed by Seddeek and Salama [64]. Sharma

and Singh [65] presented an analysis to investigate the magnetohydrodynamic flow with variable

thermal conductivity near a stagnation point past a stretching surface. Radiative flow of viscous

fluid in presence of temperature dependent thermal conductivity over non-isothermal stretched

sheet was analyzed by Vyas and Rai [66]. Aziz and Bouaziz [67] considered the fin problem with

9

thermal conductivity and heat generation/absorption. They presented the results by employing

least square method. Entropy generation analysis for steady state conduction and temperature

dependent thermal conductivity in presence of asymmetric thermal boundary conditions was

studied by Aziz and Khan [68]. Series solutions for magnetohydrodynamic flow of thixotropic

fluid with temperature dependent thermal conductivity were computed by Hayat et al. [69].

1.3 Fundamental laws

1.3.1 Law of conservation of mass

The law of conservation of mass or continuity equation can be expressed as

+∇ · (V) = 0 (1.1)

where represents the density of fluid and V the fluid velocity. Eq. (1.1) for an incompressible

fluid can be written as follows:

∇ ·V = 0 (1.2)

1.3.2 Law of conservation of linear momentum

Mathematically it can be expressed by

V

=∇ · τ+b (1.3)

For an incompressible flow τ = −pI+ S is the Cauchy stress tensor. Here is the pressure, Ithe identity tensor, S the extra stress tensor, b the body force and is the material time

derivative. The Cauchy stress tensor and the velocity field for three diemensional flow can be

written as

10

τ =

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦ (1.4)

V = [( ) ( ) ( )] (1.5)

where and are the normal stresses, and are shear

stresses and are the velocity components along the and −directions respectively.Equation (1.3) in scalar form can be expressed as

µ

+

+

+

¶=

()

+

()

+

()

+ (1.6)

µ

+

+

+

¶=

()

+

()

+

()

+ (1.7)

µ

+

+

+

¶=

( )

+

( )

+

()

+ (1.8)

in which , and show the components of body force along the and −axes,respectively.

The above equations for two-dimensional flow become

µ

+

+

¶=

()

+

()

+ (1.9)

µ

+

+

¶=

()

+

()

+ (1.10)

1.3.3 Equation of heat transfer

According to first law of thermodynamics the heat transfer equation can be written as

= τ · L−∇ · q1 + (1.11)

11

where = is the internal energy, the specific heat, the temperature, L =∇V the

velocity gradient, q1 = −∇ the heat flux, the thermal conductivity and the radiative

heating. The above equation in absence of radiative heating is given below

= τ ·∇V+∇2 (1.12)

1.4 Boundary layer equations of rate type fluids

1.4.1 Maxwell fluid

The extra stress tensor S for a Maxwell fluid can be expressed by the following relation

µ1 + 1

¶S = S+ 1

S

= A1 (1.13)

in which 1 is the relaxation time, the covariant differentiation, denotes the kinematic

viscosity andA1 the first Rivlin-Erickson tensor. The first Rivlin-Erickson tensor can be defined

as

A1 = gradV+ (gradV) 0 (1.14)

where 0 denotes the matrix transpose. For three-dimensional flow one obtains

A1 =

⎡⎢⎢⎢⎣2

+

+

+

2

+

+

+

2

⎤⎥⎥⎥⎦ (1.15)

For a tensor S of rank two, a vector b1 and a scalar we get

S

=

S

+ (V ·∇)S− S(gradV) 0 − (gradV)S (1.16)

b1

=

b1

+ (V ·∇)b1 − (gradV)b1 (1.17)

=

+ (V ·∇) (1.18)

12

Implementation of¡1 + 1

¢on Eq. (1.3), we have the following relations in the absence of

body force

µ1 + 1

¶V

= −

µ1 + 1

¶∇+

µ1 + 1

¶(∇ · S) (1.19)

By adopting the procedure as in ref. [1], we have

(∇·) = ∇ ·

µ

¶ (1.20)

Hence the above relations in absence of pressure gradient is

µ1 + 1

¶V

= (∇ ·A1) (1.21)

Components form of above equation for steady flow of Maxwell can be written as follows:

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+ 2 22

+2 2

+ 2 2

+ 2 2

⎞⎠ =

µ2

2+

2

2+

2

2

¶

(1.22)

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+2 2

2

+2 2

+ 2 2

+ 2 2

⎞⎠ =

µ2

2+

2

2+

2

2

¶

(1.23)

+

+

+ 1

⎛⎝ 2 22

+ 2 22

+ 2 22

+2 2

+ 2 2

+ 2 2

⎞⎠ =

µ2

2+

2

2+

2

2

¶

(1.24)

By using the boundary layer theory [70], the order of and is 1 and order of and is

The −momentum equation vanishes identically because it has order Hence the boundarylayer equations for three-dimensional flow of Maxwell fluid are

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+ 2 22

+2 2

+ 2 2

+ 2 2

⎞⎠ = 2

2 (1.25)

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+ 2 2

2

+2 2

+ 2 2

+ 2 2

⎞⎠ = 2

2 (1.26)

13

The boundary layer equation for two-dimensional flow of Maxwell fluid is given below

+

+ 1

µ2

2

2+ 2

2

2+ 2

2

¶=

2

2 (1.27)

1.4.2 Oldroyd-B fluid

The extra stress tensor for an Oldroyd-B fluid model can be expressed as

µ1 + 1

¶S = S+ 1

S

=

µ1 + 2

¶A1 (1.28)

where 2 denotes the retardation time and law of conservation of momentum in absence of

pressure gradient and body force can be written as

µ1 + 1

¶V

=

µ1 + 2

¶(∇ ·A1) (1.29)

The scalar forms of boundary layer equations in this case are

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+ 2 22

+2 2

+ 2 2

+ 2 2

⎞⎠=

⎛⎝2

2+ 2

⎛⎝ 32

+ 32

+ 33

−

22−

22−

22

⎞⎠⎞⎠ (1.30)

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+ 2 2

2

+2 2

+ 2 2

+ 2 2

⎞⎠=

⎛⎝2

2+ 2

⎛⎝ 32

+ 32

+ 33

−

22−

22−

22

⎞⎠⎞⎠ (1.31)

and the governing boundary layer equation for two-dimensional flow is

+

+ 1

µ2

2

2+ 2

2

2+ 2

2

¶=

⎛⎝2

2+ 2

⎛⎝ 32

+ 3

3

−

22−

22

⎞⎠⎞⎠

(1.32)

14

1.4.3 Jeffrey fluid

Extra stress tensor for a Jeffrey fluid can be mentioned below:

S =

1 + ∗

µA1 + 2

A1

¶ (1.33)

Here ∗ is the ratio of relaxation to retardation times. Further the extra stress tensor in

components form can be expressed as

=

1 + ∗

µ2

+ 2

µ

+

+

¶2

¶ (1.34)

=

1 + ∗

µµ

+

¶+ 2

µ

+

+

¶µ

+

¶¶= (1.35)

=

1 + ∗

µµ

+

¶+ 2

µ

+

+

¶µ

+

¶¶= (1.36)

=

1 + ∗

µ2

+ 2

µ

+

+

¶2

¶ (1.37)

=

1 + ∗

µµ

+

¶+ 2

µ

+

+

¶µ

+

¶¶= (1.38)

=

1 + ∗

µ2

+ 2

µ

+

+

¶2

¶ (1.39)

The law of conservation of momentum for a Jeffrey fluid model yields

µ

+

+

¶=

+

+

(1.40)

µ

+

+

¶=

+

+

(1.41)

µ

+

+

¶=

+

+

(1.42)

where the pressure gradient and body forces are neglected. By inserting the values of

and into Eqs. (1.40)-(1.42) and then utilizing the boundary

15

layer assumptions we finally get

+

+

=

1 + ∗

⎛⎝2

2+ 2

⎛⎝

2

+

2

+

22

+ 32

+ 32

+ 33

⎞⎠⎞⎠ (1.43)

+

+

=

1 + ∗

⎛⎝2

2+ 2

⎛⎝

2

+

2

+

22

+ 32

+ 32

+ 33

⎞⎠⎞⎠ (1.44)

Two-diemnsional boundary layer flow of Jeffrey fluid can be expressed by the equation

+

=

1 + ∗

µ2

2+ 2

µ

3

2+

3

3−

2

2+

2

¶¶ (1.45)

1.5 Homotopy analysis method (HAM)

Homotopy analysis method is an analytical tool to solve the nonlinear ordinary and partial dif-

ferential equations. According to Liao [71], this method distinguishes itself from other analytical

methods in the following three aspects.

1. It is valid for strongly nonlinear problems even if a given nonlinear problem does not

contain any small/large parameter.

2. It provides us with a convenient way to adjust the convergence region and rate of approx-

imation of the series solution.

3. It provides with freedom to use different base functions to approximate the solution of

nonlinear problem.

Let us consider a nonlinear differential equation

() + () = 0 (1.46)

where is a nonlinear operator, () is an unknown function to be determined and () is a

known function. The homotopic equation is

(1− )L [( )− 0()] = ~ { [( )− 0()]} (1.47)

16

in which 0() is an initial guess, L is an auxilliary linear operator, ~ is an auxilliary parameteror convergence control parameter, ∈ [0 1] is an embedding parameter and ( ) is an

unknown function. By employing Taylor’s series about one obtains

( ) = 0() +

∞X=1

() () =

1

!

( )

¯=0

(1.48)

The convergence of above series strictly depends upon ~ The value of ~ is chosen in such a

way that series solution is convergent at = 1. Substituting = 1 one obtains

() = 0() +

∞X=1

() (1.49)

The -th order deformation problems are

L [()− −1()] = ~R() (1.50)

where

=

⎧⎨⎩ 0 ≤ 11 1

(1.51)

R() =1

( − 1)! ×(

−1

−1

"0() +

∞X=1

()

#)=0

(1.52)

17

Chapter 2

Steady flow of Maxwell fluid with

convective boundary conditions

This chapter explores the steady flow of Maxwell fluid over a stretching surface. Heat transfer

is addressed using the convective boundary conditions. The arising nonlinear problems are

solved by employing homotopy analysis method (HAM). We computed the velocity, temperature

and Nusselt number. The role of embedded parameters on the velocity and temperature is

particularly analyzed. Physical interpretation is presented.

2.1 Governing problems

We consider the two-dimensional boundary layer flow of an incompressible Maxwell fluid bounded

by a continuously stretching sheet with heat transfer in a stationary fluid. We adopt that the

velocity of stretching sheet is () = (where is a real number). Further the constant

mass transfer velocity is taken as with 0 for injection and 0 for suction, respec-

tively. The convective boundary conditions are employed for the sheet. The − and −axes inthe Cartesian coordinate system are parallel and perpendicular to the sheet respectively. The

governing boundary layer equations for two-dimensional flow of Maxwell fluid are

+

= 0 (2.1)

18

+

=

2

2− 1

µ2

2

2+ 2

2

2+ 2

2

¶ (2.2)

+

=

2

2(2.3)

in which and denote the velocity components in the − and −directions, 1 the relaxationtime, the fluid temperature, the thermal diffusivity of fluid, = () the kinematic

viscosity, the density of fluid and the viscous dissipation is not accounted.

The boundary conditions are defined as

= () = = −

= ( − ) at = 0 (2.4)

= 0 = ∞ as →∞ (2.5)

where indicates the thermal conductivity of fluid, the convective heat transfer coefficient,

the wall heat transfer velocity and the convective fluid temperature below the moving

sheet.

We introduce the similarity transformations

= 0() = −√() () = − ∞ − ∞

=

r

(2.6)

Here is a constant and prime denotes the differentiation with respect to .

Equations (22)− (25) yield

000 + 00 − 02 + (2 0 00 − 2 000) = 0 (2.7)

00 + 0 = 0 (2.8)

= ∗ 0 = = 0 = −(1− (0)) at = 0 (2.9)

0 = 0 = 0 as =∞ (2.10)

where Eq. (21) is satisfied automatically and = 1 is the Deborah number ∗ = − √

is

the suction parameter, = is a parameter, =

is the Prandtl number, =

pis the

Biot number, is a constant and prime shows differentiation with respect to .

19

Expression of local Nusselt number is

=

( − ∞) (2.11)

where heat transfer is defined as

= −µ

¶=0

(2.12)

In dimensionless scale, Eq. (211) becomes

12 = −0(0)

2.2 Homotopy analysis solutions

We express and by a set of base functions [71-74]:

{ exp(−), ≥ 0 ≥ 0} (2.13)

as follows

() =

∞X=0

∞X=0

exp(−) (2.14)

() =

∞X=0

∞X=0

exp(−) (2.15)

in which and are the coefficients. We further select the following initial approximations

and auxiliary linear operators

0() = ∗ + ¡1− −

¢ 0() =

exp(−)1 +

(2.16)

L = 000 − 0 L = 00 − (2.17)

with

L (1 + 2 +3

−) = 0 L(4 +5−) = 0 (2.18)

20

where ( = 1− 5) denotes the arbitrary constants.The associated zeroth order deformation problems are

(1− )Lh(; )− 0()

i= ~N

h(; )

i (2.19)

(1− )Lh(; )− 0()

i= ~N

h(; ) ( )

i (2.20)

(0; ) = 0(0; ) = = 0(∞; ) = 0 0(0 ) = −[1− (0 )] (∞ ) = 0 (2.21)

N [( )] =3( )

3− ( )

2( )

2−Ã( )

!2

+

"2( )

( )

2( )

2− (( ))2

3( )

3

# (2.22)

N[( ) ( )] =2( )

2+Pr ( )

( )

(2.23)

Here is an embedding parameter, ~ and ~ the non zero auxiliary parameters and N and

N the nonlinear operators. Note that for = 0 and = 1 we have

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (2.24)

and when increases from 0 to 1 then ( ) and ( ) vary from 0() 0() to () and

() In view of Taylor’s series one can expand

( ) = 0() +∞P

=1

() (2.25)

( ) = 0() +∞P

=1

() (2.26)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(2.27)

where the convergence of above series strongly depends upon ~ and ~ Considering that ~

21

and ~ are selected properly so that Eqs. (225) and (226) converge at = 1 and thus one has

() = 0() +∞P

=1

() (2.28)

() = 0() +∞P

=1

() (2.29)

The problems at th-order are

L [()− −1()] = ~R () (2.30)

L[()− −1()] = ~R () (2.31)

(0) = 0(0) = 0(∞) = 0 0(0)− (0) = (∞) = 0 (2.32)

R () = 000−1() +

−1P=0

h−1− 00 − 0−1−

000

i+

−1X=0

−1−X=0

{2 0− 00 − − 000 (2.33)

R () = 00−1 +

−1P=0

0−1− (2.34)

=

⎡⎣ 0 ≤ 11 1

(2.35)

The general solutions can be expressed in the forms

() = ∗() + 1 + 2 +3

− (2.36)

() = ∗() + 4 + 5

− (2.37)

in which ∗ and ∗ indicate the special solutions.

22

2.3 Convergence of the homotopy solutions

Clearly the expressions (228) and (229) contain the nonzero auxiliary parameters ~ and ~

which can adjust and control the convergence of the homotopy solutions. For the range of

admissible values of ~ and ~ the ~−curves have been potrayed for 20-order of approxima-tions. Fig. 2.1 shows that the range of admissible values of ~ and ~ are −24 ≤ ~ ≤ −02and −21 ≤ ~ ≤ −04 The series converges in the whole region of when ~ = ~ = −14

-2.5 -2 -1.5 -1 -0.5 0Ñf, Ñq

-0.6

-0.5

-0.4

-0.3

-0.2

f''0

,q'0

Pr =1.0, a =0.3, S* =0.5, g = 1.0, b = 0.2

q'0f''0

Fig. 2.1: ~−curves for the functions () and ()

Table: 2.1. Convergence of homotopy solution for different order of approximations when

= 02 = 03 = 10 ∗ = 05 = 10 and ~ = ~ = −14

Order of approximation − 00(0) −0(0)1 0.2829900 0.4300000

5 0.2814982 0.4064811

10 0.2814950 0.4047923

20 0.2814950 0.4046587

30 0.2814950 0.4046572

35 0.2814950 0.4046572

40 0.2814950 0.4046572

23

2.4 Graphical results and discussion

In this section our main interest is to discuss the influence of emerging parameters such as

stretching parameter Deborah number suction parameter ∗ Prandtl number Pr and

Biot number on the velocity and temperature fields. The analysis of such variations is made

through the Figs. 22 − 29 Figs. 22 − 24 are displayed to see the effects of and ∗ on

the velocity field 0 As increases in Fig. 22 the flow velocity enhances. Fig. 2.3 shows the

effects of on 0 It is obvious from this Fig. that 0 is a decreasing function of This is due

to the fact that Deborah number depends upon the relaxation time and an increase in Deborah

number leads to an increase in the relaxation time. Such increase in relaxation time decrease

the fluid velocity and momentum boundary layer thickness. The same behavior is observed as

the suction parameter ∗ increases in Fig. 2.4. It is seen that the boundary layer thickness

decreases with increasing values of ∗ 0 In fact suction is an agent which resists the fluid

flow due to which the velocity is reduced. Figs. 2.5-2.9 depict the influences of ∗

and on the temperature profile Fig. 2.5 describes the effects of on Here decreases

when Pr increases. Physically, Prandtl number is the ratio of momentum to thermal diffusivity.

Higher values of Prandtl number implies the higher momentum diffusivity and lower thermal

diffusivity. This lower thermal diffusivity corresponds to a lower temperature and thinner

thermal boundary layer thickness. The proper value of Prandtl number is quite essential to

control the heat transfer in industrial processes. Fig. 2.6 indicates that is a decreasing

function of In Fig. 2.7 the variation of temperature is plotted for the different values of ∗

The temperature profile decreases by increasing ∗ Fig. 2.8 shows the influence of Biot number

on Temperature field enhances by increasing Here heat transfer coefficient is larger for

higher Biot number which gives rsie to the temperature and thermal boundary layer thickness.

Fig. 2.9 is plotted to see the effects of on temperature profile It has been seen from

this Fig. that temperature is an increasing function of Table 2.1 is computed to analyze the

convergence values of − 00(0) and −0(0) at different order of HAM approximations. This Table

depicts that less deformations are required for the velocity in comparison to the temperature

for a convergent solution. Table 2.2 includes the values of local Nusselt number for different

values and when ∗ = 05 and = 02 The values of local Nusselt number are larger

24

for higher values of and Such values are smaller for the higher values of

0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

f'h

S* = 0.5, b = 0.1

a = 1.0a = 0.6a = 0.3a = 0.0

Fig. 2.2: Influence of on 0()

0 2 4 6 8h

0

0.1

0.2

0.3

0.4

f'h

a = 0.4, S* = 0.5

b = 1.5b = 1.0b = 0.5b = 0.0


25

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

f'h

a = 0.4, b = 0.1

S* = 1.5S* = 1.0S* = 0.5S* = 0.0

Fig. 2.4: Influence of ∗ on 0()

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

qh

a = 0.4, S* = 0.5, b = 0.1, g = 1.0

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1

Fig. 2.5: Influence of on ()

26

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

0.6

qh

Pr = 0.7, S* = 0.5, b = 0.1, g = 1.0

a = 1.0a = 0.6a = 0.3a = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

qh

Pr = 0.7, a = 0.4, b = 0.1, g = 1.0

S* = 1.5S* = 1.0S* = 0.5S* = 0.0

Fig. 2.7: Influence of ∗ on ()

27

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

qh

Pr = 0.7, a = 0.4, S* = 0.5, b = 0.1

g = 1.5g = 0.8g = 0.4g = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

0.6

qh

Pr = 0.7, a = 0.4, S* = 0.5, g = 1.0

b = 1.5b = 1.0b = 0.5b = 0.0


28

Table 2.2: Values of local Nusselt number −12 for the parameters and when

∗ = 05 and = 02

−12

0.5 1.0 0.1 0.23336

1.0 0.36588

1.5 0.45796

2.0 0.52558

1.0 0. 0.3189

0.5 0.26799

1.0 0.36591

2.0 0.2039

0.1 0.36588

0.3 0.40466

0.8 0.45825

1.0 0.47254

2.5 Concluding remarks

Here we considered the effects of heat transfer in the flow of a Maxwell fluid over a stretching wall

with convective boundary conditions. The graphical results reflecting the effects of interesting

parameters are analyzed. The main results are as follows:

• By increasing the velocity field 0 increases.

• The velocity profile 0 decreases by increasing Deborah number and suction parameter∗

• Increase in Prandtl number decreases the temperature profile

• The effects of Biot number and Deborah number on are similar in a qualitative

sense.

• Increasing values of Biot number lead to higher temperature and thermal boundary layer

29

thickness.

30

Chapter 3

Flow of Maxwell fluid subject to

power law heat flux and heat source

The boundary layer flow of Maxwell fluid over a stretching sheet with power law heat flux and

heat source is studied in this chapter. An incompressible fluid fills the porous medium. The

governing partial differential equations are reduced into the ordinary differential equations by

applying similarity transformations. Series solutions of velocity and temperature are found by

adopting homotopy analysis method (HAM). Convergence of series solutions is verified. The

obtained results are examined by plotting graphs for the various parameters. Numerical values

of local Nusselt number for different parameters are computed and analyzed. It is found that

the numerical values of local Nusselt number decreases by increasing Deborah number It is

observed that effects of Prandtl number, suction/injection and heat generation parameters on

the local Nusselt number are opposite to that of the Deborah number.

3.1 Problems development

We consider the two-dimensional flow of an incompressible Maxwell fluid over a moving porous

surface with power law heat flux and heat source. A Cartesian coordinate system is chosen in

such a way that −axis is along the stretching surface and the −axis perpendicular to it. Thefluid fills the porous half space 0. In accordance with the boundary layer approximations,

31

the governing equations for flow and temperature are

+

= 0 (3.1)

+

=

2

2− 1

∙2

2

2+ 2

2

2+ 2

2

¸−

(3.2)

+

=

2

2−

( − ∞) (3.3)

where and are the velocity components in the − and −directions, 1 is the relaxationtime, = () is the kinematic viscosity, is the permeability of porous medium, is the

fluid temperature, is the density of fluid, is the thermal conductivity of fluid, is the

specific heat at constant pressure and is the heat source coefficient.

The boundary conditions are taken in the forms:

= = −0

= 2 at = 0 (3.4)

= 0 = ∞ as →∞ (3.5)

where is the temperature coefficient and ∞ is the fluid temperature far away from the sheet.

We introduce the transformations

= 0() = −√() = ∞ +

r

2() =

r

(3.6)

Here is a constant and prime denotes differentiation with respect to .

Equations (32)− (35) yield

000 + 00 − 02 + (2 0 00 − 2 000)− 0 = 0 (3.7)

00 + 0 − 2 0 − ∗ = 0 (3.8)

= ∗ 0 = 1 0 = 1 at = 0 (3.9)

0 = 0 = 0 as →∞ (3.10)

32

where Eq. (31) is satisfied automatically and = 1 is the Deborah number =

is the

permeability parameter, ∗ = 0√is the suction parameter, =

is the Prandtl number

and ∗ = is a heat generation parameter.

Expression of local Nusselt number is

=

( − ∞) (3.11)

where heat transfer can be defined as

= −µ

¶=0

(3.12)

In dimensionless form, Eq. (311) becomes

12 = − 1

(0) (3.13)


Considering a set of base functions

{ exp(−) ≥ 0 ≥ 0} (3.14)

we write

() =

∞X=0

∞X=0

exp(−) (3.15)

() =

∞X=0

∞X=0

exp(−) (3.16)

in which and are the coefficients. The initial approximations and auxiliary linear

operators are taken in the forms:

0() = ∗ + 1− exp(−) 0() = − exp(−) (3.17)

L = 000 − 0 L = 00 + 0 (3.18)

33

with

L (1 + 2 + 3

−) = 0 L(4 + 5−) = 0 (3.19)

where ( = 1− 5) represent the arbitrary constants.The zeroth order deformation problems are [75-78]:

(1− )Lh(; )− 0()

i= ~N

h(; )

i (3.20)

(1− )Lh(; )− 0()

i= ~N

h(; ) ( )

i (3.21)

(0; ) = ∗ 0(0; ) = 1 0(∞; ) = 0 0(0 ) = 1 (∞ ) = 0 (3.22)

N [( )] =3( )

3− ( )

2( )

2−Ã( )

!2

+

"2( )

( )

2( )

2− (( ))2

3( )

3

#−

( )

(3.23)

N[( ) ( )] =2( )

2+ ( )

( )

− 2( )

( )− ∗( ) (3.24)

in which is an embedding parameter, ~ and ~ the non zero auxiliary parameters and N

and N the nonlinear operators.

For = 0 and = 1 we have

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (3.25)

and when increases from 0 to 1 then ( ) and ( ) approach from 0() 0() to ()

and () By Taylor’s series one has

( ) = 0() +∞P

=1

() (3.26)

( ) = 0() +∞P

=1

() (3.27)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(3.28)

34


and ~ are selected properly so that Eqs. (326) and (327) converge at = 1 and thus we have

() = 0() +∞P

=1

() (3.29)

() = 0() +∞P

=1

() (3.30)


L [()− −1()] = ~R () (3.31)

L[()− −1()] = ~R () (3.32)

(0) = 0(0) = 0(∞) = 0 0(0)− (0) = (∞) = 0 (3.33)

R () = 000−1() +

−1P=0

h−1− 00 − 0−1−

000

i+

−1X=0

−1−X=0

{2 0− 00 − − 000 − 0−1() (3.34)

R () = 00−1 +

−1P=0

0−1− − 2−1P=0

−1− 0 − ∗ (3.35)

=

⎡⎣ 0 ≤ 11 1

(3.36)


() = ∗() + 1 + 2 + 3

− (3.37)

() = ∗() + 4 + 5− (3.38)


35


In this section, we discuss the convergence of the series given in Eqs. (329) and (330) For

this we first show that if the series (315) and (316) converge then these will converge to the

solution of the problem given by Eqs. (37)− (310) Let us suppose that ~ and ~ are selectedsuch that the series (315) and (316) converge. Therefore we have

lim→∞

() = 0, lim→∞

() = 0 (3.39)

From Eqs. (331), (332) and (336) one has

lim→∞

"~

X=1

R ()

#= lim

→∞

X=1

L [ − −1]

= lim→∞

L"

X=1

−X=1

−1

#= lim

→∞L = L lim

→∞ = 0 ∈ [0∞] (3.40)

lim→∞

"~

X=1

R ()

#= lim

→∞

X=1

L [ − −1]

= lim→∞

L"

X=1

−X=1

−1

#= lim

→∞L = L lim

→∞ = 0 ∈ [0∞] (3.41)

Equations (340) and (341) imply that the infinite sequence 1, 2, 3, , and 1, 2, 3, Ãwhere =

X=1

R () , =

X=1

R ()

!converge to zero. Now

X=1

R () =

X=1

⎧⎪⎪⎨⎪⎪⎩ 000−1 ()− 0−1 () +−1X=0

⎡⎢⎢⎣ −1− 00 − 0−1− 0

+−1−X=0

©2 0−

00 − − 000

ª⎤⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭

(3.42)

36

lim→∞

"X=1

R ()

#=

∞X=1

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ 000−1 ()− 0−1 ()

+

−1X=0

⎡⎢⎢⎣ −1− 00 − 0−1− 0

+−1−X=0

©2 0−

00 − − 000

ª⎤⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

3

3

Ã ∞X=1

−1 ()

!−

Ã ∞X=1

−1 ()

!

+

∞X=1

−1X=0

⎡⎢⎢⎣ −1− 00 − 0−1− 0

+−1−X=0

©2 0−

00 − − 000

ª⎤⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

3

3

Ã ∞X=0

()

!−

Ã ∞X=0

()

!

+

∞X=0

∞X=+1

⎡⎢⎢⎣ −1− 00 − 0−1− 0

+−1−∞X=

©2 0−

00 − − 000

ª⎤⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

3

3

Ã ∞X=0

()

!−

Ã ∞X=0

()

!+Ã ∞X

=0

()

!Ã2

2

" ∞X=0

()

#!−Ã

" ∞X=0

()

#!2−

Ã ∞X=0

()

!23

3

" ∞X=0

()

#+

Ã ∞X=0

()

!Ã

" ∞X=0

()

#!2

2

" ∞X=0

()

#

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3.43)

and

X=1

R () =

X=1

(00−1 ()− ∗−1 () + Pr

−1X=0

£0−1− − 2−1− 0

¤) (3.44)

37

lim→∞

"X=1

R ()

#=

∞X=1

⎧⎪⎪⎨⎪⎪⎩00−1 ()− ∗−1 ()

+

−1X=0

£0−1− − 2−1− 0

¤⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2

2

Ã ∞X=1

−1 ()

!− ∗

Ã ∞X=1

−1 ()

!

+

∞X=1

−1X=0

£0−1− − 2−1− 0

¤⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2

2

Ã ∞X=1

−1 ()

!− ∗

Ã ∞X=1

−1 ()

!+

∞X=0

∞X=+1

£0−1− − 2−1− 0

¤⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2

2

Ã ∞X=1

−1 ()

!− ∗

Ã ∞X=1

−1 ()

!

+

Ã ∞X=0

()

!Ã

" ∞X=0

()

#!

−2Ã

" ∞X=0

()

#!Ã ∞X=0

()

!

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(3.45)

and therefore the above equations after using Eq. (339) become

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

3

3

Ã ∞X=0

()

!−

Ã ∞X=0

()

!+

Ã ∞X=0

()

!Ã2

2

" ∞X=0

()

#!

−Ã

" ∞X=0

()

#!2−

Ã ∞X=0

()

!23

3

" ∞X=0

()

#

+

Ã ∞X=0

()

!Ã

" ∞X=0

()

#!2

2

" ∞X=0

()

#

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭= 0 (3.46)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2

2

Ã ∞X=1

−1 ()

!− ∗

Ã ∞X=1

−1 ()

!+

Ã ∞X=0

()

!Ã

" ∞X=0

()

#!

−2Ã

" ∞X=0

()

#!Ã ∞X=0

()

!⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ = 0 (3.47)

38

From Eq. (333) we now write

∞X=0

(0) = 0

∞X=0

0 (0) = 0∞X=0

0 (∞) = 0∞X=0

£0 (0)− (0)

¤= 0

∞X=0

(∞) = 0

(3.48)

Equations (346)−(348) show that if the series (315) and (316) converge, it must be a solutionof the presented problem. Thus the only requirement to choose the appropriate initial guesses

0 () 0 (), auxiliary linear operators L , L and the auxiliary parameters ~ and ~ to ensurethat the infinite series (315) and (316) are convergent.

The convergence region and rate of convergence of the series (315) and (316) strongly

depends upon the values of ~ and ~ Here the question arises that how one can select the valid

region for the values of ~ and ~ so that the solution series (315) and (316) are convergent. If

we closely look into equations (37)− (310) then for the dependent variable () there is onemissing condition 00 (0) and for the variable () the missing condition is 00(0). Therefore we

look for the convergence of the related series 00 (0) and 00 (0). If we plot these series against

the parameters ~ and ~ the curves obtained in this way are called ~-curves. We first draw

the ~-curves for the series 00 (0) and 00 (0). The portion of the ~-curves which is parallel to

the ~-axis will give the region for the admissible values of ~ and ~ and it actually gives the

values of the missing conditions for both the dependent variables. Once we get the values of

the missing conditions we can then find the solution of the problem.

For the range of admissible values of ~ and }, the ~−curves have been potrayed for14-order of approximations. Figs. 31 and 32 depict that the range of admissible values of }

and } are −15 ≤ ~ ≤ −045 and −08 ≤ ~ ≤ −04 The series converge in the whole regionof when ~ = −10 and ~ = −06 From Table 31 we see that our series solutions converge

from 20-th order of approximations Therefore 20-th order approximations are enough to find

39

the convergent solutions.

-2 -1.5 -1 -0.5 0Ñf

-1.45

-1.4

-1.35

-1.3

-1.25

-1.2

f''0

S* = 0.5, b= 0.1, l =0.2

f ''0

Fig. 3.1: ~−curve for the function ()

-1 -0.8 -0.6 -0.4 -0.2 0Ñq

-1.375

-1.35

-1.325

-1.3

-1.275

-1.25

-1.225

-1.2

q''0

Pr = 0.7, l = 0.2, b* =0.2, S* = 0.5, b =0.1

q''0



40

= 02 = 03 = 10, ∗ = 05 = 10 and } = −10 and ~ = −06


5 1.44009 1.36626

10 1.44007 1.36263

15 1.44007 1.36280

20 1.44007 1.36279

25 1.44007 1.36279

30 1.44007 1.36279

3.4 Analysis

The objective of this section is to predict the influences of different parameters ∗

and ∗ on velocity 0() and temperature () fields For this aim we plotted Figs. 33− 310for various interesting parameters on velocity and temperature fields. Figs. 33− 35 representthe variations of suction parameter ∗ Deborah number and permeability parameter on

velocity profile 0() Fig. 33 depicts the effects of ∗ on 0() From Fig. 33, we noted

that the velocity profile 0() decreases by increasing ∗ The Deborah number decreases the

velocity field 0() (see Fig. 34) Hence we can say that the velocity field 0() is a decreasing

function of Deborah number is directly proportional to relaxation time. An increasing values

of Deborah number correspond to higher relaxation time. Such higher relaxation time is caused

a reduction in the fluid velocity. Fig. 35 represents the effect of on 0() The velocity

profile decreases when is increased The permeability of porous medium is decreased with

an increase in that leads to the lower velocity and thinner boundary layer thickness. Figs.

36− 310 are drawn to see the behaviors of ∗ and ∗ on the temperature field ()

Fig. 36 describes the effects of suction parameter ∗ on () We note that ∗ leads to a

decrease in the temperature profile. Fig. 37 plots the effects of on () The temperature

field () increases by increasing The effects of permeability parameter on () have been

illustrated in Fig. 38 We see that the temperature field () increases by increasing Figs.

39 and 310 depict the effects of and on () From Figs. 39 and 310 we observed that

41

the increase in and decreases the temperature field. We conclude that both and ∗

have same qualitative effects on the temperature profile () Physically ∗ 0 implies that

∞ the supply of heat to the flow region is from the wall. Fig. 3.10 depicts that if more

fluid is injected then the temperature decreases due to a great loss of heat from hot injection.

Here the temperature is negative for all the cases. Table 3.2 presents the numerical values of

local Nusselt number for different values of embedded parameters. The local Nusselt number

increases by increasing suction parameter and Prandtl number but it decreases when we

increase Deborah number and heat generation parameter ∗

0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

f'h

b = 0.1, l = 0.2

S* = 2.0S* = 1.0S* = 0.5S* = 0.0


42

0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

f'h

S* = 0.5, l = 0.2

b = 1.2b = 0.7b = 0.3b = 0.0


0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

f'h

S* = 0.5, b = 0.1

l = 2.0l = 1.0l = 0.5l = 0.0


43

0 2 4 6 8 10h

-0.8

-0.6

-0.4

-0.2

0

qh

b = 0.1, l = 0.2, Pr = 0.7, b* = 0.2

S* = 1.5S* = 1.0S* = 0.4S* = 0.0


0 2 4 6 8 10h

-0.8

-0.6

-0.4

-0.2

0

qh

S* = 0.5, l = 0.2, Pr = 0.7, b* = 0.2

b = 1.5b = 0.7b = 0.3b = 0.0


44

0 2 4 6 8 10h

-0.8

-0.6

-0.4

-0.2

0

qh

b = 0.1, S* = 0.5, Pr = 0.7, b* = 0.2

l = 1.5l = 0.8l = 0.4l = 0.0


0 2 4 6 8 10h

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

qh

S* = 0.5, b = 0.1, l = 0.2, b* = 0.2

Pr = 1.2Pr = 0.7Pr = 0.3Pr = 0.1


45

0 2 4 6 8 10h

-0.8

-0.6

-0.4

-0.2

0

qh

S* = 0.5, b = 0.1, l = 0.2, Pr = 0.7

b* = 1.0b* = 0.6b* = 0.3b* = 0.0


Table 3.2: Values of local Nusselt number 12 for the parameters ∗ and ∗

when = 01

∗ ∗ −12

0.0 0.5 0.5 0.2 1.09743

0.2 1.05938

0.5 1.00375

0.3 0.83400

0.5 1.07838

1.0 1.63452

0.0 0.98824

0.5 1.07839

1.0 1.17777

0.0 0.91655

0.4 1.19972

0.8 1.39493

46

3.5 Final remarks

We studied the steady flow of Maxwell fluid over a stretching surface in presence of power law

heat flux and heat source. The series solutions have been developed to analyze the salient

features of this work. We noticed that the suction parameter and Deborah number have

similar effects on velocity profile 0() in a qualitative sense. Velocity field 0() decreases by

increasing permeability parameter It is observed that the temperature profile () increases

in view of an increase in and Also we have seen that the heat generation parameter ∗

leads to a decrease in the temperature ()

47

Chapter 4

On radiative flow of Maxwell fluid

with variable thermal conductivity

This chapter extends the analysis of previous chapter for variable thermal conductivity. The

governing nonlinear partial differential equations are reduced into the ordinary differential equa-

tions by appropriate transformations. The solution of temperature is presented. The variations

of various embedded parameters on the temperature are displayed and discussed. The values

of local Nusselt number are compared with the existing numerical solution in a limiting sense.

4.1 Governing problem

The energy equation in presence of thermal radiation is given by

µ

+

¶=

µ

¶−

(4.1)

In view of Rosseland approximation [35], we have = (−43∗) 4 Expanding 4 about∞ by Taylor series and neglecting higher-order terms we obtain, 4 = 4 3∞ −3 4∞ Equation(4.1) thus can be written as

µ

+

¶=

µ

¶− 16

3∞3∗

2

2 (4.2)

48

The boundary conditions are presented by

= () = ∞ +1 at = 0 (4.3)

→ ∞ as →∞ (4.4)

where is the variable thermal conductivity, the density of fluid, the specific heat at

constant pressure, the Stefan-Boltzmann constant and ∗ the mean absorption coefficient.

We consider the transformation

() = − ∞ − ∞

(4.5)

with () = ∞ +1() at = 0 and variable thermal conductivity = ∞[1 + ] (∞

is the fluid free stream conductivity) and is defined by

=( − ∞)

∞ (4.6)

in which is the fluid thermal conductivity at the wall.

The above transformations satisfy the incompressibility condition and now Eq. (42) yields

(1 + )00 + 02 +4

300 = [1

0 − 0] (4.7)

(0) = 1, (∞) = 0 (4.8)

where = 1 is the Deborah number = () is the permeability parameter, =∞ is

the Prandtl number and =4 3∞∞∗ is the radiation parameter.

The local Nusselt number with heat transfer is given by

=

( − ∞) = −

µ

¶=0

(4.9)

In dimensionless scale, Eq. (49) becomes

12 = −0(0) (4.10)

49

4.2 Solutions employing HAM

We express in the set of base function

{ exp(−) ≥ 0 ≥ 0} (4.11)

as follows

() =

∞X=0

∞X=0

exp(−) (4.12)

in which is the coefficient.

The initial approximations and auxiliary linear operators are given below

0() = (−) (4.13)

L = 00 − (4.14)

with

L(1 + 2−) = 0 (4.15)

where ( = 1 2) denotes the arbitrary constants. The following problems corresponding to

the zeroth order deformations are constructed as follows:

(1− )Lh(; )− 0()

i= ~N

h(; ) ( )

i (4.16)

(0 ) = 1 0(∞ ) = 0 (4.17)

N[( ) ( )] =

µ1 +

4

3

¶2( )

2+ ( )

2( )

2+

Ã( )

!2

−1( )( )

+ ( )( )

(4.18)

where is an embedding parameter, ~ is the non zero auxiliary parameters andN the nonlinear

50

operator. Note that for = 0 and = 1 we have

( 0) = 0() and ( 1) = () (4.19)

and when increases from 0 to 1 then ( ) varies from 0() to () By using Taylor’s series

we obtain

( ) = 0() +∞P

=1

() (4.20)

() =1

!

(; )

¯=0

(4.21)

where the convergence of above series strongly depends upon ~ Considering that ~ is selected

properly so that (420) converges at = 1 then

() = 0() +∞P

=1

() (4.22)

The problem at th-order are

L[()− −1()] = ~R () (4.23)

(0) = (∞) = 0 (4.24)

R () =

µ1 +

4

3

¶00−1 +

−1P=0

−1−00 + −1P=0

0−1−0

−1−1P=0

−1− 0 + −1P=0

−1− 0 (4.25)

=

⎡⎣ 0 ≤ 11 1

(4.26)

The general solutions are

() = ∗() +4 + 5

− (4.27)

in which ∗ denotes the special solutions.

51

4.3 Convergence analysis

We know that the expression (4.22) contains the nonzero auxiliary parameter ~ which can

adjust and control the convergence of the homotopy solutions. For admissible values of ~, the

~−curve has been potrayed for 22-order of approximations. Fig. 4.1 shows that the range foradmissible values of ~ are −12 ≤ ~ ≤ −03 The convergence of series solutions is obtainedin the whole region of when ~ = −07

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0Ñq

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

q'0

Pr = 0.7, a1 =0.5, l= 2.0, N =0.3, b =0.1, e =0.2

q'0


Table: 4.1. Convergence of homotopy solutions for different order of approximations when

= 02 1 = 10 = 10, = 03 = 20 = 02 and } = −07

Order of approximation −0(0)1 0.76667

5 0.68252

10 0.66613

20 0.65959

30 0.65834

35 0.65819

40 0.65819

50 0.65819

52

4.4 Discussion

Here our interest is just to examine the role of interesting parameters on the velocity and

temperature. Hence the Figs. 4.2-4.7 have been plotted for Deborah number permeability

parameter Prandtl number positive constant 1 radiation parameter and small pa-

rameter on the temperature profile (). Fig. 4.2 represents the effects of on () Clearly

() increases when is increased. Fig. 4.3 illustrates that the temperature field () decreases

by increasing the permeability parameter The effects of on () are plotted in Fig. 4.4.

Here the temperature field () decreases by increasing In fact the definition of Prandtl

number involves the thermal diffusivity. When we increase the Prandtl number, a lower thermal

diffusivity occurs. Such lower thermal diffusivity caused a decrease in temperature. Fig. 4.5

shows the effects of 1 on () From Fig. 4.5 we observed that the temperature field ()

decreases when 1 increases. The temperature () and thermal boundary layer thickness are

increasing function of radiation parameter (Fig. 4.6). An increase in radiation augments the

heat transfer. The fluid is heated which increases the thermal boundary layer thickness. Fig.

4.7 represents the effects of on () The temperature () and associated thermal bound-

ary layer thickness increase when is increased. From Figs. 4.6 and 4.7 it is found that ()

increases when and are increased Hence and have similar role on the temperature

field () in a qualitative sense. Physically an increase in radiation parameter provides more

heat to fluid due to which higher temperature is observed. Figs. 4.8-4.12 are plotted to see

the influences of various emerging parameters on the local Nusselt number −0(0) Fig. 4.10illustrates the effects of and on the local Nusselt number. From this Fig. it is noted that

an increase in and leads to a decrease in local Nusselt number. Influences of and 1 on

the local Nusselt number are seen in Fig. 4.11. A decrease in local Nusselt number is observed

when and 1 are increased. Figs. 4.12 and 4.13 presented the effects of (1) and ()

on −0(0). These Figs. show that −0(0) is a decreasing function of (1) and () From

Fig. 4.14 we have seen that an increase in and corresponds to an enhancement in local

Nusselt number.

Table 4.1 shows the numerical values to ensure the convergence of series solutions. One can

see that our solutions for velocity converge from 10th order of approximations. However the

solutions for temperature converge from 35th order of deformations. Table 4.2 provides the

53

comparison of values of the local Nusselt number with the previous results when = 00 This

Table indicates that our series solutions have good agreement with previous results in a limiting

case. Table 4.3 presented the numerical values of local Nusselt number for different values of

1 and From Table 4.3 we see that the local Nusselt number −0(0) decreasesby increasing and but it increases when and 1 are increased.

0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1qh

Pr = 0.7, a1= 0.5, l = 2.0, N= 0.3, e = 0.2

b = 1.0b = 0.6b = 0.3b = 0.0


0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a1 = 0.5, b = 0.1, N = 0.3, e = 0.2

l = 4.0l = 2.0l = 1.0l = 0.5


54

0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1

qh

a1= 0.5, l = 2.0, b = 0.1, N = 0.3, e = 0.2

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, l = 2.0, b = 0.1, N = 0.3, e = 0.2

a1 = 3.0a1 = 2.0a1 = 1.0a1 = 0.0

Fig. 4.5: Influence of 1 on ()

55

0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a1= 0.5, l = 2.0, b = 0.1, e = 0.2

N = 1.2N = 0.8N = 0.4N = 0.0


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a1 = 0.5, l = 2.0, b = 0.1, N= 0.3

e = 1.0e = 0.7e = 0.4e = 0.0


56

0 1 2 3 4Pr

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-q'

0

a1 = 1.0, e = 0.2, l = 2.0, N= 0.3

b = 1.2b = 0.8b = 0.4b = 0.0

Fig. 4.8: Influence of and on −0(0)

0 1 2 3 4a1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-q'

0

b = 0.1, l = 2.0, e = 0.2, N = 0.3

e = 0.6e = 0.4e = 0.2e = 0.0

Fig. 4.9: Influence of and 1 on −0(0)

57

0 1 2 3 4a1

0.4

0.6

0.8

1

1.2

-q'

0

b = 0.1, l = 2.0, e = 0.2, N = 0.3

N= 0.8N= 0.5N= 0.3N= 0.0

Fig. 4.10: Influence of and 1 on −0(0)

0 1 2 3 4b

0.2

0.4

0.6

0.8

1

-q'

0

a1 = 1.0, l = 2.0, e = 0.2, Pr = 0.7

N = 0.8N = 0.5N = 0.3N = 0.0


58

0 1 2 3 4 5 6b

0.2

0.4

0.6

0.8

1

-q'0

a1 = 1.0, e = 0.2, l = 2.0, N = 0.3

Pr = 1.6Pr = 1.2Pr = 0.8Pr = 0.4

Fig. 4.12 Influence of and on −0(0)Table: 4.2. Numerical values of local Nusselt number 0(0) compared with the Vyas and Rai

[66]

1 Vyas and Rai [66] Present results

0 0.001 1 0.5 0.023 -0.04706608006216 -0.0469873

1 0.001 1 0.5 0.023 -0.04810807780602 -0.0481857

2 0.001 1 0.5 0.023 -0.04915123655083 -0.0492375

1 0.001 1 1 0.023 -0.05813708762124 -0.0581462

1 0.001 1 2 0.023 -0.06627101076453 -0.0662179

1 0.001 2 0.5 0.023 -0.03092604001010 -0.0309465

1 0.001 3 0.5 0.023 -0.02096607234828 -0.0207548

1 0.001 5 0.5 0.023 -0.01496701292876 -0.0151473

1 0.004 1 0.5 0.023 -0.04801503596049 -0.4857646

1 0 1 0.5 0.023 -0.04814015988615 -0.0483625

1 0.001 1 0.5 0.1 -0.19563610029641 -0.1976327

1 0.001 1 0.5 0.2 -0.37208031115957 -0.3742764

Table: 4.3. Values of local Nusselt number −0(0) for different values of and when

59

1 = 10 = 02 and = 03

−0(0)0.0 2.0 0.7 0.5198

0.5 0.4892

1.0 0.4574

0.1 0.5 0.4488

1.0 0.4714

3.0 0.5217

0.5 0.3948

1.0 0.6599

1.5 0.8758

4.5 Final remarks

Thermal radiation effect in steady flow of Maxwell fluid with variable thermal conductivity is

analyzed. The main observations of presented analysis have been pointed out as follows.

• Deborah number and permeability parameter have opposite effects on the velocityfield 0()

• The temperature field () is an increasing function of

• The temperature field () decreases by increasing Prandtl number

• Increase in 1 decreases the temperature field ()

• The temperature field () increases by increasing the values of and

60

Chapter 5

On three-dimensional flow of

Maxwell fluid over a stretching

surface with convective boundary

conditions

Three-dimensional flow of non-Newtonian fluid bounded by a stretching surface has been stud-

ied. The constitutive equations of Maxwell fluid are used. The surface possesses convective

boundary conditions. Computations have been carried out for the non-linear problem. Con-

vergence of the obtained solutions is discussed. Impact of the influential parameters involved

in the heat transfer analysis is emphasized. Comparison with the previous results is shown. It

is found that effects of Deborah and Biot parameters on the Nusselt number are opposite. The

Prandtl and Biot numbers have similar impacts on the Nusselt number in a qualitative sense.


We consider the steady three-dimensional flow of an incompressible fluid over a stretched surface

at = 0 The flow takes place in the domain 0 The ambient fluid temperature is taken as

∞ while the surface temperature is maintained by convective heat transfer at a certain value

61

. The governing boundary layer equations for three-dimensional flow of Maxwell fluid are

+

+

= 0 (5.1)

+

+

=

2

2− 1

⎛⎝ 2 2

2+ 2

22

+ 2 22

+ 2 2

+2 2

+ 2 2

⎞⎠ (5.2)

+

+

=

2

2− 1

⎛⎝ 2 2

2+ 2

22

+ 2 2

2+ 2 2

+

2 2

+ 2 2

⎞⎠ (5.3)

+

+

=

2

2 (5.4)

where the respective velocity components in the − − and −directions are denoted by and , 1 shows the relaxation time, the fluid temperature, the thermal diffusivity of the

fluid, = () the kinematic viscosity, the dynamic viscosity of fluid and the density of

fluid.

The boundary conditions appropriate to the flow under consideration are

= = = 0 −

= ( − ) at = 0 (5.5)

→ 0 → 0 → ∞ as →∞ (5.6)

where indicates the thermal conductivity of fluid and and have dimension inverse of time.

Using the following variables

= 0() = 0() = −√(() + ()) () = − ∞ − ∞

=

r

(5.7)

equation (5.1) is satisfied automatically and Eqs. (52)− (57) give

000 + ( + ) 00 − 02 + [2( + ) 0 00 − ( + )2 000] = 0 (5.8)

000 + ( + )00 − 02 + [2( + )000 − ( + )2000] = 0 (5.9)

00 + ( + )0 = 0 (5.10)

62

= 0 = 0 0 = 1 0 = 0 = −(1− (0)) at = 0 (5.11)

0 → 0 0 → 0 → 0 as →∞ (5.12)

where = 1 is the Deborah number =is a parameter, =

is the Prandtl number,

=

pis the Biot number and prime shows the differentiation with respect to .

The expression for local Nusselt number with heat transfer is

=

( − ∞) = −

µ

¶=0

(5.13)

The above equation in dimensionless form can be written as

12 = −0(0) (5.14)

in which = is the local Reynolds number.

5.2 Series solutions

The initial approximations and auxiliary linear operators for homotopy analysis solutions are

chosen as

0() =¡1− −

¢ 0() =

¡1− −

¢ 0() =

exp(−)1 +

(5.15)

L = 000 − 0 L = 000 − 0 L = 00 − (5.16)

We note that the auxiliary linear operators in above equation satisfy the following properties

L (1 + 2 + 3

−) = 0 L(4 + 5 + 6

−) = 0 L(7 + 8−) = 0 (5.17)

where ( = 1− 8) are the arbitrary constants.The associated zeroth order deformation problems can be written as

(1− )Lh(; )− 0()

i= ~N

h(; ) (; )

i (5.18)

(1− )L [(; )− 0()] = ~N

h(; ) (; )

i (5.19)

63

(1− )Lh(; )− 0()

i= ~N

h(; ) (; ) ( )

i (5.20)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 0 0(0; ) = 0(∞; ) = 0

0(0 ) = −[1− (0 )] (∞ ) = 0 (5.21)

N [( ) ( )] =3( )

3−Ã( )

!2+ (( ) + ( ))

2( )

2

+

⎡⎣ 2(( ) + ( ))()

2()

2

−(( ) + ( ))23()

2

⎤⎦ (5.22)

N[( ) ( )] =3( )

3−µ( )

¶2+ (( ) + ( ))

2( )

2

+

⎡⎣ 2(( ) + ( ))()

2()

2

−(( ) + ( ))23()

2

⎤⎦ (5.23)

N[( ) ( ) ( )] =2( )

2+Pr(( ) + ( ))

( )

(5.24)

Here is an embedding parameter, ~ ~ and ~ are the non-zero auxiliary parameters and

N N and N indicate the nonlinear operators. For = 0 and = 1 we have

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (5.25)

Further when increases from 0 to 1 then ( ) ( ) and ( ) vary from 0() 0() 0()

to () () and () Using Taylor’s expansion one can write

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(5.26)

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(5.27)

64

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(5.28)

where the convergence of above series strongly depends upon ~ ~ and ~ Considering that

~ ~ and ~ are selected properly so that Eqs. (526)− (528) converge at = 1 therefore

() = 0() +∞P

=1

() (5.29)

() = 0() +∞P

=1

() (5.30)

() = 0() +∞P

=1

() (5.31)

The mth order deformation problems are

L [()− −1()] = ~R () (5.32)

L[()− −1()] = ~R () (5.33)

L[()− −1()] = ~R () (5.34)

(0) = 0(0) = 0(∞) = 0 (0) = 0(0) = 0(∞) = 0 0(0)− (0) = (∞) = 0(5.35)

R () = 000−1()−

−1P=0

0−1−0 +

−1P=0

(−1− 00 + −1− 00 )

+−1P=0

P=0

[2(−1− + −1−) 0−00

−(−1−− + −1−− + 2−1−−) 000 ] (5.36)

R () = 000−1()−

−1P=0

0−1−0 +

−1P=0

(−1−00 + −1−00)

+−1P=0

P=0

[2(−1− + −1−)0−00

−(−1−− + −1−− + 2−1−−)000 ] (5.37)

R () = 00−1 +

−1P=0

(0−1− + 0−1−) (5.38)

65

=

⎡⎣ 0 ≤ 11 1

(5.39)

Solving the corresponding mth order deformation problems we have

() = ∗() + 1 + 2 + 3

− (5.40)

() = ∗() + 4 + 5 + 6

− (5.41)

() = ∗() +7 + 8

− (5.42)

in which the ∗ ∗ and ∗ indicate the special solutions.

5.3 Convergence analysis and discussion of results

Obviously the series (529) − (531) consists of the auxiliary parameters ~ ~ and ~. Theseparameters have a key role to adjust and control the convergence of homotopic solutions. The

~−curves have been sketched at 18 order of approximations to determine the suitable rangesfor ~ ~ and ~. From Figs. 5.1 − 53 it is noted that the range of admissible values of~ ~ and ~ are −130 ≤ ~ ≤ −030 −130 ≤ ~ ≤ −025 and −140 ≤ ~ ≤ −045 Weobserved that (see Table 1) that our series solutions converge in the whole region of when

~ = ~ = −090 and ~ = −100 (see Table 5.1).Figs. 5.4-5.16 depict the behaviors of Deborah number Prandtl number and Biot

number on temperature () for different cases when = 00 05 and 10 Variations of

and are shown in the Figs 5.4−56 when = 0 It is seen that an increase in Prandtl number shows a decrease in the temperature of fluid and the thermal boundary layer thickness (see

Fig. 5.4). Figs. 5.5 and 5.6 show the variations of Deborah and Biot numbers. We conclude

from these Figs. that both the temperature profile and thermal boundary layer thickness

increase when Deborah and Biot number increase. It is also noted that the fluid temperature

is zero when the Biot number is zero. The effects of and on temperature are displayed

in the Figs. 5.7− 59 for = 05 The plotted Figs. depict that results for = 0 and = 05

are similar in a qualitative sense. The only change here that we noted is in the variation of

This can be seen by comparing the Figs. 5.5 and 5.8 The variation in temperature for the case

66

= 05 is bit smaller than = 0 Similar observations are noted in the Figs. 5.10−512 Table5.1 presents the convergence of homotpic solutions. From this Table it is concluded that we

need 20 terms for velocity and 25 order iterations for the temperature for a convergent series

solutions. Table 5.2 is prepared for the comparison between HAM results and previous existing

results in a limiting case for various values of One can see that our homotopic results have

an excellent agreement with the exact and homotopy perturbation (HPM) results in a viscous

fluid. Numerical values of local Nusselt number are analyzed in Table 5.3 The values of −0(0)decrease by increasing Deborah number. However such values increase by increasing Prandtl

and Biot numbers.

-1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñf

-1.28

-1.26

-1.24

-1.22

-1.2

f''0

a =0.4, b =0.2

f ''0


67

-1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñg

-0.58

-0.56

-0.54

-0.52

-0.5

g''0

a= 0.4, b= 0.2

g''0


-1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñq

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

q'0

a =0.4, b =0.2, Pr = 1.0, g = 0.6

q'0


68

0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

f'h

a = 0.3

b = 1.0b = 0.7b = 0.4b = 0.0


0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

f'h

b = 0.5

a = 1.0a = 0.6a = 0.3a = 0.0


69

0 1 2 3 4 5 6h

0

0.1

0.2

0.3

0.4

0.5

g'h

a = 0.5

b = 1.0b = 0.6b = 0.3b = 0.0


0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

g'h

b = 0.5

a = 1.0a = 0.7a = 0.4a = 0.0


70

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

b = 0.3, g = 0.4

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1

Fig. 5.8: Influence of on () when = 00

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, g = 0.3

b = 1.0b = 0.7b = 0.3b = 0.0


71

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

qh

Pr = 0.7, b = 0.3

g = 0.6g = 0.4g = 0.2g = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

b = 0.3, g = 0.4

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1


72

0 2 4 6 8 10h

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

qh

Pr = 1.0, g = 0.3

b = 1.0b = 0.7b = 0.3b = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b = 0.3

g = 0.6g = 0.4g = 0.2g = 0.0


73

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

b = 0.3, g = 0.4

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1


0 2 4 6 8 10h

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

qh

Pr = 1.0, g = 0.3

b = 1.0b = 0.7b = 0.3b = 0.0


74

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b = 0.3

g = 0.6g = 0.3g = 0.2g = 0.0


Table 5.1: Convergence of series solutions for different order of approximations when = 04

= 05 = 10 = 06 ~ = ~ = −09 and ~ = −10

Order of approximations − 00(0) −00(0) −0(0)1 1.232500 0.518750 0.339844

10 1.266203 0.536286 0.318905

15 1.266214 0.536300 0.318769

20 1.266215 0.536301 0.318754

25 1.266215 0.536301 0.318752

30 1.266215 0.536301 0.318752

35 1.266215 0.536301 0.318752

75

Table 5.2: Comparison for the different values of by HAM, HPM and exact solutions.

HPM [29] Exact [29] HAM

− 00(0) −00(0) − 00(0) −00(0) − 00(0) −00(0)0.0 1.0 0.0 1 0 1.0 0.0

0.1 1.02025 0.06684 1.020259 0.066847 1.020260 0.0668472

0.2 1.03949 0.14873 1.039495 0.148736 1.039495 0.148737

0.3 1.05795 0.24335 1.057954 0.243359 1.057955 0.243360

0.4 1.07578 0.34920 1.075788 0.349208 1.075788 0.349209

0.5 1.09309 0.46520 1.093095 0.465204 1.093095 0.465205

0.6 1.10994 0.59052 1.109946 0.590528 1.109942 0.590529

0.7 1.12639 0.72453 1.126397 0.724531 1.126398 0.724532

0.8 1.14248 0.86668 1.142488 0.866682 1.142489 0.866683

0.9 1.15825 1.01653 1.158253 1.016538 1.158254 1.016539

1.0 1.17372 1.17372 1.173720 1.173720 1.173721 1.173721

Table 5.3: Values of local Nusselt number −0(0) for different values of the parameters and when = 06.

−0(0)0.0 0.5 1.0 0.33040

0.3 0.32160

0.8 0.30799

1.2 0.29873

0.0 0.28908

0.4 0.31664

0.7 0.33017

1.0 0.34070

0.7 0.28279

1.2 0.34042

1.6 0.36840

2.0 0.38887

76


Three-dimensional boundary layer flow of an incompressible Maxwell fluid with convective

boundary condition is analyzed. The main findings of this chapter are listed below:

• Both the velocity components 0() and 0() are reduced for higher values of Deborah

number

• Velocity component 0() is decreased by increasing However the velocity component0() is increased.

• Temperature () is a decreasing function of Prandtl number

• An increase in Biot number enhanced the temperature () and thermal boundary layerthickness.

• Values of local Nusselt number are larger when smaller values of and are taken intoaccount.

77

Chapter 6

MHD three-dimensional flow of

Maxwell fluid with variable thermal

conductivity and heat source/sink

This chapter investigates the effects of applied magnetic field and heat transfer in flow of

Maxwell fluid with variable thermal conductivity. Three dimensional flow is induced by a

stretching surface. The thermal conductivity is taken temperature dependent. Heat transfer

analysis is carried out in the presence of heat source/sink. The series solutions are constructed

and results are interpreted through the effects of various embedded parameters. Comparison

with the previous limiting solutions is shown in an excellent agreement.

6.1 Mathematical formulation of the problems

A steady three-dimensional laminar flow of an incompressible Maxwell fluid (with variable

thermal conductivity) past a stretching surface is considered. A constant magnetic field of

strength B0 is applied. Induced magnetic field is not accounted. In addition concept of heat

source/sink in heat transfer analysis is considered. The thermal conductivity depends upon

temperature. The boundary layer assumptions are utilized in problems formulation. Thus the

78

governing equations for present flow configuration are reduced into the following equations:

+

+

= 0 (6.1)

+

+

=

2

2− 1

⎛⎝ 2 2

2+ 2

22

+ 2 22

+ 2 2

+2 2

+ 2 2

⎞⎠−

∗20

µ+ 1

¶ (6.2)

+

+

=

2

2− 1

⎛⎝ 2 2

2+ 2

22

+ 2 2

2+ 2 2

+

2 2

+ 2 2

⎞⎠−

∗20

µ + 1

¶ (6.3)

µ

+

+

¶=

µ

¶+( − ∞) (6.4)

Here the respective velocity components in the − − and −directions are represented by and , 1 the relaxation time, the fluid temperature, the variable thermal conductivity

of the fluid, = () the kinematic viscosity, the dynamic viscosity, the density, the

electrical conductivity, the specific heat and the heat source/sink parameter with 0

(heat source) and 0 (heat sink).

The boundary conditions associated to the flow consideration are

= () = = () = = 0 = at = 0 (6.5)

→ 0 → 0 → ∞ when →∞ (6.6)

in which variable thermal conductivity is

= ∞(1 + ) = − ∞

∞ (6.7)

where ∞ represents the fluid free stream conductivity and the conductivity at the wall.

79

We now employ the following transformations

= 0() = 0() = −√(() + ()) () = − ∞ − ∞

=

r

(6.8)

Now Eq. (6.1) is obviously satisfied and Eqs. (62) − (67) are presented into the followingforms:

000 + (2 + 1)( + ) 00 − 02 + ¡2( + ) 0 00 − ( + )2 000

¢−2 0 = 0 (6.9)

000 + (2 + 1)( + )00 − 02 + ¡2( + )000 − ( + )2000

¢−20 = 0 (6.10)

(1 + )00 + ( + )0 + 02 + = 0 (6.11)

= 0 = 0 0 = 1 0 = = 1 at = 0 (6.12)

0 → 0 0 → 0 → 0 as →∞ (6.13)

in which the parameters entering into above equations are (= 1) the Deborah number

(= ) the ratio of stretching rates, =

20

the Hartman number, ¡=

¢the Prandtl

number, ³=

´the heat source/sink parameter and prime the differentiation with respect

to

Local Nusselt number with heat transfer has the following definition

=

( − ∞) = −

µ

¶=0

(6.14)

Dimensionless expression of local Nusselt number is

12 = −0(0) (6.15)

where (= ) is the local Reynolds number.

80

6.2 Solutions

The initial guesses and auxiliary linear operators for the homotopy solutions can be expressed

in the forms:

0() = 1− exp(−) 0() = (1− exp(−)) 0() = exp(−) (6.16)

L = 000 − 0 L = 000 − 0 L = 00 − (6.17)

in which

L (1 + 2 + 3

−) = 0 L(4 + 5 + 6

−) = 0 L(7 + 8−) = 0 (6.18)

and ( = 1− 8) denote the arbitrary constants.The corresponding zeroth order problems are

(1− )Lh(; )− 0()

i= ~N

h(; ) (; )

i (6.19)

(1− )L [(; )− 0()] = ~N

h(; ) (; )

i (6.20)

(1− )Lh(; )− 0()

i= ~N

h(; ) (; ) ( )

i (6.21)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 0 0(0; ) = 0(∞; ) = 0

0(0 ) = (∞ ) = 0 (6.22)

N [( ) ( )] =3( )

3−Ã( )

!2+ (2 + 1)(( ) + ( ))

2( )

2

+

⎛⎝ 2(( ) + ( ))()

2()

2

−(( ) + ( ))23()

2

⎞⎠−2( )

(6.23)

81

N[( ) ( )] =3( )

3−µ( )

¶2+ (2 + 1)(( ) + ( ))

2( )

2

+

⎛⎝ 2(( ) + ( ))()

2()

2

−(( ) + ( ))23()

2

⎞⎠−2( )

(6.24)

N[( ) ( ) ( )] =³1 + ( )

´ 2( )

2+Pr(( ) + ( ))

( )

+

Ã( )

!2 (6.25)

where is an embedding parameter, ~ ~ and ~ the non-zero auxiliary parameters and N

N and N the nonlinear operators. When = 0 and = 1 then

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (6.26)

We notice that when increases from 0 to 1 then ( ) ( ) and ( ) vary from 0()

0() 0() to () () and () Using Taylor’s expansion we get

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(6.27)

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(6.28)

( ) = 0() + ∞P

=1

() () =

1

!

(; )

¯=0

(6.29)

where the convergence of series strongly depends upon ~ ~ and ~ Considering that ~ ~

and ~ are selected properly so that Eqs. (626)− (628) converge at = 1 Hence

() = 0() +∞P

=1

() (6.30)

() = 0() +∞P

=1

() (6.31)

() = 0() +∞P

=1

() (6.32)

82

The solutions of the corresponding mth order deformation problems are

() = ∗() + 1 + 2 + 3

− (6.33)

() = ∗() + 4 + 5 + 6

− (6.34)

() = ∗() +7 + 8

− (6.35)

where ∗ ∗ and ∗ denote the special solutions.

6.3 Convergence analysis and discussion

Note that the series (632)−(634) involve the auxiliary parameters ~ ~ and ~ Such parame-ters can adjust and control the convergence of series solutions. The ~−curves are displayed at16 order of approximations in order to find the ranges for ~ ~ and ~. Fig. 6.1 shows that

the ranges of admissible values of ~ ~ and ~ are −110 ≤ ~ ≤ −030 −115 ≤ ~ ≤ −020and −100 ≤ ~ ≤ −030 We note from Table 6.1 that our series solutions converge in the

whole region of when ~ = ~ = ~ = −07Figs. 6.2-6.13 have been plotted for the effects of pertinent parameters on the velocities

0() and 0() and temperature () Plots of Deborah number on the velocities 0() and

0() are shown in the Figs. 6.2 and 6.3. Both 0() and 0() have similar effects. The fluid

velocities and boundary layer thicknesses are decreased when we increased the Deborah number.

This decrease is due to the relaxation time. Deborah number involves the relaxation time. An

increase in relaxation time caused a decrease in the fluid velocities. Effects of ratio parameter

on the velocities are depicted in the Figs. 6.4 and 6.5. By increasing the opposite behaviors

for 0() and 0() are seen. The velocity 0() is decreased while 0() is increased when we

increase the values of Two-dimensional case is achieved when = 0 Effects of Hartman

number on the velocities are observed in the Figs. 6.6 and 6.7. Velocities 0() and 0()

are reduced with stronger magnetic field. Hartman number depends on the Lorentz force. An

increase in Hartman number produces larger Lorentz force. Lorentz force is an agent which

caused a reduction in velocities. We have seen from Fig. 6.8 that the temperature and thermal

boundary layer thickness rise by increasing Deborah number. In fact the relaxation time gives

83

rise to the temperature and thermal boundary layer thickness. It is also found that the effects

of Deborah number on the velocities and temperature are reversed. Effects of ratio parameter

on the temperature are analyzed in Fig. 6.9. The temperature and thermal boundary

layer thickness are decreasing functions of An increase in corresponds to a decrease in

temperature. From Fig. 6.10 we observed that larger values of Prandtl number lead to lower

the temperature and thermal boundary layer thickness. This occurs due to a decrease in thermal

diffusivity. Increase in Prandtl number implies lower thermal diffusivity and smaller thermal

diffusivity shows lower temperature. Fig. 6.11 illustrates that larger values of Hartman number

corresponds to increase in temperature and thermal boundary layer thickness. Fig. 6.12 shows

the behavior of on temperature () The temperature becomes stronger for the larger values

of Fig. 6.13 clearly indicates that heat source/sink parameter gives rise to the temperature

and thermal boundary layer thickness. Table 6.1 provides the convergence values of series

solutions of velocities and temperature. This Table indicates that we need less approximations

for velocities in comparison to the temperature. Table 6.2 is computed just to compare the

analysis with the previous results in a limiting sense for various values of It is found that our

homotopic solutions have an excellent agreement with the previous results for different values

of

-1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñf , Ñg, Ñq

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

f''0

,g

''0,q

'0

b =0.4, S= 0.2, M= 0.7, Pr = 1.5, a =0.5, e = 0.3

q'0g''0f''0

84

Fig. 6.1: ~−curves for the functions () () and ()

2 4 6 8 10h

0.2

0.4

0.6

0.8

1

f'h

b = 0.0, 0.2, 0.4, 0.6, 0.8

Fig. 6.2: Influence of on the velocity 0() when = 06 and = 07

2 4 6 8 10h

0.1

0.2

0.3

0.4

0.5

0.6

g'h

b =0.0, 0.2, 0.4, 0.6,0.8

85


2 4 6 8 10h

0.2

0.4

0.6

0.8

1

f'h

a= 0.1, 0.3, 0.5, 0.7, 1.0


2 4 6 8 10h

0.2

0.4

0.6

0.8

1

g'h

a= 0.1, 0.3, 0.6, 0.8, 1.0

86


2 4 6 8 10h

0.2

0.4

0.6

0.8

1

f'h

M= 0.0, 0.3, 0.6, 0.9, 1.2


2 4 6 8 10h

0.1

0.2

0.3

0.4

0.5

0.6

g'h

M=0.0, 0.3, 0.6, 0.9, 1.2

87


2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

b =0.0, 0.2, 0.4, 0.6, 0.8

Fig. 6.8: Influence of on the temperature () when = 06 = 07 = 12 = 02

and = 03

2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

a =0.1, 0.3, 0.5, 0.7, 1.0


88

and = 03

2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

Pr =0.4, 0.8, 1.2, 1.6, 2.0


and = 03

2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

M= 0.0, 0.3, 0.6, 0.9, 1.2


89

and = 03

2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

e =0.0, 0.3, 0.6, 0.9, 1.2


and = 03

2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

S= 0.0, 0.3, 0.6, 0.8, 1.0


and = 02

Table 6.1: Convergence of series solutions for different order of approximations when

90

= 03 = 05 = 07 = 15 = 03, = 02 and ~ = ~ = ~ = −07


5 1.477280 0.658060 0.50817

15 1.477327 0.658282 0.47558

20 1.477328 0.658282 0.47349

25 1.477328 0.658282 0.47242

30 1.477328 0.658282 0.47242

40 1.477328 0.658282 0.47242

Table 6.2: Comparison for different values of by HAM, HPM and exact solutions.

[29] [25] Present solutions

− 00(0) −00(0) − 00(0) −00(0) − 00(0) −00(0)0.0 1.0 0.0 1.0 0 1.0 0.0

0.10 1.02025 0.06684 – – 1.020260 0.0668472

0.20 1.03949 0.14873 – – 1.039495 0.148737

0.25 – – 1.048813 0.194564 1.04881 0.19457

0.30 1.05795 0.24335 – – 1.057955 0.243360

0.40 1.07578 0.34920 – – 1.075788 0.349209

0.50 1.09309 0.46520 1.093097 0.465205 1.093095 0.465205

0.60 1.10994 0.59052 – – 1.109942 0.590529

0.70 1.12639 0.72453 – – 1.126398 0.724532

0.75 – – 1.134485 0.794622 1.13450 0.79462

0.80 1.14248 0.86668 – – 1.142489 0.866683

0.90 1.15825 1.01653 – – 1.158254 1.016539

1.00 1.17372 1.17372 1.173720 1.173720 1.173721 1.173721

91

6.4 Final remarks

Three-dimensional stretched flow of magnetohydrodynamic (MHD) Maxwell fluid is investigated

via homotopy analysis method (HAM). Heat transfer analysis is presented when the thermal

conductivity of the fluid varies linearly with temperature. The main observations are as follows:

• Effects of Deborah number on the velocities 0() and 0() and temperature () are

quite reverse.

• Influence of on 0() and 0() is quite opposite.

• Increase in Hartman number gives rise to the temperature and thermal boundary layerthickness.

• Temperature and thermal boundary layer thickness are increasing functions of

• Temperature rise when we increase the heat source/sink parameter.

92

Chapter 7

Hydromagnetic steady flow of

Maxwell fluid over a bidirectional

stretching surface with prescribed

surface temperature and prescribed

surface heat flux

This chapter investigates the effects of heat transfer in steady hydromagnetic three-dimensional

boundary layer flow of Maxwell fluid by a bidirectional stretching surface. Two cases are

considered. Explicitly heat transfer process is analyzed due to prescribed surface temperature

(PST) and prescribed heat flux (PHF). Results are plotted and examined. Convergence for

the solutions is presented for the temperatures. Comparison of PST and PHF cases with the

existing solutions in a limiting sense is given and illustrated.

93

7.1 Flow model

The considered energy equation with heat source/sink is given below:

+

+

=

2

2+

( − ∞) (7.1)

Fig. 7.1: Physical model

The associated boundary conditions are defined as follows.

Type i. Prescribed surface temperature (PST)

= ( ) = ∞ + at = 0

→ ∞ as →∞ (7.2)

Type ii. Prescribed surface heat flux (PHF)

−

= at = 0

→ ∞ as →∞ (7.3)

Here is the thermal conductivity of the fluid, ∞ the constant temperature outside the thermal

boundary layer, and the positive constants. The power indices and determine how the

temperature or the heat flux varies in the −plane.Following [79,80] the temperature similarity variables take different forms depending on the

94

boundary conditions being considered. These are

For PST: () = ( )− ∞( )− ∞

and for PHF: ( )− ∞ =

r

() (7.4)

Eqs. (7.1)-(7.3) take the following forms:

00 + ( + )0 + ( − 0 − 0) = 0 (7.5)

00 + ( + )0 + ( − 0 − 0) = 0 (7.6)

= 1 0 = −1 at = 0

→ 0 → 0 as →∞ (7.7)

where = the Prandtl number, the thermal diffusivity and =

the internal heat

parameter.


In this section, we solve the problem consisting of Eqs. (7.5) and (7.6) with boundary conditions

in Eq. (7.7). For that the initial guesses and auxiliary linear operators are taken as follows:

0() = exp(−) 0() = exp(−) (7.8)

L = 00 − L = 00 − (7.9)

subject to the properties

L(7 + 8−) = 0 L(9 + 10

−) = 0 (7.10)

where ( = 1− 10) are the arbitrary constants.

95

At zeroth order the problems satisfy

(1− )L³(; )− 0()

´= ~N

³(; ) (; ) (; )

´ (7.11)

(1− )L³(; )− 0()

´= ~N

³(; ) (; ) (; )

´ (7.12)

(0; ) = 1 (∞ ) = 0 0(0 ) = 0 (∞ ) = 0 (7.13)

N[( ) ( ) ( )] =2( )

2+Pr(( ) + ( ))

( )

+Pr

Ã −

( )

−

( )

!( ) (7.14)

N[( ) ( ) ( )] =2( )

2+Pr(( ) + ( ))

( )

+Pr

Ã −

( )

−

( )

!( ) (7.15)

In above expressions shows the embedding parameter, ~ and ~ the non-zero auxiliary

parameters and N and N the nonlinear operators. When = 0 and = 1 then we obtain

( 0) = 0() ( 0) = 0()

( 1) = () ( 1) = () (7.16)

It should be pointed out that when increases from 0 to 1 then ( ) and ( ) vary

from 0() 0() to () and () Using Taylors’ expansion we write

( ) = 0() +∞P

=1

() (7.17)

( ) = 0() +∞P

=1

() (7.18)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(7.19)

96

where the parameters ~ and ~ have a key role in the convergence of series solutions. The

values of parameters are chosen in such a manner that Eqs. (719) and (720) converge at = 1

Hence Eqs. (719) and (720) give

() = 0() +∞P

=1

() (7.20)

() = 0() +∞P

=1

() (7.21)

The general solutions are arranged as follows.

() = ∗() +7 + 8

− (7.22)

() = ∗() + 9 + 10

− (7.23)

in which the special solutions are denoted by ∗ and ∗

7.3 Convergence of series solutions and discussion

It is well known fact that the homotopy analysis method has a great freedom to choose the

auxiliary parameters ~ and ~ for adjusting and controlling the convergence of series solu-

tions. To determine the appropriate convergence interval of the constructed series solutions,

the ~−curves at 17-order of approximations are sketched. Figs. 7.1 and 7.2 clearly show thatthe range of admissible values of ~ and ~ are −140 ≤ ~ ≤ −04 and −135 ≤ ~ ≤ −025

The results are displayed graphically to see the effects of and on

the prescribed surface temperature and prescribed surface heat flux. We denote temperature

variation for PST case by () and for PHF situation by () in the Figs. 7.3-7.16. Figs. 7.3

and 7.4 illustrate the variations of Deborah number on () and () From these Figs. we have

seen that both () and () are increased with an increase in Deborah number is based

on the relaxation time. When Deborah number increases then relaxation time increases. This

increase in relaxation time causes an increase in () and () Comparison of Figs. 7.3 and 7.4

show that has similar effects on () and () Figs. 7.5 and 7.6 are plotted to see the effects

of magnetic parameter on () and () Clearly the thermal boundary layer thicknesses

97

are increased for larger values of magnetic parameter. In fact the magnetic parameter involves

the Lorentz force. Larger values of magnetic parameter corresponds to the stronger Lorentz

force. This stronger Lorentz force give rise to the thermal boundary layer thicknesses. Figs.

7.7 and 7.8 illustrate the variations of on () and () From these Figs. it is noticed

that both () and () are reduced when we increased the values of Also the thermal

boundary layer becomes thinner for higher values of This reduction in thermal boundary

layer for larger values of is due to the entertainment of cooler to ambient fluid. The power

indices and control the non-uniformity of the surface temperature in the prescribed surface

temperature situation. Figs. 7.9 and 7.10 depict that () and () are decreasing functions

of Also we noted that () reduces rapidly in comparison to () Effect of on () and

() are seen in the Figs. 7.11 and 7.12. The values of () and () are reduced when

values of are increased. It is concluded that the non-uniformity of the sheet temperature

has prominent effect on the temperature fields for the reduction in temperature and thinner

thermal boundary layer. Comparison of Figs. 7.11 and 7.12 illustrates that the variations in

() are more pronounced when compared to the variations in () Also we examined that ()

at the wall is reduced rapidly when the values of are larger. Figs. 7.13 and 7.14 depict the

variations of heat generation/absorption parameter on () and () Both () and () are

increased by increasing values of heat generation/absorption parameter. Physically an increase

in heat generation/absorption parameter produced more heat due to which the temperature of

fluid increases. Such increase in temperature gives rise to () and () The effects of Prandtl

number on () and () are analyzed in the Figs. 7.15 and 7.16. These Figs. clearly show that

() () and their related thermal boundary layer thicknesses are reduced for the larger values

of Prandtl number Obviously the Prandtl number depends upon the thermal diffusivity.

Larger values of Prandtl number give smaller thermal diffusivity and consequently the values

of () and () decrease.

Table 7.1 has been prepared to analyze the convergent values of () and () We have

seen that our solutions for velocities converge from 16th order of approximations whereas one

needs 25th order of deformations for () and () Hence we need less deformations for the

velocities in comparison to temperatures for a convergent solution. Table 7.2 provides the values

of temperature gradient 0(0) for different values of and when = = 0 and = 10

98

One can see that our solutions has an excellent agreement with the previous results in a limiting

case. Further it is observed that the temperature gradient at surface 0(0) becomes positive and

it reduces for = −20 and = 0 and negative for = 0 and = −20 Table 7.3 presents thenumerical values of 0(0) and (0) for different values of and when = = 0 = = 10

and = 025 From this Table we noted that our series solutions have very good agreement

with the previous available results in the literature.

-1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñq

-1.4

-1.2

-1

-0.8

-0.6

q'0

q'0

Fig. 7.1: ~−curve for the function () when = 01 = 07 = 05 = 14 = = 04

and = 03

-1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñf

0

0.2

0.4

0.6

0.8

1

f''0

f''0

99

Fig. 7.2: ~−curve for the function () when = 01 = 07 = 05 = 14 = = 04

and = 03

2 4 6 8h

0.2

0.4

0.6

0.8

1

qh

b = 0.0, 0.3, 0.6, 1.0, 1.4

Fig. 7.3: Influence of on () when = 07 = 05 = 15 = 03 = 04 and = 04

2 4 6 8h

0.25

0.5

0.75

1

1.25

1.5

fh

b = 0.0, 0.3, 0.6, 1.0, 1.4

100


2 4 6 8h

0.2

0.4

0.6

0.8

1

qh

M=0.0, 0.4, 0.8, 1.2, 1.6


2 4 6 8h

0.2

0.4

0.6

0.8

1

1.2

fh

M=0.0, 0.4, 0.8, 1.2, 1.6

101


2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

a= 0.1, 0.3, 0.5, 0.7, 1.0


2 4 6 8 10h

0.25

0.5

0.75

1

1.25

1.5

fh

a= 0.1, 0.3, 0.5, 0.7, 1.0

102


2 4 6 8h

0.2

0.4

0.6

0.8

1

qh

s=0.0, 0.6, 1.2, 1.8, 2.5


2 4 6 8h

0.2

0.4

0.6

0.8

1

1.2

1.4

fh

s=0.0, 0.6, 1.2, 1.8, 2.5

Fig. 7.10: Influence of on () when = 02 = 07 = 15 = 03 = 05 and

103

= 04

2 4 6 8h

0.2

0.4

0.6

0.8

1

qh

r = 0.0, 0.6, 1.2, 1.8, 2.5


= 04

2 4 6 8h

0.25

0.5

0.75

1

1.25

1.5

fh

r =0.0, 0.6, 1.2, 1.8, 2.5


104

= 04

2 4 6 8 10h

0.2

0.4

0.6

0.8

1

qh

S= 0.0, 0.3, 0.6, 0.9, 1.2


= 03

2 4 6 8 10h

0.5

1

1.5

2

2.5

3

3.5

fh

S= 0.0, 0.3, 0.6, 0.9, 1.2


105

= 03

2 4 6 8h

0.2

0.4

0.6

0.8

1

qh

Pr =0.4, 0.8, 1.2, 1.6, 2.0


= 04

2 4 6 8h

0.25

0.5

0.75

1

1.25

1.5

1.75

fh

Pr =0.4, 0.8, 1.2, 1.6, 2.0


= 04

Table 7.1: Convergence analysis of series solutions by numerical data for different order

of deformations when = 01 = 07 = 05 = 14 = = 04 = 03 and

106

~ = ~ = −09Order of deformations 0(0) 00(0)

1 -0.92800 0.55000

10 -0.84012 0.50038

16 -0.83823 0.50111

25 -0.83775 0.50128

30 -0.83775 0.50128

35 -0.83775 0.50128

40 -0.83775 0.50128

Table 7.2: Temperature gradient at surface 0(0) for different values of and with = 00

and = 10

= = 0 = −2 = 0 = 2 = 0 = 0 = −2 = 0 = 2

[79] = 025 -0.665933 0.554512 -1.364890 -0.413111 -0.883125

[80] -0.665927 0.554573 -1.364890 -0.413101 -0.883123

Present -0.66593 0.55457 -1.36489 -0.41310 -0.88312

[79] = 050 -0.735334 0.308578 -1.395356 -0.263381 -1.106491

[80] -0.735333 0.308590 -1.395357 -0.263376 -1.106500

Present -0.73533 0.30858 -1.39536 -0.26338 -1.10649

[79] = 075 -0.796472 0.135471 -1.425038 -0.126679 -1.292003

[80] -0.696470 0.135470 -1.425037 -0.216679 -1.292010

Present -0.79472 0.13547 -1.42504 -0.12667 -1.29200

Table 7.3: Temperature gradient at surface 0(0) and (0) for different values of and

107

when = = 0 = = 10 and = 05

0(0) for PST (0) for PHF

= −02 = 00 = 02 = −02 = 00 = 02

Ref. [79] = 10 -1.348064 -1.255781 -1.148932 0.741805 0.796317 0.870355

Ref. [80] -1.348064 -1.255780 -1.148934 0.741808 0.796318 0.870372

Present -1.34806 -1.25578 -1.14893 0.74180 0.76632 0.87037

Ref. [79] = 50 -3.330392 -3.170979 -3.002380 0.300265 0.315360 0.333069

Ref. [80] -3.330394 -3.170981 -3.002384 0.3002657 0.315363 0.333071

Present -3.33039 -3.17098 -3.00238 0.30028 0.31537 0.33308

Ref. [79] = 100 -4.812149 -4.597141 -4.371512 0.207807 0.217527 0.228754

Ref. [80] -4.812151 -4.597143 -4.371516 0.207809 0.217529 0.228756

Present -4.81215 -4.59714 -4.37152 0.20781 0.21753 0.22876


Here the MHD three-dimensional flow of Maxwell fluid generated by bidirectional stretching

surface is investigated for the two cases namely the prescribed surface temperature (PST) and

prescribed surface heat flux (PHF). Interesting observations of this study can be mentioned

below:

• Effects of Deborah number on () and () are similar in a qualitative manner.

• Both () and () are increasing functions of magnetic parameter

• Increase in ratio parameter reduces the temperatures and their boundary layer thick-nesses.

• Temperature for () decreases rapidly in comparison to () when larger values of and are employed.

• An increase in heat generation/absorption parameter enhances the temperatures () and()

108

• Our series solutions have an excellent agreement with the previous results in limitingcases.

109

Chapter 8

Three-dimensional flow of an

Oldroyd-B fluid over a surface with

convective boundary conditions

The present chapter addresses the three-dimensional flow of an Oldroyd-B fluid over a stretching

surface with convective boundary conditions. Problems formulation is presented using conser-

vation laws of mass, momentum and energy. The solutions to the dimensionless problems are

computed. Convergence of series solutions by homotopy analysis method (HAM) is discussed.

The graphs are plotted for various parameters of the temperature profile. Numerical values of

local Nusselt number are analyzed.

8.1 Formulation

We consider the steady three-dimensional flow of an incompressible Oldroyd-B fluid over a

stretched surface at = 0 The flow takes place in the domain 0 The ambient fluid

temperature is taken as ∞ while the surface temperature is maintained by convective heat

transfer at a certain value . The equations for the steady flow of an incompressible fluid with

heat transfer are

+

+

= 0 (8.1)

110

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+ 2 22

+ 2 2

+2 2

+ 2 2

⎞⎠=

⎛⎝2

2+ 2

⎛⎝ 32

+ 32

+ 33−

22

−

22−

22

⎞⎠⎞⎠ (8.2)

+

+

+ 1

⎛⎝ 2 2

2+ 2

22

+ 2 2

2+ 2 2

+

2 2

+ 2 2

⎞⎠=

⎛⎝2

2+ 2

⎛⎝ 32

+ 32

+ 33−

22

−

22−

22

⎞⎠⎞⎠ (8.3)

+

+

=

2

2 (8.4)

where the respective velocity components in the − − and −directions are denoted by and , 1 and 2 show the relaxation and retardation times respectively, the fluid

temperature, the thermal diffusivity of the fluid, = () the kinematic viscosity, the

dynamic viscosity of fluid and the density of fluid.

The convective boundary conditions are

= = = 0 −

= ( − ) at = 0 (8.5)

→ 0 → 0 → ∞ as →∞ (8.6)

where indicates the thermal conductivity of fluid and and have dimension inverse of time.

Using the following new variables

= 0() = 0() = −√(() + ()) () = − ∞ − ∞

=

r

(8.7)

equation (8.1) is satisfied automatically and Eqs. (82)− (86) give

000+( + ) 00− 02+1[2( + ) 0 00− ( + )2 000] +2[(00+ 00) 00− ( + ) 0000] = 0 (8.8)

111

000 + ( + )00 − 02 + 1[2( + )000 − ( + )2000] + 2[(00 + 00)00 − ( + )0000] = 0 (8.9)

00 + ( + )0 = 0 (8.10)

= 0 = 0 0 = 1 0 = , 0 = −(1− (0)) at = 0 (8.11)

0 → 0 0 → 0 → 0 as →∞ (8.12)

where 1 = 1 and 2 = 2 are the Deborah numbers =is a parameter, =

is the

Prandtl number, =

pis the Biot number and prime shows the differentiation with respect

to .

Expression for local Nusselt number through heat transfer is represented by

=

( − ∞) = −

µ

¶=0

(8.13)

The above equation in dimensionless form can be written as

12 = −0(0) (8.14)

in which = is the local Reynolds number.


Initial approximations and auxiliary linear operators are

0() = 1− exp(−) 0() = (1− exp(−)) 0() = exp(−)1 +

(8.15)

L = 000 − 0 L = 000 − 0 L = 00 − (8.16)

We note that the auxiliary linear operators in above equation satisfy the following properties

L (1 + 2 + 3

−) = 0 L(4 + 5 + 6

−) = 0 L(7 + 8−) = 0 (8.17)

with ( = 1− 8) being the arbitrary constants.

112

The associated zeroth order deformation problems are

(1− )Lh(; )− 0()

i= ~N

h(; ) (; )

i (8.18)

(1− )L [(; )− 0()] = ~N

h(; ) (; )

i (8.19)

(1− )Lh(; )− 0()

i= ~N

h(; ) (; ) ( )

i (8.20)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 0 0(0; ) = 0(∞; ) = 0

0(0 ) = −[1− (0 )] (∞ ) = 0 (8.21)

N [( ) ( )] =3( )

3−Ã( )

!2+ (( ) + ( ))

2( )

2

+1

⎛⎝ 2(( ) + ( ))()

2()

2

−(( ) + ( ))23()

2

⎞⎠+2

⎛⎝ ³2()

2+

2()

2

´2()

2

−(( ) + ( ))4()

4

⎞⎠ (8.22)

N[( ) ( )] =3( )

3−µ( )

¶2+ (( ) + ( ))

2( )

2

+1

⎡⎣ 2(( ) + ( ))()

2()

2

−(( ) + ( ))23()

2

⎤⎦+2

⎛⎝ ³2()

2+

2()

2

´2()

2

−(( ) + ( ))4()

4

⎞⎠ (8.23)

N[( ) ( ) ( )] =2( )

2+Pr(( ) + ( ))

( )

(8.24)

Here is an embedding parameter, ~ ~ and ~ are the non-zero auxiliary parameters and

113

N N and N indicate the nonlinear operators. For = 0 and = 1 we have

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (8.25)

Further when increases from 0 to 1 then ( ) ( ) and ( ) vary from 0() 0() 0()

to () () and () Using Taylor’s series expansion one can write

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(8.26)

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(8.27)

( ) = 0()∞P

=1

() () =

1

!

(; )

¯=0

(8.28)

where the convergence of above series strongly depends upon ~ ~ and ~ Considering that

~ ~ and ~ are selected properly so that Eqs. (825)− (827) converge at = 1 and therefore

() = 0() +∞P

=1

() (8.29)

() = 0() +∞P

=1

() (8.30)

() = 0() +∞P

=1

() (8.31)

The general solutions can be expressed as

() = ∗() + 1 + 2 + 3

− (8.32)

() = ∗() + 4 + 5 + 6

− (8.33)

() = ∗() +7 + 8

− (8.34)

in which the ∗ ∗ and ∗ indicate the special solutions.

114

8.3 Convergence analysis and discussion of results

We note that the series (828) − (830) have the auxiliary parameters ~ ~ and ~. Theseparameters have a key role to adjust and control the convergence of series solutions. The

~−curves have been sketched at 18 order of approximations to determine the suitable rangesfor ~ ~ and ~. Fig. 81 showed that the range of admissible values of ~ ~ and ~ are

−130 ≤ ~ ≤ −030 −130 ≤ ~ ≤ −025 and −140 ≤ ~ ≤ −045 We observed that ourseries solutions converge in the whole region of when ~ = ~ = ~ = −06 (see Table 8.1).

Figs. 8.2-8.4 illustrate the variations of Deborah numbers 1 2 and ratio parameter

on the velocity component 0() By increasing the values of Deborah number 1, there is a

decrease in 0() and momentum boundary layer thickness (see Fig. 8.2). Fig. 8.3 shows

that the velocity component 0() and its related boundary layer thickness is higher for the

larger values of 2 Effects of on the velocity component 0() are analyzed in Fig. 8.4. We

examined that the velocity component 0() is decreasing by increasing the values of Figs.

8.5-8.7 show the influences of 1 2 and on the velocity component 0() From Figs. 8.5

and 8.6 we analyzed that the effects of 1 and 2 on the velocity component 0() are similar

to that of 0() Fig. 8.7 illustrates that an increase in leads to an increase in the velocity

component 0() and its related boundary layer thickness.

Figs. 8.8-8.19 are plotted to see the variations of Deborah numbers 1 and 2 Prandtl

number and Biot number on the fluid temperature () when = 00 = 05 and = 10

Fig. 8.8 illustrates the effect of Deborah number 1 on the temperature field when = 00 Here

both the fluid temperature and thermal boundary layer thickness are increased by increasing

1 Physically this is due to the fact that Deborah number 1 contains the relaxation time

1. The increase in relaxation time leads to an increase in temperature and thermal boundary

layer thickness. Fig. 8.9 shows the influence of Deborah number 2 on the temperature field

= 00 The effects of 2 on temperature and thermal boundary layer thickness are opposite

to that of 1. This is due to the reason that retardation time provides resistance which causes

reduction in temperature and thermal boundary layer thickness. Fig. 8.10 clearly depicts that

the larger Prandtl number corresponds to the lower temperature and thermal boundary layer

thickness. In fact the larger Prandtl number means that thermal diffusivity is lower. Such

decrease in thermal diffusivity leads to a decrease in temperature and its associated boundary

115

layer thickness. Fig. 8.11 presents the variations of Biot number on the temperature profile for

= 00. An increase in Biot number give rise to the temperature and thermal boundary layer

thickness. We also observed that the temperature and thermal boundary layer thickness are

increasing functions of Biot number. Further it is noticed that the peak temperature occurs in

the thermal boundary layer in the region near the surface. Figs. 8.12-8.15 are plotted to see the

influences of different parameters on the temperature () for = 05 From Fig. 8.12 one can

see that 1 has same effects on the temperature as in the case of = 00 The only difference

we noticed that the increase in temperature is more dominant for = 05 in comparison to

= 00 By making a comparison of Figs. 8.9 and 8.13, we conclude that 2 have a similar

effects for = 00 and = 05 Fig. 8.14 clearly shows that the variations in temperature

due to an increase in Prandtl number for = 05 are large when compared with = 00 The

effects of Biot number on the temperature are similar in a qualitative sense (see Figs. 8.11 and

8.15). Fig. 8.16 is plotted to see the effects of 1 on the temperature for = 10 It shows

that the fluid temperature and thermal boundary layer thickness are increasing functions of 1

when = 10 There is a decrease in temperature and thermal boundary layer thickness with

an increase in 2 for = 10 (see Fig. 8.17). The effects of Prandtl and Biot numbers on the

fluid temperature are similar to that of = 00 and = 05 (see Figs. 8.18 and 8.19).

To see the convergent values of velocity and temperature, Table 8.1 is provided. From this

Table we made an argument that 20th order deformations are enough for the convergent series

solutions. Table 8.2 provides the numerical values of local Nusselt number for different values

of 1 2 and for = 00 and = 05. We noticed that the values of Nusselt number are

small when = 00 in comparison to the values of = 05 This means that the values of local

116

Nusselt number are increased for larger

-1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñf ,Ñg,Ñq

-1

-0.8

-0.6

-0.4

-0.2

f''0

,g''0

,q'0

b1 = 0.3, b2 = 0.4, Pr = 1.0, g = 0.8, a = 0.5

q'0g''0f ''0

Fig. 8.1: ~-curves for the functions () () and ()

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.4, a = 0.5

b1 = 1.0b1 = 0.7b1 = 0.3b1 = 0.0

Fig. 8.2: Influence of 1 on 0()

117

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

f'h

b1 = 0.4, a = 0.5

b2 = 1.0b2 = 0.7b2 = 0.3b2 = 0.0


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

f'h

b1 = 0.4= b2

a = 1.0a = 0.7a = 0.4a = 0.0


118

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

g'h

b2 = 0.4, a = 0.5

b1 = 1.0b1 = 0.7b1 = 0.3b1 = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

g'h

b1 = 0.4, a = 0.5

b2 = 1.0b2 = 0.7b2 = 0.3b2 = 0.0


119

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

g'h

b1 = 0.4= b2

a = 1.0a = 0.7a = 0.4a = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b2 = 0.4, g = 0.6

b1 = 3.0b1 = 2.0b1 = 1.0b1 = 0.0

Fig. 8.8: Influence of 1 on () when = 00

120

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b1 = 0.4, g = 0.6

b2 = 3.0b2 = 2.0b2 = 1.0b2 = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

qh

b1 = b2 = 0.4, g = 0.6

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1

Fig. 8.10: Influence of Pr on () when = 00

121

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b1 = b2 = 0.4

g = 0.6g = 0.4g = 0.2g = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b2 = 0.4, g = 0.6

b1 = 3.0b1 = 2.0b1 = 1.0b1 = 0.0


122

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b1 = 0.4, g = 0.6

b2 = 3.0b2 = 2.0b2 = 1.0b2 = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

qh

b1 = b2 = 0.4, g = 0.6

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1


123

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b1 = b2 = 0.4

g = 0.6g = 0.4g = 0.2g = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b2 = 0.4, g = 0.6

b1 = 3.0b1 = 2.0b1 = 1.0b1 = 0.0


124

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b1 = 0.4, g = 0.6

b2 = 3.0b2 = 2.0b2 = 1.0b2 = 0.0


0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

qh

b1 = b2 = 0.4, g = 0.6

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1


125

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, b1 = b2 = 0.4

g = 0.6g = 0.4g = 0.2g = 0.0


Table 8. 1: Convergence of series solutions for different order of approximations when

1 = 03 2 = 04 = 10 = 08 = 05 and ~ = ~ = ~ = −06


10 0.96460 0.40614 0.38771

15 0.96449 0.40619 0.38791

20 0.96450 0.40622 0.38790

25 0.96450 0.40622 0.38790

30 0.96450 0.40622 0.38790

35 0.96450 0.40622 0.38790

Table 8.2: Values of local Nusselt number −0(0) for different values of the parameters 1 2

126

and .

1 2 −0(0) = 00 = 05

0.0 0.4 1.0 0.8 0.34759 0.39658

0.5 0.33651 0.38228

1.0 0.32636 0.36997

0.4 0.0 0.32613 0.36761

0.5 0.34094 0.38793

1.0 0.34963 0.39552

0.5 0.24839 0.29221

0.8 0.30916 0.35506

1.3 0.37300 0.41932

0.3 0.19856 0.21365

0.6 0.29677 0.33187

1.0 0.36993 0.42614

8.4 Conclusions

An analysis is presented for the three-dimensional flow of an Oldroyd-B fluid subject to con-

vective type surface condition. Series solutions are obtained for the velocity and temperature

profiles. The main observations of this analysis are given below:

• The variations of temperature by increasing 1 are dominant for = 10 in comparisonto = 00 and = 05

• The fluid temperature and thermal boundary layer thickness are decreased rapidly for = 10 when compared with the temperature for = 00 and = 05

• Effects of Biot number on the temperature and thermal boundary layer thickness are quitesimilar for = 00 = 05 and = 10

• Numerical values of local Nusselt number are increased by increasing

127

• Values of local Nusselt number are smaller for = 00 when compared with = 05 (see

Table 8.2).

128

Chapter 9

Radiative flow of Jeffrey fluid in a

porous medium with power law heat

flux and heat source

The aim of this chapter is to examine the flow of an incompressible Jeffrey fluid over a stretching

surface. In addition the heat transfer process subject to power law heat flux and heat source

is addressed. Mathematical analysis has been carried out in the presence of thermal radiation.

Homotopic solutions for the velocity and temperature are developed. The related convergence

analysis is made very carefully. The plotted results are discussed for the temperature and heat

transfer characteristics.


We consider the two-dimensional flow of an incompressible Jeffrey fluid over a moving porous

surface in the presence of power law heat flux and heat source. Thermal radiation effects are

also accounted. A Cartesian coordinate system is chosen in such a way that −axis is alongthe stretching surface and the −axis perpendicular to it. The fluid fills the porous half space 0. The boundary layer equations for flow and temperature are given by

+

= 0 (9.1)

129

+

=

1 + ∗

∙2

2+ 2

µ

3

2−

2

2+

2

+

3

3

¶¸−

(9.2)

+

=

2

2−

( − ∞)−

(9.3)

where and are the velocity components in the − and −directions, ∗ and 2 are the ratioof relaxation to retardation times and retardation time respectively, = () is the kinematic

viscosity, is the permeability of porous medium, is the fluid temperature, is the density of

fluid, is the thermal conductivity of fluid, is the specific heat at constant pressure, is the

heat source coefficient and is the radiative heat flux. By using the Rosseland approximation

we have

= −4∗

31

4

(9.4)

where ∗ is the Stefan-Boltzmann constant and 1 is the mean absorption coefficient. Taylor’s

series helps in writing the following expressions

4 ∼= 4 3∞ − 3 4∞ (9.5)

One can write now after employing the Eqs. (93)− (95) as follows

+

=

2

2−

( − ∞) +

16 3∞3∗

2

2 (9.6)

The boundary conditions are defined by

= = −0

= 2 at = 0 (9.7)

= 0 = ∞ as →∞ (9.8)

where is the temperature coefficient and ∞ is the ambient temperature.

We introduce the similarity transformations

= 0() = −√() = ∞ +

r

2() =

r

(9.9)

130

Here is a constant and prime denotes differentiation with respect to .

Using Eq. (98) we have

000 + ( 002 − 0000) + (1 + ∗)( 00 − 02)− 1(1 + ∗) 0 = 0 (9.10)

(1 +4

3)00 + 0 − 2 0 − ∗ = 0 (9.11)

= ∗ 0 = 1 0 = −1 at = 0 (9.12)

0 = 0 = 0 as →∞ (9.13)

where Eq. (91) is satisfied automatically and = 2 is the Deborah number =

is the

permeability parameter, ∗ = 0√is the suction parameter, =

is the Prandtl number

and ∗ = is a heat generation parameter.

Expression of local Nusselt number with heat transfer is

=

( − ∞) = −

µ

¶=0

(9.14)

Dimensionless form of Eq. (914) gives

12 = − 1

(0) (9.15)


We can express and by a set of base functions

{ exp(−) ≥ 0 ≥ 0} (9.16)

in the forms

() =

∞X=0

∞X=0

exp(−) (9.17)

() =

∞X=0

∞X=0

exp(−) (9.18)

131

in which and are the coefficients. We further choose the following initial approxima-

tions and auxiliary linear operators

0() = ∗ + 1− exp(−) 0() = − exp(−) (9.19)

L = 000 − 0 L = 00 + 0 (9.20)

with

L (1 + 2 + 3

−) = 0 L(4 + 5−) = 0 (9.21)

where ( = 1− 5) represent the arbitrary constants.The zeroth order deformation problems are

(1− )Lh(; )− 0()

i= ~N

h(; )

i (9.22)

(1− )Lh(; )− 0()

i= ~N

h(; ) ( )

i (9.23)

(0; ) = ∗ 0(0; ) = 1 0(∞; ) = 0 0(0 ) = 1 (∞ ) = 0 (9.24)

N [( )] =3( )

3+ (1 + ∗)( )

2( )

2− (1 + ∗)

Ã( )

!2

+2

"2( )

( )

2( )

2− (( ))2

3( )

3

#− 1

(1 + )

( )

(9.25)

N[( ) ( )] =

µ1 +

4

3

¶2( )

2+( )

( )

−2( )

( )−∗( )

(9.26)

in which is an embedding parameter, ~ and ~ the non zero auxiliary parameters and N

and N the nonlinear operators.

For = 0 and = 1 we have

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (9.27)

and when increases from 0 to 1 then ( ) and ( ) vary from 0() 0() to () and

132

() By Taylor’s’ series one has

( ) = 0() +∞P

=1

() (9.28)

( ) = 0() +∞P

=1

() (9.29)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(9.30)


and ~ are selected properly so that Eqs. (922) and (923) converge at = 1 and thus we have

() = 0() +∞P

=1

() (9.31)

() = 0() +∞P

=1

() (9.32)


L [()− −1()] = ~R () (9.33)

L[()− −1()] = ~R () (9.34)

(0) = 0(0) = 0(∞) = 0 0(0) = (∞) = 0 (9.35)

R () = 000−1() + (1 + ∗)

−1P=0

h−1− 00 − 0−1−

000

i+2

−1X=0

−1−X=0

{2 0− 00 − − 000 − (1 + ∗)1

0−1() (9.36)

R () = 00−1 +

−1P=0

0−1− − 2−1P=0

−1− 0 − ∗ (9.37)

=

⎡⎣ 0 ≤ 11 1

(9.38)

133


() = ∗() + 1 + 2 +3

− (9.39)

() = ∗() + 4 + 5− (9.40)



As we know that the auxiliary parameters ~ and ~ play indispensable role to adjust and

control the convergence of the homotopy solutions. For the range of admissible values of ~

and ~ we plot the ~−curves for 16-order of approximations. Figs. 91 and 92 show thatthe range of admissible values of ~ and ~ are −13 ≤ ~ ≤ −025 and −085 ≤ ~ ≤ −025Further the series converges in the whole region of when ~ = −10 and ~ = −06 FromTable 91 we see that our series solutions converge from 20 order of approximations Hence

25 order approximations are enough for the convergent solutions.

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0Ñf

-1.7

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

f''0

N=0.4, S* = 0.5, Pr =1.0, l = 1.0, b2 =0.1, b* = 0.2, l* =0.2

f''0


134

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0Ñq

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

q''0

N=0.4, S* =0.5, Pr =1.0, l = 1.0, b2 = 0.1, b* = 0.2, l* = 0.2

q''0



2 = 01 ∗ = 02 = 10, ∗ = 05 ∗ = 02 = 20 = 04 and ~ = −08 and~ = −06


5 1.485427 1.512662

10 1.485505 1.514593

15 1.485505 1.514741

25 1.485505 1.514758

30 1.485505 1.514758

35 1.485505 1.514758


This section highlights the influence of pertinent parameters on the velocity, temperature and

surface heat transfer. Fig. 93 displays the velocity profile for the suction parameter ∗ on the

velocity 0() The velocity and boundary layer thickness are decreasing function of ∗. Fig.

9.4 illustrates the effect of Deborah number 2 on 0() An increase in the Deborah number

increases the velocity profile 0() The effect of porosity parameter is seen in Fig. 9.5.

An increase in corresponds to an increase in velocity and boundary layer thickness. The

135

effect of parameter ∗ on velocity 0() is presented in Fig. 9.6. The boundary layer thickness

decreases with an increase in ∗ The outcome of an increase in Prandtl number Pr is observed

in Fig. 9.7. There is a lower value of thermal conductivity when Prandtl number increases.

Consequently a rapid increase in the Prandtl number Pr decreases the thermal boundary layer

thickness. The feature of ∗ on thermal boundary layer thickness is similar to Pr (see Fig.

9.8). Fig. 99 clearly shows that larger values of increase the temperature and the thermal

boundary layer thickness. The larger values of ∗ result in a decrease in the thermal boundary

layer thickness (Fig. 9.10). It is clear from Fig. 9.11 that there is an increase in thickness

of the thermal boundary layer and the temperature distribution also increases when radiation

parameter increases.

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.1, l = 2.0, l* = 0.2

S* = 1.5S* = 1.0S* = 0.5S* = 0.0


136

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

S* = 0.5, l = 2.0, l* = 0.2

b2 = 1.0b2 = 0.6b2 = 0.3b2 = 0.0


0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

S* = 0.5, b2 = 0.1, l* = 0.2

l = 3.0l = 2.0l = 1.0l = 0.5


137

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

S* = 0.5, b2 = 0.1, l = 2.0

l* = 1.4l* = 0.8l* = 0.4l* = 0.0


0 2 4 6 8 10 12h

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

qh

S* = 0.5, l = 2.0, b2 = 0.1, b* = 0.2, N = 0.4, l* = 0.2

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1

Fig. 9.7: Influence of Pr on ()

138

0 2 4 6 8 10h

-0.8

-0.6

-0.4

-0.2

0

qh

Pr = 1.0, l = 2.0, b2 = 0.1, b* = 0.2, N = 0.4, l* = 0.2

S* = 1.5S* = 1.0S* = 0.5S* = 0.0


0 2 4 6 8 10h

-0.8

-0.6

-0.4

-0.2

0

qh

Pr = 1.0, S* = 0.5, b2 = 0.1, b* = 0.2, N= 0.4, l* = 0.2

l = 3.0l = 2.0l = 1.0l = 0.5


139

0 2 4 6 8 10 12h

-0.8

-0.6

-0.4

-0.2

0

qh

Pr = 1.0, S* = 0.5, l = 2.0, b2 = 0.2, N = 0.3, l* = 0.2

b* = 1.2b* = 0.8b* = 0.4b* = 0.0


0 2 4 6 8 10 12h

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

qh

Pr = 1.0, S* = 0.5, l = 2.0, b2 = 0.1, b* = 0.2, l* = 0.2

N= 1.4N= 0.9N= 0.4N= 0.0


140

Table 9.2: Values of local Nusselt number 12 for the different values of parameters

∗ 2 ∗ and when ∗ = 05 and = 04

∗ 2 ∗ −12

0.5 0.2 0.1 0.2 1.0 1.13518

1.0 1.17909

2.0 1.20704

2.0 0.0 1.22933

0.4 1.18641

0.8 1.14936

0.0 1.17970

0.3 1.24496

0.6 1.28144

0.0 1.11523

0.5 1.31994

1.0 1.47567

0.4 0.71664

0.8 1.05025

1.5 1.57240

9.5 Final remarks

Steady flow of Jeffrey fluid in a porous medium is discussed. Analysis has been carried out in

presence of power law heat flux, heat source and thermal radiation. Important points can be

summed up as follows:

• Suction parameter ∗ and Deborah number 2 have similar effects on the velocity profile 0()

• Velocity field 0() increases when the porosity parameter increases

• The temperature profile () increases in view of an increase in Pr

• The heat generation parameter ∗ leads to a decrease in ()

141

Chapter 10

Radiative flow of Jeffrey with

variable thermal conductivity in

porous medium

This chapter considers the thermal radiation effect in the flow of a Jeffrey fluid with variable

thermal conductivity. Similarity transformations are employed to convert the partial differential

equation into the ordinary differential equation. The resulting equation has been computed for

the series solution of temperature. The numerical values of local Nusselt numbers are also

computed. Comparison with the numerical solutions of 0(0) is presented. The obtained results

are displayed and physical aspects have been examined in detail.

10.1 Mathematical analysis

Consider the flow of an incompressible Jeffrey fluid over a linearly stretching sheet in a porous

medium. The thermal conductivity is not constant. Two equal and opposite forces are applied

along the sheet due to which the wall is stretched keeping the position of origin unchanged. We

suppose that the wall temperature () ∞ where ∞ denotes temperature of the fluid for

away from the sheet. Further both fluid and the porous medium are in local thermal equilibrium.

The − and −axes in the Cartesian coordinate system are chosen along and normal to the

142

sheet respectively. The energy equation subject to radiative effect can be expressed in the form

∙

+

¸=

∙

¸−

(10.1)

and the subjected boundary conditions are

= () = ∞ +∗at = 0 (10.2)

→ ∞ as →∞ (10.3)

where , are the flow velocities in the − and −directions respectively, ∗ the ratio ofrelaxation to retardation times, 2 the retardation time, the kinematic viscosity, the

permeability, the temperature, the variable thermal conductivity, the density of the fluid,

the specific heat at constant pressure and the radiative heat flux. By making use of

Rosseland approximation, the radiative heat flux is given by

= −4∗

31

4

(10.4)

where ∗ is the Stefan-Boltzmann constant and 1 is the mean absorption coefficient. In view

of Taylor’s series, the term 4 can be written as

4 ∼= 4 3∞ − 3 4∞ (10.5)

By making use of Eqs. (104) and (105) Eq. (101) becomes

∙

+

¸=

∙

¸+16 3∞31

2

2 (10.6)

The similarity transformations are defined as follows

= 0() = −√() =

r

() =

− ∞ − ∞

(10.7)

where is the variable wall temperature and () is the non-dimensional form of the temper-

ature. We consider = () = ∞+() at = 0 The variable thermal conductivity is

143

= ∞[1 + ] (here and are positive constants, ∞ is the fluid free stream conductivity)

and is given by

= − ∞

∞ (10.8)

where is a constant, is the thermal conductivity at the wall and prime denotes the differ-

entiation with respect to .

Eqs. (101)− (106) reduce to the following expressions

(1 + )00 + 02 +4

300 = [∗ 0 − 0] (10.9)

(0) = 1 and (∞) = 0 (10.10)

where =

∞ is the Prandtl number and =4 3∞∞ is the radiation parameter. The local

Nusselt number is defined as follows

=

( − ∞) (10.11)

with heat transfer given by

= −µ

¶=0

(10.12)

Dimensionless expression of Eq. (1011) is

12 = −0(0) (10.13)

The problems consisting of Eqs. (109) and (1010) can be computed by a homotopy analysis

method (HAM). For that we express in the set of base function

{ exp(−) ≥ 0 ≥ 0} (10.14)

by

() =

∞X=0

∞X=0

exp(−) (10.15)

with and as the coefficients. The initial approximations and auxiliary linear operators

144

can be written as

0() = (−) (10.16)

L = 00 − (10.17)

L(1 + 2−) = 0 (10.18)

where ( = 1− 2) are the arbitrary constants.The zeroth order deformation problems may be expressed as follows:

(1− )Lh(; )− 0()

i= ~N

h(; ) ( )

i (10.19)

(0 ) = 1 0(∞ ) = 0 (10.20)

N[( ) ( )] =

µ1 +

4

3

¶2( )

2+ ( )

2( )

2+

Ã( )

!2

−∗( )( )

+ ( )( )

(10.21)

where is the embedding parameter, ~ the non-zero auxiliary parameter and N the nonlinear

operator. For = 0 and = 1 one has

( 0) = 0() and ( 1) = () (10.22)

and when increases from 0 to 1 then ( ) varies from 0() to () Taylor’s series expansion

allows the following relations

( ) = 0() +∞P

=1

() (10.23)

() =1

!

(; )

¯=0

(10.24)

where the convergence of above series depends upon ~ Considering that ~ are selected prop-

145

erly so that (1023) converges at = 1 and thus

() = 0() +∞P

=1

() (10.25)

The th-order problems are given by

L[()− −1()] = ~R () (10.26)

0(0)− (0) = (∞) = 0 (10.27)

R () =

µ1 +

4

3

¶00−1 +

−1P=0

−1−00 + −1P=0

0−1−0

−∗−1P=0

−1− 0 + −1P=0

−1− 0 (10.28)

=

⎡⎣ 0 ≤ 11 1

(10.29)

The general solutions may be written as

() = ∗() +4 + 5

− (10.30)

where ∗ stands as the special solution.


We found that the expression (1025) has the non-zero auxiliary parameter ~ Such auxiliary

parameters play key role in the analysis of convergence for the obtained series solutions. In

order to define the adequate values of ~, the ~−curve has been potrayed for 20-order ofapproximations. From Fig. 10.1 it is noted that the range of admissible values of ~ are

146

−115 ≤ ~ ≤ −035 The series converges in the whole region of when ~ = −07

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Ñq

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

q'0

l = 2.0, N=0.3, l* =0.2, b2 = 0.1, Pr = 1.0, a* = 1, e = 0.2

q'0



= 01 = = 10, = 03 = 20 = = 02 and } = } = −07

Order of approximation −0(0)1 0.76667

5 0.68968

10 0.67553

20 0.67017

30 0.66923

35 0.66908

40 0.66908

50 0.66908

10.3 Discussion

In this section we plot Figs. 102 − 108 for the effects of Deborah number 2 permeabilityparameter ratio of relaxation time over retardation time ∗ Prandtl number positive

constant ∗ radiation parameter and small parameter on the temperature (). Fig. 102

147

represents the effects of on () By increasing the temperature () decreases. From Fig.

103 we observed that the temperature field () decreases by increasing the values of ∗ Fig.

104 plots the variations of on () The temperature field () decreases when increases

Fig. 105 shows the effects of 2 on () From Fig. 105 we observed that the temperature field

() decreases when 2 increases. Here Deborah number is directly proportional to retardation

time. An increase in Deborah number implies to the larger retardation time. This larger

retardation time is responsible to a reduction in the temperature and thermal boundary layer

thickness. Fig. 106 shows that the temperature profile () increases when increases. The

larger radiation parameter correspond to higher temperature and smaller radiation parameter

shows lower temperature. An increase in radiation parameter give more to the fluid which

results an enhancement in the temperature and related thermal boundary layer thickness. Fig.

107 plots the effects of on () The temperature field () increases when is increased.

Here is the thermal conductivity parameter. It is a fact that fluid possesses stronger thermal

conductivity has higher temperature. From Figs. 106 and 107 it is obvious that and have

similar effects on the temperature field () in a qualitative sense Fig. 108 shows the effects

of ∗ on temperature profile. We see that () increases by increasing ∗

0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1

qh

a* = 0.5, l = 2.0, b2 = 0.1, l* = 0.2, N = 0.3, e = 0.2

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1

Fig. 10.2: Influence of Pr on ()

148

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, l = 2.0, b2 = 0.1, l* = 0.2, N = 0.3, e = 0.2

a* = 3.0a* = 2.0a* = 1.0a* = 0.0


0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a* = 0.5, b2 = 0.1, l* = 0.2, N = 0.3, e = 0.2

l = 4.0l = 2.0l = 1.0l = 0.5


149

0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a* = 0.5, l = 2.0, l* = 0.2, N = 0.3, e = 0.2

b2 = 1.0b2 = 0.6b2 = 0.3b2 = 0.0


0 2 4 6 8 10 12h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a* = 0.5, l = 2.0, l* = 0.2, b2 = 0.1, e = 0.2

N = 1.2N = 0.8N = 0.4N = 0.0


150

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a* = 0.5, l = 2.0, l* = 0.2, b2 = 0.1, N = 0.3

e = 1.5e = 1.0e = 0.5e = 0.0


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

Pr = 0.7, a* = 0.5, l = 2.0, N = 0.3, b2 = 0.1, e = 0.2

l* = 0.9l* = 0.6l* = 0.3l* = 0.0



Radiative flow of Jeffrey fluid with variable thermal conductivity is studied. The thermal

conductivity varies linearly with the temperature. The key points of present study are:

• By increasing the permeability parameter the temperature field () decreases.

• The temperature profile () decreases by increasing

• The permeability parameter has quite opposite effects on the velocity and temperature

151

profiles.

• Numerical values of local Nusselt number decreases by increasing ∗ but it increases byincreasing 2 and

152

Chapter 11

Influence of thermophoresis and

Joule heating on the radiative flow of

Jeffrey fluid with mixed convection

This chapter addresses the magnetohydrodynamic (MHD) radiative flow of an incompress-

ible Jeffrey fluid over a linearly stretched surface. Heat and mass transfer characteristics are

accounted in the presence of Joule heating and thermophoretic effects. Series solutions by ho-

motopy analysis method are constructed for the velocity, temperature and concentration fields.

Convergence criteria for the series solutions is established. Numerical values of skin friction

coefficient, local Nusselt and Sherwood numbers are computed and analyzed.

11.1 Flow formulation

We consider Cartesian coordinate system in such a way that −axis is along the stretchingsurface and −axis is perpendicular to the −axis. Magnetohydrodynamic boundary layer flowof Jeffrey fluid is considered. A uniform magnetic field B0 is applied parallel to the −axis.Induced magnetic field is neglected for small magnetic Reynolds number. Heat and mass transfer

characteristics are taken into account in the presence of thermal radiation and thermophoresis

effects. Uniform temperature of the surface is larger than the ambient fluid temperature

∞ The species concentration at the surface and ambient concentration ∞ are constants.

153

Invoking Rosseland approximation, the resulting equations take the following forms

+

= 0 (11.1)

+

=

1 + ∗

µ2

2+ 2

µ

3

2+

3

3−

2

2+

2

¶¶−

∗20

+ [ ( − ∞) + ( −∞)] (11.2)

+

=

2

2+16 3∞3∗

2

2+

µ

¶2+

∗20

2 (11.3)

+

=

2

2−

() (11.4)

Here ( ) are the velocity components along the − and −axes, ∗ and 2 are the ratio

of relaxation/retardation times and retardation time, respectively, the dynamic viscosity,

the density of fluid, ∗ the electrical conductivity, the gravitational acceleration, and

the thermal expansion coefficients, the temperature, the specific heat, the Stefan-

Boltzmann constant, ∗ the mean absorption coefficient, the diffusion coefficient and the

thermophoretic velocity.

The associated boundary conditions are

= = = 0 = () = () at = 0

→ 0

→ 0 → ∞ → ∞ as →∞ (11.5)

where is the stretching velocity, is the temperature at the wall, is the concentration

at the wall and ∞ and ∞ are the ambient fluid temperature and concentration, respectively.

The term in Eq. (11.4) can be defined as follows:

= −1

(11.6)

in which 1 is the thermophoretic coefficient and is the reference temperature. A ther-

154

mophoretic parameter is defined by the following relation

= −1( − ∞)

(11.7)

The wall temperature and concentration fields are

= ∞ + = ∞ + (11.8)

where and are the positive constants.

Through the following transformations

= 0() = −√() =

r

() = − ∞ − ∞

() = − ∞ − ∞

(11.9)

equation (11.1) is automatically satisfied and the Eqs. (11.2)-(11.4) are reduced as follows:

000 + (1+ ∗)( 00 − 02) + 2(002 − 0000)− (1 + ∗)2 0 + (1+ ∗)1(+ 2) = 0 (11.10)

(1 +4

3)00 + [0 − 0] + 002 +2 02 = 0 (11.11)

00 + (0 − 0)− (00 − 00) = 0 (11.12)

= 0 0 = 1 = 1 = 1 at = 0

0 → 0 00 → 0 → 0 → 0 as →∞ (11.13)

In the above expressions, 2 = 2 is the Deborah number 2 = ∗20 the Hartmann

number, 1 =2

the local buoyancy parameter, = (−∞)32

222

the local Grashof

number, 2 =(−∞) (−∞) the constant dimensionless concentration buoyancy parameter, =

the Prandtl number, =

4∗ 3∞∗ the radiation parameter, =

2(−∞) the Eckert

number and = the Schmidt number.

Skin friction coefficient, local Nusselt number and local Sherwood number are transformed

155

to the following

12 =1

1 + ∗( 00(0) + 2

00(0)), −12 = −0(0) and −12 = −0(0) (11.14)


The homotopic solutions for , and in a set of base functions

{ exp(−) ≥ 0 ≥ 0} (11.15)

can be written to the following expressions

() =

∞X=0

∞X=0

exp(−) (11.16)

() =

∞X=0

∞X=0

exp(−) (11.17)

() =

∞X=0

∞X=0

exp(−) (11.18)

in which , and are the coefficients. Initial guesses and auxiliary linear operators

are

0() = (1− exp(−) 0() = exp(−), 0() = exp(−), (11.19)

L = 000 − 0 L = 00 − L = 00 − (11.20)

The operators in Eq. (11.19) satisfy the following properties

L (1 + 2 + 3

−) = 0, L(4 + 5−) = 0 L(6 + 7

−) (11.21)

where ( = 1 − 7) denote the arbitrary constants. The zeroth order deformation problemsare expressible in the form

(1− )L³(; )− 0()

´= ~N

³(; ) (; ) (; )

´ (11.22)

156

(1− )L³(; )− 0()

´= ~N

³(; ) (; ) (; )

´ (11.23)

(1− )L³(; )− 0()

´= ~N

³(; ) (; ) (; )

´ (11.24)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 00(∞; ) = 0

(0; ) = 1, (∞; ) = 0 (0; ) = 1 and (∞; ) = 0 (11.25)

Here shows embedding parameter, ~ , ~ and ~ the non-zero auxiliary parameters and the

nonlinear operators N , N and N are given by

N [( ) (; ) (; )] =3( )

3+ (1 + ∗)

⎛⎝( )2( )

2−Ã( )

!2⎞⎠+2

⎛⎝Ã2( )

2

!2− ( )

4( )

4

⎞⎠− (1 + ∗)2( )

+(1 + ∗)³1(( ) + 2(; )

´ (11.26)

N[( ) (; ) (; )] =

µ1 +

4

3

¶2( )

2+

Ã2( )

2

!2+2

Ã( )

!2

−( )( )

+ ( )( )

(11.27)

N[( ) (; ) (; )] =2( )

2+

Ã( )

( )

− ( )

( )

!

−Ã( )

( )

− ( )

2( )

2

! (11.28)

When = 0 and = 1 one has

(; 0) = 0(); (; 1) = () (11.29)

(; 0) = 0(); (; 1) = () (11.30)

157

(; 0) = 0(); (; 1) = () (11.31)

Note that when increases from 0 to 1 then ( ), ( ) and ( ) approach 0() to (),

0() to () and 0() to () According to Taylor’s series one has

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(11.32)

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(11.33)

( ) = 0() +∞P

=1

() () =

1

!

(; )

¯=0

(11.34)

The convergence of series (11.22)-(11.24) is closely associated with ~ ~ and ~ The values

of ~ , ~ and ~ are chosen such that the series (11.22)-(11.24) converge at = 1. Thus

() = 0() +∞P

=1

() (11.35)

() = 0() +∞P

=1

() (11.36)

() = 0() +∞P

=1

() (11.37)

The general solutions , and in terms of special solutions ∗, ∗ and ∗ are

() = ∗() + 1 + 2 + 3

− (11.38)

() = ∗() +4 + 5

− (11.39)

() = ∗() +6 + 7

− (11.40)

11.3 Convergence analysis and discussion

The auxiliary parameters ~ , ~ and ~ play important role in controlling and adjusting the

convergence of series solutions. To find the suitable values for each of these auxiliary parameters,

the ~−curves at 19 order of approximations are displayed. Fig. 11.1 indicates that the

158

admissible values of ~ , ~ and ~ are −110 ≤ ~ ≤ −020, −120 ≤ ~ ≤ −010 and −120 ≤~ ≤ −020 Our series solutions converge in the whole region of for ~ = ~ = ~ = −07.

Figs. 11.2-11.17 plot the behaviors of various interesting parameters on the velocity 0(),

temperature () and concentration (). The fluid velocity and momentum boundary layer

thickness increase with an increase in Deborah number 2 (see Fig. 11.2). In Fig. 11.3,

the influence of ratio of relaxation to retardation times is sketched for the fluid velocity. It

shows that the fluid velocity decreases by increasing ∗ Fig. 11.4 shows that an increase in

local buoyancy parameter 1 yields an increase in the velocity. In fact the local buoyancy

parameter depends upon the buoyancy force and increase in buoyancy force gives rise to the

fluid velocity. An increase in Hartmann number decreases the fluid velocity (see Fig. 11.5).

Hartmann number is a consequence of the Lorentz force. Obviously an increase in Lorentz

force opposes the flow and thus the fluid velocity decreases. Fig. 11.6 shows that the velocity

and momentum boundary layer thickness are decreasing functions of Prandtl number Fig.

11.7 displays the effects of ratio of buoyancy parameter 2. We analyzed that the effects of

ratio of buoyancy parameter are similar to that of 1 in a qualitative way. The difference

we noticed is that the fluid velocity increases more rapidly in case of increasing local buoyancy

parameter when compared with the ratio of buoyancy parameter. Effects of different parameters

on the temperature are seen in the Figs. 11.8-11.13. Fig. 11.8 depicts the variations of local

buoyancy parameter on the temperature field. It is noticed that the temperature field and

thermal boundary layer thickness are reduced with an increase in 1 Fig. 11.9 depicts that the

temperature increases when Hartmann number is increased. In fact that larger corresponds

to lower permeability and hence the temperature and thermal boundary layer thickness increase.

Figs. 11.10 and 11.11 illustrate the influences of Eckert number and Prandtl number

on the temperature. It is observed from these Figs. that and have quite opposite

effects on velocity and associated thermal boundary layer thickness. The temperature profile

decreases more quickly for in comparison to The ratio of buoyancy parameter and

radiation parameter corresponds to the decrease and increase in the temperature respectively

(see Figs. 11.12 and 11.13). The variations of 1 , 2 and on concentration profile are

seen through the Figs. 11.14-11.17. Fig. 11.14 illustrates that the concentration and boundary

layer thickness are decreasing functions of 1 It is found from Fig. 11.15 that the Hartmann

159

number increases the concentration profile and associated boundary layer thickness. The effects

of 2 and on () are similar. Increase in 2 and decreases the concentration and related

boundary layer thickness (see Figs. 11.16 and 11.17).

Table 11.1 is prepared to analyze the convergence of series solutions through computations

of numerical values. This table witnesses that our series solutions converge from 20th-order of

deformations for velocity and temperature and 25th-order of approximations for the concentra-

tion. Table 11.2 includes the numerical values of skin-friction coefficient for the different values

of 2 ∗ 1 and when 2 = 03 and = 04 Skin-friction coefficient increases

by increasing 2 and It decreases when ∗ 1 and are increased. Interestingly

the variation in values of skin-friction coefficient is very small by increasing thermophoretic

parameter . Table 11.3 consists of the values of local Nusselt and Sherwood numbers. It is

observed that the values of local Nusselt and Sherwood numbers increase slowly in comparison

to an increase in the values of skin-friction coefficient when 2 increases. Increase in the values

of local Nusselt number is very small by increasing but the increase in the values of local

Sherwood number is large. A comparative study of Tables 11.2 and 11.3 show that the values

of skin-friction coefficient and local Sherwood number are larger than the values of local Nusselt

numbss.

-1.25 -1 -0.75 -0.5 -0.25 0Ñf , Ñq , Ñf

-2.5

-2

-1.5

-1

-0.5

f''0

,q'0

,f'0

b2 = 0.2, l* = Ec = 0.5, g1 =0.4, t = M=0.6, Sc=0.7, Pr = 1.0, g2 = 0.3, N= 0.4

q'0g'0f''0

Fig. 11.1: ~−curves for the functions () () and ()

160

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

l* = g1 = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3, N = 0.4

b2 = 2.0

b2 = 1.3

b2 = 0.7

b2 = 0.0

Fig. 11.2: Variations of 2 on velocity 0()

0 1 2 3 4 5 6h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.2, g1 = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3, N = 0.4

l* = 1.5

l* = 1.0

l* = 0.5

l* = 0.0

Fig. 11.3: Variations of ∗ on velocity 0()

161

0 1 2 3 4 5 6 7h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.2, l* = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3, N = 0.4

g1 = 1.2

g1 = 0.8

g1 = 0.4

g1 = 0.0


0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.2, l* = g1 = 0.4, t = 0.3, Sc= 0.7, Ec = 0.5, Pr = 1.0, g2 = 0.3, N= 0.4

M= 1.2

M= 0.8

M= 0.4

M= 0.0

Fig. 11.5: Variations of on velocity 0()

162

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, g2 = 0.3, N = 0.4

Pr = 2.0

Pr = 1.5

Pr = 1.0

Pr = 0.5

Fig. 11.6: Variations of Pr on velocity 0()

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M= 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, N = 0.4

g2 = 1.8

g2 = 1.2

g2 = 0.6

g2 = 0.0


163

0 1 2 3 4 5 6 7h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.2, l* = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3, N = 0.4

g1 = 1.2

g1 = 0.8

g1 = 0.4

g1 = 0.0

Fig. 11.8: Variations of 1 on temperaure ()

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3, N = 0.4

M= 1.2

M= 0.8

M= 0.4

M= 0.0

Fig. 11.9: Variations of on temperaure ()

164

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M= 0.6, Sc= 0.7, Pr = 1.0, g2 = 0.3, N = 0.4

Ec= 3.0

Ec= 2.0

Ec= 1.0

Ec= 0.0


0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M= 0.6, Sc= 0.7, Ec= 0.5, g2 = 0.3, N = 0.4

Pr = 2.0

Pr = 1.5

Pr = 1.0

Pr = 0.5

Fig. 11.11: Variations of Pr on temperaure ()

165

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M= 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, N = 0.4

g2 = 1.8

g2 = 1.2

g2 = 0.6

g2 = 0.0

Fig. 11.12: Variations of 2 on temperaure ()

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3

N = 1.2

N = 0.8

N = 0.4

N = 0.0


166

0 1 2 3 4 5 6 7h

0

0.2

0.4

0.6

0.8

1

fh

b2 = 0.2, l* = 0.4, t = 0.3, M= 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3, N= 0.4

g1 = 1.2

g1 = 0.8

g1 = 0.4

g1 = 0.0

Fig. 11.14: Variations of 1 on concentration ()

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

fh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3, N= 0.4

M= 1.2

M= 0.8

M= 0.4

M= 0.0

Fig. 11.15: Variations of on concentration ()

167

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

fh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, N = 0.4

g2 = 1.8

g2 = 1.2

g2 = 0.6

g2 = 0.0

Fig. 11.16: Variations of 2 on concentration ()

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

fh

b2 = 0.2, l* = g1 = 0.4, t = 0.3, M = 0.6, Sc= 0.7, Ec= 0.5, Pr = 1.0, g2 = 0.3

N = 1.2

N = 0.8

N = 0.4

N = 0.0

Fig. 11.17: Variations of on concentration ()

Table 11.1: Convergence of series solutions for different order of approximations when 2 =

02 1 = 04 = = 06 = 07 ∗ = = 05 = 10 2 = 03 = 04 and

168

~ = ~ = ~ = −07


5 0.99364 0.61878 0.85224

10 0.99306 0.61970 0.84720

20 0.99305 0.61974 0.84646

25 0.99305 0.61974 0.84643

30 0.99305 0.61974 0.84643

35 0.99305 0.61974 0.84643

Table 2: Numerical values of skin-friction coefficient for different values of 2 ∗ 1

169

and when 2 = 03 and = 04

2 ∗ 1 11+∗ (

00(0) + 200(0))

00 05 03 06 05 07 05 10 071413

05 090356

08 10027

03 00 10491

04 086561

07 077473

00 10408

05 070657

10 041345

00 083167

10 083249

20 083348

00 071759

07 093320

10 11251

05 082715

10 083743

15 084318

04 083355

07 082934

10 082517

08 082329

14 084533

20 085857

Table 3: Numerical values of local Nusselt number −0(0) and local Sherwood number −0(0)

170

for different values of 2 ∗ 1 and when 2 = 03 and = 04

2 ∗ 1 −0(0) −0(0)00 05 03 06 05 07 05 10 059817 083428

05 064563 086222

08 066635 087576

03 00 067517 088177

04 063701 085682

07 061469 084353

00 051553 077548

05 067418 088124

10 074023 093082

00 062941 081539

10 062886 087961

20 062821 096235

00 070545 086946

07 056261 083748

10 043885 081203

05 063443 066407

10 062439 11064

15 062040 14825

04 065805 084409

07 057159 086747

10 048634 088986

08 055885 086466

14 074998 082693

20 089921 079074

171

11.4 Closing remarks

In this study we discussed the effects of thermal radiation, Joule heating and thermophoresis

in stretched flow of Jeffrey fluid. The main observations are as follows:

• Effects of 2 and ∗ on the fluid velocity are quite opposite.

• An increase in ratio of buoyancy parameter 2 corresponds to an increase in 0()

• The fluid temperature rises with an increase in Eckert number

• Effects of ratio of buoyancy and radiation parameters are qualitatively similar.

• Variations in temperature are more pronounced than that in concentration when in-

creases.

• Increase in the values of skin-friction coefficient and local Nusselt number is very small incomparison to an increase in local Sherwood number when increases.

• Numerical values of local Nusselt number increase and the values of local Sherwood num-ber decrease by increasing the Prandtl number

172

Chapter 12

Three-dimensional flow of Jeffrey

fluid with convective surface

boundary conditions

Three-dimensional flow of Jeffrey fluid over a stretched surface with convective boundary con-

dition is examined in this chapter. The equations governing this flow are modeled. The series

solutions of nonlinear equations are constructed. Results of velocity and temperature are ana-

lyzed. Further the numerical values of Nusselt number are computed and discussed. The present

analysis in a limiting sense is compared with the previous results. An excellent agreement is

noted.

12.1 Statement of the problems

We consider three-dimensional boundary layer flow of an incompressible Jeffrey fluid over a

stretching surface at = 0. The flow occupies the domain 0 We denote the ambient fluid

temperature by ∞. The surface temperature (to be determined later) is the result of convective

heating process which is characterized by a temperature and a heat transfer coefficient

The equations which can govern the flow in present situation are

+

+

= 0 (12.1)

173

+

+

=

1 + ∗

⎛⎝2

2+ 2

⎛⎝

2

+

2

+

22

+ 32

+ 32

+ 33

⎞⎠⎞⎠ (12.2)

+

+

=

1 + ∗

⎛⎝2

2+ 2

⎛⎝

2

+

2

+

22

+ 32

+ 32

+ 33

⎞⎠⎞⎠ (12.3)

+

+

=

2

2 (12.4)

In the above equations and are the velocity components in the − − and −directions,respectively, the fluid temperature, the thermal diffusivity of the fluid, = () the

kinematic viscosity, the density of fluid, the dynamic viscosity of fluid, 2 the retardation

time and ∗ is the ratio of relaxation and retardation times..

The suitable boundary conditions are prescribed as

= () = = () = = 0 −

= ( − ) at = 0 (12.5)

→ 0 → 0 → ∞ as →∞ (12.6)

where indicates the thermal conductivity of fluid and and have the dimension (time)−1.

Invoking the following variable

= 0() = 0() = −√(() + ()) () = − ∞ − ∞

=

r

(12.7)

equation (12.1) is automatically satisfied while the Eqs. (12.2)-(12.6) give

000 + (1 + ∗)(( + ) 00 − 02) + 2(002 − ( + ) 0000 − 0 000) = 0 (12.8)

000 + (1 + ∗)(( + )00 − 02) + 2(002 − ( + )0000 − 0000) = 0 (12.9)

00 + ( + )0 = 0 (12.10)

= 0 = 0 0 = 1 0 = , 0 = −(1− (0)) at = 0 (12.11)

0 → 0 0 → 0 → 0 as →∞ (12.12)

174

where 2 = 2 is the Deborah number =is a ratio of stretching rates, =

is the

Prandtl number and =

pis the Biot number.

The local Nusselt number in non-dimensional form can be written as

12 = −0(0) (12.13)

in which = () is the local Reynolds number.


In order to proceed for homotopy analysis solutions, we write the following initial approxima-

tions and auxiliary linear operators

0() =¡1− −

¢ 0() =

¡1− −

¢ 0() =

exp(−)1 +

(12.14)

L = 000 − 0 L = 000 − 0 L = 00 − (12.15)

with

L (1 + 2 + 3

−) = 0 L(4 + 5 + 6

−) = 0 L(7 + 8−) = 0 (12.16)

where ( = 1− 8) are the arbitrary constants.The zeroth order problems can be expressed as

(1− )Lh(; )− 0()

i= ~N

h(; ) (; )

i (12.17)

(1− )L [(; )− 0()] = ~N

h(; ) (; )

i (12.18)

(1− )Lh(; )− 0()

i= ~N

h(; ) (; ) ( )

i (12.19)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 0 0(0; ) = 0(∞; ) = 0(12.20)

0(0 ) = −[1− (0 )] (∞ ) = 0 (12.21)

175

N [( ) ( )] =3( )

3+ (1 + ∗)

⎡⎣{(( ) + ( )}2( )

2−Ã( )

!2⎤⎦+2

⎡⎣ ³2()2

´2− ()

3()

3

−{( ) + ( )}4()4

⎤⎦ (12.22)

N[( ) ( )] =3( )

3+ (1 + ∗)

"{(( ) + ( )}

2( )

2−µ( )

¶2#

+2

⎡⎣ ³2()2

´2− ()

3()

3

−{( ) + ( )}4()4

⎤⎦ (12.23)

N[( ) ( ) ( )] =2( )

2+Pr(( ) + ( ))

( )

(12.24)

in which indicate an embedding parameter, the non-zero auxiliary parameters are denoted by

and and N N and N show the nonlinear operators. For = 0 and = 1 one has

(; 0) = 0() ( 0) = 0() ( 0) = 0()

(; 1) = () ( 1) = () ( 1) = () (12.25)

and when increases from 0 to 1 then ( ) ( ) and ( ) vary from 0() 0() 0()

to () () and () By Taylor’s series expansion we can write

( ) = 0() +∞P

=1

() (12.26)

( ) = 0() +∞P

=1

() (12.27)

( ) = 0() +∞P

=1

() (12.28)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(12.29)

Note that the convergence analysis in the above series strongly depends upon ~ ~ and ~

We choose ~ ~ and ~ in such a way that (12.26)-(12.28) converge at = 1. Hence Eqs.

176

(12.26)-(12.28) are reduced as

() = 0() +∞P

=1

() (12.30)

() = 0() +∞P

=1

() (12.31)

() = 0() +∞P

=1

() (12.32)

The problems at order deformations satisfy the expressions given below

L [()− −1()] = ~R () (12.33)

L[()− −1()] = ~R () (12.34)

L[()− −1()] = ~R () (12.35)

(0) = 0(0) = 0(∞) = 0 (0) = 0(0) = 0(∞) = 0 0(0)− (0) = (∞) = 0(12.36)

R () = 000−1() + (1 + ∗)

−1P=0

[(−1− + −1−) 00 − 0−1−0]

+2

−1P=0

[ 00−1−00 − 0−1−

000 − (−1− + −1−) 0000 ] (12.37)

R () = 000−1() + (1 + ∗)

−1P=0

[(−1− + −1−)00 − 0−1−0]

+2

−1P=0

[00−1−00 − 0−1−

000 − (−1− + −1−)0000 ] (12.38)

R () = 00−1 +

−1P=0

(0−1− + 0−1−) (12.39)

=

⎡⎣ 0 ≤ 11 1

(12.40)

177


() = ∗() + 1 + 2 +3

− (12.41)

() = ∗() + 4 + 5 + 6

− (12.42)

() = ∗() + 7 + 8

− (12.43)

where ∗ ∗ and ∗ are the special solutions.


Homotopy analysis method provides us a freedom to choose the auxiliary parameters ~ ~

and ~ to adjust and control the convergence of series solutions. Hence to find the appropriate

convergence region, we plotted the ~-curves at 21-order of approximations. It is found from

the Figs. 121−123 that the range of admissible values of ~ ~ and ~ are −09 ≤ } ≤ −02,−085 ≤ ~ ≤ −025 and −12 ≤ ~ ≤ −03. More our series solutions converge in the wholeregion of when ~ = ~ = −06 and ~ = −07

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0Ñf

-1.125

-1.12

-1.115

-1.11

-1.105

-1.1

f''0

l* = 0.3, a = 0.4, b2 = 0.2

f ''0

Fig. 12.1: ~−curve for the function

178

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0Ñg

-0.4675

-0.465

-0.4625

-0.46

-0.4575

-0.455

-0.4525

-0.45

g''0

l* = 0.3, a= 0.4, b2 = 0.2

g''0


-1.5 -1.25 -1 -0.75 -0.5 -0.25 0Ñq

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

q'0

l* = 0.3, b2 = 0.2, a= 0.4, Pr =1.0, g = 0.6

q'0


179

Table: 12. 1. Convergence of homotopy solution for different order of approximations when

= 04 2 = 02 ∗ = 03 = 10, = 06 ~ = ~ = −06 and ~ = −07

Order of approximation − 00(0) −00(0) −0(0)1 1.090000 0.424000 0.348750

10 1.120311 0.461683 0.320161

20 1.120273 0.461715 0.319996

25 1.120274 0.461717 0.319996

30 1.120274 0.461717 0.319996

35 1.120274 0.461717 0.319996


The graphical results for various emerging parameters are discussed in this section. Figs.

124 − 1212 are plotted to see the variations of Deborah number 2 ratio of relaxation toretardation time ∗ the parameter Prandtl number and Biot number on the velocity

components 0() 0() and temperature () Fig. 12.4 depicts the influence of 2 on the

velocity 0() This Fig. clearly indicates that there is an increase in the boundary layer

thickness. It is due to the fact that 2 is dependent on the retardation time 2 The retardation

time increases the fluid velocity. The effects of ∗ and on 0() are seen in the Figs. 12.5

and 12.6. The velocity 0 decreases when ∗ and are increased. Figs. 12.7 − 129 show theinfluences of 2

∗ and on 0(). The effects of 2 and ∗ on 0() are quite similar to

that of 0() but the effect of on 0() is quite different. From Fig. 12.9 we observed that

the fluid velocity is zero when = 0 Further both the fluid velocity and thermal boundary

layer thickness increase when increases. The effects of Prandtl number on temperature

profile can be seen in Fig. 12.10. An increase in Prandtl number always decreases the thermal

boundary layer thickness and fluid temperature. Here we observed that the larger Prandtl

number fluids have lower thermal diffusivity and smaller Prandtl number fluids have stronger

thermal diffusivity. From Fig. 12.11 it can be seen that fluid temperature is zero for = 0

and increase in Biot number increases the fluid temperature. Biot number is dependent on

the heat transfer coefficient. This heat transfer coefficient has major role for the variation in

180

temperature. The larger heat transfer coefficient correspond to higher temperature. Fig. 12.12

shows the variations of stretching parameter on () By comparing Figs. 12.6 and 12.12, we

conclude that the parameter has same effects on 0() and () qualitatively but decrease in

() is slightly greater when compared with 0() Table 12.2 presents the numerical values of

local Nusselt number for different values of and 2 when ∗ = 03 It is found that the

local Nusselt number increases when 2 and are increased.

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

l* = 0.3, a = 0.4

b2 = 1.0b2 = 0.6b2 = 0.3b2 = 0.0

Fig. 12.4: Influence of 2 on 0().

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

b2 = 0.2, a = 0.4

l* = 1.5l* = 1.0l* = 0.5l* = 0.0

Fig. 12.5: Influence of ∗ on 0().

181

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

f'h

b2= 0.2, l* = 0.3

a = 1.0a = 0.6a = 0.3a = 0.0

Fig. 12.6: Influence of on 0().

0 2 4 6 8h

0

0.1

0.2

0.3

0.4

g'h

l* = 0.3, a = 0.4

b2 = 1.0b2 = 0.6b2 = 0.3b2 = 0.0

Fig. 12.7: Influence of 2 on 0().

182

0 2 4 6 8h

0

0.1

0.2

0.3

0.4

g'h

b2 = 0.2, a = 0.4

l* = 1.5l* = 1.0l* = 0.5l* = 0.0

Fig. 12.8: Influence of ∗ on 0().

0 2 4 6 8h

0

0.2

0.4

0.6

0.8

1

g'h

b2= 0.2, l* = 0.3

a = 1.0a = 0.6a = 0.3a = 0.0

Fig. 12.9: Influence of on 0().

183

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

0.5

qh

a = 0.4, g = 0.5, b2 = 0.2, l* = 0.3

Pr = 1.5Pr = 1.0Pr = 0.5Pr = 0.1

Fig. 12.10: Influence of on ().

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 1.0, a = 0.4, b2 = 0.2, l* = 0.3

g = 0.6g = 0.4g = 0.2g = 0.0

Fig. 12.11: Influence of on ().

184

0 2 4 6 8 10h

0

0.1

0.2

0.3

0.4

qh

Pr = 0.7, a = 0.4, g = 0.2, l* = 0.3

b2 = 1.5b2 = 1.0b2 = 0.5b2 = 0.0

Fig. 12.12: Influence of 2 on ().

Table 12.2: Numerical values of −0(0) for different values of and 2 when ∗ = 03

2 −0(0)0.0 0.6 1.0 0.2 0.293238

0.3 0.313806

0.7 0.336750

1.0 0.351085

0.3 0.1 0.086805

0.4 0.248756

0.8 0.361009

1.3 0.436825

0.8 0.290453

1.2 0.332210

1.6 0.359781

2.0 0.379821

0.0 0.308166

0.3 0.315891

0.5 0.319381

0.7 0.322297

185


Effects of convective surface boundary condition in the three-dimensional flow of Jeffrey fluid

over a stretched surface are analyzed. The main observations are summarized in the following

points.

• Deborah number 2 and parameter ∗ have quite opposite effects on the velocity compo-nents 0() and 0()

• The velocity component 0() decreases while the velocity component 0() increases whenratio of stretching rates increases

• Increase in Prandtl number decreases the thermal boundary layer thickness and tem-perature.

• The Biot number increases the temperature.

• The local Nusselt number increases by increasing the Biot number.

186

Chapter 13

Three-dimensional flow of Jeffrey

fluid over a bidirectional stretching

surface with heat source/sink

This chapter provides generalization to the contents of previous chapter in the presence of vari-

able thermal conditions and heat source/sink. Two cases of heat transfer namely the prescribed

surface temperature (PST) and prescribed surface heat flux (PHF) are examined. Concept of

heat source/sink is employed. Homotopy analysis method (HAM) is adopted for the devel-

opment of series solutions. Limiting solutions available in the literature are deduced as the

special cases of the present results. Plots are prepared and discussed for the involved pertinent

parameters.

13.1 Heat transfer analysis

Consider three-dimensional boundary layer flow of an incompressible Jeffrey fluid induced by

bidirectional stretching surface (at = 0) with prescribed wall temperature and prescribed

surface heat flux. Flow of an incompressible fluid is considered for 0 The flow is considered

in the presence of heat source/sink parameter. The equations governing the present flow are

+

+

=

2

2+

( − ∞) (13.1)

187

where and are the velocity components in the − − and −directions, the fluid

temperature, the thermal diffusivity of the fluid, = () the kinematic viscosity, the

density of fluid, the dynamic viscosity of fluid, the specific heat at constant pressure of the

fluid and the heat source/sink parameter with 0 (heat source) and 0 (heat sink).

The boundary conditions corresponding to temperature are as follows.

Type I. Prescribed surface temperature (PST) [79,80]:

= ( ) = ∞ + at = 0

→ ∞ as →∞ (13.2)

Type II. Prescribed surface heat flux (PHF) [79,80]:

−

= at = 0

→ ∞ as →∞ (13.3)

In the above expressions is the thermal conductivity of the fluid, ∞ the constant temperature

outside the thermal boundary layer, and the positive constants. The power indices and

determine how the temperature or the heat flux varies in −plane.On setting

PST: () = ( )− ∞( )− ∞

PHF: ( )− ∞ =

r

() (13.4)

the above equations are reduced to the following forms:

00 + ( + )0 + ( − 0 − 0) = 0 (13.5)

00 + ( + )0 + ( − 0 − 0) = 0 (13.6)

= 1 0 = −1 at = 0

→ 0 → 0 as →∞ (13.7)

188

where = is the Prandtl number, the thermal diffusivity and =

the internal heat

parameter.


Initial approximations and auxiliary linear operators for the homotopy solutions are considered

in the following forms:

0() = exp(−) 0() = exp(−) (13.8)

L = 00 − L = 00 − (13.9)

with the properties

L(1 + 2−) = 0 L(3 + 4

−) = 0 (13.10)

where ( = 1− 4) are the arbitrary constants.The zeroth order problems can be constructed as follows:

(1− )Lh(; )− 0()

i= ~N

h(; ) (; ) ( )

i (13.11)

(1− )Lh(; )− 0()

i= ~N

h(; ) (; ) ( )

i (13.12)

(0; ) = 1 (∞ ) = 0 0(0 ) = 0 (∞ ) = 0 (13.13)

N[( ) ( ) ( )] =2( )

2+Pr(( ) + ( ))

( )

+Pr

Ã −

( )

−

( )

!( ) (13.14)

N[( ) ( ) ( )] =2( )

2+Pr(( ) + ( ))

( )

+Pr

Ã −

( )

−

( )

!( ) (13.15)

189

in which indicates the embedding parameter, the non-zero auxiliary parameters are denoted

by ~ and ~ and N and N show the nonlinear operators. For = 0 and = 1 we have

( 0) = 0() ( 0) = 0()

( 1) = () ( 1) = () (13.16)

and when increases from 0 to 1 then ( ) ( ) vary from 0() 0() to () and ()

By Taylor’s series expansion we have

( ) = 0() +∞P

=1

() (13.17)

( ) = 0() +∞P

=1

() (13.18)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(13.19)

The convergence analysis in the series strongly depends upon ~ and ~ We choose ~ and ~

in such a way that Eqs. (1317) and (1318) converge at = 1. Hence Eqs. (1317) and (1318)

give

() = 0() +∞P

=1

() (13.20)

() = 0() +∞P

=1

() (13.21)

The subjected problems for th order deformations are given by

L[()− −1()] = ~R () (13.22)

L[()− −1()] = ~R () (13.23)

(0) = (∞) = 0 0(0) = (∞) = 0 (13.24)

R () = 00−1 +

−1P=0

(0−1− + 0−1−) + −1()

−−1P=0

0−1− − −1P=0

0−1− (13.25)

190

R () = 00−1 +

−1P=0

(0−1− + 0−1−) + −1()

−−1P=0

0−1− − −1P=0

0−1− (13.26)

=

⎡⎣ 0 ≤ 11 1

(13.27)


() = ∗() + 7 + 8

− (13.28)

() = ∗() +9 + 10

− (13.29)

where ∗ and ∗ are the special solutions.


Homotopy analysis method provides us a freedom about the selection of the auxiliary parameters

~ and ~ in order to adjust and control the convergence of series solutions. Hence to find the

appropriate convergence region, we have plotted the ~−curves at 24-order of approximations.It is found from the Figs. 1 and 2 that the range of admissible values of ~ and ~ are

191

−08 ≤ ~ ≤ −025 and −065 ≤ ~ ≤ −015

-1 -0.8 -0.6 -0.4 -0.2 0Ñq

-1.265

-1.26

-1.255

-1.25

-1.245

-1.24

q'0

b2 = 0.3, l* = 0.4, a= 0.5, Pr =1.0, r =0.4, s= 0.5, S=0.6

q'0


-1 -0.8 -0.6 -0.4 -0.2 0Ñf

0.99

0.991

0.992

0.993

0.994

0.995

0.996

f''0

b2 = 0.3, l* =0.4, a =0.5, Pr = 1.0, r = 0.4, s= 0.5, S= 0.6

f''0



Here we have an interest to describe the effects of different embedding parameter on prescribed

surface temperature and prescribed surface heat flux. Figs. 13.3-13.16 are prepared to analyze

the behaviors of 2, , ∗, Pr, , and for prescribed surface temperature () (PST) and

192

prescribed surface heat flux () (PHF). The Deborah number 2 has similar effects for PST

and PHF in a qualitative sense. An increase in 2 corresponds to a decrease in PST and PHF

cases. The variations in PHF are significant in comparison to the variation in PST (see Figs.

13.3 and 13.4). Furthermore this decrease is due to an increase in retardation time. Figs. 13.5

and 13.6 analyzed that both () and () are decreasing functions of . The thermal boundary

layer thickness for PST and PHF cases are reduced with an increase in . Figs. 13.7 and 13.8

show that the boundary layer thickness increase by increasing ∗. Effects of Prandtl number

Pr are seen in the Figs. 13.9 and 13.10. These Figs. depict that temperatures in both the

PST and PHF cases are decreasing functions of Prandtl number. This occurs due to the fact

that an increase in Pr corresponds to a lower thermal diffusivity due to which such decrease

arises. The heat generation parameter gives rise to the temperatures in PST and PHF cases

(see Figs. 13.11 and 13.12). This is due to the fact that heat generation creates the thicker

thermal boundary layer. Figs. 13.13 and 13.14 depict that (), () and their boundary layer

thickness reduce with an increase in . Also we noted that the decrease in PHF is larger in

comparison to a decrease for PST case. The values of () and () are decreasing functions of

. Also such decrease in PHF case is more dominant (see Figs. 13.15 and 13.16). The effects of

power indices and on local Nusselt number −0(0) are seen in Fig. 13.17. The local Nusseltnumber increases by increasing and The local Nusselt number is an increasing function of

ratio parameter and Prandtl number (see Fig. 13.18). Fig. 13.19 shows that an increase in

and 2 leads to an increase in the local Nusselt number. Tables 1 and 2 are computed for

the values correspond to the different values of in a limiting case. From these Tables, we

have seen that our series solutions have a complete agreement with the previous solutions for

193

different values of

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

a = 0.5, l* = 0.4, Pr = 1.0, S= 0.3, s = 0.5, r = 0.4

b2 = 1.0b2 = 0.6b2 = 0.3b2 = 0.0


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

1.2

fh

a = 0.5, l* = 0.4, Pr = 1.0, S= 0.3, s= 0.5, r = 0.4

b2 = 1.0b2 = 0.6b2 = 0.3b2 = 0.0


194

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.5, l* = 0.4, Pr = 1.0, S= 0.3, s= 0.5, s= 0.4

a = 1.0a = 0.6a = 0.3a = 0.0


0 2 4 6 8 10h

0

0.25

0.5

0.75

1

1.25

1.5

fh

b2 = 0.5, l* = 0.4, Pr = 1.0, S= 0.3, s = 0.5, r = 0.4

a = 1.0a = 0.6a = 0.3a = 0.0


195

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.5, a = 0.4, Pr = 1.0, S= 0.3, s= 0.5, r = 0.4

l* = 1.5l* = 1.0l* = 0.5l* = 0.0


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

1.2

1.4

fh

b2 = 0.5, a = 0.4, Pr = 1.0, S= 0.3, s= 0.5, r = 0.4

l* = 1.5l* = 1.0l* = 0.5l* = 0.0


196

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.5, a = 0.4, l* = 0.4, S= 0.3, s= 0.5, r = 0.4

Pr = 1.6Pr = 1.2Pr = 0.8Pr = 0.4


0 2 4 6 8 10h

0

0.25

0.5

0.75

1

1.25

1.5

fh

b2 = 0.5, a = 0.4, l* = 0.4, S= 0.3, s= 0.5, r = 0.4

Pr = 1.6Pr = 1.2Pr = 0.8Pr = 0.4


197

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.5, a = 0.4, l* = 0.4, Pr = 1.0, s = 0.5, r = 0.4

S= 0.8S= 0.6S= 0.3S= 0.0


0 2 4 6 8 10h

0

0.25

0.5

0.75

1

1.25

1.5

1.75

fh

b2 = 0.5, a = 0.4, l* = 0.4, Pr = 1.0, s= 0.5, r = 0.4

S= 0.8S= 0.6S= 0.3S= 0.0


198

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.5, a = 0.4, l* = 0.4, Pr = 1.0, S= 0.3, r = 0.4

s= 1.5s= 1.0s= 0.5s= 0.0


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

1.2

1.4

fh

b2 = 0.5, a = 0.4, l* = 0.4, Pr = 1.0, S= 0.3, r = 0.4

s= 1.5s= 1.0s= 0.5s= 0.0


199

0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

qh

b2 = 0.5, a = 0.4, l* = 0.4, Pr = 1.0, S= 0.3, s= 0.5

r = 1.2r = 0.8r = 0.4r = 0.0


0 2 4 6 8 10h

0

0.2

0.4

0.6

0.8

1

1.2

1.4

fh

b2 = 0.5, a = 0.4, l* = 0.4, Pr = 1.0, S= 0.3, s = 0.5

r = 1.2r = 0.8r = 0.4r = 0.0


200

0 0.5 1 1.5 2s

1.1

1.2

1.3

1.4

1.5

1.6

-q'

0

b2 = 0.3, l* = 0.4, a = 0.5, Pr = 1.0, S= 0.6

r = 1.0r = 0.7r = 0.3r = 0.0


0.25 0.5 0.75 1 1.25 1.5 1.75 2Pr

0.5

0.75

1

1.25

1.5

1.75

2

-q'

0

b2 = 0.3, l* = 0.4, a = 0.5, r = 0.4, s= 0.5

S= 1.2S= 0.8S= 0.4S= 0.0


201

0 0.5 1 1.5 2b2

1.15

1.2

1.25

1.3

1.35

1.4

1.45

-q'

0

l* = 0.4, Pr = 1.0, r = 0.4, s= 0.5, S= 0.6

a = 1.0a = 0.7a = 0.4a = 0.1

Fig. 13.19: Influence of and on −0(0)Table 13.2: Numerical values of 00(0) 00(0) (∞) and (∞) for different values of

when 1 = 1 = 0

Wang [25] Present results

00(0) 00(0) (∞) (∞) 00(0) 00(0) (∞) (∞)0.0 -1 0 1 0 -1 0 1 0

0.25 -1.048813 -0.194564 0.907075 0.257986 -1.04881 -0.19457 0.907047 0.25790

0.50 -1.093097 -0.465205 0.842360 0.451671 -109309 -0.46522 0.84293 0.45169

0.75 -1.134485 -0.794622 0.792308 0.612049 -1.13450 -0.79462 0.79231 0.61214

1.0 -1.173720 -1.173720 0.751527 0.751527 -1.17372 -1.17372 0.75149 0.75149


Effects of prescribed surface temperature and prescribed surface heat flux in three-dimensional

flow of Jeffrey fluid is discussed in the presence of heat source/sink. The main observations are

as follows.

• The prescribed surface temperature and prescribed surface heat flux are decreasing func-tions of

• The variations in prescribed surface heat flux are more dominant when compared with

202

that of prescribed surface temperature.

• Both PST and PHF are reduced when we increase the values of Pr

• An increase in the values of leads to a decrease in both PST and PHF.

203

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Department of Mathematics Quaid-i-Azam University