contentsprr.hec.gov.pk/jspui/bitstream/123456789/9547/1/thesis.pdf · [52] investigated the heat...

Contents

1 Literature review about boundary layer flows and description of solution

procedure 5

1.1 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Heat transfer analysis in flow of Walters’ B fluid with convective boundary

condition 12

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Homotopy analysis solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Convergence of the homotopy solutions . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Flow of Walters’ B fluid by an inclined unsteady stretching sheet with ther-

mal radiation 30

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30



3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6 Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Flow of Powell-Eyring fluid by an exponentially stretching surface with non-

1

linear thermal radiation 54

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54




4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Radiation effects in the flow of Powell-Eyring fluid past an unsteady inclined

stretching sheet with non-uniform heat source/sink 69

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Mathematical modeling and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Series solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72


5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.6 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 MHDmixed convection flow of Burgers’ fluid in a thermally stratified medium 87

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2 Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4 Development of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.5 Convergence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.7 Main points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7 Flow of Burgers fluid through an inclined stretching sheet with heat and

mass transfer 104

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 Development of the problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


2

7.4 Convergence of the solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.5.1 Dimensionless velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.5.2 Dimensionless temperature field . . . . . . . . . . . . . . . . . . . . . . . . 114

7.5.3 Dimensionless concentration field . . . . . . . . . . . . . . . . . . . . . . . 114

7.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8 Stretched flow of Casson fluid with nonlinear thermal radiation and viscous

dissipation 127

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.2 Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.3 Construction of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129


8.5 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134


9 Thermal radiation and viscous dissipation effects in flow of Casson fluid due

to a stretching cylinder 144

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.2 Problem formulation and flow equations . . . . . . . . . . . . . . . . . . . . . . . 144

9.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9.4 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148


9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10 Flow of variable thermal conductivity fluid due to inclined stretching cylinder

with viscous dissipation and thermal radiation 158

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

10.2 Flow description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

10.3 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

10.4 Convergence of the developed solutions . . . . . . . . . . . . . . . . . . . . . . . . 163

3

10.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165


11 MHD stagnation-point flow of Jeffrey fluid over a convectively heated stretch-

ing sheet 176

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

11.2 Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

11.3 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177


11.5 Convergence of the series solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 182


11.6.1 Dimensionless velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . 185

11.6.2 Dimensionless temperature profile . . . . . . . . . . . . . . . . . . . . . . 185

11.6.3 Dimensionless skin friction coefficient . . . . . . . . . . . . . . . . . . . . 186

11.6.4 Dimensionless heat transfer rate . . . . . . . . . . . . . . . . . . . . . . . 186


12 Radiative flow of Jeffrey fluid through a convectively heated stretching cylin-

der 195

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

12.2 Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196


12.4 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

12.5 Result and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

12.5.1 Dimensionless velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . 203

12.5.2 Dimensionless temperature profile . . . . . . . . . . . . . . . . . . . . . . 204

12.5.3 Dimensionless skin friction coefficient . . . . . . . . . . . . . . . . . . . . 205

12.5.4 Dimensionless Nusselt number . . . . . . . . . . . . . . . . . . . . . . . . 205

12.6 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

4

Chapter 1

Literature review about boundary

layer flows and description of

solution procedure

The study of boundary layer flows over moving surfaces has gained major interest in different

industries and technologies. Mostly such flows occur in heat treatment of material traveling

between a feed roll and wind-up roll or a conveyer belt, spinning of fibers, glass blowing, cooling

of a large metallic plate in a bath, aerodynamic extrusion of plastic sheets, continuous moving of

plastic films etc. The extrude from a die is usually drawn and at the same time stretched into a

sheet which is then solidified through regular make cold by direct contact. In such processes the

quality of final product greatly depends on the rate of cooling which is fixed by structure of the

boundary lay near the moving strip. Sakiadis [1] initiated the pioneering work on boundary layer

flows over a moving surface. After Sakiadis, Crane [2] found closed from solution of boundary

layer flow of viscous incompressible fluid over a stretching surface. He considered elastic sheet

that stretched with velocity linearly proportional to the distance from the origin. This idea

was later used by various investigators for effects of suction/injection, porosity, slip, MHD and

heat and mass transfer (see [3 − 6]). Mcleod and Rajagopal [7] discussed the boundary layerflow of Navier-Stokes fluid over a moving stretching surface. Wang [8] explored the rotating

boundary layer flow generated by a stretching sheet. Andersson et al. [9] computed exact

5

solution of viscous fluid in the presence of slip effects. Axisymmetric boundary layer flow of

viscous fluid over a stretching surface with partial slip was examined by Ariel [10]. Liu and

Wang [11] explored the derivation of Prandtl boundary layer with small viscosity limit. The 2D

Navier-Stokes equations settled in a curved domain. Boundary layer flow of Carreau fluid over a

convectively heated surface has been described by Hayat et al. [12]. Rosali et al. [13] examined

the time-dependent boundary layer stagnation point flow in porous medium. Boundary layer

flow of micropolar fluid with second order slip velocity over a shrinking surface was studies by

Rosca and Pop [14]. Analytical solution of MHD free convective boundary layer flow over a

vertical porous surface has been investigated by Raju et al. [15]. They also considered thermal

radiation, chemical reaction and constant suction. Zhang and Zhang [16] found similarity

solutions of boundary layer flow of power law fluid. Awais et al. [17] reported 3D boundary

layer flow of Maxwell fluid. Effect of uniform magnetic field on free convective boundary layer

flow of nanofluid has been studied by Freidoonimehr et al. [18]. Sheikholeslami et al. [19]

have explored the effects of radiation and MHD in flow of nanofluid. Hydrothermal behavior

of nanofluid in the presence of variable magnetic field is examined by Sheikholeslami and Ganji

[20].

It is now established argument that the fluids in numerous technological and biological

applications do not follow the commonly assumed relationship between the stress and the rate

of strain at a point. Such fluids have come to be known as the non-Newtonian fluids. Materials

like molten plastics, polymers, shampoos, certain oils, personal care products, pulps, mud, ice,

foods and fossil fuels, display non-Newtonian behavior. In recent times there has been a great

deal of interest in understanding the behavior of viscoelastic fluids. Such fluids are of great

interest to the applied mathematicians and engineers from the points of view of theory and

simulation of differential equations. On the other hand in applied sciences such as rheology

or physics of the atmosphere, the approach to fluid mechanics is in an experimental set up

leading to the measurement of material coefficients. In theoretically studying how to predict

the weather, the ordinary differential equations represent the main tool. Since the failures

in the predictions are strictly related to a chaotic behavior, one may find it unessential to

ask whether the fluids are really Newtonian. Constitutive equations describe the rheological

properties of viscoelastic fluids. Most of the models or constitutive equations are empirical

6

or semi-empirical. Also the extra rheological parameters in such relations add more nonlinear

terms to the corresponding differential systems. The order of the differential systems involving

non-Newtonian fluids is higher in general than the Navier-Stokes equations. For this reason, a

variety of non-Newtonian fluid models (exhibiting different rheological effects) are available in

the literature [21− 30]. Boundary layer flow of Powell-Eyring fluid over a moving flat late wasanalyzed by Hayat et al. [31]. Mehmood et al. [32] studied the influence of heat transfer on

peristaltic motion of Walters’ B fluid. Series solutions for flow of Jeffrey and Casson fluids over

a surface with convective boundary condition have been computed by Hayat et al. [33, 34].

Time-dependent flow of Casson fluid over a moving surface is investigated by Mukhopadhyay et

al. [35]. Steady flow of Powel- Eyring fluid by an exponentially stretching sheet was numerically

investigated by Mushtaq et al. [36]. The flow and heat transfer of Powell- Eyring fluid thin film

over unsteady stretching sheet are examined by Khader and Megahed [37]. Impact of uniform

suction/injection in unsteady Couette flow of Powel-Eyring fluid is explored by Zaman et al

[38]. Effect of variable thermal conductivity in boundary layer flow of Casson fluid is examined

by Hayat et al. [39]. Shafiq et al. [40] presented the influence of MHD on third grade fluid

between two rotating disk. Stagnation point flow of Jeffrey fluid towards a stretching/shrinking

surface was reported by Turkyilmazoglu and Pop [41]. Hayat et al. [42] studied the combines

effect of Joule heating and thermal radiation in flow of third grade fluid. Two-dimensional

flow of viscoelastic fluid in a vertical channel was examined by Marinca and Herisanu [43].

Turkyilmazoglu [44] found the exact solution for flow of electrically conducting couple stress

fluid. Effect of variable thermal conductivity in flow of couple stress fluid has been reported

by Asad et al. [45]. Mathematical modelling for axisymmetric MHD flow of Jeffrey fluid in

rotating channel was addressed by Alla and Dahab [46]. In melting process the steady flow of

Burgers’ fluid over a stretching surface was examined by Awais et al. [47]. Effect of non-uniform

external magnetic field in stagnation point flow of micropolar fluid was studied by Borrelli et al.

[48]. Javed et al. [49] examined the effect of heat transfer in peristalsis of Burgers fluid through

compliant channels walls. Pramanik [50] discussed boundary layer flow of incompressible Casson

fluid by an exponentially stretching sheet. He also considered the effect of suction or blowing

at the surface.

The heat transfer over a moving surface is one of the hot areas of research due to its extensive

7

applications in science and engineering disciplines especially in chemical engineering processes.

In such processes the fluid mechanical properties of the end product mainly depend on two

factors, one is the cooling liquid property and other the rate of stretching. Many practical situ-

ations demand for physical properties with variable fluid characteristics. Thermal conductivity

is one of such properties which are assumed to vary linearly with the temperature [51]. Chiam

[52] investigated the heat transfer with variable conductivity in stagnation point flow towards

a stretching sheet. Tsai et al. [53] investigated the flow and heat transfer over an unsteady

stretching surface with non-uniform heat source. They considered the flow analysis subject

to variable thermal conductivity, thermal radiation, chemical reaction, Ohmic dissipation and

suction/injection. Recently the variable thermal conductivity effect in boundary layer flow of

viscous fluid due to a circular cylinder is analyzed by Oyem and Koriko [54]. Turkyilmazoglu

and Pop [55] considered the heat and mass transfer effects in the flow of viscous nanofluid past

a vertical infinite flat plate with radiation effect. Also the radiative effects on heat transfer

characteristics are important in space technology, physics and engineering disciplines. These

effects cannot be neglected as gas cooled nuclear reactors and power plants. Su et al. [56] stud-

ied the MHD mixed convective flow and heat transfer of an incompressible fluid by a stretching

permeable wedge. They analyzed the flow in the presence of thermal radiation and Ohmic heat-

ing. The effects of thermal radiation on the flow of micropolar fluid past a porous shrinking

sheet is investigated by Bhattacharyya et al. [57]. Thermal radiation effects in magnetohydro-

dynamic free convection heat and mass transfer from a sphere has been reported by Prasad et

al. [58]. Das [59] examined the effect of radiation on electrically conducting boundary layer

flow over a moving sheet. Natural convection flow over a semi-infinite irregular surface with

radiation effect has been studied by Siddiqa et al. [60]. Ara et al. [61] reported the influence

of thermal radiation in flow of an Eyring-Powell by an exponentially stretching surface. Very

few researchers investigated the flow and heat transfer analysis with nonlinear thermal radia-

tion. For example, Mushtaq et al. [62] explored nonlinear radiation effects in the stagnation

point flow of upper-convective Maxwell (UCM) fluid. Cortell [63] numerically investigated the

fluid flow and radiative nonlinear heat transfer in flow of viscous fluid over a stretching sheet.

Nonlinear radiative heat transfer in the flow of nanofluid has been discussed by Mushtaq et al.

[64]. Three dimensional flow of Jeffrey fluid with non-linear thermal radiation is discussed by

8

Shehzed et al. [65].

The heat transfer analysis in the past has been analyzed extensively either through the

prescribed surface temperature or surface heat flux. Now it is known that heat transfer un-

der convective boundary conditions is useful in processes such as thermal energy storage, gas

turbines, nuclear plants, heat exchangers etc. For convection heat transfer to have a physical

meaning, there must be a temperature difference between the heated surface and moving fluid.

Blasius and Sakiadis flows with convective boundary conditions have been described by Cortell

[66]. Aziz [67] numerically investigated the viscous flow over a flat plate with convective bound-

ary conditions. Ishak [68] provided the numerical solution for flow and heat transfer over a

permeable stretching sheet with convective boundary condition. Makinde [69] investigated the

hydromagnetic flow by a vertical plate with convective surface boundary condition. Makinde

and Aziz [70] considered the MHD mixed convection from a vertical plate embedded in a porous

medium. They considered convective boundary condition. Makinde [71] examined the MHD

heat and mass transfer in flow over a moving vertical plate with a convective surface boundary

condition. Series solution for flow of second grade fluid with convective boundary conditions

has been computed by Hayat et al. [72]. Mustafa et al. [73] explored the axisymmetric flow

of nanofluid over a convectively heated radially stretching sheet. Investigation of heat and

mass transfer together with thermal convective condition has been made by Hamad et al. [74].

Rashad et al. [75] studied mixed convection boundary layer flow in the presence of the thermal

convective condition. The flow of electrically conducting nanofluid over a heated stretching

with convective condition has been numerically computed by Das et al. [76]. Ramesh and

Gireesha [77] examined numerical solution for the flow of Maxwell fluid over a surface with

convective boundary condition. Patil et al. [78] analyzed influence of convective condition in

the two-dimensional double diffusive mixed convection flow. Hayat et al. [79] discussed MHD

nanofluid flow with convective boundary condition over an exponentially stretching surface.

Effects of magnetic field, hydrodynamic slip and convective condition in entropy generated flow

was studied by Ibanez [80].

Although the flow and heat transfer due to stretching cylinder is important in wire draw-

ing, hot rolling and fiber. However not much has been said about it yet through stretching

cylinder. The study of steady flow of an incompressible viscous fluid outside the stretching

9

cylinder in an ambient fluid at rest is made by Wang [81]. He found exact similarity solution

of nonlinear ordinary differential problem. Burded [82] investigated the exact solutions of the

Navier-Stokes equations which describe the steady axsymmetric motion of an incompressible

fluid near a stretching infinite circular cylinder. Later, Ishak et al. [83 − 85] extended thework of Wang [81] in different physical situation. They presented a numerical solution for

flow and heat transfer due to stretching cylinder. The boundary layer flow past a stretching

cylinder and heat transfer with variable thermal conductivity has been reported by Rangi and

Ahmad [86]. They found that the curvature of stretching cylinder plays a key role in flow and

temperature field. Mukhopadhyay and Ishak [87] studied the laminar boundary layer mixed

convection flow of viscous incompressible fluid towards a stretching cylinder immersed in a ther-

mally stratified medium. They concluded that the temperature gradient is smaller for flow in

a thermally stratified medium when compared to that of an unstratified medium. Chemically

reactive solute transfer in boundary layer slip flow along a stretching cylinder has been exam-

ined by Mukhopadhyay[88]. MHD flow with heat transfer characteristics at a non-isothermal

stretching cylinder has been examined by Vajravelu et al. [89]. They also considered internal

heat generation/absorption and temperature dependent thermal conductivity. Slip effects in

flow by stretching cylinder was studies by Mukhopadhyay [90]. Si et al. [91] investigated flow

and heat transfer analysis over an impermeable stretching cylinder. Time-dependent flow and

heat transfer over a contracting cylinder is studied by Zaimi et al. [92]. Ahmed et al. [93]

reported the influence of heat source/sink over a permeable stretching tube. Mishra and Singh

[94] found the dual solutions of momentum and thermal slip at a shrinking cylinder. Hayat et

al. [95] presented stagnation point flow of an electrically conducting second grade fluid over a

stretching cylinder.

1.1 Solution procedure

Flow equations of non-Newtonian fluids in general are highly nonlinear. Therefore it is very

difficult to find the exact solution of such equations. Usually perturbation, Adomian decompo-

sion and homotopy perturbation methods are used to find the solution of nonlinear equations.

But these methods have some drawback through involvement of large/small parameters in the

10

equations and convergence. Homotopy analysis method (HAM) [96− 110] is one while is inde-pendent of small/large parameters. This method also gives us a way to adjust and control the

convergence region (i.e. by plotting h-curve). It also provides exemption to choose different

sets of base functions.

11

Chapter 2

Heat transfer analysis in flow of

Walters’ B fluid with convective

boundary condition

2.1 Introduction

This chapter addresses the radiative heat transfer in the steady two-dimensional flow of Wal-

ters’ B fluid. An incompressible fluid is bounded by a stretching porous surface. Convective

boundary condition is used for the thermal boundary layer problem. Mathematical analysis is

carried out in the presences of non-uniform heat source/sink. The relevant equations are first

simplified under usual boundary layer assumptions and then transformed into similar form by

suitable transformations. Convergent series solutions of velocity and temperature are derived

by homotopy analysis method (HAM). The dimensionless velocity and temperature gradients

at the wall are calculated and discussed.

2.2 Problem formulation

We consider the steady two-dimensional flow of an incompressible Walters’ B fluid over a porous

stretching surface. In addition, heat transfer analysis is considered with radiation effects and

non-uniform heat source/sink. Further we consider −axis parallel to the stretching sheet and

12

− normal to it. The extra stress tensor S for Walters’ B fluid is defined by

S =20A1 − 20A1

(2.1)

A1

=

A1

+V∇A1 −A1∇V − (∇V) A1 (2.2)

where A1 is the rate of strain tensor, V is the velocity field of fluid, A1

is the covariant

derivative of the rate of strain tensor in relation to the material in motion, and 0 and 0 are

the limiting viscosities at small shear rate and short memory coefficient respectively. These are

defined as follows:

0 =

Z ∞

0

() (2.3)

0 =

Z ∞

0

() (2.4)

with () being as the distribution function with relaxation time By taking short memory

into account the terms involving

Z ∞

0

() > 2 (2.5)

are neglected in case of Walters’ B fluid [22].

The equations of continuity, momentum and energy are

divV = 0 (2.6)

V

= div τ (2.7)

= ∇2 + τ (gradV) (2.8)

Here is the fluid density, the time, V the velocity, the fluid temperature, τ the Cauchy

stress tensor, the specific heat, the thermal conductivity of the material and ∇2 =³2

2+ 2

2

´

The velocity and temperature fields under the boundary layer approximations are governed

13

by the following equations

+

= 0 (2.9)

+

=

0

µ2

2

¶− 0

µ

3

2+

3

3+

2

2−

2

¶ (2.10)

µ

+

¶=

2

2−

+ 00 (2.11)

= () = = −

= ( − ) at = 0 (2.12)

→ 0 → ∞ as →∞ (2.13)

where and represent the velocity components in the and directions, respectively,

is the velocity of the stretching sheet, is the constant mass transfer velocity with 0

for injection and 0 for suction, is the fluid temperature, ∞ is the ambient fluid

temperature, is the thermal conductivity of the fluid, = () is the kinematic viscosity,

and 00 is the non-uniform heat generated (00 0) or absorbed (00 0) per unit volume. For

non-uniform heat source/sink, 00 is modeled by the following expression:

00 =()

£( − ∞) 0 + ( − ∞)

¤ (2.14)

in which and are the coefficients of space and temperature-dependent heat source/sink,

respectively. Here two cases arise. For internal heat generation 0 and 0 and for

internal heat absorption, we have 0 and 0 is the density of the fluid, is the thermal

conductivity of fluid, is the convective heat transfer coefficient, and is the convective fluid

temperature below the moving sheet.

We define the similarity transformations through

= √() () =

− ∞ − ∞

=

r

(2.15)

where prime denotes the differentiation with respect to , is the dimensionless stream function,

14

is the dimensionless temperature, and is the stream function given by

=

= −

(2.16)

Now Eq. (29) is identically satisfied and Eqs. (210− 213) yield

000 + 00 − 02 +( 002 − 2 0 000 + ) = 0 (2.17)

(1 +4

3)00 +

¡0¢+ 0 + = 0 (2.18)

= 0 = 1 0 = −[1− (0)] at = 0 (2.19)

0 = 0 = 0 at =∞ (2.20)

Here 0 00 000 and are the derivatives of the stream function with respect to = 00

is the local Weissenberg number, = − √is the mass transfer parameter with 0 for

suction and 0 for injection, = is the Prandtl number, = 4∗

3∞∗ is the radiation

parameter and =

pis the Biot number.

The skin friction coefficient and local Nusselt number are

=122

=

( − ∞) (2.21)

where the skin friction and the heat transfer from the plate are

= 0

µ

¶+ 0

½2

+ 2

+

2

2

¾¯=0

(2.22)

= −

¯=0

− 16∗ 3∞3∗

¯=0

(2.23)

Dimensionless form of skin friction coefficient and local Nusselt number are

12 = (1−) 00(0)− 000(0) 12 = −(1 + 4

3)0(0) (2.24)

In the above equations the Reynolds number Re =q

15

2.3 Homotopy analysis solutions

Initial approximations and auxiliary linear operators are chosen below:

0() = + (1− −) (2.25)

0() = exp(−)1 +

(2.26)

L = 000 − 0 (2.27)

L = 00 − (2.28)

with properties

L (1 + 2 + 3

−) = 0 (2.29)

L(4 + 5−) = 0 (2.30)

where ( = 1 − 5) are the arbitrary constants determined from the boundary conditions.

If ∈ [0 1] denotes an embedding parameter, and represent the non-zero auxiliary

parameters then the zeroth order deformation problems are defined by

(1− )Lh(; )− 0()

i= N

h(; )

i (2.31)

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (2.32)

(0; ) = 0(0; ) = 1 0(∞; ) = 0 0(0; ) = −[1− (0; )] (∞; ) = 0 (2.33)

where N and N are the nonlinear operators defined in the forms

N [( )] =3( )

3+ ( )

2( )

2−Ã( )

!2

+

⎡⎣Ã2( )

2

!2+

Ã( )

4( )

4

!− 2( )

3( )

3

⎤⎦ (2.34)

16

N[( ) ( )] =

µ1 +

4

3

¶2( )

2+

Ã( )

( )

!

+( )

+( ) (2.35)

When = 0 and = 1 then one has

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (2.36)

and when variation is taken from 0 to 1 then ( ) and ( ) approach 0() and 0() to

() and () Now and in Taylor’s series can be expanded as follows:

( ) = 0() +∞P

=1

() (2.37)

( ) = 0() +∞P

=1

() (2.38)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(2.39)

where the convergence depends on and By appropriately choosing and the series

(237) and (238) converge for = 1 and so

() = 0() +∞P

=1

() (2.40)

() = 0() +∞P

=1

() (2.41)

The -order deformation problems can be provided as follows:

L [()− −1()] = R () (2.42)

L[()− −1()] = R () (2.43)

(0) = 0(0) = 0(∞) = 0 0(0)−(0) = (∞) = 0 (2.44)

17

R () = 000−1() +

−1P=0

£−1− 00 − 0−1−

0

¤+

−1X=0

( 00−1−00 + −1− − 2 0−1− 000 ) (2.45)

R () = (1 +

4

3)00−1 +

−1P=0

0−1−

+ 0−1 +−1 (2.46)

=

⎧⎨⎩ 0 ≤ 11 1

(2.47)

The general solutions of Eqs. (242) and (243) can be expressed by the following equations:

() = ∗() + 1 + 2 + 3

− (2.48)

() = ∗() +4 + 5

− (2.49)

where ∗ and ∗ are the particular solutions of Eqs. (248) and (249) and constants

( = 1− 5) can be determined by the boundary conditions (244). They are given by

2 = 4 = 0 3 =∗()

¯=0

1 = −3 − ∗(0) (2.50)

5 =1

+ 1

"∗()

¯=0

−∗(0)

# (2.51)

2.4 Convergence of the homotopy solutions

Series (240) and (241) have nonzero auxiliary parameters } and } that help us to control

and adjust the convergence of series solutions. For admissible values of } and } we plotted

the } − curves of 00(0) and 0(0). Figs. 21 and 22 show that the ranges of admissible values

of } and } are −18 ≤ } ≤ −03 and −19 ≤ } ≤ −055 respectively. The series convergein the whole region of when } = −15 and } = −13 Table 21 displays the convergence ofhomotopy solutions for different orders of approximations.

18

Fig. 21. ~−curve for velocity field.

Fig. 22. ~−curve for temperature field.

19

Table 21: Convergence of HAM solutions for different orders of approximations when =

04 = 02 = = 01 Pr = 1 = 07 and = 01

Order of approximation − 00(0) −0(0)1 151000 0047359

5 154617 0027114

10 154727 0030756

15 154728 0031432

20 154728 0031568

25 154728 0031601

30 154728 0031608

35 154728 0031611

40 154728 0031611

50 154728 0031611

2.5 Results and discussion

In this section we focus on the behaviors of different physical parameters entering into the series

solutions. In particular the plots for dimensionless velocity gradient and heat transfer rate at

the bounding surface are displayed and discussed. Here the velocity decreases with the increase

of which corresponds to a thinner momentum boundary layer. The viscoelasticity produces

tensile stress which contracts the boundary layer and consequently the velocity decreases (see

Fig. 23). An increase in suction parameter moves the velocity distribution towards the bound-

ary, indicating a decrease in the thickness of boundary layer (see Fig. 24). Fig. 25 shows the

influence of viscoelastic parameter on temperature. We found that the temperature and

the thermal boundary layer thickness decrease as viscoelastic effects strengthens. Influence of

Biot number is shown in Fig. 26. As expected, an increase in results in the stronger

convective heating which increases the surface temperature. This allows the thermal effect to

penetrate more deeply into the quiescent fluid. The variation of temperature distribution with

the Prandtl number Pr is presented in Fig. 27. From the physical point of view, the larger

Prandtl number coincides with the weaker thermal diffusivity and thinner boundary layer. This

is because a higher Prandtl number fluid has a relatively low thermal conductivity which re-

20

duces conduction and thereby increases the variations of thermal characteristics. Figure 28

shows that temperature appreciably rises when the thermal radiation effect is strengthened.

However suction reduces the thermal boundary layer thickness as can be seen from Fig. 29.

Figs. 210 and 211 show the influences of heat source/sink parameters on temperature . As

expected, the larger heat source parameter enhances the temperature distribution to a great

extent and thus largely contributes to the growth of thermal boundary layer. Energy is also

absorbed for smaller values of 0 and 0 and this causes the magnitude of temperature

to decrease whereas non-uniform heat sink corresponding to 0 and 0 can contribute

to quenching the heat from stretching sheet effectively.

Table 22 provides the numerical values of dimensionless velocity gradient at the wall for

different values of suction parameter and viscoelastic parameter. It is clear that increases

of and reduce the dimensionless velocity gradient on the stretching surface. This is very

useful result from the industrial point of view since the power generation involved in displacing

the fluid over the sheet is reduced by assuming sufficiently large values of these parameters.

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

have been tabulated (see Table 23). Here larger convective heating at the bounding surface

generates larger heat flux from the surface which eventually increases the magnitude of local

Nusselt number It is clear from Fig. 24 that although thermal boundary layer thins but profiles

become increasingly steeper with the increase of the value of Pr. The Nusselt number, being

proportional to the increase of initial slope with the augment of the value of Pr. On the other

hand the magnitude of local Nusselt number decreases as viscoelastic effect strengthens.

21

Fig. 23. Influence of on 0()

Fig. 24. Influence of on 0()

22

Fig. 25. Influence of on ()


23

Fig. 27. Influence of Pr on ()


24



25


Fig. 2.12. Influence of 12 on Pr

26

Fig. 2.13. Influence of 12 on

Table 22: Values of skin friction coefficient 12 for the parameters and

−12

00 02 0870726

02 0849053

04 0712725

06 0364968

00 12198

01 101937

02 0712725

27

Table 23: Values of local Nusselt number 12 for the parameters , , Pr, and

12

04 10 02 04 01 0223253

06 0280333

08 0321423

07 11 0323167

12 0340928

13 0356453

00 0313797

01 0309528

02 0302425

02 0257777

03 0281064

05 0321888

00 0322187

01 0302425

02 0284298

2.6 Main points

Boundary layer flow of Walters’ B fluid over a porous stretching sheet with radiation and

non-uniform heat source/sink is investigated here. The series solutions of the governing math-

ematical problems are obtained by using HAM and appropriately choosing the convergence

control parameters. Thus the convergent solutions are obtained through plots of }−curves. Itis noticed that velocity and wall shear stress decrease with the increase of Weissenberg number

whereas temperature () is an increasing function of Increase in the suction para-

meter also reduces the momentum boundary layer. The temperature () and rate of heat

transfer at the sheet increase for larger convective heating at the surface. For sufficiently large

values of Biot number the results for the case of constant wall temperature ((0) = 1) can

be recovered. Moreover the present finding for the Newtonian fluid case can be retrieved by

28

choosing = 0

29

Chapter 3

Flow of Walters’ B fluid by an

inclined unsteady stretching sheet

with thermal radiation

3.1 Introduction

This chapter deals with the boundary layer flow of Walters’ B liquid by an inclined unsteady

stretching sheet. Simultaneous effects of heat and mass transfer are considered. Analysis has

been carried out in the presence of thermal radiation and viscous dissipation. Convective con-

dition is employed for the heat transfer process. Resulting problems are computed for the

convergent solutions of velocity, temperature and concentration fields. The effects of different

physical parameters on the velocity, temperature and concentration fields are discussed. Nu-

merical values of skin friction coefficient, local Nusselt number and local Sherwood number are

computed and analyzed.

3.2 Mathematical formulation

Here we investigate the flow of an incompressible Walters’ B fluid due to an inclined stretching

sheet with convective boundary condition. The effects of viscous dissipation and thermal radi-

ation are present. The sheet is inclined at an angle with the vertical axis i.e. − and

30

− is taken normal to it. The subjected boundary layer equations are given by

+

= 0 (3.1)

+

+

=

0

µ2

2

¶− 0

∙3

2+

3

2+

3

3+

2

2

−

2

¸+ 0 ( − ∞) cos (3.2)

µ

+

+

¶=

2

2−

+ 0

µ

¶2− 20

∙

2

+

2

+

2

2

¸ (3.3)

+

+

=

2

2 (3.4)

= () = 0 −

= ( − ) , = at = 0 (3.5)

→ 0 → ∞ → ∞ as →∞ (3.6)

in which and represent the velocity components along and normal to the sheet, =1−

is the velocity of the stretching sheet, is the fluid temperature, = () is the kinematic

viscosity, is the density of fluid, 0 is the acceleration due to gravity, is the volumetric

coefficient of thermal expansion, is the thermal conductivity of fluid, is the convective

heat transfer coefficient, = ∞ + 2

2(1 − )−32 is the surface temperature, is the

concentration of fluid, is the effective diffusion coefficient and = ∞ + 1− is the fluid

concentration.

We introduce

=

µ

1−

¶−12() () =

− ∞ − ∞

() = − ∞ − ∞

=

µ

(1− )

¶12 (3.7)

and the velocity components

=

= −

31

where is the stream function. Now Eq. (31) is identically satisfied and Eqs. (32− 36) arereduced into the following forms:

000 + 00 − 02 +−µ 0 +

1

2 00

¶+

µ

µ2 00 +

1

2

¶+ 002 − 2 0 000 +

¶

+ cos = 0 (3.8)

(1 +4

3)00 − Pr

µ1

2¡3 + 0

¢− 0 + 2 0¶+Pr

002

−Pr

µ1

2¡3 00 + 000

¢+ 0 002 − 00 000

¶= 0 (3.9)

00 + ¡0 − 0

¢−

µ+

1

20¶= 0 (3.10)

= 0 0 = 1 0 = −[1− (0)] = 1 at = 0 (3.11)

0 → 0 → 0 → 0 at →∞ (3.12)

Here = 00

1− is the Weissenberg number, =

is the Prandtl number, =

is the

ratio parameter, =0 (−∞)3

22

is the mixed convection parameter =2

(−∞) is

the Eckert number, = 4∗3∞

∗ is the radiation parameter, =

q(1−)

is the Biot number

and = is the Schmidt number.

The skin friction coefficient , local Nusselt number and local Sherwood number

can be expressed in the following forms:

=122

=

( − ∞) =

( − ∞) (3.13)

whence

= 0

¯=0

− 0

½2

+

2

− 2

+

2

2

¾¯¯=0

= −

¯=0

− 16∗ 3∞3∗

µ

¶¯=0

= −

¯=0

(3.14)

Dimensionless forms of skin friction coefficient , local Nusselt number and local Sher-

32

wood number Sh can be represented by the relations

= 2 (Re)−12

∙(1 + 3) 00()−

1

2

¡3 00() + 000()

¢¸¯=0

12 = −(1 + 43)0(0) 12 = −0(0) (3.15)

in which (Re)−12 =

q.


Initial approximations and auxiliary linear operators for homotopy analysis solutions can be

put into the forms

0() = (1− −) 0() = exp(−)1 +

0 = − (3.16)

L = 000 − 0 L = 00 − L = 00 − (3.17)

with properties mentioned below:

L (1 + 2 + 3

−) = 0 (3.18)

L(4 + 5−) = 0 (3.19)

L(6 + 7−) = 0 (3.20)

where ( = 1− 7) are the arbitrary constants. If ∈ [0 1] denotes an embedding parameterand and the non-zero auxiliary parameters then the zeroth order deformation problems

are

(1− )Lh(; )− 0()

i= N

h(; )

i (3.21)

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (3.22)

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (3.23)

33

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

(0; ) = 1 (∞; ) = 0 (3.24)

The involved nonlinear operators N N and N in above equations are

N [( )] =3( )

3+ ( )

2( )

2−Ã( )

!2−

Ã( )

+1

22( )

2

!

+

⎡⎣Ã23( )3

+1

24( )

4

!+

Ã2( )

2

!2

+

Ã( )

4( )

4

!− 2( )

3( )

3

#+ cos (3.25)

N[( ) ( )] =

µ1 +

4

3

¶2( )

2+

"1

2

Ã3( ) +

( )

!

−( )( )

− 2( )( )

#+Pr

2( )

2

−Pr

⎧⎨⎩12Ã32( )

2+

3( )

3

!+

( )

Ã2( )

2

!2

−( )2( )

23( )

3

)# (3.26)

N[( ) ( )] =2( )

2+

Ã( )

( )

− ( )

( )

!

−Ã( ) +

1

2( )

! (3.27)

For = 0 and = 1 we can write

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

(; 1) = () (; 1) = () (; 1) = () (3.28)

34

and when variation is taken from 0 to 1 then ( ) ( ) and ( ) approach 0() 0()

and 0() to () () and () Now and in Taylor’s series are

( ) = 0() +∞P

=1

() (3.29)

( ) = 0() +∞P

=1

() (3.30)

( ) = 0() +∞P

=1

() (3.31)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(3.32)

where the convergence depends upon and By proper choices of and the

series (329− 331) converge for = 1 and therefore

() = 0() +∞P

=1

() (3.33)

() = 0() +∞P

=1

() (3.34)

() = 0() +∞P

=1

() (3.35)

The -order deformation problems are

L [()− −1()] = R () (3.36)

L[()− −1()] = R () (3.37)

L[()− −1()] = R () (3.38)

(0) = 0(0) = 0(∞) = 0 0(0)−(0) = (∞) = 0 (0) = (∞) = 0 (3.39)

35

with

R () = 000−1() +

−1P=0

£−1− 00 − 0−1−

0

¤−

µ2 0−1 +

1

2 00−1

¶+

µ2 000−1 +

1

2 −1

¶+

−1X=0

( 00−1−00

+−1− − 2 0−1− 000 ) +−1 cos (3.40)

R () = (1 +

4

3)00−1 +Pr

µ1

2¡3−1 + 0−1

¢¶+Pr

−1P=0

¡0−1− − 2 0−1−

¢+Pr

00−1

−Pr−1P=0

µ1

2¡3 00−1 + 000−1

¢+ 0−1−

00 − −

−P=0

00−000

¶(3.41)

R () = 00−1 +

−1P=0

¡0−1− − 0−1−

¢−

µ−1 +

1

20−1

¶ (3.42)

=

⎧⎨⎩ 0 ≤ 11 1

The general solutions of Eqs. (336− 338) can be expressed in the forms:

() = ∗() + 1 + 2 + 3

− (3.43)

() = ∗() +4 + 5

− (3.44)

() = ∗() +6 + 7

− (3.45)

where ∗ ∗ and ∗ are the particular solutions.

36


It is well known that the convergence of series (336− 338) depends on the auxiliary parameters} } and } These parameters are useful to control and adjust the convergence of series

solutions. In order to find the admissible values of } } and } the } − curves of 00(0) 0(0)and 0(0) are displayed Figs. (31 − 33) depict that the range of admissible values of } }and } are −125 ≤ } ≤ −026 −123 ≤ } ≤ −027 and −126 ≤ } ≤ −04 The seriesconverge in the whole region of when } = } = −08 and } = −09 Table 31 displaysthe convergence of homotopy solutions for different order of approximations. Tabulated values

clearly indicate that 25 order of approximations are enough for the convergent solutions.

Fig. 3.1. ~−curve for velocity field.

37

Fig. 3.2. ~−curve for temperature field.

Fig. 3.3. ~−curve for concentration field.

38

Table 31: Convergence of HAM solutions for different order of approximations when =

= 03 = 02 = 4 Pr = 10 = 04 = 02 = 09 and = 01

Order of approximations − 00(0) −0(0) −0(0)1 113084 0302136 104613

5 118308 0295175 101901

10 118324 0295141 101677

15 118324 0295143 101657

20 118324 0295144 101654

25 118324 0295144 101653

30 118324 0295144 101653

35 118324 0295144 101653

3.5 Discussion

The arrangement of this section is to disclose the impact of different physical parameters,

including ratio parameter, the Biot number, We the Weissenberg number, Pr the Prandtl

number, the Sherwood number, the Eckert number, the mixed convection parameter,

the angle of inclination and the radiation parameter. The variation of aforementioned

parameters are seen for the velocity, temperature and concentration fields. It is observed that

the effects of and on the velocity field are quite opposite (see Figs. 34 and 35). Velocity

field and boundary layer thickness decay for larger values of Influence of Weissenberg number

on 0() is shown in Fig. 36. Here the velocity field is decreasing function of. For large

values of mixed convection parameter the velocity field increases (see Fig. 37). Effect of

radiation parameter and Eckert number are qualitatively similar for the velocity field 0()

(see Figs. 38 and 39) It is also noticed that the variation of between 0 to 1 is insignificant

on the temperature. Fig. 310 depicts the influence of on the temperature field. Apparently

both the temperature field and thermal boundary layer thickness are decreased via . Fig. 311

gives the influence of Weissenberg number on the temperature field. Here the temperature

field is increasing function of Fig. 312 shows that the temperature field decays very slowly

when mixed convection parameter increases. Fig. 313 presents the influence of on the

temperature field. Both the temperature field and thermal boundary layer thickness decrease

39

when increases. Fig. 314 witnesses that the temperature field is more pronounced when

radiation effects strengthen. Fig. 315 depicts the variation of Pr on temperature field. For

larger values of Pr the temperature and thermal boundary layer thickness decrease. This is due

to the fact that enhancement in Pr causes the reduction in thermal conductivity. Effect of Eckert

number is displayed in Fig. 316. When we increase the Eckert number then the fluid kinetic

energy increases and thus temperature field increases for larger values of Eckert number. Fig.

317 shows the influence of Biot number on temperature field. Larger Biot numbers correspond

to more convection than conduction and this leads to increase in temperature as well as thermal

boundary layer thickness. Influence of parameter on the concentration field is displayed in Fig.

318. It is noticed that the concentration field decreases when is increased. Fig. 319 shows

the influence of Weissenberg number on the concentration field. It is clearly seen from this

Fig. that the concentration field increases when there is an increase in Weissenberg number

. Fig. 320 gives the influence of on the concentration field. The angle of inclination

increases the concentration field. Effect of on the concentration field is plotted in Fig. 321.

The concentration field decreases when increases. Here larger number corresponds to

lower molecular diffusivity.

Tables 32−34 include the numerical values of skin friction coefficient, local Nusselt numberand Sherwood number respectively. The magnitude of skin friction coefficient decreases for

larger values of , , and . However it increases when We, and Pr are increased. It

is noticed that the heat transfer at the wall −(0) increases for larger values of , Pr, and it decreases for larger We, , and . Table 33 shows that the local Sherwood number

increases when radiation parameter and Schmidt number Sc are increased.

40

Fig. 3.4. Influence of on 0()


41



42



43

Fig. 3.10. Influence of on ()


44



45


Fig. 3.15. Influence of Pr on ()

46



47



48



49

Table 32: Values of skin friction coefficient 12 for the parameters Pr

and

Pr −12

12

00 03 02 4 01 10 02 04 222135

02 213488

04 204054

03 00 108510

02 173555

04 246015

03 00 212112

02 208842

04 205607

00 207498

6 208112

3 209796

00 209047

03 208459

06 207940

05 207403

09 208657

11 208999

00 209390

03 208569

06 207760

02 209969

06 207946

10 206611

50

Table 3.3: Values of local Nusselt number 12 for the parameters , , ,

Pr and

Pr − ¡1 + 43¢0 (0)

00 03 02 4 01 10 02 04 0286182

03 0295144

06 0301292

03 00 0295985

02 0295379

05 0294885

00 0294608

03 0295403

05 0295907

00 0295359

6 0295261

3 0294989

00 0299395

01 0295144

03 0287178

07 0281292

11 0298509

14 0306404

00 0308526

02 0295144

06 0268771

02 0166532

05 0349078

09 0517163

51

Table 3.4: Values of local Sherwood number Sh for the parameters R, We, and Sc.

−0(0)00 094588

03 094707

09 094930

00 096707

02 095358

03 094628

00 094783

6 094713

4 094628

4 07 087195

09 094628

11 114703

3.6 Main points

Boundary layer flow of Walters’ B liquid by an inclined stretching sheet is discussed in the

presence of viscous dissipation and thermal radiation. The main observations can be mentioned

below.

• Velocity field 0() is decreasing function of parameter

• Effects of on the temperature and concentration fields are qualitatively similar.

• Weissenberg number decreases both the velocity and associated boundary layer thick-

ness.

• Weissenberg number increases the temperature and concentration fields.

• Velocity field 0 is decreasing function through larger

• Effects of , and on velocity field are qualitatively similar.

52

• There is enhancement of temperature for larger values of Eckert number , thermal

radiation and Biot number

• Variations of and on the concentration field are qualitatively similar.

• Magnitude of skin friction coefficient is decreasing function of , and .

• Influences of and on the temperature gradient at the surface are qualitatively

similar.

• Temperature gradient at the surface decreases when , and are enhanced.

53

Chapter 4

Flow of Powell-Eyring fluid by an

exponentially stretching surface with

nonlinear thermal radiation

4.1 Introduction

The present chapter concerns with the flow of Powell-Eyring fluid due to exponentially stretching

sheet with thermal radiation effect. Here sheet is considered inclined and making an angle

with the vertical axis. By using the similarity transformations, the nonlinear partial differential

equations are transformed into ordinary differential equations. The series solutions of ordinary

differential equations are obtained by using homotopy analysis method (HAM). In order to

analyze the variations of physical parameters on temperature field, the effect of heat transfer

characteristics at wall are tabulated. Also the variations of parameters on the velocity and

temperature fields are analyze graphically.

4.2 Mathematical formulation

We consider steady non-Newtonian two-dimensional incompressible flow by an exponentially

stretching sheet. We assume that sheet stretch with velocity 0 and makes the angle with

− where 0 is the reference velocity and is the characteristics length. Heat transfer

54

analysis are also investigated with non-linear thermal radiation effects. The stress tensor in the

Eyring-Powell model is

=

+1

sinh−1

µ1

¶ (4.1)

where is the viscosity coefficient, and are the characteristics of Powell-Eyring Model

We take and axes along and perpendicular to the sheet respectively. Here the velocity

components and depend on the and coordinates only. The resultant boundary layer

equations with subjected boundary conditions are

+

= 0 (4.2)

+

=

µ +

1

¶2

2− 1

23

µ

¶22

2+ 0 ( − ∞) cos (4.3)

µ

+

¶=

∙µ +

16∗ 3

3∗

¶

¸

(4.4)

= 0 = 0 = ∞ + 0

2 at = 0 (4.5)

→ 0 → 0 as →∞

where = () is the kinematic viscosity, is the thermal conductivity of the fluid, is

the specific heat, ∗ is the mean absorption coefficient, ∗ is the Stefan-Boltzmann constant,

is the fluid density, is the fluid temperature, 0 is the acceleration due to gravity, is

the volumetric coefficient of thermal expansion, is the specific heat and 0 and ∞ are the

temperature at the plate and far away from the plate. Defining the dimensionless variables as

= 0 0() = −

r0

22

£() + 0()

¤

= 02() =

r0

22 (4.6)

Invoking Eq. (46) into Eqs. (42− 45) Eq. (42) is identically satisfied and Eqs. (43− 45)yield

(1 + Γ) 000 − 2 02 + 00 − Γ 002 000 + cos = 0 (4.7)

55

hn1 + (1 + ( − 1))3

o0i0+Pr

£0 − 0

¤= 0 (4.8)

= 0 0 = 1 = 1 at = 0

0 → 0 → 0 as →∞ (4.9)


is the dimensionless temperature and the dimensionless numbers are Γ = 1

, =

4∗ 3∞∗ ,

=

2 =

30 3

42 = and Pr =

. Here Γ and are the dimensionless material

parameters, is the mixed convection parameter, is the radiation parameter, is the

temperature ratio parameter and Pr is the Prandtl number.

The local Nusselt number is defined as

=

(0 − ∞)with = −

µ

¶=0

+ () and so

= −0r

0

2(1 +3 )

0(0) (4.10)

where is heat transfer from the plate.


Initial approximations and auxiliary linear operators are defined by

0() = 1− − 0() = − (4.11)

L = 000 − 0 L = 00 − (4.12)

with properties

L (1 +2 + 3

−) = 0L(4 + 5−) = 0 (4.13)

where ( = 1− 5) are the constants to be determined by the boundary conditions.

56

The zeroth order deformation problems are

(1− )Lh(; )− 0()

i= N

h(; )

i (4.14)

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (4.15)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 1 (∞; ) = 0 (4.16)

If ∈ [0 1] denotes an embedding parameter, and the non-zero auxiliary parameters,

then the nonlinear differential operators N and N are

N [( )] = (1 + Γ)3( )

3+ ( )

2( )

2− 2

Ã( )

!2(4.17)

−ΓÃ2( )

2

!23( )

3+ cos (4.18)

N[( ) ( )] =

"½1 +

³1 + ( − 1)( )

´3¾ ( )

#

+Pr

Ã( )

( )

− ( )

( )

! (4.19)

Obviously when = 0 and = 1 then

(; 0) = 0() ( 0) = 0() (4.20)

(; 1) = () ( 1) = ()

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

By using Taylor series expansion one has

( ) = 0() +∞P

=1

() (4.21)

( ) = 0() +∞P

=1

() (4.22)

57

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(4.23)

The convergence of above series strongly depends upon and By proper choices of and

the series (421) and (422) converge for = 1 and so

() = 0() +∞P

=1

() (4.24)

() = 0() +∞P

=1

() (4.25)

The resulting problems at order are

L [()− −1()] = R () (4.26)

L[()− −1()] = R () (4.27)

(0) = 0(0) = 0(∞) = (0) = (∞) = 0 (4.28)

R () = (1 + ) 000−1() +

−1P=0

³−1− 00 − 0−1−

0

− 00−1−1P=0

00−000

¶+−1 cos (4.29)

R () = (1 +) 00−1() + Pr

−1P=0

©0−1− − 0−1−

ª+( − 1)

−1P=0

∙( − 1)2 −1−

½P=0

−P

=0

−00

+3 ( − 1)P=0

00− + 300

¾¸+ 3 ( − 1)

∙−1P=0

0−1−0½

1 +P=0

− + 2 ( − 1) ¾¸

(4.30)

=

⎧⎨⎩ 0 ≤ 11 1

58

The general solutions are

() = ∗() + 1 + 2 + 3

− (4.31)

() = ∗() +4 + 5

− (4.32)

where ∗ and ∗ are the particular solutions and the constants ( = 1− 5) are given by

2 = 4 = 0 3 =∗()

¯=0

1 = −3 − ∗(0) 5 = ∗()|=0


The convergent rate and region of the series (426) and (427) depend on the auxiliary parameters

and So we plot ~−curves for the functions 00(0) and 0(0) at 12 order of approximations.The range for the admissible values of and are −13 ≤ } ≤ −02 and −14 ≤ } ≤ −04We find that series given in the equation (424) and (425) converge in the whole region of

for different values of pertinent parameters by choosing the admissible values of } = −06 and} = −07

Fig. 4.1. ~−curve for velocity field.

59

Fig. 4.2. ~−curve for temperature field.Table 4.1: Convergence of series solutions for different order of approximations when =

4 = 04 Γ = 05 = 03 = 02 = 102 Pr = 10 } = −06 and } = −07


5 097413 085600

10 097455 085769

15 097460 085771

20 097459 085770

25 097459 085770

30 097459 085770

35 097459 085770

40 097459 085770

60


The variations of velocity and temperature profiles for different physical parameters are high-

lighted in this suction. Different parameters include Γ and the fluid parameters, Pr the

Prandtl number, the radiation parameter, the temperature ratio parameter and the

mixed convection parameter. Figs. 43 and 44 are plotted for the non-Newtonian fluid para-

meter Γ and . Velocity profile 0() is an increasing function of Γ. Slight decrease is observed

for larger values of . Fig. 45 shows the variation of angle of inclination on 0(). As

increases, sheet moves from vertical to horizontal direction and buoyancy force decreases.

Ultimate the velocity and momentum boundary layer thickness decay. Buoyancy parameter

enhances the velocity profile (see Fig. 46). Here we considered 0 (Buoyancy aid flow)

therefore temperature gradient at the wall is larger in comparison to the ambient. Fig. 47

depicts the effect of thermal radiation parameter on 0(). Near the wall the variation is

negligible but at some distance from the wall small increment is observed in velocity profile.

Fluid parameter Γ decreases the thermal boundary layer thickness (see Fig. 48). Radiation

parameter enhances the temperature field significantly (see Fig. 49). Larger values corre-

spond to more surface heat flux. Effect of temperature ratio parameter is more pronounced

near the wall (see Fig. 410). In fact its larger values corresponds to stronger heat transfer at

the wall than far away. Variation in temperature profile for larger values of Pr is displayed in

Fig. 411. Larger values of Pr decay the thermal diffusivity and hence the temperature and

thermal boundary layer thickness decrease.

Table 42 gives the comparison between numerical and HAM solutions in special case (Γ =

= 0). Here we found an excellent agreement. Table 43 shows the effect of local Nusselt

number for different values of physical parameters Γ, and Pr Here we observe that the

magnitude of Γ and on local Nusselt number are quite opposite. Local Nusselt number

decreases for large values of .

61

Fig. 4.3. Influence of Γ on velocity field.

Fig. 4.4. Influence of on velocity field.

62



63


Fig. 4.8. Influence of Γ on temperature field.

64

Fig. 4.9. Influence of on temperature field.

Fig. 4.10. Influence of on temperature field.

65

Fig. 4.11. Influence of Pr on temperature field.

Table 42: Comparison between numerical and HAM solutions in a special case when Γ =

= 0

Pr [4] [6]

10 00 −095478 −08548 −0954802 −081664

20 00 −14715 −14715 −1471402 −1274505 −10735 −10735

66

Table 43: Values of heat transfer characteristics at wall −0(0) for the parameters Γ, , and Pr

Γ Pr −(1 +3 )0(0)

00 03 05 10 03 11006

6 10962

2 10629

4 00 10931

02 10916

04 10900

03 00 10483

04 10835

06 10984

05 11 11973

14 13329

18 01 10483

02 10908

03 11308

4.6 Conclusions

This chapter investigated the flow of Powell-Eyring fluid by an inclined exponentially stretching

sheet. The major conclusions with nonlinear thermal radiation effects are summarized below:

• Influence of Γ on the velocity and temperature fields are quite opposite

• Velocity field 0() is decreasing function of

• Temperature field () and thermal boundary layer thickness are increased for larger

thermal radiation parameter .

• Angle of inclination decays the velocity profile 0()

• Effects of and on temperature field are qualitatively similar.

• It is found that () decreases when Pr increases.

67

• Temperature ratio parameter enhances the temperature and thermal boundary layer thick-ness.

• Behaviors of Γ and Pr on the magnitude of local Nusselt number are qualitatively similar.

• Magnitude of local Nusselt number decreases for larger values of and

68

Chapter 5

Radiation effects in the flow of

Powell-Eyring fluid past an unsteady

inclined stretching sheet with

non-uniform heat source/sink

5.1 Introduction

This chapter addresses time-depentent flow of Powell-Eyring fluid past an inclined stretching

sheet. Unsteadiness in the flow is due to the time-dependence of stretching velocity and wall

temperature. Mathematical analysis is performed in the presence of thermal radiation and non-

uniform heat source/sink. The relevant boundary layer equations are reduced into self-similar

forms by using suitable transformations. The analytic convergent solutions are constructed in

the series forms. The convergence interval of the auxiliary parameters is obtained. Graphical

results displaying the impacts of interesting parameters are given. Numerical values of skin

friction coefficient and local Nusselt number are computed and analyzed.

69

5.2 Mathematical modeling and analysis

We consider unsteady two-dimensional incompressible flow of Powell-Eyring fluid past a stretch-

ing sheet. The sheet makes an angle with the vertical direction. The − and −axes are takenalong and perpendicular to the sheet respectively. In addition the effects of thermal radiation

and non-uniform heat source/sink are considered. The boundary layer equations comprising

the balance laws of mass, linear momentum and energy can be written into the forms:

+

= 0 (5.1)

+

+

=

µ +

1

¶2

2− 1

23

µ

¶22

2+ 0 ( − ∞) cos (5.2)

µ

+

+

¶=

2

2−

+ 00 (5.3)

In the above expressions is the time, = () is the kinematic viscosity, is the thermal

conductivity of the fluid, is the fluid density, is the fluid temperature, is the specific heat,

is the acceleration due to gravity, is the volumetric coefficient of thermal exponential,

= −16∗3∞3∗

[56 − 65] is the linearized radiative heat flux ∗ is the mean absorptioncoefficient, ∗ is the Stefan-Boltzmann constant, 00 is the non-uniform heat generated (00 0)

or absorbed (00 0) per unit volume. The non-uniform heat source/sink 00 is modeled by the

following expression:

00 =( )

£( − ∞) 0 + ( − ∞)

¤ (5.4)


respectively. Here two cases arise. For internal heat generation 0 and 0 and for

internal heat absorption we have 0 and 0

The surface velocity is denoted by ( ) =

(1−) whereas the surface temperature ( ) =

∞+2

2(1−)−32. Here (stretching rate) and are the positive constants with dimen-

sion time−1 Also is a constant reference temperature. We note that the temperature of

stretching sheet is larger than the free stream temperature ∞

70

The boundary conditions are taken as follows:

= ( ) = 0 = ( ) at = 0 (5.5)

→ 0 → ∞ as →∞

Introducing

=

(1− ) 0() = −

s

(1− )()

= − ∞ − ∞

=

s

(1− ) (5.6)

equation (51) is identically satisfied and Eqs. (52− 55) become

(1 + Γ) 000 − 02 + 00 − Γ 002 000 − ( 0 +1

2 00) + cos = 0 (5.7)

µ1 +

4

3

¶00 +Pr

µ0 − 2 0 − 1

2(3 + 0)

¶+ 0 + = 0 (5.8)

= 0 0 = 1 = 1 at = 0

0 → 0 → 0 as →∞ (5.9)


is the dimensionless temperature and the dimensionless numbers are

Γ =1

=

4∗ 3∞∗

=32

= ( − ∞)32

222

=

Re2 =

, =

(5.10)

Here Γ and are the dimensionless material fluid parameters, is the radiation parameter,

is the unsteady parameter, is the mixed convection parameter and Pr is the Prandtl number.

Local Nusselt number is defined by

=

( − ∞); = −

µ

¶=0

+ () (5.11)

71

Re−12 = −µ1 +

4

3

¶0(0)

where Re =is the local Reynolds number.

5.3 Series solutions

Initial approximations and auxiliary linear operators are taken as

0() = 1− − 0() = − (5.12)

L = 000 − 0 L = 00 − (5.13)

subject to the properties

L (1 +2 + 3

−) = 0L(4 + 5−) = 0 (5.14)

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

(1− )Lh(; )− 0()

i= N

h(; )

i (5.15)

(1− )Lh(; )− 0()

i= N

h(; ) ( )

i (5.16)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 1 (∞; ) = 0 (5.17)

If ∈ [0 1] indicates the embedding parameter, and the non-zero auxiliary parameters

then the nonlinear differential operators N and N are given by

N [( )] = (1 + Γ)3( )

3+ ( )

2( )

2−Ã( )

!2

−ΓÃ2( )

2

!23( )

3−

Ã( )

+1

22( )

2

!+( ) cos (5.18)

72

N[( ) ( )] =

µ1 +

4

3

¶2( )

2+

ÃPr ( )

( )

− 2( )( )

−Ã3( ) +

( )

!!+

( )

+( ) (5.19)

We have for = 0 and = 1 the following equations

(; 0) = 0() ( 0) = 0() (5.20)

(; 1) = () ( 1) = () (5.21)

It is noticed that when varies from 0 to 1 then ( ) and ( ) approach from 0() 0()

to () and () The series of and through Taylor’s expansion are chosen convergent for

= 1 and thus

() = 0() +∞P

=1

() () =1

!

(; )

¯=0

(5.22)

() = 0() +∞P

=1

() () =1

!

(; )

¯=0

(5.23)

The resulting problems at order can be presented in the following forms:

L [()− −1()] = R () (5.24)

L[()− −1()] = R () (5.25)

(0) = 0(0) = 0(∞) = (0) = (∞) = 0 (5.26)

R () = (1 + Γ) 000−1() +

−1P=0

"−1− 00 − 0−1−

0 − Γ 00−1

−X=0

00−000

−µ 0−1 +

1

2 00−1

¶¸+−1 cos (5.27)

73

R () =

µ1 +

4

3

¶00−1() +

−1P=0

∙0−1− − 2 0−1− −

1

2¡3−1 + 0−1

¢¸+ 0−1 +−1 (5.28)

=

⎧⎨⎩ 0 ≤ 11 1

The general solutions ( ) comprising the special solutions (∗

∗) are

() = ∗() + 1 + 2 + 3

− (5.29)

() = ∗() +4 + 5

− (5.30)


It is now well established argument that the convergence of series solutions (524) and (525)

depends upon the auxiliary parameters. The admissible range of values of and (for some

fixed values of parameters) lie along the line segment parallel to and −axes. For examplein Figs. 51 and 52 the permissible ranges of values of and are −126 ≤ ≤ −024and −14 ≤ ≤ −025 respectively when = 06. The series solutions converge for the whole

region of when = −09 and = −08.

74

Fig. 5.1. − curves for the velocity field.

Fig. 5.2. − curves for the temperature field.

75

Table 51: Convergence of series solutions for different order of approximations when = 4

= 05 Γ = 02 = 02 = 06 = 03 Pr = 10 = = 01 } = −08 and } = −07


5 104402 135252

10 104401 135252

15 104401 135252

20 104401 135252

30 104401 135252

5.5 Discussion

This section is aimed to examine the effects of different physical parameters on the velocity

and temperature fields. Hence Figs. (53− 515) are plotted. Fig. 53 elucidates the behaviorof inclination angle on the velocity and boundary layer thickness. Here = 0 corresponds

to velocity profiles in the case of vertical sheet for which the fluid experiences the maximum

gravitational force. On the other hand when changes from 0 to 2 i.e when the sheet

moves from vertical to horizontal direction then, the strength of buoyancy force decreases and

consequently the velocity and the boundary layer thickness decrease. Fig. 54 indicates that the

velocity field 0 is an increasing function of . This is because of the fact that larger values of

yield a stronger buoyancy force which leads to an increase in the −component of velocity. Theboundary layer thickness also increases by increasing Variation in 0 through can be seen

from Fig. 55. It is noticed that 0 decreases and boundary layer thins when is increased

Influence of unsteady parameter on the velocity field is displayed in Fig. 56. Increasing

values of indicates smaller stretching rate in the −direction which eventually decreases theboundary layer thickness. Interestingly the velocity enhances by increasing at sufficiently large

distance from the sheet. Variation in the − component of velocity with an increase in thefluid parameter Γ can be described from Fig. 57. In accordance with Mushtaq et al. [36],the

velocity field 0 increases when Γ is increased. Radiation effects on the velocity and temperature

distributions can be perceived through Figs. 58 and 59. An increase in enhances the heat

76

flux from the sheet which gives rise to the fluid’s velocity and temperature. Wall slope of the

temperature function therefore increases with an increase in . Fig. 510 portrays the effect of

Prandtl number on the thermal boundary layer. From the definition of Pr given in Eq. (510),

it is obvious that increasing values of Pr decreases conduction and enhances pure convection or

the transfer of heat through unit area. That is why the temperature and thermal boundary layer

thickness decrease for larger Pr. Such reduction in the thermal boundary layer accompanies

with the larger heat transfer rate from the sheet. Temperature profiles for different values of

Γ are shown in Fig. 511. It is seen that temperature is an increasing function of Γ. Fig.

512 indicates that an increase in the strength of buoyancy force due to temperature gradient

decreases the temperature and thermal boundary layer thickness. Influence of heat source/sink

parameter on the thermal boundary layer is shown in the Figs. 513 and 514 As expected the

larger heat source (corresponding to 0 and 0) rises the temperature of fluid above the

sheet However the non-uniform heat sink corresponding to 0 and 0 can contribute in

quenching the heat from stretching sheet effectively. Fig. 518 depicts that temperature is a

decreasing function of the unsteady parameter .

Table 52 shows the comparison of present work with Tsai et al. [53] in a special case. A

very good agreement is found for the results of wall temperature gradient. Table 53 shows the

effect of embedded parameters on heat transfer characteristics at the wall −0(0). Since in thepresent case the sheet is hotter than the fluid i.e ∞ thus heat flows from the sheet to

the fluid and hence 0(0) is negative. From this table we observe that with an increase in ,

and the wall heat transfer rate¯0(0)

¯decreases. However it increases when Γ and Pr are

increased.

77

Fig. 53. Variation of on velocity field.


78



79

Fig. 57. Variation of Γ on velocity field.


80

Fig. 59. Variation of on temperature field.

Fig. 510. Variation of Pr on temperature field.

81

Fig. 511. Variation of Γ on temperature field.


82



83


Table 52: Comparison between numerical solution Tsai et. al. [53] and HAM solution in a

special case when = = = Γ = = = 0

Pr Present study Tsai et.[53]

10 −10 00 −171094 −1710937−20 −10 −236788

20 −10 00 −225987−20 −10 −2486000 −2485997

84

Table 53: Values of heat transfer characteristics at wall −0(0) for different emerging para-meters when = −08 and = −07.

Γ Pr −(1 + 43)0(0)

0 02 05 06 03 02 10 135702

6 135498

3 134926

4 00 169881

04 172556

07 174109

09 174984

00 171555

05 171319

09 171114

00 110162

04 161674

06 171319

00 169868

05 172227

08 173515

00 154046

03 171319

06 200303

12 191058

15 217721

19 249323

−01 180783

00 176059

01 171319

85

5.6 Closing remarks

This chapter addressed the radiation effects in the unsteady boundary layer flow of Powell-

Eyring fluid past an unsteady inclined stretching sheet with non-uniform heat source/sink.

The important findings are listed below.

1. The strength of gravitational force can be varied by changing the inclination angle which

the sheet makes with the vertical direction. The velocity decreases with an increase in

2. Velocity field 0 and temperature are decreasing function of the unsteady parameter

3. Velocity increases and temperature decreases when the fluid parameter Γ is increased

4. Increase in the radiation parameter enhances the heat flux from the plate. As a conse-

quence the fluid velocity and temperature both increases.

5. The analysis for the case of viscous fluid can be obtained by choosing Γ = = 0 Further

the results for horizontal stretching sheet are achieved when = 2

86

Chapter 6

MHD mixed convection flow of

Burgers’ fluid in a thermally

stratified medium

6.1 Introduction

This chapter explores the effects of nonlinear thermal radiation and magnetohydrodynamics

(MHD) in mixed convection flow of Burgers’ fluid over a stretching sheet embedded in a ther-

mally stratified medium. Transformation method has been employed to reduce the nonlinear

PDE into the nonlinear ODE. The resulting nonlinear coupled ordinary differential equations

are solved for the convergent series solutions. Convergence of derived series solutions is shown

explicitly explored. Physical interpretation of different parameters through graphs and numer-

ical values of local Nusselt number are discussed.

6.2 Flow equations

The extra stress tensor for Burgers’ fluid satisfies [49]:

S+1S

+ 2

2S

2=

∙A1 + 3

A1

¸ (6.1)

87

in which is the dynamic viscosity, A1 is the rate of strain tensor, 1 and 2 are the relaxation

times, 3 is the retardation time and the upper convected derivativeis

S

=

S

− LS− SL (6.2)

in which is the material time derivative and L is the velocity gradient. It should be

noted that Burgers’ fluid model reduces to the special cases of an Oldroyd-B model, Maxwell

model and the Newtonian fluid model when (2 = 0) (2 = 3 = 0) and (1 = 2 = 3 = 0)

respectively.

6.3 Problem statement

Here we consider MHD mixed convection flow of Burgers’ fluid in a thermally stratified medium.

An incompressible fluid is electrically conducting in the presence of constant applied magnetic

field 0. Electric and induced magnetic fields are assumed negligible. The flow is because of

linear stretching sheet along the − . The − is taken normal to the surface with

constant temperature. Also thermal radiation effects in heat transfer analysis are present.

The equations for velocity and temperature in boundary layer regime are

+

= 0 (6.3)

+

+ 1

∙2

2

2+ 2

2

2+ 2

2

¸+ 2

∙3

3

3+ 3

3

3

+2µ

2

2−

2

2+ 2

2

¶+ 32

µ

2

2+

2

¶+3

µ

3

2+

3

2

¶+ 2

µ2

2

2+

2

2+

2

¶¸=

2

2+ 3

∙

3

2+

3

3−

2

2−

2

2

¸+ 0 ( − ∞)

−20

µ+ 1

+ 2

½

−

+

2

+ 2

2

2

¾¶ (6.4)

µ

+

¶=

∙µ +

16∗ 3

3∗

¶

¸ (6.5)

88

with the subjected conditions

= () = = 0 = , at = 0 (6.6)

→ 0 → ∞ as →∞ (6.7)

where and represent the velocity components in the and directions respectively, is

the fluid temperature, = 0 + is the prescribed surface temperature, ∞ = 0 + is

the variable ambient fluid temperature, = () is the kinematic viscosity, is the density

of fluid, is the convective heat transfer coefficient, is the effective diffusion coefficient,

is the specific heat at constant pressure and is the thermal conductivity of the fluid. Making

use of following change of variables

= ()−12 () () = − ∞ − 0

= ³

´12 (6.8)

=

= −

(6.9)

equation (66) is identically satisfied and Eqs. (67− 610) yield

000 + 00 − 02 + 1¡2 0 00 − 2 000

¢+ 2

¡3 0000 − 2 02 00 − 32 002¢

+3¡ 002 − 0000

¢+ −2

¡ 0 − 1

00 + 22 000

¢= 0 (6.10)hn

1 + (1 + ( − 1))3o0i0+Pr

h0 − 0 − 0

i (6.11)

= 0 0 = 1 = 1− at = 0

0 → 0 → 0 as →∞ (6.12)

where 1 = 1 and 2 = 2 are the Deborah numbers in terms of relaxation times respectively,

3 = 3 is the Deborah number in term of retardation times, is the stream function,

=20

is the magnetic parameter, =∞ is the Prandtl number, =

02

is the mixed

convection parameter = 4∗3∞

∞∗ is the radiation parameter, is the temperature ratio

parameter and = is the stratification parameter.

89

The local Nusselt number is defined by

=

( − ∞) with = −

¯=0

+ () (6.13)

Dimensionless form of local Nusselt number is

12 = − ¡1 +3¢0(0)

where (Re)−12 =

q

is the local Reynolds number .

6.4 Development of series

Initial approximations and auxiliary linear operators for the HAM solution expressions are

0() = (1− −) 0() = (1− )− (6.14)

L = 000 − 0 L = 00 − (6.15)

L (1 + 2 + 3

−) = 0 (6.16)

L(4 + 5−) = 0 (6.17)

where ( = 1 − 7) are the arbitrary constants, ∈ [0 1] denotes an embedding parameterand and the non-zero auxiliary parameters. The zeroth order deformation problems are

(1− )Lh(; )− 0()

i= N

h(; )

i (6.18)

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (6.19)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 1− (∞; ) = 0 (6.20)

90

N [( )] =3( )

3+ ( )

2( )

2−Ã( )

!2

+1

"2( )

( )

2( )

2−³( )

´2 3( )3

#

+2

⎡⎣⎛⎝³( )´3 4( )4

− 2( )Ã( )

!22( )

2

⎞⎠−3³( )

´2Ã2( )

2

!2⎤⎦+3

⎡⎣Ã2( )

2

!2− ( )

4( )

4

⎤⎦+( )

−2

Ã( )

− 1( )

2( )

2+ 2

³( )

´2 2( )2

! (6.21)

N[( ) ( )] =

"½1 +

³1 + ( − 1)( )

´3¾ ( )

#

+Pr

Ã( )

( )

−

( )

! (6.22)

When = 0 and = 1 then

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

(; 1) = () (; 1) = () (6.23)

When variation is taken from 0 to 1 then ( ) and ( ) approach 0() and 0() to ()

and () Now and in Taylor’s series can be expanded in the forms

( ) = 0() +∞P

=1

() (6.24)

( ) = 0() +∞P

=1

() (6.25)

91

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(6.26)

where the convergence depends upon and For = 1 the series (627) and (628) yield

() = 0() +∞P

=1

() (6.27)

() = 0() +∞P

=1

() (6.28)

The deformation problems corresponding to -order are

L [()− −1()] = R () (6.29)

L[()− −1()] = R () (6.30)

(0) = 0(0) = 0(∞) = 0 (0) = (∞) = 0 (6.31)

R () = 000−1() +

−1P=0

©−1− 00 − 0−1−

0

ª+1

−1X=0

(2 00−1−

X=0

0−00 − −1−

X=0

− 000

)

+2

−1X=0

⎧⎨⎩−1−X=0

−X

=0

00−0000 − 2−1−

X=0

0−00 − 3−1−

X=0

− 0000

⎫⎬⎭+3

( 00−1−

−1X=0

00 − −1−−1X=0

0000

)+−1

−2

Ã 0−1 − 1

−1X=0

− 00−1− + 2

−1X=0

−1−X=0

− 000

! (6.32)

92

R () = (1 +) 00−1() + Pr

−1P=0

n0−1− − 0−1− − 0−1

o+( − 1)

−1P=0

∙( − 1)2 −1−

½P=0

−P

=0

−00

+3 ( − 1)P=0

00− + 300

¾¸+ 3 ( − 1)

∙−1P=0

0−1−0½

1 +P=0

− + 2 ( − 1) ¾¸

(6.33)

=

⎧⎨⎩ 0 ≤ 11 1

The general solutions of Eqs. (632− 633) are

() = ∗() + 1 + 2 + 3

− (6.34)

() = ∗() +4 + 5

− (6.35)

where ∗ and ∗ indicate the particular solutions.

6.5 Convergence of solutions

Obviously the series (632) and (633) consist of the auxiliary parameters } and ~. Such

parameters have key role for the convergence of series solutions. We plot the ~−curves at 17

order of approximations (see Figs. 61 and 62) It is clearly seen from these Figs. that the

admissible values of } and } are [−13−06] and [−13−06] respectively Moreover Table61 depicts that the series solutions are convergent up to five decimal places.

93

Fig. 6.1. ~− curve for the velocity field.

Fig. 6.1. ~− curve for the temperature field.

94

Table 61 : Convergence of homotopic solution for different order of approximations.

Order of approximations − 00(0) −0(0)1 089000 066803

5 092132 066595

10 092121 066614

15 092125 066623

20 092125 066621

25 092125 066621

30 092125 066621

35 092125 066621

40 092125 066621

6.6 Discussion

To provide a physical insight into the computed analysis, the velocity and temperature fields

are described for various values of pertinent variables. Here the influence of Deborah numbers

on the velocity field is presented in the Figs. (63 − 65). Velocity field is found to decreasefor larger values of 1 and 2 (see Fig. 63 and 64). As Deborah number exhibits both

viscous and elastic effects therefore velocity always retards when we increase this parameter.

Fig. 65 depicts the variation of 3 on the velocity field. A small increment is observed in the

velocity field when 3 enhances. This is due to the fact that larger 3 corresponds to the large

retardation time. Therefore velocity and associated boundary layer thickness increase. Impact

of magnetic parameter on the velocity field is displayed in Fig. 66. It is revealed that

both the velocity and momentum boundary layer thickness decrease when magnetic parameter

is increased. In fact through magnetic field, the apparent viscosity enhances to the point

of becoming viscoelastic solid. Also variation in magnetic field intensity is used to control the

yield stress and boundary layer thickness. Consequently the fluid ability to transmit the force is

controlled. Fig. 67 shows that an increase in mixed convection parameter corresponds to an

increase in the velocity field. This is because of the reason that buoyancy forces induce a pressure

gradient in the boundary layer which acts to modify the flow field. Fig. 68 shows the variation

of magnetic parameter on the temperature field. Here the temperature increases when

95

is increased. Physically larger magnetic field corresponds to much Lorentz force therefore

temperature and thermal boundary layer thickness are increased. Temperature field decays

for larger values of mixed convection parameter (see Fig. 69). The influence of radiation

parameter is displayed in Fig. 610. Enhancement in radiation parameter leads to increase

in kinetic energy that rises the fluid temperature and thermal boundary layer thickness. Effect

of temperature ratio parameter is displayed in Fig. 611. It is clearly seen from this Fig.

that the temperature field increases when there is an increase in temperature ratio parameter.

Thermal stratification parameter decays the temperature most rapidly near the wall but after

some large distance from the wall the variation of is insignificant (see Fig. 612). Physically

this is due to the fact that with the increase in the stratification parameter, the buoyancy

factor reduces within the boundary layer and so the temperature decays. Fig. 613 illustrates

the effect of Prandtl number on temperature field. Higher values of Prandtl number show a

decrease in the thermal diffusivity. As a result both the temperature and thermal boundary

layer thickness decrease.

Table 62 shows the effect of local Nusselt number for different values of physical parameters

1, 2 3 and Pr Here we observe that the effects of 1 and 3 on local Nusselt number

are quite opposite. Local Nusselt number decreases for larger values of 3, and Pr.

Fig. 63. Variation of 1 on velocity field.

96



97



98



99



100


Fig. 613. Variation of Pr on temperature field.

101

Table 62 : Values of local Nusselt number 12 for different parameters.

1 2 3 Pr 12

00 03 02 01 10 03 104322

02 101932

04 099778

02 00 104396

02 102749

03 101934

00 099122

01 10059

03 10318

01 090120

03 096389

06 10449

10 096390

12 10850

15 12523

00 11055

02 10119

05 086558

101 096492

12 095004

15 094931

6.7 Main points

The effects of MHD, nonlinear radiation and viscous dissipation in mixed convection flow in a

thermally stratified medium are studied. The following result are worthmentioning.

1. Velocity field decays slowly for larger values of martial parameters 1 and 2.

102

2. Deborah number corresponding to retardation time 3 increases both the velocity and

momentum boundary layer thickness.

3. The velocity field via mixed convection parameter and temperature via magnetic para-

meter behave in a similar fashion.

4. Temperature and thermal boundary layer thickness are enhanced for larger values of

radiation parameter and temperature ratio parameter .

5. Thermal stratification parameter causes reduction in the temperature as well as thermal

boundary layer thickness.

6. Local Nusselt number decreases when and are increased.

103

Chapter 7

Flow of Burgers fluid through an

inclined stretching sheet with heat

and mass transfer

7.1 Introduction

Effects of heat and mass transfer in the flow of Burgers fluid by an inclined sheet are discussed

in this chapter. Problems formulation and relevant analysis are given in the presence of ther-

mal radiation and non-uniform heat source/sink. Thermal conductivity is taken temperature

dependent. The nonlinear partial differential equations are simplified using boundary layer ap-

proximations. The resultant nonlinear ordinary differential equations are solved for the series

solutions. The convergence of series solutions is obtained by plotting the ~ - curves for the

velocity, temperature and concentration fields. Effects of various physical parameters on the

velocity, temperature and concentration fields are studied. Numerical values of local Nusselt

number and local Sherwood number are computed and analyzed.

7.2 Development of the problems

Here we have considered the effects of heat and mass transfer in the flow of Burgers fluid over

an inclined stretching sheet. Thermal radiation, variable thermal conductivity and non-uniform

104

heat source/sink effects are taken into account. The stretching sheet makes an angle with the

vertical axis i.e. − The − is taken normal to the − . Under the boundary

layer approximation the velocity, temperature and the concentration fields are governed by the

following equations

+

= 0 (7.1)

+

+ 1

∙2

2

2+ 2

2

2+ 2

2

¸+ 2

∙3

3

3+ 3

3

3

+2µ

2

2−

2

2+ 2

2

¶+ 32

µ

2

2+

2

¶+3

µ

3

2+

3

2

¶+ 2

µ2

2

2+

2

2+

2

¶¸=

2

2+ 3

∙

3

2+

3

3−

2

2−

2

2

¸+ 0 ( − ∞) cos (7.2)

µ

+

¶=

µ

¶−

+ 00 (7.3)

+

=

2

2 (7.4)

= () = = 0 = , = at = 0 (7.5)

→ 0 → ∞ → ∞ as →∞ (7.6)

where and represent the velocity components in the and directions respectively, is

the fluid temperature, is the specific heat, ∞ is the ambient fluid temperature, ∞ is the

ambient fluid concentration, 0 is the acceleration due to gravity, is the volumetric coefficient

of thermal expansion, = () is the kinematic viscosity, is the density of fluid, is the

convective heat transfer coefficient, is the concentration of fluid, is the effective diffusion

coefficient and the variable thermal conductivity of the fluid is adopted in the form:

= ∞

µ1 +

− ∞∆

¶ (7.7)

in which is the small parameter, ∞ is the thermal conductivity of the fluid far away from

the surface, ∆ = − ∞ and 00 is the non-uniform heat generated (00 0) or absorbed

105

(00 0) per unit volume. The non-uniform heat source/sink 00 is modeled by the following

expression

00 =( )

£( − ∞) 0 + ( − ∞)

¤ (7.8)


respectively. It should be noted that for internal heat generation 0 and 0 and for

internal heat absorption we have 0 and 0 We introduce the change of variables as

follows:

= ()12 () () = − ∞ − ∞

() = − ∞ − ∞

= ³

´12(7.9)

and the velocity components are

=

= −

Here is the stream function. Now Eq. (71) is identically satisfied and Eqs. (72− 76) yield

000 + 00 − 02 + 1¡2 0 00 − 2 000

¢+ 2

¡3 0000 − 2 02 00 − 32 002¢

+3¡ 002 − 0000

¢+ cos = 0 (7.10)

(1 +4

3+ )00 +Pr 0 + 02 + (1 + )

¡ 0 +

¢= 0 (7.11)

00 + ¡0 − 0

¢= 0 (7.12)

= 0 0 = 1 = 1 = 1 at = 0 (7.13)

0 → 0 → 0 → 0 as →∞ (7.14)

where 1 = 1 and 2 = 2 are the Deborah numbers in terms of relaxation times respectively,

3 = 3 is the Deborah number in terms of retardation times, =∞ is the Prandtl number,

=0 (−∞)3

22

is the mixed convection parameter = 4∗3∞

∞∗ is the radiation parameter

and = is the Schmidt number.

106

The local Nusselt number and local Sherwood number are

=

( − ∞) =

( − ∞) (7.15)

with

= −

¯=0

+ () = −

¯=0

Dimensionless form of local Nusselt number and local Sherwood number are

12 = −µ1 +

4

3

¶0(0) 12 = −0(0)

in which the local Reynolds number (Re)−12 =

q

.


Initial approximations and auxiliary linear operators are chosen as follows:

0() = (1− −) 0() = − 0 = − (7.16)

L = 000 − 0 L = 00 − L = 00 − (7.17)

with properties

L (1 + 2 + 3

−) = 0 (7.18)

L(4 + 5−) = 0 L(6 + 7

−) = 0 (7.19)

where ( = 1− 7) are the arbitrary constants. If ∈ [0 1] denotes an embedding parameterand and the non-zero auxiliary parameters then the zeroth order deformation problems

are

(1− )Lh(; )− 0()

i= N

h(; )

i (7.20)

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (7.21)

107

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (7.22)

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

(0; ) = 1 (∞; ) = 0 (7.23)

where the nonlinear operators N N and N are

N [( )] =3( )

3+ ( )

2( )

2−Ã( )

!2

+1

"2( )

( )

2( )

2−³( )

´2 3( )3

#(7.24)

+2

⎡⎣⎛⎝³( )´3 4( )4

− 2( )Ã( )

!22( )

2

⎞⎠−3³( )

´2Ã2( )

2

!2⎤⎦ (7.25)

+3

⎡⎣Ã2( )

2

!2− ( )

4( )

4

⎤⎦+( ) cos (7.26)

N[( ) ( )] =

µ1 +

4

3+

¶2( )

2+

Ã( )

( )

!+³( )

´2+ (1 + )

¡ 0 +

¢ (7.27)

N[( ) ( )] =2( )

2+

Ã( )

( )

− ( )

( )

! (7.28)

For = 0 and = 1 one has

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

(; 1) = () (; 1) = () (; 1) = () (7.29)

108

When variation is taken from 0 to 1 then ( ) ( ) and ( ) approach 0() 0() and

0() to () () and () Now and in Taylor’s series can be expanded in the forms

( ) = 0() +∞P

=1

() (7.30)

( ) = 0() +∞P

=1

() (7.31)

( ) = 0() +∞P

=1

() (7.32)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(7.33)

where the convergence depends upon and By proper choices of and the

series (730− 732) converge for = 1 and hence

() = 0() +∞P

=1

() (7.34)

() = 0() +∞P

=1

() (7.35)

() = 0() +∞P

=1

() (7.36)

The -order deformation problems can be provided in the forms

L [()− −1()] = R () (7.37)

L[()− −1()] = R () (7.38)

L[()− −1()] = R () (7.39)

(0) = 0(0) = 0(∞) = 0 (0) = (∞) = 0 (0) = (∞) = 0 (7.40)

109

in which

R () = 000−1() +

−1P=0

©−1− 00 − 0−1−

0

ª+1

−1X=0

(2 00−1−

X=0

0−00 − −1−

X=0

− 000

)

2

−1X=0

⎧⎨⎩−1−X=0

−X

=0

00−0000 − 2−1−

X=0

0−00 − 3−1−

X=0

− 0000

⎫⎬⎭3

( 00−1−

−1X=0

00 − −1−−1X=0

0000

)+−1 cos (7.41)

R () = (1 +

4

3+)00−1 + −1−

−1X=0

00 +

½−1P=0

¡0−1−

¢¾

+−1−−1X=0

+¡ 0−1 +−1

¢+ −1−

−1X=0

0

+−1−−1X=0

(7.42)

R () = 00−1 +

−1P=0

¡0−1− − 0−1−

¢ (7.43)

=

⎧⎨⎩ 0 ≤ 11 1

The general solutions of Eqs. (737− 739) can be written as follows:

() = ∗() + 1 + 2 + 3

− (7.44)

() = ∗() +4 + 5

− () = ∗() +6 + 7

− (7.45)

where ∗ ∗ and ∗ are the particular solutions.

110

7.4 Convergence of the solutions

It is fairly understandable that the resultant series solutions contain the auxiliary parameters

} ~ and ~. Such parameters have key role for the convergence of series solutions. For

permissible values of auxiliary parameters, we plot the ~−curves at 14 order of approximations(see Figs. 71− 73) It is clearly seen from these Figs. that the meaningful values of } }

and } are [−13−06] [−125−03] and [−13−06] respectively Moreover Table 71 depictsthat the series solutions are convergent up to six decimal places.


111


Fig. 7.3. ~− curve for the concentration field.

112

Table 71: Convergence of HAM solution for different order of approximations.

Order of approximation − 00(0) −0(0) −0(0)1 113084 0412012 104613

5 118308 0365947 101901

10 118324 0365564 101677

15 118324 0365553 101657

20 118324 0365551 101654

25 118324 0365550 101653

30 118324 0365550 101653

35 118324 0365550 101653

40 118324 0365550 101653

50 118324 0365550 101653

7.5 Discussion

Considered problem contains different physical parameters like 1 2 3 Pr

and . Graphical results of such parameters are displayed and discussed in the following

subsections.

7.5.1 Dimensionless velocity field

Fig. 74 presents the influence of on the velocity field. It is seen that larger values of

cause a reduction in the velocity profile. In fact an increase in corresponds to more gravity

effect. Figs. (75− 77) are plotted for various values of Deborah numbers (1 2 and 3).

An increase in 1 and 2 causes a reduction in velocity profile and associated boundary layer

thickness. Effect of 3 on the fluid motion is quite opposite to that of 1 and 2 It is also

clear from Fig. 77 that the velocity field is more pronounced near the wall but after some large

distance it vanishes. Fig. 78 illustrates that an increase in enhances in the velocity and

boundary layer thickness.

113

7.5.2 Dimensionless temperature field

Fig. 79 depicts the variation of on the temperature profile. In fact the increase in variable

thermal conductivity parameter result in the increase of the temperature and thermal bound-

ary layer thickness. Figs. 710 and 711 illustrate the effects of 1 and 2 on temperature profile.

Deborah numbers related to relaxation times (1 and 2) increase the temperature profile and

thermal boundary layer thickness. Influence of 3 (Deborah number related to retardation

times) and (mixed convection parameter) are qualitatively similar (see Figs. 712 and 713).

Both parameters decrease the temperature and thermal boundary layer thickness. Fig.714

illustrates the effect of thermal radiation on the temperature profile. Temperature profile

decreases rapidly when increases. Physically the larger values of imply larger surface heat

flux and thereby making the fluid warmer which enhances the temperature profile. The effect

of Prandtl number Pr on the temperature distribution is displayed in Fig. 715. It can be seen

that the temperature decreases with an increase of Pr. It implies that the momentum boundary

layer is thicker than the thermal boundary layer. This is due to the fact that for higher Prandtl

number, the fluid has a relatively low thermal conductivity which reduces conduction. For large

values of heat generation/absorption parameters and the temperature distribution across

the boundary layer increases (see Figs. 716 and 717).

7.5.3 Dimensionless concentration field

Figs. (718− 720) show the variation of Deborah numbers (1 2 and 3) on the concentrationfield. Here concentration field shows gradual increase when 1 and 2 are increased. However

the concentration field is decreasing function of 3 (see Fig. 720) Fig. 721 illustrates the

effect of mixed convection parameter on the concentration field. For larger values of the

concentration field decreases. Fig. 722 depicts the influence of on the concentration field. It

is seen that larger values of decrease the concentration field. Fig. 723 shows the variation of

on () Concentration field decreases rapidly when increases.

114

Fig. 74 Influence of on velocity field.

Fig. 75 Influence of 1 on velocity field.

115



116



117

Fig. 710 Influence of 1 on temperature field.


118


Fig. 713 Influence of on temperature field.

119


Fig. 715 Influence of Pr on temperature field.

120



121

Fig. 718 Influence of 1 on concentration field.


122


Fig. 721 Influence of on concentration field.

123



124

Table 72: Values of local Nusselt number 12 for different parameters.

1 2 3 Pr 12

00 03 02 4 01 10 0763096

02 0672509

03 0634519

02 00 0650943

01 0642600

03 0626717

00 0644663

01 0639651

02 0634519

01 0626474

02 0634519

03 0641885

00 0712749

01 0678452

02 0634519

11 0694574

13 0791845

14 0880650

125

Table 73: Values of local Sherwood number for different parameters.

1 2 3 −0(0)00 02 02 11 11135

03 10914

06 10716

00 11066

03 10944

05 10862

00 10748

01 10836

02 10913

00 082259

01 10295

03 12071

7.6 Concluding remarks

Here series solutions are developed to solve the nonlinear problem of Burgers fluid flow by an

inclined stretching sheet with heat and mass transfer. Results presented describe the role of

different physical parameters involved in the problem. The Deborah numbers corresponding

to relaxation times (1 and 2) and angle of inclination () decrease the fluid velocity and

concentration field. Deborah number corresponding to retardation times (3) has opposite

behavior on the velocity and temperature fields. Concentration field decays as (3) and mixed

convection parameter () increase. Local Nusselt number decreases when 1 2 and are

larger. It is also found that 0(0) increases with an increase in (3)

126

Chapter 8

Stretched flow of Casson fluid with

nonlinear thermal radiation and

viscous dissipation

8.1 Introduction

The purpose of this chapter is to examine the steady flow of Casson fluid over an inclined

exponential stretching surface. The heat transfer analysis is considered in the presence of

nonlinear thermal radiation and viscous dissipation. The governing partial differential equations

are reduced to ordinary differential equations by employing adequate transformations. Series

solutions of the resulting problems are computed. The effects of physical parameter Prandtl

number Pr, temperature ratio parameter , Eckert number and radiation parameter

on the velocity, temperature and Nusselt number are investigated. Comparison of the present

analysis is made with the case of viscous fluid.

8.2 Flow equations

We consider the flow of an incompressible non-Newtonian fluid by an exponentially stretching

sheet which makes an angle with −axis. The flow is confined to 0. Also the −axis isperpendicular to −axis. The heat transfer is considered in the presence of viscous dissipation

127

and nonlinear thermal radiation. The extra stress tensor in Casson model is given by [34]:

=

⎧⎨⎩ 2¡+

√2¢

2¡+

√2¢

(8.1)

where is the plastic dynamic viscosity, is the yield stress of fluid, is the product of

the component of deformation rate with itself, is a critical value of this product based on

the non-Newtonian model, = and is the ( ) component of the deformation rate.

The governing boundary layer equations are

+

= 0 (8.2)

+

=

µ1 +

1

¶2

2+ 0 ( − ∞) cos (8.3)

µ

+

¶=

∙µ +

16∗ 3

3∗

¶

¸

+

µ1 +

1

¶µ

¶2 (8.4)

where and represent the velocity components along the flow (−direction) and normal tothe flow (−direction), = () is the kinematic viscosity, is the fluid density, 0 is the

acceleration due to gravity, is the volumetric coefficient of thermal expansion, =√2

is

the Casson fluid parameter, is the fluid temperature, is the thermal conductivity of fluid,

∗ is the mean absorption coefficient, ∗ is the Stefan-Boltzmann constant and is the specific

heat. The boundary conditions for the velocity and temperature fields are

= 0 = 0 = ∞ +

2 at = 0 (8.5)

→ 0 → 0 as →∞

where and ∞ are the temperatures at the plate and far away from the plate while is a

constant and 0 is the reference velocity. Defining variables

128

= 0 0() = −

r0

22

£() + 0()

¤

() = − ∞ − ∞

=

r0

22 (8.6)


is the dimensionless temperature with = ∞(1 + ( − 1)) and =∞ , Eq. (82) is

identically satisfied and Eqs. (83− 85) yieldµ1 +

1

¶ 000 − 2 02 + 00 + cos = 0 (8.7)

hn1 + (1 + ( − 1))3

o0i0+Pr

∙0 − 0 +

µ1 +

1

¶

002¸= 0 (8.8)

= 0 0 = 1 = 1 at = 0

0 → 0 → 0 as →∞ (8.9)

in which =16∗ 3∞3∗ is the radiation number, =

20

is the Eckert number, =0 (−∞)3

22

is the mixed convection parameter and Pr =is the Prandtl number.

The local Nusselt number is defined below:

=

( − ∞)with = −

µ

¶=0

+ () and so = −¡1 +3

¢0(0)

(8.10)

where is heat transfer from the plate.

8.3 Construction of solutions

Initial approximations and auxiliary linear operators are defined below

0() = 1− − 0() = − (8.11)

L = 000 − 0 L = 00 − (8.12)

129

with

L (1 +2 + 3

−) = 0L(4 + 5−) = 0 (8.13)

where ( = 1− 5) are the integral constants to be determined by the boundary conditions.The zeroth order deformation problems are

(1− )Lh(; )− 0()

i= N

h(; )

i (8.14)

(1− )Lh(; )− 0()

i= N

h(; ) ( )

i (8.15)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 1 (∞; ) = 0 (8.16)

If ∈ [0 1] denotes an embedding parameter, and the non-zero auxiliary parameters,

then the nonlinear differential operators N and N are

N [( )] =

µ1 +

1

¶3( )

3+ ( )

2( )

2− 2

Ã( )

!2+( ) cos (8.17)

N[( ) ( )] =

"½1 +

³1 + ( − 1)( )

´3¾ ( )

#

+Pr

Ã( )

( )

− ( )

( )

!

+Pr

µ1 +

1

¶Ã2( )

2

!2 (8.18)

Obviously when = 0 and = 1 then

(; 0) = 0() ( 0) = 0() (8.19)

(; 1) = () ( 1) = ()

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

According to Taylor series

( ) = 0() +∞P

=1

() (8.20)

130

( ) = 0() +∞P

=1

() (8.21)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(8.22)

and convergence of above series strongly depends upon and By proper choices of and

the series (820) and (821) converge for = 1 and so

() = 0() +∞P

=1

() (8.23)

() = 0() +∞P

=1

() (8.24)

The resulting problems at order are

L [()− −1()] = R () (8.25)

L[()− −1()] = R () (8.26)

(0) = 0(0) = 0(∞) = (0) = (∞) = 0 (8.27)

R () =

µ1 +

1

¶ 000−1() +

−1P=0

£−1− 00 − 0−1−

0

¤+ cos−1 (8.28)

R () = (1 +) 00−1() + Pr

−1P=0

©0−1− − 0−1−

ª+Pr

µ1 +

1

¶ 00−1−

00

+( − 1)−1P=0

∙( − 1)2 −1−

½P=0

−P

=0

−00

+3 ( − 1)P=0

00− + 300

¾¸+ 3 ( − 1)

∙−1P=0

0−1−0½

1 +P=0

− + 2 ( − 1) ¾¸

(8.29)

=

⎧⎨⎩ 0 ≤ 11 1

131

The general solutions are

() = ∗() + 1 + 2 + 3

− (8.30)

() = ∗() +4 + 5

− (8.31)

where ∗ and ∗ are the particular solutions and the constants ( = 1− 5) are given by

2 = 4 = 0 3 =∗()

¯=0

1 = −3 − ∗(0) 5 = ∗(0)


Our series solutions (825) and (826) contain the auxiliary parameters and which influ-

ence the convergence rate and regions of these two series. We choose proper values of and

from ~−curves at 14 order approximations. It is noticed that meaningful values of and are −02 ≤ } ≤ −09 and −01 ≤ } ≤ −07 We find that series given in the equations(825) and (826) converge in the whole region of for different values of pertinent parameters

when } = −06 and } = −05


132

Fig. 8.2. ~− curve for the temperature field.Table 81: Convergence of homotopy solutions for different order of approximations when

= 02 = 4 = 04 = 102 = 03 Pr = 10 = 07 } = −06 and } = −05


5 0948613 0752558

10 0948896 0752877

15 0948876 0752846

20 0948877 0752850

25 0948877 0752849

30 0948877 0752849

35 0948877 0752849

40 0948877 0752849

50 0948877 0752849

133

8.5 Interpretation of results

This section discloses the variations of emerging parameters on the velocity and temperature

fields. In particular, the variations of Casson fluid parameter mixed convection parameter ,

angle of inclination radiation parameter temperature ratio parameter Prandtl number

Pr and Eckert number are studied. In order to analyze the salient features of the emerging

parameters on the velocity and temperature profiles, we have plotted the Figs. (83− 813) The effects of Casson fluid parameter have been depicted in the Figs. 83 and 84. These Figs.

show that both velocity components and 0 decrease with an increase in Fig. 85 presents

the variation of angle of inclination on the velocity profile. Angle of inclination decreases the

velocity and corresponding boundary layer thickness. Fig. 86 demonstrates the effect of mixed

convection parameter on 0(). Mixed convection parameter enhances the velocity profile.

This is due to fact that the larger is associated with a stronger buoyancy force. A boundary

layer is laminar for smaller . Fig. 87 shows the influence of radiation parameter on the

velocity profile. As larger values of correspond to more heat flux from the wall therefore

velocity and monument boundary layer thickness increase. Fig. 88 is prepared to examine the

profiles of () for different values of We find that increases with an increase in Fig. 89

depicts the influence of on the temperature profile. Mixed convection parameter decreases

the temperature and thermal boundary layer thickness. When 1, the viscous force is

negligible in comparison to the buoyancy and inertial forces. When buoyant forces overcome

the viscous forces, the flow starts a transition to the turbulent regime. The effect of radiation

parameter on () has been shown in Fig. 810. It clearly shows that by increasing ()

increases. Fig. 811 is prepared to perceive the effect of ratio parameter on temperature profile.

Clearly the temperature increases for larger values of because wall temperature is larger

than ambient fluid temperature. Fig. 812 shows the change in () with respect to the Eckert

number . It is noted that () increases for large Because larger values of Eckert number

enhances the kinetic energy that heat up the fluid. To observe the variations of temperature

profile for different values of Pr we have prepared Fig. 813. As Prandtl number is used to adjust

the rate of cooling in conducting flows therefore thermal boundary layer thickness decreases by

increasing Prandtl number Pr.

Tables 82 and 83 show the effect of heat transfer characteristics at wall −0(0) for different

134

values of the parameters and Pr. In Table 82 the values of heat transfer

characteristics −0(0) at the wall are compared with the existing numerical solution in thespecial case ( → ∞) for viscous fluid. We found an excellent agreement in the HAM and

numerical solutions. Table 83 shows that the influences of and on −0(0) arequalitatively similar. Increase in decreases the heat transfer coefficient at the wall −0(0).


135



136



137



138



139



140

Table 82: Comparison between numerical solutions [6] and HAM solutions in the special

case ( →∞)

= 0 = 02 = 09

00 [6]

00

05 [6]

05

10 [6]

10

Pr = 1 Pr = 2

09548 14714

09548 14715

06765 10735

06765 10735

05315 08627

05313 08628

Pr = 1 Pr = 2

08622 13055

08623 103055

06177 09656

06173 09654

04877 07818

04574 07818

Pr = 1 Pr = 2

05385 07248

05386 07248

04101 05869

04101 05870

03343 04984

03340 04985

141

Table 83: Values of heat transfer characteristics at wall for the parameters , , and

Pr

Pr −(1 +3 )0(0)

07 03 02 10 10059

15 099776

20 099438

20 00 086813

03 099254

06 10986

00 11332

03 099253

07 11310

03 10 099253

15 12375

18 13638

10 101 099637

102 099254

104 098475


The flow of Casson fluid through nonlinear thermal radiation and viscous dissipation has been

studied. The major conclusions have been summarized as follows.

• Velocity field is decreasing function of non-Newtonian fluid parameter .

• Angle of inclination and mixed convection parameter have opposite behavior on the

velocity field 0()

• Temperature field () increases for larger values of .

• The effect of radiation parameter and Eckert number on temperature field () are

qualitatively similar.

142

• Temperature profile decreases for larger values of whereas it increases with temperatureratio parameter

• Increase in Pr decreases the temperature profile ().

• Influences of and on −0(0) are similar in a qualitative sense.

143

Chapter 9

Thermal radiation and viscous

dissipation effects in flow of Casson

fluid due to a stretching cylinder

9.1 Introduction

The purpose of present chapter is to examine the boundary layer flow by a stretching cylinder.

An incompressible Casson fluid is considered. The flow analysis is formulated through convective

boundary condition. Effects of thermal radiation and viscous dissipation are present. The

ordinary differential equations are computed for the convergent series solutions of velocity and

temperature. The velocity and temperature fields are analyzed for Casson fluid parameter,

curvature parameter, Prandtl number, Biot number, radiation parameter and Eckert number.

Skin friction coefficient and local Nusselt number are analyzed through numerical values.

9.2 Problem formulation and flow equations

We consider steady and two-dimensional incompressible flow of Casson fluid by a stretching

cylinder. The − and − are taken parallel and perpendicular to the cylinder respec-

tively. In addition, the effects of thermal radiation and viscous dissipation are considered. The

boundary layer equations comprising the balance laws of mass, linear momentum and energy

144

can be written as follows:

+

= 0 (9.1)

+

=

µ1 +

1

¶µ2

2+1

¶ (9.2)

∙

+

¸=

2

2+

− 1

() +

µ1 +

1

¶µ

¶2 (9.3)

In the above expressions = () is the kinematic viscosity, is the thermal conductivity of

the fluid, is the fluid density, is the fluid temperature, is the specific heat, = −16∗ 3∞3∗

is the radiative heat flux ∗ is the mean absorption coefficient and ∗ is the Stefan-Boltzmann

constant. The boundary conditions are taken as follows:

= 0

−

= ( − ) at = (9.4)

→ 0 → ∞ as →∞ (9.5)

Introducing

=0

0() = −

µ0

¶12(), (9.6)

() = − ∞ − ∞

=2 − 2

2

µ0

¶12 (9.7)

equation (91) is identically satisfied and Eqs. (92− 95) becomeµ1 +

1

¶¡(1 + 2) 000 + 2 00

¢− 02 + 00 = 0 (9.8)

(1 +4

3)¡(1 + 2) 00 + 20

¢+Pr 0 +Pr (1 + 2)

µ1 +

1

¶ 002 = 0 (9.9)

= 0 0 = 1 0 = −[1− (0)] at = 0 (9.10)

0 = 0 = 0 as →∞ (9.11)


is the dimensionless temperature and the dimensionless numbers are =√2

, =

4∗ 3∞∗ ,

145

=q

20

=

q0 Pr =

and =

20 ()2

(−∞). Here is the dimensionless material

parameters, is the radiation parameter, is the curvature parameter, is the Biot number,

Pr is the Prandtl number and is the Eckert number.

The skin friction coefficient is defined as

=122

, =

µ

¶=

and Re12 =

µ1 +

1

¶ 00(0)

Local Nusselt number is given by

=

( − ∞), = −

µ

¶=

+ () and Re12 = −µ1 +

4

3

¶0(0) (9.12)

9.3 Solutions

Initial approximations and auxiliary linear operators are chosen in the forms

0() = 1− − 0() =−

1 + (9.13)

L = 000 − 0 L = 00 − (9.14)

with the properties

L (1 +2 + 3

−) = 0L(4 + 5−) = 0 (9.15)

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

(1− )Lh(; )− 0()

i= N

h(; )

i (9.16)

(1− )Lh(; )− 0()

i= N

h(; ) (; )

i (9.17)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 0(0; ) = −[1− (0; )] (∞; ) = 0 (9.18)

If ∈ [0 1] indicates the embedding parameter, and the non-zero auxiliary parameters

146

then the nonlinear differential operators N and N are given by

N [( )] =

µ1 +

1

¶Ã(1 + 2)

3( )

3+ 2

2( )

2

!

+( )2( )

2−Ã( )

!2 (9.19)

N[( ) ( )] =

µ1 +

4

3

¶Ã(1 + 2)

2( )

2+ 2

( )

!

+Pr ( )( )

+Pr

µ1 +

1

¶(1 + 2)

Ã2( )

2

!2(9.20)

We have for = 0 and = 1 the following equations

(; 0) = 0() ( 0) = 0() (9.21)

(; 1) = () ( 1) = () (9.22)



= 1 and thus

() = 0() +∞P

=1

() () =1

!

(; )

¯=0

(9.23)

() = 0() +∞P

=1

() () =1

!

(; )

¯=0

(9.24)

The resulting problems at order can be presented in the following forms:

L [()− −1()] = R () (9.25)

L[()− −1()] = R () (9.26)

(0) = 0(0) = 0(∞) = 0(0)−(0) = (∞) = 0 (9.27)

147

R () =

µ1 +

1

¶¡(1 + 2) 000−1 + 2

00−1

¢+

−1P=0

¡−1− 00 − 0−1−

0

¢(9.28)

R () =

µ1 +

4

3

¶¡(1 + 2) 00−1() + 2

0−1

¢+

−1P=0

0−1−

+Pr

µ1 +

1

¶(1 + 2)

−1P=0

00−1−00 (9.29)

=

⎧⎨⎩ 0 ≤ 11 1

The general solutions ( ) comprising the special solutions (∗

∗) are

() = ∗() + 1 + 2 + 3

− (9.30)

() = ∗() +4 + 5

− (9.31)

1 = −3 − ∗(0) 2 = 4 = 0 3 =∗()

¯=0

5 =

∗()

¯=0− ∗()|=01 +

9.4 Convergence analysis

It is now well established argument that the convergence of series solutions (925) and (926)

depend upon the auxiliary parameters ~. It is clear from Figs. 91 and 92 that the admissible

values of and are −126 ≤ ≤ −024 and −11 ≤ ≤ −045 respectively. The seriessolutions converge for the whole region of when = −09 and = −08.

148



149

Table 91: Convergence of series solutions for different order of approximations when = 20

= 03 = 01 = 01 Pr = 10 = 04 } = −065 and } = −08


5 085415 034825

10 085327 036023

15 085312 036127

20 085307 036136

25 085304 036135

30 085303 036134

35 085303 036134

40 085303 036134

45 085303 036134


This section is aimed to examine the effects of different physical parameters on the velocity

and temperature fields. Hence the Figs. (93− 910) are plotted. Fig. 93 represents thevariation of on the velocity field. This Fig. shows that the velocity field decreases when

Casson fluid parameter is large. Also the velocity vanishes at some large distance from the

sheet (at = 12). Fig. 94 depicts the effect of on 0(). Here 0() is increasing function

of curvature parameter The boundary layer thickness decreases as increases. The velocity

component approaches to zero asymptotically when = 0. Influence of parameter on the

temperature field is displayed in Fig. 95. It is noticed that () decreases near the wall,

when is increased but at some large distance from the wall it vanishes. Fig. 96 presents the

influence of on the temperature field. We see that the temperature profile decreases when

1 thereafter, for large values of curvature parameter the temperature () increases. Also

when = 0 physically the outer surface of cylinder behaves like a flat surface. Fig. 97 shows

that temperature field is decreasing function of Prandtl number Pr when 1 and increasing

function of Pr when 1. Effect of Biot number on temperature field is plotted in Fig.

150

98. Temperature field increases when increases. Here we considered the Biot number less

than one due to uniformity of the temperature field in the fluid. A lower Biot number means

the conductive heat transfer is much faster than the convective heat transfer. Fig. 99 depicts

the variation of on () For large values of the temperature field increases. Fig. 910 gives

the influence of Eckert number on temperature field. It is clearly seen from this Fig. that

temperature field increases when there is an increase in the Eckert number Physically, the

larger Eckert number leads to enhancement in kinetic energy which enhances the temperature

rises. When 6 the temperature field vanishes asymptotically. Fig. 911 shows the effect of

Biot number on temperature and temperature gradient. Temperature field increases near the

wall most rapidly and it vanishes far away the wall where as temperature gradient has opposite

behavior for large values of Biot number.

The study of Table 92 indicates the effects of and on wall shear stress. From this

table it is clear that shear stress at wall is negative here. Physically, negative sign shows that

surface exerts a dragging force on the fluid. Table 93 shows the effect of physical parameters

on heat transfer characteristics at the wall −(0). We observe through tabular values that forlarge values of and Pr the magnitude of heat transfer characteristics at the wall −(0)increases. However it decreases for and .

Fig. 93 Influence of on 0()

151


Fig. 95 Influence of on ()

152


Fig. 97 Influence of Pr on ()

153



154


Fig. 911. Influence of on () and 0()

155

Table 92: Values of skin friction coefficient 12 for the parameters and

−12

11 01 14514

15 13519

21 12689

20 00 12247

01 12796

015 13067

Table 93: Values of heat transfer characteristics at wall −0(0) for different emergingparameters when = −065 and = −08.

Pr − ¡1 + 43¢0(0)

10 02 05 06 03 02 01394

13 01403

18 01412

20 00 01367

01 01414

02 01453

07 01345

10 01414

20 01473

01 006068

04 01696

06 02119

00 01457

02 01371

04 01288

00 01995

03 01559

06 01122

156

9.6 Conclusions

Here the radiative effect in the flow of Casson fluid due to stretching cylinder is modeled.

Convective boundary condition is used. The important findings are listed below.

• Effects of and on the velocity field are quite opposite.

• Temperature field () is decreasing function of

• Prandtl number Pr and curvature parameter have monotonic behavior on the temper-ature field

• Influences of radiation parameter and Eckert number on temperature field are

qalitatively similar.

• Temperature field enhances when Biot number is increased.

• Behaviors of and curvature parameter on skin friction coefficient are quite opposite.

• Magnitude of temperature gradient increases when curvature parameter and Biot num-ber are increased.

157

Chapter 10

Flow of variable thermal

conductivity fluid due to inclined

stretching cylinder with viscous

dissipation and thermal radiation

10.1 Introduction

The aim of present chapter is to investigate the flow of variable thermal conductivity Casson

fluid by an inclined stretching cylinder. Heat transfer analysis has been carried out in the

presence of thermal radiation and viscous dissipation effects. The relevant equations are first

simplified under usual boundary layer assumptions and then transformed into ordinary differ-

ential equations by suitable transformations. The transformed ordinary differential equations

are computed for the series solutions of velocity and temperature. Convergence analysis is

shown explicitly. Velocity and temperature fields are discussed for different physical parame-

ters through graphs and numerical values. It is found that the velocity decreases when angle

of inclination increases but it increases for larger values of mixed convection parameter. Fur-

ther an enhancement in the thermal conductivity and radiation effects corresponds to higher

fluid temperature. It is also found that heat transfer is more pronounced in a cylinder when

158

compared to a flat plate. Thermal boundary layer thickness enhances with increasing values of

Eckert number. Radiation and variable thermal conductivity decreases the heat transfer rate

at the surface.

10.2 Flow description

We consider steady two-dimensional incompressible flow of Casson fluid by an inclined stretching

cylinder. The stretching cylinder makes an angle with the vertical axis i.e. − and −is taken normal to it. In addition, the energy equation is considered with combined effects of

thermal radiation and viscous dissipation. The thermal conductivity is taken temperature

dependent. The boundary layer equations through the balance laws of mass, linear momentum

and energy can be written as follows:

()

+

()

= 0 (10.1)

+

=

µ1 +

1

¶µ2

2+1

¶+ 0 ( − ∞) cos (10.2)

∙

+

¸=1

µ

¶− 1

() +

µ1 +

1

¶µ

¶2 (10.3)

In the above expressions is the kinematic viscosity, is the fluid density, is the fluid

temperature, is the specific heat, = −16∗ 3∞3∗

is the radiative heat flux ∗ is the mean

absorption coefficient, 0 is the acceleration due to gravity, is the volumetric coefficient

of thermal exponential, ∗ is the Stefan-Boltzmann constant and is the variable thermal

conductivity of the fluid defined as

= ∞

µ1 +

− ∞∆

¶ (10.4)

in which is the small parameter, ∞ is the thermal conductivity of the fluid far away from

the surface and ∆ = − ∞ The boundary conditions are expressed as follows:

= 0

= 0 = at = (10.5)

159

→ 0 → ∞ as →∞ (10.6)

Introducing the relations

=0

0() = −

µ0

¶(), (10.7)

() = − ∞ − ∞

=2 − 2

2

µ0

¶ (10.8)

equation (101) is identically satisfied and Eqs. (102− 106) becomeµ1 +

1

¶¡(1 + 2) 000 + 2 00

¢− 02 + 00 + cos = 0 (10.9)

(1 +4

3)£(1 + 2) 00 + 20

¤+

£(1 + 2)

¡00 + 02

¢+ 20

¤+Pr 0

+Pr (1 + 2)

µ1 +

1

¶ 002 = 0 (10.10)

= 0 0 = 1 = 1 at = 0 (10.11)

0 = 0 = 0 as →∞ (10.12)


is the dimensionless temperature and the dimensionless numbers are =√2

, =

4∗ 3∞∗ ,

=q

20

=0 (−∞)32

222

, Pr =

and =20 ()

2

(−∞) . Here is the Casson

fluid parameter, is the radiation parameter, is the curvature parameter, Pr is the Prandtl

number, is the mixed convection parameter and is the Eckert number.

The skin friction coefficient is

=122

, =

µ

¶=

and Re12 =

µ1 +

1

¶ 00(0) (10.13)

Local Nusselt number is

=

( − ∞), = −

µ1 +

16∗ 3∞3∗

¶µ

¶=

and Re12 = −(1 + 43)0(0)

(10.14)

160

10.3 Analytical solutions

Here we select the following values of initial approximations and auxiliary linear operators:

0() = 1− − 0() = − (10.15)

L = 000 − 0 L = 00 − (10.16)

subject to the properties

L (1 +2 + 3

−) = 0L(4 + 5−) = 0 (10.17)

where ( = 1− 5) are the constants.If ∈ [0 1] indicates the embedding parameter, the zeroth order deformation problems are

constructed as follows:

(1− )Lh(; )− 0()

i= ~N

h(; )

i (10.18)

(1− )Lh(; )− 0()

i= ~N

h(; ) (; )

i (10.19)

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 (0; ) = 1 (∞; ) = 0 (10.20)

Here and are the non-zero auxiliary parameters and the nonlinear operators N and N

are given by

N [( )] =

µ1 +

1

¶Ã(1 + 2)

3( )

3+ 2

2( )

2

!+ ( )

2( )

2

−Ã( )

!2+ cos( )

161

N[( ) ( )] =

µ1 +

4

3

¶Ã(1 + 2)

2( )

2+ 2

( )

!

+

⎡⎣(1 + 2)⎛⎝Ã( )

!2+( )

2( )

2

!+2( )

( )

#

+Pr ( )( )

+Pr

µ1 +

1

¶(1 + 2)

Ã2( )

2

!2 (10.21)

For = 0 and = 1 the following equations are obtained:

(; 0) = 0() ( 0) = 0() (10.22)

(; 1) = () ( 1) = ()



= 1 and thus

() = 0() +∞P

=1

() () =1

!

(; )

¯=0

(10.23)

() = 0() +∞P

=1

() () =1

!

(; )

¯=0

(10.24)

The resulting problems at order can be presented in the following forms

L [()− −1()] = ~R () (10.25)

L[()− −1()] = ~R () (10.26)

(0) = 0(0) = 0(∞) = (0) = (∞) = 0 (10.27)

R () =

µ1 +

1

¶¡(1 + 2) 000−1 + 2

00−1

¢+

−1P=0

¡−1− 00 − 0−1−

0

¢+ cos−1 (10.28)

162

R () =

µ1 +

4

3

¶£(1 + 2) 00−1() + 2

0−1

¤+

∙(1 + 2)

−1P=0

¡0−1−

0 + −1−00

¢+ 2

−1P=0

−1−0

¸+Pr

−1P=0

0−1− +Prµ1 +

1

¶(1 + 2)

−1P=0

00−1−00 (10.29)

=

⎧⎨⎩ 0 ≤ 11 1

The general solutions ( ) consisting of the special solutions (∗

∗) are

() = ∗() + 1 + 2 + 3

− (10.30)

() = ∗() +4 + 5

− (10.31)

10.4 Convergence of the developed solutions

It is well recognized fact that the convergence of series solutions (1025) and (1026) depends

upon the non-zero auxiliary parameter ~. To find values of auxiliary parameters ensuring the

convergence we have plotted the ~-curves for velocity and temperature profiles. It is clear from

Figs. 101 and 102 that the admissible values of ~ and ~ are −126 ≤ ~ ≤ −024 and−11 ≤ ~ ≤ −045 respectively. These series solutions converge for the whole region of when~ = −07 and ~ = −07. Table 101 shows the convergence of homotopy solutions. It isobvious that 25th-order approximations are enough for the convergent series solutions.

163

Fig. 101. ~− curve for the velocity field.

Fig. 102. ~− curve for the temperature field.

164

Table 101: Convergence of series solutions for different order of approximations when

= 20 = = = 01 Pr = 10 = = 04 } = −07 and } = −07


5 069386 015647

10 069374 015914

15 069392 015925

20 069388 015919

25 069387 015918

30 069387 015918

35 069387 015918

40 069387 015918

45 069387 015918

10.5 Discussion

Our intention in this section is to analyze the velocity and temperature profiles for different

physical parameters Casson fluid parameter, curvature parameter, angle of inclination, mixed

convection parameter, Eckert number, Prandtl number, radiation parameter and variable ther-

mal conductivity parameter. The salient features of Casson fluid and curvature parameters are

displayed in the Figs. 103 and 104. It is observed that the velocity field is decreasing function

of . On the other hand, the velocity is found to increase for larger curvature parameter .

Also it shows that the rate of transport decreases with the increasing distance from the cylinder

and vanishes at some large distance from the cylinder. Fig. 105 exhibits the variation of angle

of inclination. As expected, an increase in the values of decreases the velocity. This is due

to the fact that the angle of inclination increases the influence of the buoyancy force due to

thermal decrease by the factor of cos(). Fig. 106 presents the effects of mixed convection

parameter on velocity profile. For mixed convection parameter, velocity decays very slowly

to the ambient but the fluid motion can infiltrate quite deeply into the ambient fluids. Interest-

ingly if is positive, it means heating of fluid or cooling of the boundary surface whereas for

165

negative values of means cooling of the fluid or heating of the boundary surface and when

= 0 it corresponds to the absence of free convection current. When 9 the temperature

field vanishes asymptotically. The effect of curvature parameter on the temperature profile

is displayed in Fig. 107. When = 0 the outer surface of the cylinder behaves like a flat

plate. Temperature profile increases most rapidly when curvature parameter increases. Also

for 0 ≤ ≤ 1 no variation can be observed for various values of Temperature and thermalboundary layer thickness decrease for large values of Pr. The fluid with higher values of Prandtl

number implies more viscous fluid and lower Prandtl number corresponds to less viscous fluid.

Fluids with higher viscosity have lower temperature and fluids with lower viscosity have higher

temperature. This leads to decrease in temperature and boundary layer thickness (see Fig.

108). Fig 109 illustrates the effect of on the temperature profile. Temperature profile is

larger when thermal conductivity varies linearly with the temperature whereas for constant

thermal conductivity ( = 0) the temperature profile decays. As we assume that the thermal

conductivity varies linearly with temperature which causes reduction in the magnitude of the

transverse velocity, therefore temperature profile increases with an increase in variable thermal

conductivity parameter. The enhancement in accelerates the temperature as well as bound-

ary layer thickness (see Fig. 1010). Because larger values of imply higher surface heat flux

and thereby making the fluid more warmer which enhances the temperature profile. Here we

noted that the temperature increases most rapidly in the region 0 ≤ ≤ 4 Fig. 1011 plotsthe temperature distribution versus the Eckert number . It is seen that fluid temperature

rises when the Eckert number becomes pronounced. Physically the Eckert number depends

on the kinetic energy. When we increase the values of Eckert number then the kinetic energy

enhances. This enhancement in the kinetic energy leads to an increase in the temperature and

thermal boundary layer thickness. In all cases the velocity is maximum at the surface of the

cylinder and it starts decreasing as →∞ Figs (1012− 1015) show the influences of , , and Pr on the temperature distribution, for the flat plate ( = 0) and stretching cylinder

( = 07) respectively. It is noticed that the temperature profile increases remarkably when

= 07 in comparison to flat plate when = 0 (see Figs. 1012 − 1014). The magnitude ofboundary layer thickness in case of stretching cylinder is larger than the flat plate. Fig. 1015

illustrates that for larger values of Pr the temperature profile decays. Also the thermal bound-

166

ary layer thickness reduces. It is also noted that the influence of Pr is much more prominent for

stretching flat plate when compared with stretching cylinder. Fluid with lower Prandtl number

possess higher thermal conductivity so that heat can diffuse from the cylinder faster than for

higher Pr fluid.

The magnitude of skin friction coefficient is enhanced when curvature parameter and

angle of inclination are increased (see Table 102). Table 103 shows the local Nusselt number

for different set of values of the involved parameters. The effects of and on local Nusselt

number are similar. The magnitude of local Nusselt number decreases for larger values of Eckert

number , radiation parameter and variable thermal conductivity parameter .


167



168



169



170



171



172



173

Table 102: Values of skin friction coefficient 12 for different parameters

−12

10 01 12347

15 11082

21 10310

20 00 09966

01 10409

02 10850

00 10064

6 10409

4 10824

6 01 12165

03 10976

06 09311

174

Table 103: Values of local Nusselt number for different emerging parameters.

−Re12

10 02 03 02 05276

14 05316

18 05336

20 00 05442

012 05336

019 05279

00 04375

02 04898

04 05331

00 06424

03 05446

07 04187

00 05739

02 05308

03 05123


An analysis in this attempt has been carried out for flow and heat transfer by a stretching

cylinder in presence of variable thermal conductivity and thermal radiation effects. The present

study indicated that due to increase in curvature parameter (), the velocity and temperature

are increased. Variable thermal conductivity enhances the temperature profile and associated

boundary layer thickness. The magnitude of temperature in case of stretching cylinder rises

when Eckert number (), radiation parameter () and variable thermal conductivity parame-

ter () are increased. Such magnitude of temperature is smaller for flat plate. The growth of

the boundary layers with distance from the leading edge for stretching cylinder is greater than

the flat plate. The local Nusselt number decreases when radiation () and variable thermal

conductivity parameters () are increased.

175

Chapter 11

MHD stagnation-point flow of

Jeffrey fluid over a convectively

heated stretching sheet

11.1 Introduction

This chapter deals with two-dimensional stagnation-point flow of Jeffrey fluid over an expo-

nentially stretching sheet. Convective boundary condition is used for the analysis of thermal

boundary layer. In addition the combined effects of thermal radiation and applied magnetic

field are taken into consideration. The developed nonlinear problems have been solved for the

series solution. The convergence of the series solutions is carefully analyzed. The behaviors of

various physical parameters such as viscoelastic parameter (3), magnetic field parameter (),

radiation parameter (), Biot number () and velocity ratio parameter (∗) are examined

through graphical and numerical results of velocity and temperature distributions.

11.2 Flow equations

The extra stress tensor for Jeffrey fluid is [41]

S =

1 +

∙A1 + 3

A1

¸ (11.1)

176

In above expressions is the dynamic viscosity, − is the indeterminate part of the stresstensor, is the ratio of relaxation to retardation times, 3 is the retardation time, A1 is the

first Rivlin-Erickson tensor, is the material derivative define as

=

+ (V5) (11.2)

in which Eq. (111) reduces to a Newtonian fluid when 1 = 2 = 0.

11.3 Problem formulation

We consider the steady two-dimensional MHD stagnation point flow of an incompressible Jeffrey

fluid by an exponentially stretching sheet. Constant magnetic field 0 is applied normal to the

plate. Induced magnetic field is assumed negligible in comparison to applied magnetic field

for small magnetic Reynolds number. In addition heat transfer analysis is considered with

radiation effects. Further we consider − parallel to the sheet and − normal to it

(see Fig. 111).

Fig. 11.1. Physical model and coordinate system.

177

The velocity and temperature fields subject to boundary layer approximations are governed by

the following equations

+

= 0 (11.3)

+

= 0

0

+

1 +

∙2

2+ 3

µ

2

+

3

2

−

2

2+

3

3

¶¸− 20

(− 0) (11.4)

µ

+

¶=

2

2−

(11.5)

= () = = 0 −

= ( − ) at = 0 (11.6)

= 0 () = = ∞ as →∞ (11.7)

where and represent the velocity components along the and −axes respectively, 0 isthe strain velocity, is the ratio of relaxation to retardation times, 3 is the retardation time,

is the stretching/shrinking velocity, is the fluid temperature, is the thermal diffusivity

of the fluid, = () is the kinematic viscosity, is the density of the fluid, is the thermal

conductivity of fluid, is the convective heat transfer coefficient and is the convective fluid

temperature below the moving sheet.

We define the following transformations as

=√2()2 () =

− ∞ − ∞

=

r

22 (11.8)

where and are constants and prime denotes the differentiation with respect to , is the

dimensionless stream function, is the dimensionless temperature and is the stream function

given by

=

= −

Now Eq. (113) is identically satisfied and (115− 118) yield

000 + (1 + )( 00 − 2 02 + 2) + 3(3

2 002 + 0 000 − 1

2 )− 2(1 + )2( 0 − 1) = 0 (11.9)

178

(1 +4

3)00 +Pr

¡0¢= 0 (11.10)

= 0 0 = ∗ = 0 = −[1− (0)] at = 0 (11.11)

0 = 1 = 0 at =∞ (11.12)

where 3 =3

is a local dimensionless parameter ∗ =

is a ratio parameter with ∗ 1

for assisting flow and ∗ 1 for opposing flow (i.e. when free stream velocity exceeds the

stretching velocity), =20

is the magnetic parameter, Pr =

is the Prandtl number,

= 4∗3∞

∗ is a radiation parameter and =

pis the Biot number. The skin friction

coefficient and local Nusselt number are

=122

=

( − ∞) (11.13)

where the skin friction and the heat transfer from the plate are

=

1 +

∙µ

¶+ 3

½2

+

2

2+

2

2

¾¸=0

= −µ1 +

16∗ 3∞3∗

¶µ

¶=0


= (2Re)−12 ∗−2

∙1

1 +

½ 00(0) + 3

µ3

2 00(0)

¶¾¸ (11.14)

12 = −(1 + 43)0(0) (11.15)


The definitions of initial guesses and auxiliary linear operators are

0() = (∗ − 1) + + (1− ∗)− (11.16)

0() = exp(−)1 +

(11.17)

179

L = 000 − 0 (11.18)

L = 00 − (11.19)

L (1 + 2 + 3

−) = 0 (11.20)

L(4 + 5−) = 0 (11.21)

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

(1− )Lh(; )− 0()

i= N

h(; )

i (11.22)

(1− )Lh(; )− 0()

i= N

h(; ) ( )

i (11.23)

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

(11.24)

N [( )] =3( )

3+ (1 + )

⎡⎣( )2( )2

− 2Ã( )

!2+ 2

⎤⎦+3

⎡⎣32

Ã2( )

2

!2− 12

Ã( )

4( )

4

!+

( )

3( )

3

⎤⎦−2(1 + )2

Ã( )

− 1! (11.25)

N[( ) ( )] =

µ1 +

4

3

¶2( )

2+Pr

Ã( )

( )

! (11.26)

When = 0 and = 1 then one has

(; 0) = 0() (; 0) = 0() and (; 1) = () (; 1) = () (11.27)

and when variation is taken from 0 to 1 then ( ) and ( ) approach 0() 0() to ()

and () Now and in Taylor’s series can be expanded in the series

( ) = 0() +∞P

=1

()

180

( ) = 0() +∞P

=1

() (11.28)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(11.29)

where the convergence depends upon and By proper choices of and the series

(1128) and (1129) converge for = 1 and so

() = 0() +∞P

=1

() (11.30)

() = 0() +∞P

=1

() (11.31)


L [()− −1()] = R () (11.32)

L[()− −1()] = R () (11.33)

(0) = 0(0) = 0(∞) = 0 0(0)−(0) = (∞) = 0 (11.34)

R () = 000−1() + (1 + )

−1P=0

£−1− 00 − 2 0−1− 0 + 2

¤+3

−1X=0

(3

2 00−1−

00 −

1

2−1− + 0−1−

000 )

−22(1 + )( 0−1 − 1) (11.35)

R () = (1 +

4

3)00−1 +Pr

−1P=0

0−1− (11.36)

=

⎧⎨⎩ 0 ≤ 11 1

The general solutions of Eqs. (1132) and (1133) can be expressed by the following equations

() = ∗() + 1 + 2 + 3

− (11.37)

181

() = ∗() +4 + 5

− (11.38)

where ∗ and ∗ are the particular solutions and the constants ( = 1−5) can be determined

by the boundary conditions (1134). These are given by

2 = 4 = 0 3 =∗()

¯=0

1 = −3 − ∗(0)

5 =1

+ 1

"∗()

¯=0

−∗(0)

#

11.5 Convergence of the series solutions

As pointed out by Liao [96], the non-zero auxiliary parameter ~ can adjust and accelerate the

convergence of the series solutions. To obtain an appropriate value of such parameters the

so-called ~−curves have been plotted (see Figs. 112 and 113). The admissible ranges of ~can be obtained from the line segment parallel to ~−axis. It is found that the interval ofconvergence is [−12−05]. The range for ~-curve shrinks as 3 is increased. This is becauselarger values of 3 yields strongly nonlinear differential equation. Obviously one requires a

higher order approximation for reasonably convergent series solution. Table 111 guarantees

the convergence of homotopy solutions. It is clear that 20-order approximations are sufficient

for convergent series solution.

182

Fig. 112. The ~−curves of 00(0) and 0(0)

Fig. 113. The ~−curves of 00(0) and 0(0)

183

Table 111: Convergence of HAM solution for different order of approximations when =

02, ∗ = 09, 3 = 02 = 01Pr = 1 = 03 and = 02

Order of approximation 00(0) 0(0)

1 0164878 02274

5 0208556 02179

10 0209506 02117

20 0209480 02085

25 0209472 02061

30 0209471 02058

35 0209471 02058

40 0209471 02058

50 0209471 02058

Table 112: Convergence of HAM solutions for different order of approximations when = 02,

∗ = 12, 3 = 03 = 01Pr = 1 = 03 and = 01

Order of approximation − 00(0) 0(0)

1 02754 02281

5 03694 02206

10 03614 02158

15 03823 02136

20 03822 02126

30 03822 02124

35 03822 02124


The presented series solutions can be easily used to explore the effects of various interesting

parameters on the velocity and temperature distributions.

184

11.6.1 Dimensionless velocity profile

The parameter measures the effect of applied magnetic field. It can take values in the range

0 ∞. The larger , the stronger will be magnetic field strength. Figs. 114 and 115indicate that presence of magnetic field creates a bulk known as Lorentz force which causes a

huge restriction to the fluid velocity and, therefore decreases the momentum boundary layer .

The parameter ∗ compares the ratio of the stretching sheet velocity to the velocity of external

stream. The larger values of ∗ indicates larger stretching sheet velocity compared to the free

stream velocity. It is clear from Figs. 116 and 117 that velocity field 0 is a strong function of

ratio parameter ∗ and it increases with an increase in ∗ The boundary layer layer thickness

decreases with an increase in ∗ when ∗ 1. The case ∗ 1 represents the situation when

velocity of the stretching sheet () exceeds the external stream velocity ∞() and the flow

has an inverted boundary layer structure. Moreover the boundary layer is not formed when

∗ = 1. Fig. 118 shows that velocity decreases and profiles move away from the bounding

surface indicating an increase in the momentum boundary layer when ∗ 1 On the other

hand the boundary layer thins when 3 is increased and ∗ 1 This outcome is similar to

that accounted in the case of second grade fluid. Irrespective of the chosen value of ∗ the

horizontal velocity increases by increasing the ratio The boundary layer thickness decreases

when ∗ 1 and it increases for ∗ 1 (see Fig. 119)

11.6.2 Dimensionless temperature profile

Fig. 1110 displays the effect of Prandtl number Pr for a fixed value of convective heating

parameter (Biot number) . Specifically, Prandtl number Pr = 072 10 and 70 correspond

to air, electrolyte solution such as salt water and water respectively. We have arbitrarily selected

Pr = 1 to retrieve the graphical results. It is seen that temperature is a decreasing function

of Pr. Physically a larger Prandtl number fluid has a relatively lower thermal conductivity.

Thus an increase in Pr decreases conduction and thereby increases the variations of thermal

characteristics. This results in the reduction of the thermal boundary layer thickness and

an increase in the heat transfer rate at the bounding surface. As expected, the temperature

significantly rises when the thermal radiation effect strengthens. The thermal boundary layer

also grows when is increased (see Fig. 1111). Fig. 1112 plots the temperature distribution

185

versus the ratio parameter ∗ It is seen that fluid’s temperature rises when the ratio parameter

∗ becomes pronounced. The variation of temperature with an increase in can be seen

in Fig. 1113. Here a gradual increase in results in the larger convection at the stretching

sheet which rises the temperature.

11.6.3 Dimensionless skin friction coefficient

The numerical values of dimensionless skin friction coefficient for various parametric values have

been given in the Table 113. We noticed that skin friction coefficient increases with an increase

in while it decreases when ∗ and 3 are increased. In other words the power generation

involved in stretching the sheet can be reduced by considering the fluids which display strong

viscoelastic effects.

11.6.4 Dimensionless heat transfer rate

In Table 114 the values of dimensionless heat transfer rate at the boundary are computed.

We have already witnessed in the graphical results that profiles become increasingly steeper

by increasing Pr and Thus local Nusselt number 0(0), being proportional to the slope of

temperature at = 0, increases with an increase in Pr and It is found that¯0(0)

¯also

increases with an increase in ∗ This outcome leads to the conclusion that heat transfer rate

at the sheet is enhanced by increasing the velocity of the stretching sheet. A slight variation

in¯0(0)

¯is noticed by increasing the viscoelastic parameter 3 It can be clearly seen from

Fig. 1111 that slope of temperature function continuously decreases when the radiation effect

intensifies. Thus magnitude of local Nusselt number increases when is increased.

186

Fig. 11.4. Influence of M on 0() (for opposing flow).

Fig. 11.5. Influence of M on 0() (for assiting flow).

187

Fig. 11.6. Influence of ∗ on 0() (for opposing flow).

Fig. 11.7. Influence of ∗ on 0() (for assiting flow).

188

Fig. 11.8. Influence of 3 on 0()


189



190

Fig. 11.12. Influence of ∗ on ()


191

Table 113: Values of skin friction coefficient 12 for the parameters ∗ and 3

∗ 3 12

00 09 05 02 015065

03 015271

07 016085

05 00 17912

04 070973

07 037457

00 016105

03 013585

06 011776

00 020090

03 015158

05 013125

192

Table 114: Values of local Nusselt number 12 for the parameters , ∗, , Pr,

and 3

∗ Pr 3 12

02 09 01 10 03 02 02 01532

05 02834

07 03381

04 01986

09 02044

12 02095

00 02059

04 02058

09 02059

08 02033

10 02058

12 02103

00 02059

04 02057

08 02056

00 02057

05 02058

08 02059

00 02119

02 02044

04 02009


Heat transfer characteristics in the MHD stagnation-point flow of Jeffrey fluid toward a stretch-

ing surface with convective condition are studied. The convergent expressions of velocity and

temperature are constructed. It is seen that for sufficiently large Biot number the analysis for

constant wall temperature case can be recovered. The velocity ratio (∗) has a dual behavior

193

on the momentum boundary layer and the skin friction coefficient. We found that temperature

decreases with an increase in Pr. This decrease is associated with the larger rate of heat

transfer at the sheet. Surface heat transfer rate also enhances by increasing convective heating

at the sheet. The present findings for the case of Newtonian fluid can be recovered by choosing

3 = 0.

194

Chapter 12

Radiative flow of Jeffrey fluid

through a convectively heated

stretching cylinder

12.1 Introduction

This chapter studied the two-dimensional flow of Jeffrey fluid by an inclined stretching cylinder

with convective boundary condition. In addition the combined effects of thermal radiation and

viscous dissipation are taken into consideration. The developed nonlinear partial differential

equations are reduced into the ordinary differential equations. The governing equations are

solved for the series solutions. The convergence of the series solutions for velocity and temper-

ature fields are carefully analyzed. The effects of various physical parameters such as ratio of

relaxation to retardation times, Deborah number, radiation parameter, Biot number, curvature

parameter, mixed convection parameter, Prandtl number, Eckert number and angle of incli-

nation are examined through graphical and numerical results of the velocity and temperature

distributions.

195

12.2 Flow equations

We consider the steady two-dimensional flow of an incompressible Jeffrey fluid by an inclined

stretching cylinder. We also considered thermal radiation and viscous dissipation effects. The

convective boundary condition for a cylinder is employed. The stretching cylinder makes an

angle with vertical axis i.e. r-axis. Here x-axis is taken normal to the r-axis.

The relevant boundary layer equations are

()

+

()

= 0 (12.1)

+

=

1 +

∙2

2+1

+ 3

µ

2

+

2

2

+

2

+

3

2+

2

2+

3

3

¶¸+ 0 ( − ∞) cos(12.2)

µ

+

¶=

µ

¶+16∗ 3∞3∗

µ1

+

2

2

¶+

1 +

∙2

2+ 3

½

2

+

2

2

¾¸ (12.3)

= = 0

= 0 −

= ( − ) at = (12.4)

→ 0 → ∞ as →∞ (12.5)

in which and represent the velocity components along the and directions respectively, 0

is the reference velocity, is the ratio of relaxation to retardation times, 3 is the retardation

time, is the characteristic length, is the acceleration due to gravity, is the volumetric

coefficient of thermal exponential, is the fluid temperature, is the thermal diffusivity of the

fluid, = () is the kinematic viscosity, is the density of fluid, is the thermal conductivity

of fluid, ∗ is the mean absorption coefficient, is the convective heat transfer coefficient and

is the convective fluid temperature. Note that the thermal radiation term in Eq. (123) is

written using Rosselands’ approximation. It should be pointed out that convection condition in

Eq. (124) represents the heat transfer rate through the surface being proportional to the local

196

difference in temperature with the ambient conditions. Such configuration occurs in several

engineering devices for instance in heat exchangers, where the conduction in the solid tube wall

is greatly influenced by the condition in the fluid flowing over it.

We introduce the similarity transformations in the forms

=0

0() = −

µ0

¶12(),

() = − ∞ − ∞

=2 − 2

2

µ0

¶12 (12.6)

With the help of above transformations, Eq. (121) is identically satisfied and Eqs. (122− 125)yield

(1 + 2) 000 + 2 00 + (1 + )( 00 − 02) + 3(0 00 − 3 000)

+3(1 + 2)(002 − ) + cos = 0 (12.7)

(1 +4

3)¡(1 + 2) 00 + 20

¢+Pr 0 +Pr

1

1 +

£(1 + 2) 002

+3©(1 + 2)

¡ 00 − 00 000

¢− 00ª¤= 0 (12.8)

= 0 0 = 1 0 = −[1− (0)] at = 0

0 = 0 = 0 at =∞ (12.9)

where 3 =10is a Deborah number =

q

20is a curvature parameter, =

is the

Prandtl number, =4∗ 3∞∗ is a radiation parameter, =

20 ()2

(−∞)is the Eckert number,

=0 (−∞)3

20 ()22

is the mixed convection parameter and =

q0is the Biot number.

The skin friction coefficient and local Nusselt number are

=122

=

( − ∞) (12.10)

197

where and are

=

1 +

∙µ

¶+ 3

½2

+

2

2

¾¸=

= −µ

¶=

− 16∗ 3∞3∗

µ

¶=


= 2 (Re)−12

∙1

1 + (1 + 3)

00(0)¸ 12 = −(1 + 4

3)0(0) (12.11)


Initial approximations and auxiliary linear operators are chosen as follows:

0() = 1− − (12.12)

0() = exp(−)1 +

(12.13)

L = 000 − 0 (12.14)

L = 00 − (12.15)

The operators satisfy the following properties

L (1 + 2 + 3

−) = 0 (12.16)

L(4 + 5−) = 0 (12.17)

where i( = 1 − 5) are the arbitrary constants determined from the boundary conditions. If

∈ [0 1] denotes an embedding parameter, and the non-zero auxiliary parameters then

the zeroth order deformation problems are

(1− )Lh(; )− 0()

i= N

h(; )

i (12.18)

(1− )Lh(; )− 0()

i= N

h(; ) ( )

i (12.19)

198

(0; ) = 0 0(0; ) = 1 0(∞; ) = 0 0(0; ) = −[1− (0; )] (∞; ) = 0 (12.20)

N [( )] = (1 + 2)3( )

3+ 2

2( )

2+ (1 + )

⎡⎣( )2( )2

−Ã( )

!2⎤⎦+3

"( )

2( )

2− 3( )

3( )

3

#

+3 (1 + 2)

Ã2( )

2− ( )

4( )

4

!+ cos (12.21)

N[( ) ( )] =

µ1 +

4

3

¶Ã(1 + 2)

2( )

2+ 2

( )

!+Pr

Ã( )

( )

!

+Pr1

1 +

"(1 + 2)

2( )

2+ 3

((1 + 2)

Ã( )

2( )

2

−( )2( )

23( )

3

!)− ( )

2( )

2

# (12.22)

When = 0 and = 1 then one can easily write

(; 0) = 0() ( 0) = 0() and (; 1) = () ( 1) = () (12.23)

When variation is taken from 0 to 1 then ( ) and ( ) approach 0() 0() to () and

() Now and in Taylor’s series can be expanded as follows:

( ) = 0() +∞P

=1

() (12.24)

( ) = 0() +∞P

=1

() (12.25)

() =1

!

(; )

¯=0

() =1

!

(; )

¯=0

(12.26)

199

where the convergence depends upon and By proper choices of and the series

(1229) and (1230) converge for = 1 and hence

() = 0() +∞P

=1

() (12.27)

() = 0() +∞P

=1

() (12.28)


L [()− −1()] = R () (12.29)

L[()− −1()] = R () (12.30)

(0) = 0(0) = 0(∞) = 0 0(0)−(0) = (∞) = 0 (12.31)

R () = (1 + 2) 000−1() + 2

00−1 − (1 + )

−1P=0

£−1− 00 − 0−1−

0

¤+3

−1X=0

( 0−1−00 − 3−1− 000 )

+3 (1 + 2)

−1X=0

( 00−1−00 − −1− 0000 ) +−1 cos (12.32)

R () = (1 +

4

3)¡(1 + 2) 00−1 + 2

0−1

¢+Pr

µ−1P=0

0−1−

¶+Pr

1

1 +

"(1 + 2)

−1X=0

( 00−1−00 + 3

((1 + 2)

−1X=0

( 00−1−00

−−1−X=0

0−000 )

)−

−1X=0

(−1− 00

# (12.33)

=

⎧⎨⎩ 0 ≤ 11 1

200

The general solutions of Eqs. (1233) and (1234) are given by

() = ∗() + 1 + 2 + 3

− (12.34)

() = ∗() +4 + 5

− (12.35)

where ∗ and ∗ are the particular solutions.

12.4 Convergence analysis

Convergence of series solution strongly depends upon the non-zero auxiliary parameters } and

} For this purpose we plot the } − curves for the velocity 00(0) and temperature 0(0) fields.Figs. 121 and 122 depict the } − curves for different values of physical parameters. Here

admissible range of } thus is −13 } −01 and for } it is −12 } −05. Theauxiliary parameter } can be regarded as an iteration factor that is usually used in numerical

computations. It is well known that a properly chosen iteration factor can ensure the conver-

gence of iteration. In fact the mentioned convergence interval are the values for which the plots

of physical quantities 00(0) and 0(0) are parallel to horizontal axis [96]. It is found that the

series solutions converge in the whole region of when } = −11 and } = −09. Table 121guarantees the convergence of HAM solutions. It is obvious that 25th-order approximations are

enough for the convergent series solutions.

201

Fig. 121. ~− curve for the velocity field.

Fig. 122. ~− curve for the temperature field.

202

Table 121 : Convergence of HAM solution for different order of approximations when 3 =

02 = 03 = 6, = 01, Pr = 1, = = 01, = 02 and = 03.


5 10150 01100

10 10125 01173

15 10120 01183

20 10119 01185

25 10118 01186

30 10118 01186

40 10118 01186

45 10118 01186

12.5 Result and discussion

Here the nonlinear analysis is computed for the homotopy solutions. The presented analytically

solutions can be easily used to explore the influence of various physical parameters on the

velocity and temperature fields.

12.5.1 Dimensionless velocity profile

The velocity field is illustrated in Fig. 123 for different values of (ratio of relaxation to

retardation times). The horizontal velocity decreases for large values of with the increasing

distance () of the cylinder. It is because of the fact that being the viscoelastic parameter

exhibits both viscous and elastic characteristics. Thus the fluids always retard whenever vis-

cosity or elasticity increases. The velocity increases with increasing values of 3 (see Fig. 124).

As in the case of polymers, a high Deborah number means the flow is strong enough that the

polymer becomes highly oriented in one direction and stretched. For most flows this occurs

when the time it takes for the polymer to relax is long compared to the rate at which the flow

is deforming it. A very small Deborah number can be obtained for a fluid with extremely small

relaxation time if the Deborah number is very large then fluid behaves like a solid. With the

203

increasing the horizontal velocity is found to decrease because the effect of buoyancy force

increases due to thermal decrease (see Fig. 125). The graph of mixed convection parameter

on the velocity field 0() is displayed in Fig. 126. It is clearly seen from this Fig. that

for large values of shows that the velocity field increases. The positive values of cool the

wall or heat the fluid. When 1, the viscous force is negligible when compared to the

buoyancy and inertial forces. When buoyant forces overcome the viscous forces, the flow starts

a transition to the turbulent regime. In Fig. 127 the horizontal velocity profile is shown for

different values of curvature parameter The effect of curvature parameter near the cylinder

wall is negligible but far away it increases for larger values of This is due to the fact that

larger values of curvature parameter leads to increase the radius of cylinder.

12.5.2 Dimensionless temperature profile

Fig. 128 depicts the influence of on temperature field () Here temperature field increases

rapidly for larger values of (ratio of relaxation to retardation times). Fig. 129 depicts the

variations of Deborah number 3 on The temperature profile is decreasing function of 3.

In fact small Deborah numbers correspond to situations where the material has time to relax

while high Deborah numbers represents the cases where the material behaves rather elastic. Fig.

1210 illustrates the effect of curvature parameter on temperature field. Through an increase

in the temperature increases and maximum temperature is found near the wall. Figs. 1211

and 1212 show the effect of Prandtl number Pr and Eckert number Ec on temperature field.

The temperature () and thermal boundary layer thickness are decreased via increase in Pr.

The higher Prandtl number fluid has a lower thermal conductivity (or a higher viscosity) which

results in thinner thermal boundary layer and hence higher heat transfer rate at the surface.

Whereas opposite behavior is noted for Eckert number. Physically this is due to the reason that

the fluid takes the kinetic energy from the motion of the fluid and transforms it into internal

energy. Therefore the fluid temperature rises. Obviously this process is more pronounced when

the fluid is more viscous or it is flowing at high speed. Fig. 1213 demonstrates the effect of

thermal radiation parameter on temperature field () It is noted that thermal radiation

increases the temperature field and associated thermal boundary layer thickness. Physically the

larger values of imply larger surface heat flux and thereby the fluid become warmer which

204

enhances the temperature profile. Also the effect of thermal radiation in thermal boundary layer

is equivalent to that an increased thermal diffusivity in Prandtl number. Fig 1214 depicts the

effect of Biot number on () Biot number is a ratio of the temperature fall in the solid

material and the temperature fall the solid and the fluid. So for smaller Biot number, most

of the temperature drop is in the fluid and the solid may be considered isothermal. Therefore

Biot number enhances both the temperature field and thermal boundary layer thickness. Here

we considered the Biot number less than one due to uniformity of the temperature field in the

fluid. A lower Biot number means that the conductive heat transfer is much faster than the

convective heat transfer.

12.5.3 Dimensionless skin friction coefficient

Table 122 shows the numerical values of dimensionless skin friction coefficient for various phys-

ical parameters. We notice that skin friction coefficient decreases with an increase in and

3We see that when increases then the viscous forces start decreasing and consequently the

skin friction coefficient increases. Skin friction coefficient increases for large values of angle of

inclination

12.5.4 Dimensionless Nusselt number

In Table 123 the values of dimensionless Nusselt number are computed. It is seen that Nusselt

number is positive therefore heat transfer is from hot fluid to the sheet. Fluid parameters

and 3 increase the local Nusselt number at the wall Thus local Nusselt number increases with

an increase in Pr and . Hence Prandtl number can be used to adjust the rate of cooling in

conducting flows. It is found that Nusselt number also increases when is increased A slight

variation in local Nusselt number is noticed by increasing the mixed convection parameter

It is found that slope of temperature function continuously increases when the convective heat

transfer intensifies. Thus local Nusselt number enhances when is increased. As curvature

parameter increases, the local Nusselt number is increased.

205


Fig. 12.4. Influence of 3 on 0()

206



207



208

Fig. 12.9. Influence of 3 on ()


209



210



211

Table 122: Values of skin friction coefficient 12 for the parameters 3 and

3 −12

00 02 6 01 11390

03 099430

05 092350

02 00 094337

01 099085

03 10800

00 10037

9 10364

6 10363

01 10363

015 10543

02 10721

212

Table 123 : Values of local Nusselt number 12 for the parameters 3 = 02 = 03

= 6, = 01 Pr = 1 and = = 01

3 Pr 12

00 02 6 01 01 10 01 01788

04 01854

08 01878

02 00 01786

03 01856

04 01882

02 00 01831

6 01832

4 01830

6 00 01824

02 01835

04 01840

00 01955

01 01832

02 01712

10 01832

15 01909

20 02025

00 01587

01 01832

213

12.6 Final remarks

Heat transfer characteristics in the flow of Jeffrey fluid with convective boundary condition

are explored. The effects of thermal radiation and viscous dissipation are taken into account.

It is seen that the curvature of stretching cylinder is essential parameter for the velocity and

temperature fields. Curvature parameter enhances the velocity as well as the temperature of

fluid. Fluid parameters (ratio of relaxation to retardation times) and 3 (Deborah number)

have opposite behavior on the velocity and temperature fields. The local Nusselt number

increases for sufficiently large Biot number . Radiation parameter and Eckert number

also accelerate the temperature field. Thermal boundary layer thickness reduces for larger

values of Prandtl number (Pr).

214

Bibliography

[1] B. C. Sakiadis, Boundary layer behavior on continuous solid surface: I. Boundary-layer

equations for two-dimensional and axisymmetric flow. AIChE 7 (1961) 26− 28

[2] L. J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970) 645− 647

[3] P. D. Ariel, MHD flow of viscoelastic fluid past a stretching sheet with suction, Acta

Mech. 105 (1994) 49− 56

[4] E. Magyari and B. Keller, Heat and mass transfer in the boundary on an exponentially

stretching continuous surface, J. Phys. D. Appl. Phys. 32 (1999) 577− 585

[5] G. Degan, H. Beji, P. Vasseur and L. Robillaxd, Effect of anisotropy on the development

of convective boundary layer flow in porous media, Int. Comm. Heat Mass Transfer 25

(1998) 1159− 1168

[6] B. Bidin and R. Nazar, Numerical solution of the boundary layer flow over an exponen-

tially stretching sheet with thermal radiation, Eurp. J. Sci. Res. 33 (2009) 710− 717

[7] J. B. McLeod and K. R. Rajagopal, On the uniqueness of flow of Navier-Stokes fluid due

to a stretching boundary. Arch. Rat. Mech. Anal. 98 (1987) 385− 393

[8] C. Y. Wang, Stretching surface in a rotating fluid. J. Appl. Phys. 39 (1988) 177− 185

[9] H. I. Andersson, Slip flow past a stretching surface. Acta Mech. 158 (2002) 121− 125

[10] P. D. Ariel, Axisymmetric flow due to a stretching sheet with partial slip. Comput. Math.

Appl. 54 (2007) 1169− 1183

215

[11] C. Jie Liu and Ya-G. Wang, Derivation of Prandtl boundary layer equations for the

incompressible Navier-Stokes equations in a curved domain. Appl. Math. Lett. 34 (2014)

81− 85

[12] T. Hayat, S. Asad, M. Mustafa and A. Alsaedi, Boundary layer flow of Carreau fluid over

a convectively heated stretching sheet. Appl. Math. Comput. 246 (2014) 12− 22

[13] H. Rosali, A. Ishak, R.Nazar, J. H. Merkin and I. Pop, The effect of unsteadiness on mixed

convection boundary-layer stagnation-point flow over a vertical flat surface embedded in

a porous medium. Int. J. Heat Mass Transfer 77 (2014) 147− 156

[14] N. C. Rosca and I. Pop, Boundary layer flow past a permeable shrink sheet in a micropalor

fluid with a second order slip flow model. European J. Mech. B/Fluids 48 (2014) 115−122

[15] M. C. Raju, N. A. Reddy and S. V. K. Varma, Analytical study of MHD free convection,

dissipative boundary layer flow past a porous vertical surface in the presence of thermal

radiation, chemical reaction and constant suction. Ain Shams Eng. J. (2014) (in press).

[16] Z. Zhang and C. Zhang, Similarity solutions of a boundary layer problem with a negative

parameter arising in steady two-dimensional flow for power-law fluids. Nonlinear Analysis

102 (2014) 1− 13

[17] M. Awais, T. Hayat, A. Alsaedi and S. Asghar, Time-dependent three-dimensional bound-

ary layer flow of a Maxwell fluid. Computer & Fluids 91 (2014) 21− 27

[18] N. Freidoonimehr, M. M. Rashidi and S. Mahmud, Unsteady MHD free convective flow

past a permeable stretching vertical surface in a nano-fluid. Int. J. Thermal Sci. 87 (2015)

136− 145

[19] M. Sheikholesalami, D. D. Ganji, M. Y. Javed and R. Ellahi, Effect of radiation on

magnetohydrodynamic nanofluid flow and heat transfer by means of two phase model. J.

Magnetism and Magnetic Materials 374 (2015) 36− 43

[20] M. Sheikholesalami and D. D. Ganji, Nanofluid flow and heat transfer between parallel

plates considering Brownian motion using DTM, Comput. Methods Appl. Mech. Eng.

283 (2015) 651− 663

216

[21] R. E. Powell and H. Eyring, Nature (1944) London.

[22] D. W. Beard and K. Walters: Elasto-viscous boundary layer flow. Proc. Camb. Philos.

Soc. 60 (1964) 667− 674

[23] V. M. Soundalgekar, Flow of an elastico-viscous fluid past an oscillating plate. Czechoslo-

vak J. Phy. B 28 (1978) 1217− 1220

[24] J. Harris, Rheology and non-Newtonian flow. Longman (1977)

[25] R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of Polymeric

liquids. Wiley (1987)

[26] K. R. Rajagopal, T. Y. Na and A. S. Gupta, Flow of a viscoelastic fluid over a stretching

sheet, Rheo. Acta 23 (1984) 213− 215

[27] V. V. Shelukhin, Existence of periodic solutions of generalized Burger system:, J. Appl.

Math. Mech. 43 (1979) 1073− 1079

[28] M. Nakamura and T. Sawada, Numerical study on the flow of a non-Newtonian fluid

through an axisymmetric stenosis. ASME J. Biomechanical Eng. 110 (1988) 137− 143

[29] N. T. M. Eldabe, A. A. Hassan and M. A. A. Mohamed, Effect of couple stresses on the

MHD of a non-Newtonian unsteady flow between two parallel porous plates. Z. Natur-

forsch 58 (2003) 204− 210.

[30] A. Ishak, R. Nazar and I. Pop, Heat transfer over a stretching surface with variable heat

flux in micropolar fluid. Phys. Lett. A 372 (2008) 559− 561

[31] T. Hayat, Z. Iqbal, M. Qasim and S. Obaidat, Steady flow of an EyringPowell fluid over a

moving surface with convective boundary conditions. Int. J. Heat Mass Transfer 55 (2012)

1817− 1822

[32] O. Ullah Mehmood, N. Mustupha and S. Shafie, Heat transfer on peristaltically induced

Walters’ B fluid. Heat Transfer 41 (2012) 690− 699

[33] T. Hayat, S. Asad, M. Qasim and Awatif A. Hendi, Boundary layer flow of a Jeffrey fluid

with convective boundary conditions. Int. J. Num. Meth. Fluids 69 (2012) 1350− 1362

217

[34] T. Hayat, S. A. Shehzad, A. Alsaedi and M. S. Alhothuali, Mixed convection stagnation

point flow of Casson fluid with convective boundary conditions. Chin. Phys. Lett. 29

(2012) 114704.

[35] S. Mukhopadhyay, P. Ranjan De, K. Bhattacharyya and G. C. Layek, Casson fluid flow

over an unsteady stretching surface. Ain Shams Eng. J. 4 (2013) 933− 938

[36] A. Mushtaq, M. Mustafa, T. Hayat, M. Rahi and A. Alsaedi, Exponentially stretching

sheet in a Powell-Eyring fluid: Numerical and series solutions. Z. Naturforsch 68 (2013)

791− 798

[37] M. M. Khader and A. M. Megahed, Numerical studies for flow and heat transfer of

the Powell-Eyring fluid thin film over an unsteady stretching sheet with internal heat

generation using the chebyshev finite difference method. J. Applied Mechanics Technical

Phys. 54 (2013) 440− 450

[38] H. Zaman, Unsteady incompressible Couette flow problem for the Eyring-Powell model

with porous walls. American J. Computational Math. 3 (2013) 313− 325

[39] T. Hayat, M. Farooq, Z. Iqbal and Awatif A. Hendi, Stretched flow of Casson fluid with

variable thermal conductivity, J. Sci. Tech. 10 (2013) 181− 190

[40] A. Shafiq, M. Nawaz, T. Hayat and A. Alsaedi, Magnetohydrodynamic axisymmetric

flow of a third-grade uid between two porous disks. Brazilian J. Chem. Eng. 30 (2013)

599− 609

[41] M. Turkyilmazoglu and I. Pop, Exact analytical solutions for the flow and heat transfer

near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid. Int. J. Heat

and Mass Transfer 57 (2013) 82− 88.

[42] T. Hayat, A. Shafiq and A. Alsaedi, Effect of Joule heating and thermal radiation in flow

of third-grade fluid over radiative surface. PLOS One 9 (2014) 83153

[43] V. Marinca and N. Herisanu, On the flow of a Walters-type B’ viscoelastic fluid in a

vertical channel with porous wall. Int. J. Heat Mass Transfer 79 (2014) 146− 165

218

[44] M. Turkyilmazoglu, Exact solutions for two dimensional laminar flow over a continuously

stretching or shrinking sheet in an electrically conducting quiescent couple stress fluid.

Int. J. Heat Mass Transfer 72 (2014) 1− 8

[45] S. Asad, A. Alsaedi and T. Hayat, Flow of couple stress fluid with variable thermal

conductivity. Appl. Math. and Mech. (in press).

[46] A. M. Abd-Alla and S. M. Abo-Dahab, Magnetic field and rotation effects on peristaltic

transport of a Jeffrey fluid in an asymmetric channel. J. Magnetism Magnetic Materials

374 (2015) 680− 689

[47] M. Awais, T. Hayat and A. Alsaedi, Investigation of heat transfer in flow of Burgers’ fluid

during a melting process. J. Egyptian Math. Society (2014) (in press).

[48] A. Borrelli, G. Giantesio and M.C. Patria, An exact solution for the 3D MHD stagnation-

point flow of a micropolar fluid. Commun. Nonlinear Sci. Numerical Simul. 20 (2015)

121− 135

[49] M. Javed, T. Hayat and A. Alsaedi, Peristaltic flow of Burgers’ fluid with compliant walls

and heat transfer. App. Math. Comput. 244 (2014) 654− 671

[50] S. Pramanik, Casson fluid flow and heat transfer past an exponentially porous stretching

surface in presence of thermal radiation. Ain Shams Eng. J. 5 (2014) 205− 212

[51] W. M. Kay, Convective heat and mass transfer. Mc-Graw Hill, New York (1966).

[52] T. C. Chiam, Heat transfer with variable conductivity in stagnation-point flow towards a

stretching sheet. Int. Commun. Heat Mass Transfer 23 (1966) 239− 248

[53] R. Tsai, K. H. Huang and J. S. Huang, Flow and heat transfer over an unsteady stretching

surface with non-uniform heat source. Int. Commun. Heat Mass Transfer 35 (2008) 1340−1343

[54] O. O. Anselm and Koriko O. K., Thermal conductivity and its effects on compressible

boundary layer flow over a circular cylinder. IJRRAS 15 (2013) 1− 12

219

[55] M. Turkyilmazoglu and I. Pop, Heat and mass transfer of unsteady natural convection

flow of some nanofluids past a vertical infinite flat plate with radiation effect. Int. J. Heat

Mass Transfer 59 (2013) 167− 171

[56] X. Su, L. Zheng, X. Zhang and J. Zhang, MHD mixed convective heat transfer over a

permeable stretching wedge with thermal radiation and Ohmic heating. Chem. Eng. Sci.

78 (2012) 1− 8

[57] K. Bhattacharyya, S. Mukhopadhyay, G. C. Layek and I. Pop, Effects of thermal radiation

on micropolar fluid flow and heat transfer over a porous shrinking sheet. Int. J. Heat Mass

Transfer 55 (2012) 2945− 2952

[58] R. Prasad, B. Vasu, O. A. Bég and R. D. Parshad, Thermal radiation effects on magneto-

hydrodynamic free convection heat and mass transfer from a sphere in a variable porosity

regime. Commun. Nonlinear Sci. Numer Simul. 17 (2012) 654− 671

[59] K. Das, Radiation and melting effects on MHD boundary layer flow over a moving surface.

Ain Shams Eng. J. 5 (2014) 1207− 1214

[60] S. Siddiqa, M. A. Hossain and S. C. Saha, The effect of thermal radiation on the natural

convection boundary layer flow over a wavy horizontal surface. Int. J. Thermal Sci. 84

(2014) 143150

[61] A. Ara, N. A. Khan, H. Khan and F. Sultan, Radiation effect on boundary layer flow of

an Eyring—Powell fluid over an exponentially shrinking sheet. Ain Shams Eng. J. 5 (2014)

1337− 1342

[62] A. Mushtaq, M. Mustafa, T. Hayat and A. Alsaedi, About thermal radiation surface effects

on the stagnation-point flow of uper-convected Maxwell (UCM) fluid over a stretching

sheet. J. Aerosp. Eng. DOI:10.1061/(ASCE)AS.1943− 5525− 0000361

[63] R. Cortell, Fluid flow and radiative nonlinear heat transfer over a stretching sheet. J.

King Saud University-Sci. 26 (2014) 161− 167

220

[64] A. Mushtaq, M. Mustafa, T. Hayat and A. Alsaedi, Nonlinear radiative heat transfer in

the flow of nanofluid due to solar energy: A numerical study, J. Taiwan Inst. Chem. Eng.

(2013) (in press).

[65] S. A. Shehzad, T. Hayat, A. Alsaedi and M. Ali Obid, Nonlinear thermal radiation in

three-dimensional flow of Jeffrey nanofluid: A model for solar energy. Appl. Math. Com-

put. 248 (2014) 273− 286

[66] R. Cortell, Radiation effects for the Blasius and Sakiadis flows with a convective surface

boundary condition. Appl. Math. Comput. 206 (2008) 832− 840

[67] A. Aziz, A similarity solution for laminar thermal boundary layer over a flat plate with

a convective surface boundary condition, Comm. Nonlinear Sci. Num. Simul. 14 (2009)

1064− 1068.

[68] A. Ishak, Similarity solutions for flow and heat transfer over a permeable surface with

convective boundary condition. Appl. Math. Comput. 217 (2010) 837− 842

[69] O. D. Makinde, Similarity solution of hydromagnetic heat and mass transfer over a vertical

plate with a convective surface boundary condition. Int. J. Phys. Sci. 5 (2010) 700− 710

[70] O. D. Makinde and A. Aziz, MHD mixed convection from a vertical plate embedded in

a porous medium with convective boundary condition. Int. J. Thermal Sci. 49 (2010)

1813− 1820

[71] O. D. Makinde, On MHD heat and mass transfer over a moving vertical plate with a

convective surface boundary condition, Canadian J. Chem. Eng. 88 (2010) 1− 8

[72] T. Hayat, S. A. Shehzad, M. Qasim and S. Obaidat, Flow of a second grade fluid with

convective boundary condition. Therm. Sci. 15 (2011) 253− 261.

[73] M. Mustafa, T. Hayat and A. Alsaedi, Axisymmetric flow of a nanofluid over a radially

stretching sheet with convective boundary conditions. Curr. Nanosci. 8 (2012) 328− 334.

[74] M. A. A. Hamad, Md. J. Uddin and A. I. Md. Ismail, Investigation of combined heat and

mass transfer by Lie group analysis with variable diffusivity taking into account hydro-

221

dynamic slip and thermal convective boundary conditions. Int. J. Heat Mass Transfer 55

(2012) 1355− 1362

[75] A. M. Rashad, A. J. Chamkha and M. Modather, Mixed convection boundary-layer flow

past a horizontal circular cylinder embedded in a porous medium filled with a nanofluid

under convective boundary condition. Computers & Fluids 86 (2013) 380− 388

[76] K. Das, P. R. Duari and P. K. Kundu, Numerical simulation of nanofluid flow with

convective boundary condition. J. Egyp. Math. Soci. (2014) (in press).

[77] G. K. Ramesh and B. J. Gireesha, Influence of heat source/sink on a Maxwell fluid over

a stretching surface with convective boundary condition in the presence of nanoparticles.

Ain Shams Eng. J. (2014) (in press).

[78] P. M. Patil, E. Momoniat and S. Roy, Influence of convective boundary condition on

double diffusive mixed convection from a permeable vertical surface. Int. J. Heat Mass

Transfer 70 (2014) 313− 321

[79] T. Hayat, M. Imtiaz, A. Alsaedi and R. Mansoor, MHD flow of nanofluid over an expo-

nentially stretching sheet in a porous medium with convective boundary conditions. Chin.

Phys. B 23 (2014) 054701

[80] G. Ibanez, Entropy generation in MHD porous channel with hydrodynamic slip and con-

vective boundary conditions. Int. J. Heat Mass Transfer 80 (2015) 274− 280

[81] C. Y. Wang, Fluid flow due to stretching cylinder. Phys. Fluids 31 (1988) 466− 468

[82] H. I. Burde, On the motion of fluid near stretching circular cylinder. J. Appl. Math. Mech.

53 (1989) 271− 273

[83] A. Ishak, R. Nazar and I.Pop, Magnetohydrodynamic (MHD) flow and heat transfer due

to a stretching cylinder. Energy Conv. Manag. 49 (2008) 3265− 3269

[84] A. Ishak, R. Nazar and I. Pop, Uniform suction/blowing effect on flow and heat transfer

due to a stretching cylinder. Appl. Math. Model. 32 (2008) 2059− 2066

222

[85] A. Ishak, Mixed convection boundary layer flow over a vertical cylinder with prescribed

surface heat flux. J. Phys. A: Math. and Theor. 42 (2009) 1− 8.

[86] R. R. Rangi and N. Ahmad, Boundary layer flow past a stretching cylinder and heat

transfer with variable thermal conductivity. Appl. Math. 3 (2012) 205− 209.

[87] S. Mukhopadhyay and A. Ishak, Mixed convection flow along a stretching cylinder in a

thermally stratified medium. J. Appl. Math. 2012 (2012) 1− 8.

[88] S. Mukhopadhyay, Chemically reactive solute transfer in a boundary layer slip flow along

a stretching cylinder. Chem. Sci. Eng. 5 (2011) 385− 391

[89] K. Vajravelu, K. V. Prasad and S. R. Santhi, Axisymmetric magnetohydrodynamic

(MHD) flow and heat transfer at a non-isothermal stretching cylinder. Appl. Math. Com-

put. 219 (2012) 3993− 4005

[90] S. Mukhopadhyay, MHD boundary layer slip flow along a stretching cylinder. Ain Shams

Eng. J. 4 (2013) 317− 324

[91] X. Si, L. Li, L. Zheng, X. Zhang and B. Liu, The exterior unsteady viscous flow and heat

transfer due to a porous expanding stretching cylinder. Computer & Fluids 105 (2014)

280− 284

[92] K. Zaimi, A. Ishak and I. Pop, Unsteady flow due to a contracting cylinder in a nanofluid

using Buongiorno’s model. Int. J. Heat Mass Transfer 68 (2014) 509− 513

[93] S. E. Ahmed, A. K. Hussein, H. A. Mohammad and S. Sivasankaran, Boundary layer flow

and heat transfer due to permeable stretching tube in the presence of heat source/sink.

Appl. Math. Comput. 238 (2014) 149− 162

[94] U. Mishra and G. Singh, Dual solutions of mixed convection flow with momentum and

thermal slip flow over a permeable shrinking cylinder. Computer & Fluids 93 (2014)

107− 115

[95] T. Hayat, M. S. Anwar, M. Farooq and A. Alsaedi, MHD stagnation point flow of second

grade fluid over a stretching cylinder with heat and mass transfer. Int. J. Numer. Simul.

15 (2014) 365− 376

223

[96] S. J. Liao and A. T. Chwang , Apllication of homotopy analysis method in nonlinear

oscillations. ASME J. Appl. Mech. 65 (1998) 914− 922

[97] S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems.

Comm. Non-linear Sci. Num. Simul. 14 (2009) 983− 997.

[98] S. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equa-

tions. Comm. Nonlinear Sci. Num. Simul. 15 (2010) 2003− 2016.

[99] S. J. Liao, On the relationship between the homotopy analysis method and Euler trans-

form. Comm. Nonlinear Sci. Numer. Simul. 15 (2010) 1421− 1431

[100] S. Abbasbandy and A. Shirzadi, A new application of the homotopy analysis method:

Solving the Sturm—Liouville problems. Comm. Nonlinear Sci. Num. Simul. 16 (2011)

112− 126

[101] S. Abbasbandy, E. Shivanian and K. Vajravelu, Mathematical properties of ~-curve in the

frame work of the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul.16

(2011) 4268− 4275

[102] M. Mustafa, T. Hayat and S. Obaidat, On heat and mass transfer in the unsteady squeez-

ing flow between parallel plates. Mecc. 47 (2012) 1581− 1589.

[103] M. M. Rashidi, T. Hayat, E. Erfani, S. A. M. Pour and A. A. Hendi, Simultaneous effects

of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow

due to a rotating disk. Comm. Nonlinear Sci. Numer. Simul. 16 (2011) 4303− 4317

[104] M. M. Rashidi, S. Abelman and F. N. Mehr, Entropy generation in steady MHD flow due

to a rotating disk in a nanofluid. Int. J. Heat Mass Transfer 62 (2013) 515− 525

[105] S. Abbasbandy, M. S. Hashemi and I. Hashim, On convergence of homotopy analysis

method and its application to fractional integro-differential equations. Quaestiones Math.

36 (2013) 93− 105.

[106] M. Maleki, S. A. M. Tonekaboni and S. Abbasbandy, A homotopy analysis solution to

large deformation of beams under static arbitrary distributed load. Applied Math. Modell.

38 (2014) 355− 368

224

[107] B. Rostami, M. M. Rashidi, P. Rostami, E. Momoniat and N. Rreidoonimehr, Analytical

investigation of laminar viscoelastic fluid flow over a wedge in the presence of buoyancy

force effects. Abstract and Appl. Analysis 2014 (2014) 1− 11

[108] M. Sheikholeslami, R. Ellahi, H. R. Ashorynejad, G. Domairry and T. Hayat. Effect of

heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium.

J. Comp. and Theo. Nanoscience 11 (2014) 486− 496

[109] M. M. Rashidi, M. Ali, N. Freidoonimehr, B. Rostami and A. Hossian, Mixed convection

heat transfer for viscoelastic fluid flow over a porous wedge with thermal radiation. Adv.

Mech. Eng. 2014 (2014) 1− 10

[110] T. Hayat, S. Asad and A. Alsaedi, Analysis for flow of Jeffrey fluid with nanoparticles.

Chin. Phys. B 24 (2015) 044702

225

contentsprr.hec.gov.pk/jspui/bitstream/123456789/9547/1/thesis.pdf · [52] investigated the heat...

Documents