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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
3.3 Homotopy analysis solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Convergence of the homotopy solutions . . . . . . . . . . . . . . . . . . . . . . . . 37
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
4.3 Homotopy analysis solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Convergence of the homotopy solutions . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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
5.4 Convergence of the homotopy solutions . . . . . . . . . . . . . . . . . . . . . . . . 74
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
7.3 Homotopy analysis solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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
8.4 Convergence of the homotopy solutions . . . . . . . . . . . . . . . . . . . . . . . . 132
8.5 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
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
9.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
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
10.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
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
11.4 Homotopy analysis solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.5 Convergence of the series solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
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
11.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12 Radiative flow of Jeffrey fluid through a convectively heated stretching cylin-
der 195
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
12.2 Flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
12.3 Homotopy analysis solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
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 ()
Fig. 26. Influence of on ()
23
Fig. 27. Influence of Pr on ()
Fig. 28. Influence of on ()
24
Fig. 29. Influence of on ()
Fig. 210. Influence of on ()
25
Fig. 211. Influence of on ()
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.
3.3 Homotopy analysis solutions
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
3.4 Convergence of the homotopy solutions
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()
Fig. 3.5. Influence of on 0()
41
Fig. 3.6. Influence of on 0()
Fig. 3.7. Influence of on 0()
42
Fig. 3.8. Influence of on 0()
Fig. 3.9. Influence of on 0()
43
Fig. 3.10. Influence of on ()
Fig. 3.11. Influence of on ()
44
Fig. 3.12. Influence of on ()
Fig. 3.13. Influence of on ()
45
Fig. 3.14. Influence of on ()
Fig. 3.15. Influence of Pr on ()
46
Fig. 3.16. Influence of on ()
Fig. 3.17. Influence of on ()
47
Fig. 3.18. Influence of on ()
Fig. 3.19. Influence of on ()
48
Fig. 3.20. Influence of on ()
Fig. 3.21. Influence of on ()
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)
where prime denotes the differentiation with respect to , is the dimensionless stream function,
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.
4.3 Homotopy analysis solutions
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
4.4 Convergence of the homotopy solutions
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
Order of approximation − 00(0) −0(0)1 098765 087631
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
4.5 Results and discussion
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
Fig. 4.5. Influence of on velocity field.
Fig. 4.6. Influence of on velocity field.
63
Fig. 4.7. Influence of on velocity field.
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)
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
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)
where prime denotes the differentiation with respect to , is the dimensionless stream function,
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)
5.4 Convergence of the homotopy solutions
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
Order of approximation − 00(0) −0(0)1 103515 133250
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.
Fig. 54. Variation of on velocity field.
78
Fig. 55. Variation of on velocity field.
Fig. 56. Variation of on velocity field.
79
Fig. 57. Variation of Γ on velocity field.
Fig. 58. Variation of 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.
Fig. 512. Variation of on temperature field.
82
Fig. 513. Variation of on temperature field.
Fig. 514. Variation of on temperature field.
83
Fig. 515. Variation of on temperature field.
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
à 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
Fig. 64. Variation of 2 on velocity field.
Fig. 65. Variation of 3 on velocity field.
97
Fig. 66. Variation of on velocity field.
Fig. 67. Variation of on velocity field.
98
Fig. 68. Variation of on temperature field.
Fig. 69. Variation of on temperature field.
99
Fig. 610. Variation of on temperature field.
Fig. 611. Variation of on temperature field.
100
Fig. 612. Variation of on temperature field.
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)
in which and are the coefficients of space and temperature-dependent heat source/sink,
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
.
7.3 Homotopy analysis solutions
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.
Fig. 7.1. ~− curve for the velocity field.
111
Fig. 7.2. ~− curve for the temperature field.
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
Fig. 76 Influence of 2 on velocity field.
Fig. 77 Influence of 3 on velocity field.
116
Fig. 78 Influence of on velocity field.
Fig. 79 Influence of on velocity field.
117
Fig. 710 Influence of 1 on temperature field.
Fig. 711 Influence of 2 on temperature field.
118
Fig. 712 Influence of 3 on temperature field.
Fig. 713 Influence of on temperature field.
119
Fig. 714 Influence of on temperature field.
Fig. 715 Influence of Pr on temperature field.
120
Fig. 716 Influence of on temperature field.
Fig. 717 Influence of on temperature field.
121
Fig. 718 Influence of 1 on concentration field.
Fig. 719 Influence of 2 on concentration field.
122
Fig. 720 Influence of 3 on concentration field.
Fig. 721 Influence of on concentration field.
123
Fig. 722 Influence of on concentration field.
Fig. 723 Influence of on concentration field.
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)
where prime denotes the differentiation with respect to , is the dimensionless stream function,
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)
8.4 Convergence of the homotopy solutions
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
Fig. 8.1. ~− curve for the velocity field.
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
Order of approximation − 00(0) −0(0)1 0953530 0705974
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).
Fig. 8.3. Influence of on ()
135
Fig. 8.4. Influence of on 0()
Fig. 8.5. Influence of on 0()
136
Fig. 8.6. Influence of on 0()
Fig. 8.7. Influence of on 0()
137
Fig. 8.8. Influence of on ()
Fig. 8.9. Influence of on ()
138
Fig. 8.10. Influence of on ()
Fig. 8.11. Influence of on ()
139
Fig. 8.12. Influence of on ()
Fig. 8.13. Influence of Pr on ()
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
8.6 Concluding remarks
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)
where prime denotes the differentiation with respect to , is the dimensionless stream function,
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)
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
(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
Fig. 9.1. ~− curve for the velocity field.
Fig. 9.2. ~− curve for the temperature field.
149
Table 91: Convergence of series solutions for different order of approximations when = 20
= 03 = 01 = 01 Pr = 10 = 04 } = −065 and } = −08
Order of approximation − 00(0) −0(0)1 086875 023961
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
9.5 Results and discussion
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. 94 Influence of on 0()
Fig. 95 Influence of on ()
152
Fig. 96 Influence of on ()
Fig. 97 Influence of Pr on ()
153
Fig. 98 Influence of on ()
Fig. 99 Influence of on ()
154
Fig. 910 Influence of on ()
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)
where prime denotes the differentiation with respect to , is the dimensionless stream function,
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) = ()
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
(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
Order of approximations − 00(0) −0(0)1 075626 018067
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 .
Fig. 103 Influence of on 0()
167
Fig. 104 Influence of on 0()
Fig. 105 Influence of on 0()
168
Fig. 106 Influence of on 0()
Fig. 107 Influence of on ()
169
Fig. 108 Influence of Pr on ()
Fig. 109 Influence of on ()
170
Fig. 1010 Influence of on ()
Fig. 1011 Influence of on ()
171
Fig. 1012 Influence of on ()
Fig. 1013 Influence of on ()
172
Fig. 1014 Influence of on ()
Fig. 1015 Influence of Pr on ()
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
10.6 Concluding remarks
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
Dimensionless form of skin friction coefficient and local Nusselt number are
= (2Re)−12 ∗−2
∙1
1 +
½ 00(0) + 3
µ3
2 00(0)
¶¾¸ (11.14)
12 = −(1 + 43)0(0) (11.15)
11.4 Homotopy analysis solutions
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)
The -order deformation problems are
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
11.6 Results and discussion
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()
Fig. 11.9. Influence of on 0()
189
Fig. 11.10. Influence of Pr on ()
Fig. 11.11. Influence of on ()
190
Fig. 11.12. Influence of ∗ on ()
Fig. 11.13. 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
11.7 Concluding remarks
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
¶=
Dimensionless form of skin friction coefficient and local Nusselt number are
= 2 (Re)−12
∙1
1 + (1 + 3)
00(0)¸ 12 = −(1 + 4
3)0(0) (12.11)
12.3 Homotopy analysis solutions
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)
The -order deformation problems are
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.
Order of approximations − 00(0) −0(0)1 10420 003216
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. 12.3. Influence of on 0()
Fig. 12.4. Influence of 3 on 0()
206
Fig. 12.5. Influence of on 0()
Fig. 12.6. Influence of on 0()
207
Fig. 12.7. Influence of on 0()
Fig. 12.8. Influence of on ()
208
Fig. 12.9. Influence of 3 on ()
Fig. 12.10. Influence of on ()
209
Fig. 12.11. Influence of Pr on ()
Fig. 12.12. Influence of on ()
210
Fig. 12.13. Influence of on ()
Fig. 12.14. Influence of on ()
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
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