**1. Introduction**

Over the past several decades, the Navier-Stokes equations have been studied frequently in the literature. This is due to the fact that the use of the Newtonian fluid model in numerous industrial applications to predict the behavior of many real fluids has been adopted. However, there are many materials of industrial importance (e.g. polymeric liquids, molten plastics, lubricating oils, drilling muds, biological fluids, food products, personal care products, paints, greases and so forth) are non-Newtonian. That is, they might exhibit dramatic deviation from Newtonian behavior and display a range of non-Newtonian characteristics. A few points of non-Newtonian characteristic are the ability of the fluid to exhibit relaxation and retardation, shear dependent viscosity, shear thinning or shear thickening, yield stress, viscoelasticity and many more. Thus, it has been now well recognized in technology and industrial applications that non-Newtonian fluids are more appropriate than the Newtonian fluid. Consequently, the theory of non-Newtonian fluids has become an active field of research for the last few years.

Unlike, the Newtonian fluid, it is very difficult to provide a universal constitutive model for non-Newtonian fluids as they possess very complex structure. However, there are some classes of fluids that cannot be classified as Newtonian or purely non-Newtonian such as water-borne coating etc. This situation demands some more general models which can be utilized for analysis of both Newtonian and non-Newtonian behaviors. For this purpose, some models have been proposed in the literature including generalized Newtonian fluids. The Sisko fluid model [1] is a subclass of the generalized Newtonian fluids which is considered as the most appropriate model for lubricating oils and greases [2]. The Sisko fluid model is of much importance due to its adequate description of a few non-Newtonian fluids over the most important range of shear rates. The appropriateness of the Sisko fluid model has been successfully extended to the shear thinning rheological behavior of concentrated non-

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Newtonian slurries [3]. The three parameters Sisko fluid model, which can be considered as a generalized power-law model that includes Newtonian component, has not been given due attention in spite of its diverse industrial applications. A representative sample of the recent literature on the Sisko fluid is provided by references [4-10].

Investigations of the boundary layer flow and heat transfer of non-Newtonian fluids over a stretching surface are important due to immense applications in engineering and science. A great number of investigations concern the boundary layer behavior on a stretching surface. Many manufacturing processes involve the cooling of continuous sheets. To be more specific, examples of such applications are wire drawing, hot rolling, drawing of plastic films, paper production, and glass fiber etc. In all these situations, study of the flow and heat transfer is of significant importance as the quality of the final products depends to the large extent on the skin friction and heat transfer rate at the surface. In view of these, the boundary layer flows and heat transfer over a stretching surface have been studied extensively by many researchers. Crane [11] was first to investigate the boundary layer flow of a viscous fluid over a stretching sheet when the sheet is stretched in its own plane with velocity varies linearly with the distance from a fixed point on the sheet. Dutta *et al*. [12] examined the heat transfer in a viscous fluid over a stretching surface with uniform heat flux. Later on, this problem was extended by Chen and Char [13] by considering the variable heat flux. Grubka and Bobba [14] analyzed the heat transfer over a stretching surface by considering the non-isothermal wall that is varying as a power-law with the distance. Cortell [15] investigated the flow and heat transfer of a viscous fluid over nonlinear stretching sheet by considering the constant surface temperature and prescribed surface temperature. It seems that Schowalter [16] was the first who has obtained the similarity solutions for the boundary layer flow for power-law pseudoplastic fluids. Howel *et al.* [17] considered the laminar flow and heat transfer of a power-law fluid over a stretching sheet. Hassanien *et al.* [18] investigated the heat transfer to a power-law fluid flow over a nonisothermal stretching sheet. Abel *et al*. [19] studied the flow and heat transfer of a power-law fluid over a stretching sheet with variable thermal conductivity and non-uniform heat source. Prasad and Vajravelu [20] analyzed the heat transfer of a power-law fluid over a non-isother‐ mal stretching sheet. Khan and Shahzad [21,22] have considered the boundary layer theory of the Sisko fluid over the planer and radially stretching sheets and found the analytic solutions; however, they only considered the integral values of the power-law index in their flow problems. The integral values of the power-law index are inadequate to completely compre‐ hend the shear thinning and shear thickening effects of the Sisko fluid. Moreover, a literature survey also indicates that no work has so far been available with regards to heat transfer to Sisko fluid flow over a stretching sheet in presence of viscous dissipation.

The objective of this chapter is to analyze the flow and heat transfer characteristics of Sisko fluid over a radially stretching sheet with the stretching velocity *cr <sup>s</sup>* in the presence of viscous dissipation. In the present work we have spanned the value of the power-law index from highly shear thinning to shear thickening Sisko fluid (0.2≤*n* ≤1.9). The modeled partial differential equations are reduced to a system of nonlinear ordinary differential equations using the appropriate transformations. The resulting equations are then solved numerically by implicit finite difference method in the domain 0, *∞*). The numerical results for the velocity and temperature fields are graphically depicted and effects of the relevant parameters are dis‐ cussed in detail. In addition, the skin friction coefficient and the local Nusselt number for different values of the pertaining parameters are given in tabulated form. Moreover, numerical results are compared with exact solutions as special cases of the problem. Furthermore, the present results for the velocity field are also validated by comparison with the previous pertinent literature.

#### **2. Governing equations**

Newtonian slurries [3]. The three parameters Sisko fluid model, which can be considered as a generalized power-law model that includes Newtonian component, has not been given due attention in spite of its diverse industrial applications. A representative sample of the recent

Investigations of the boundary layer flow and heat transfer of non-Newtonian fluids over a stretching surface are important due to immense applications in engineering and science. A great number of investigations concern the boundary layer behavior on a stretching surface. Many manufacturing processes involve the cooling of continuous sheets. To be more specific, examples of such applications are wire drawing, hot rolling, drawing of plastic films, paper production, and glass fiber etc. In all these situations, study of the flow and heat transfer is of significant importance as the quality of the final products depends to the large extent on the skin friction and heat transfer rate at the surface. In view of these, the boundary layer flows and heat transfer over a stretching surface have been studied extensively by many researchers. Crane [11] was first to investigate the boundary layer flow of a viscous fluid over a stretching sheet when the sheet is stretched in its own plane with velocity varies linearly with the distance from a fixed point on the sheet. Dutta *et al*. [12] examined the heat transfer in a viscous fluid over a stretching surface with uniform heat flux. Later on, this problem was extended by Chen and Char [13] by considering the variable heat flux. Grubka and Bobba [14] analyzed the heat transfer over a stretching surface by considering the non-isothermal wall that is varying as a power-law with the distance. Cortell [15] investigated the flow and heat transfer of a viscous fluid over nonlinear stretching sheet by considering the constant surface temperature and prescribed surface temperature. It seems that Schowalter [16] was the first who has obtained the similarity solutions for the boundary layer flow for power-law pseudoplastic fluids. Howel *et al.* [17] considered the laminar flow and heat transfer of a power-law fluid over a stretching sheet. Hassanien *et al.* [18] investigated the heat transfer to a power-law fluid flow over a nonisothermal stretching sheet. Abel *et al*. [19] studied the flow and heat transfer of a power-law fluid over a stretching sheet with variable thermal conductivity and non-uniform heat source. Prasad and Vajravelu [20] analyzed the heat transfer of a power-law fluid over a non-isother‐ mal stretching sheet. Khan and Shahzad [21,22] have considered the boundary layer theory of the Sisko fluid over the planer and radially stretching sheets and found the analytic solutions; however, they only considered the integral values of the power-law index in their flow problems. The integral values of the power-law index are inadequate to completely compre‐ hend the shear thinning and shear thickening effects of the Sisko fluid. Moreover, a literature survey also indicates that no work has so far been available with regards to heat transfer to

literature on the Sisko fluid is provided by references [4-10].

342 Heat Transfer Studies and Applications

Sisko fluid flow over a stretching sheet in presence of viscous dissipation.

The objective of this chapter is to analyze the flow and heat transfer characteristics of Sisko fluid over a radially stretching sheet with the stretching velocity *cr <sup>s</sup>* in the presence of viscous dissipation. In the present work we have spanned the value of the power-law index from highly shear thinning to shear thickening Sisko fluid (0.2≤*n* ≤1.9). The modeled partial differential equations are reduced to a system of nonlinear ordinary differential equations using the appropriate transformations. The resulting equations are then solved numerically by implicit finite difference method in the domain 0, *∞*). The numerical results for the velocity and

This section comprises the governing equations and the rheological model for the steady twodimensional flow and heat transfer of an incompressible and inelastic fluid Sisko fluid in the cylindrical polar coordinates. To derive the governing equations we make use of fundamental laws of fluid mechanics, namely conservations of mass, linear momentum and energy, including the viscous dissipation

$$
\nabla \cdot \mathbf{V} = 0,\tag{1}
$$

$$
\rho(\mathbf{V}\cdot\mathbf{V})\mathbf{V}=-\nabla p\_1+\nabla\cdot\mathbf{S},\tag{2}
$$

$$
\rho c\_p (\mathbf{V} \cdot \nabla) \mathbf{T} = -\nabla \cdot \mathbf{q} + \mathbf{S} \cdot \mathbf{L}.\tag{3}
$$

In the above equations **V** is the velocity vector, *ρ* the density of fluid, *cp* the specific heat at constant pressure, *p*1 the pressure, *T* the temperature, **S** the extra stress tensor and **q** the heat flux given by

$$\mathbf{q} = -\kappa(\nabla T),\tag{4}$$

where *κ* is the thermal conductivity of the fluid and ∇ the gradient operator.

The extra stress tensor **S** for an incompressible fluid obeys the Sisko rheological model. This model mathematically can be expressed as [4]

$$\mathbf{S} = \left[ a \mathbf{+} b \left| \sqrt{\frac{1}{2} \operatorname{tr} \left( \mathbf{A}\_1^2 \right)} \right|^{n-1} \right] \mathbf{A}\_{1'} \tag{5}$$

where **A**1 is the rate of deformation tensor or the first Rivlin-Erickson tensor defined as

$$\mathbf{A}\_1 = \mathbf{L} + \mathbf{L}^T, \ \mathbf{L} = \mathbf{V} \mathbf{V}\_1 \tag{6}$$

with *a* the dynamic viscosity, *b* the Sisko fluid parameter or the flow consistency index, (*n* ≥0) the power-law index or the flow behavior index (a non-negative real number) and **T** stands for transpose.

The quantity

$$\mu\_{\rm eff} = \left[ a \star b \left| \sqrt{\frac{1}{2} \operatorname{tr}(\mathbf{A}\_1^2)} \right|^{\nu - 1} \right] \tag{7}$$

represents an apparent or effective viscosity as a function of the shear rate. If *a* =0 **and** *n* =1 (**or** *b* =0) the equations for Newtonian fluid, *a* =0 for the power-law model and *n* =0 with *b* as yield stress for the Bingham plastic model are obtained.

For the steady two-dimensional axisymmetric flow, we assume the velocity, temperature and stress fields of the form

$$\mathbf{V} = \left[ \mu(r, z), 0, w(r, z) \right], \; T = T(r, z), \; \mathbf{S} = \mathbf{S}(r, z), \tag{8}$$

when (*r*, *z*) denotes the cylindrical polar coordinates along the sheet and vertical to it, *u* **and** *w* the velocity components in the *r* − **and** *z* − directions, respectively.

The steady two-dimensional and incompressible equations of motion (2) including conserva‐ tion of mass (1) and thermal energy (3) can be written as

$$\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = 0,\tag{9}$$

$$
\rho \left\{ u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} \right\} = -\frac{\partial p\_1}{\partial r} + \frac{\partial S\_{rr}}{\partial r} + \frac{\partial S\_{rz}}{\partial z} + \frac{S\_{rr} - S\_{\phi\phi}}{r}, \tag{10}
$$

$$
\rho \left\{ u \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z} \right\} = -\frac{\partial p\_{\perp}}{\partial z} + \frac{1}{r} \frac{\partial}{\partial r} \left( r S\_{\tau z} \right) + \frac{\partial S\_{zz}}{\partial z}, \tag{11}
$$

$$\begin{split} \rho c\_{\rho} \left\{ u \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} \right\} &= \kappa \left\{ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right) + \frac{\partial^2 T}{\partial z^2} \right\} + \iota \left[ 2 \left( \frac{\partial u}{\partial r} \right)^2 + \left( \frac{\partial u}{\partial z} \right)^2 + 2 \frac{\partial u}{\partial z} \frac{\partial w}{\partial r} + 2 \frac{\partial w}{\partial z} \frac{\partial w}{\partial r} + 2 \frac{\mu^2}{r^2} \right] \\ + \iota \left[ 2 \left( \frac{\partial u}{\partial r} \right)^2 + \left( \frac{\partial u}{\partial z} \right)^2 + 2 \frac{\partial u}{\partial z} \frac{\partial w}{\partial r} + 2 \frac{\partial w}{\partial z} \frac{\partial w}{\partial r} + 2 \frac{\mu^2}{r^2} \right] \left| \sqrt{\frac{1}{2} r A\_1^2} \right|^{\nu = 1} \end{split} \tag{12}$$

where

$$\frac{1}{2}tr\left\{\mathbf{A}\_{\mathrm{i}}^{2}\right\} = 2\left(\frac{\partial u}{\partial r}\right)^{2} + 2\left(\frac{\partial w}{\partial z}\right)^{2} + 2\left(\frac{\partial u}{\partial z}\right)^{2} + \frac{\partial u}{\partial z}\frac{\partial w}{\partial r} + 2\frac{u^{2}}{r^{2}}.\tag{13}$$

In view of Eq. (8) the stress components are inserted into the equations of motion and the usual boundary layer approximations are made, the equations of motion characterizing the steady boundary layer flow and heat transfer take the form

$$
\rho \left( u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial p\_1}{\partial r} + a \frac{\partial^2 u}{\partial z^2} + b \frac{\partial}{\partial z} \left( \left| \frac{\partial u}{\partial z} \right|^{\nu - 1} \frac{\partial u}{\partial z} \right), \tag{14}
$$

$$0 = -\frac{\mathfrak{d}\wp\_1}{\mathfrak{d}z} \tag{15}$$

$$u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} = a\frac{\partial^2 T}{\partial z^2} + a\left(\frac{\partial u}{\partial z}\right)^2 + b\left(\frac{\partial u}{\partial z}\right)^2 \left|\frac{\partial u}{\partial z}\right|^{n-1},\tag{16}$$

where *<sup>α</sup>* <sup>=</sup> *<sup>κ</sup> ρcp* is the thermal diffusivity with *κ* as the thermal conductivity.

#### **3. Mathematical formulation**

#### **3.1. Flow analysis**

The quantity

344 Heat Transfer Studies and Applications

stress fields of the form

1

û **S = S** = (8)

<sup>+</sup> <sup>+</sup> <sup>=</sup> ¶ ¶ (9)

ff

<sup>ç</sup> <sup>+</sup> <sup>÷</sup> <sup>=</sup> - <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>è</sup> ¶ ¶ <sup>ø</sup> ¶ ¶ ¶ (10)

<sup>æ</sup> ¶ ¶ <sup>ö</sup> ¶ ¶ ¶ <sup>ç</sup> <sup>+</sup> <sup>÷</sup> <sup>=</sup> - <sup>+</sup> <sup>+</sup> <sup>è</sup> ¶ ¶ <sup>ø</sup> ¶ ¶ ¶ (11)

2 2 <sup>2</sup> <sup>2</sup> 2 2

( ) <sup>1</sup> <sup>1</sup> , *zz rz*

2

**A** -

*n*

**<sup>A</sup>** <sup>æ</sup> ¶ <sup>ö</sup> <sup>æ</sup> ¶ <sup>ö</sup> <sup>æ</sup> ¶ <sup>ö</sup> ¶ ¶ <sup>=</sup> <sup>ç</sup> <sup>÷</sup> <sup>+</sup> <sup>ç</sup> <sup>÷</sup> <sup>+</sup> <sup>ç</sup> <sup>÷</sup> <sup>+</sup> <sup>+</sup> <sup>è</sup> ¶ <sup>ø</sup> <sup>è</sup> ¶ <sup>ø</sup> <sup>è</sup> ¶ <sup>ø</sup> ¶ ¶ (13)

<sup>1</sup> 2 22 2

2 1

(7)

(12)

2 1 <sup>1</sup> () , <sup>2</sup> *n*

*eff* m

with *b* as yield stress for the Bingham plastic model are obtained.

the velocity components in the *r* − **and** *z* − directions, respectively.

tion of mass (1) and thermal energy (3) can be written as

*u w*

r

r

 k

*p*

r

where

**V=** é

*a b tr* **A**


represents an apparent or effective viscosity as a function of the shear rate. If *a* =0 **and** *n* =1 (**or** *b* =0) the equations for Newtonian fluid, *a* =0 for the power-law model and *n* =0

For the steady two-dimensional axisymmetric flow, we assume the velocity, temperature and

ë*urz wrz T Tr r* ( , ),0, ( , ) , ( ,z), ( ,z), ù

when (*r*, *z*) denotes the cylindrical polar coordinates along the sheet and vertical to it, *u* **and** *w*

The steady two-dimensional and incompressible equations of motion (2) including conserva‐

0, *uu w rr z* ¶ ¶

*r z rr z r*

*w w p S*

*r z z rr z*

*T T T T u u uw ww u cu w <sup>r</sup> <sup>a</sup> r z rr r z r r z zr zr*

é ù æ ¶ ¶ ö æ ¶ æ ¶ ö ¶ ö æ ¶ ö æ ¶ ö ¶ ¶ ¶ ¶ <sup>+</sup> <sup>=</sup> <sup>+</sup> <sup>+</sup> <sup>ê</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>ú</sup> <sup>ç</sup> <sup>÷</sup> <sup>ç</sup> <sup>ç</sup> <sup>÷</sup> <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>è</sup> ¶ ¶ <sup>ø</sup> <sup>è</sup> ¶ <sup>è</sup> ¶ <sup>ø</sup> ¶ <sup>ø</sup> êë <sup>è</sup> ¶ <sup>ø</sup> <sup>è</sup> ¶ <sup>ø</sup> ¶ ¶ ¶ ¶ úû

( ) 2 22 <sup>2</sup>

*u w u uw u tr*

1 2 <sup>1</sup> 22 2 2 .

In view of Eq. (8) the stress components are inserted into the equations of motion and the usual boundary layer approximations are made, the equations of motion characterizing the steady

*r z z zr r*

æ ¶ ¶ ö ¶ ¶ ¶ -

*u w rS*

<sup>1</sup> 2 2 <sup>2</sup>

<sup>1</sup> 2 22 2 , <sup>2</sup>

*u u uw ww u <sup>b</sup> tr r z zr zr <sup>r</sup>*

é ù æ ¶ ö æ ¶ ö ¶ ¶ ¶ ¶ <sup>+</sup> <sup>ê</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>+</sup> <sup>ú</sup> <sup>ç</sup> <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> êë <sup>è</sup> ¶ <sup>ø</sup> <sup>è</sup> ¶ <sup>ø</sup> ¶ ¶ ¶ ¶ úû

2

boundary layer flow and heat transfer take the form

2

<sup>1</sup> , *rr rz rr u u <sup>p</sup> S S S S*

Consider the steady, two-dimensional and incompressible flow of Sisko fluid over a nonlinear radially stretching sheet. The fluid is confined in the region *z* >0, and flow is induced due to stretching of the sheet along the radial direction with velocity *Uw* <sup>=</sup>*cr <sup>s</sup>* with *c* and *s* are positive real numbers pertaining to stretching of the sheet. We assume that the constant temperature of the sheet is *Tw*, while *T∞* is the uniform ambient fluid temperature with *Tw* >*T∞* For mathe‐ matical modeling we take the cylindrical polar coordinate system (*r*, *ϕ*, *z*). Due to the rotational symmetry, all the physical quantities are independent of *θ*. Note that if the streamwise velocity component *u* increases with the distance *z* from the moving surface, the velocity gradient and therefore the shear rate are positive; however, if *u* decreases with increasing *z* the velocity gradient and therefore shear rate are negative. In the present problem within the boundary layer the shear rate is assumed to be negative since the streamwise velocity component *u* decreases monotonically with increasing *z* from the moving boundary (stretching sheet). Thus, under these assumptions, the flow is governed by the following equation:

$$
\rho \left\{ u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} \right\} = a \frac{\partial^2 u}{\partial z^2} - b \frac{\partial}{\partial z} \left\{ - \frac{\partial u}{\partial z} \right\}^\nu. \tag{17}
$$

The boundary conditions associated to flow field are

$$u = cr^s, \; w = 0 \quad \text{at} \; z = 0,\tag{18}$$

$$
u \xrightarrow{} 0 \quad \text{as} \ z \xrightarrow{} \mathfrak{o}.\tag{19}$$

**Figure 1.** Physical model and coordinate system.

We define the following variables

$$\eta = \frac{z}{r} \text{Re}\_{\overline{b}^\#}^{\frac{1}{\mathfrak{q}^\#}} \text{ and } \wp\left\{r, z\right\} = -r^2 U \text{Re}\_{\overline{b}^\#}^{\frac{\omega}{\mathfrak{q}^\#}} f\left\{\eta\right\}. \tag{20}$$

where *ψ*(*r*, *z*) is the Stokes stream function defined by *u* = − <sup>1</sup> *r* ∂ *ψ* <sup>∂</sup> *<sup>z</sup>* and *w* = <sup>1</sup> *r* ∂ *ψ* <sup>∂</sup> *<sup>r</sup>* giving

$$u = \mathsf{U}f'\{\eta\} \text{ and } w = -\mathsf{U}\mathrm{Re}\_{\mathfrak{s}}^{\frac{\omega}{n+1}} \left[ \frac{s\{2n-1\} + n + 2}{n+1} f\{\eta\} - \frac{s\{n-2\} + 1}{n+1} \eta f'\{\eta\} \right]. \tag{21}$$

On employing the above transformations, Eqs. (17) to (19) take the form [21]

$$A f''' + n(-f'')^{n-1} f''' + \left(\frac{s\{2n-1\} + n + 2}{n+1}\right) f''' - s(f')^2 = 0,\tag{22}$$

$$f\begin{Bmatrix}0\\f\end{Bmatrix} = 0, \ f^\*\begin{Bmatrix}0\\f^\*\end{Bmatrix} = 1, \ f^\*\begin{Bmatrix}\infty\end{Bmatrix} = 0.\tag{23}$$

where prime denotes differentiation with respect to *η* and

$$\text{Re}\_a \equiv \rho r \text{LI} / a \text{, } \text{Re}\_b \equiv \rho r \text{"} \text{U}^{2\text{-n}} / b \text{ and } A \equiv \text{Re}\_b^{\frac{2}{\text{n} \Theta}} / \text{Re}\_a \tag{24}$$

The physical quantity of major interest is the local skin friction coefficient and is given by [21]

$$\frac{1}{2}\operatorname{Re}\_{\flat}^{\perp}\operatorname{C}\_{\flat} = A\gamma'''\{0\} - \left[-\gamma''\{0\}\right]^{\*}.\tag{25}$$

#### **3.2. Heat transfer analysis**

**Figure 1.** Physical model and coordinate system.

*r*  **and** hy

where *ψ*(*r*, *z*) is the Stokes stream function defined by *u* = − <sup>1</sup>

where prime denotes differentiation with respect to *η* and

r

1

+

*n b*

( ) ( ) 1 1 1 1 <sup>2</sup> Re , Re , *n n b b <sup>z</sup> rz rU f*

( ) ( ) ( ) ( ) ( ) <sup>1</sup>

<sup>é</sup> - <sup>+</sup> <sup>+</sup> - <sup>+</sup> <sup>ù</sup> <sup>=</sup> ¢ <sup>=</sup> - <sup>ê</sup> - ¢ <sup>ú</sup> <sup>ê</sup> <sup>+</sup> <sup>+</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

*u Uf w U f f*

On employing the above transformations, Eqs. (17) to (19) take the form [21]

*<sup>n</sup> sn n Af n f f ff s f*

*s n n sn*

( ) 1 2 21 2 ( ) ( ) 0, <sup>1</sup>

*n* - <sup>æ</sup> - <sup>+</sup> <sup>+</sup> <sup>ö</sup> ¢¢¢ <sup>+</sup> - ¢¢ ¢¢¢ <sup>+</sup> <sup>ç</sup> <sup>÷</sup> ¢¢ - ¢ <sup>=</sup> <sup>ç</sup> <sup>+</sup> <sup>÷</sup> <sup>è</sup> <sup>ø</sup>

Re / Re <sup>2</sup> <sup>1</sup> / Re / Re *n n <sup>n</sup> a b b a*

 r

 h - <sup>+</sup> <sup>+</sup> = = - (20)

> *r* ∂ *ψ*

*ff f* (0 0, 0 1, 0, ) = ¢( ) = ¢(¥) = (23)

2

*rU a r U b A* **, and** - <sup>+</sup> = = = (24)

21 2 21 Re . 1 1


h

*n n*

<sup>∂</sup> *<sup>z</sup>* and *w* = <sup>1</sup>

*r* ∂ *ψ* <sup>∂</sup> *<sup>r</sup>* giving

h h (21)

(22)

We define the following variables

346 Heat Transfer Studies and Applications

h **and**  In the assumption of boundary layer flow, the energy equation for the non-Newtonian Sisko fluid taking into account the viscous dissipation effects and neglecting the heat generation effects for the temperature field *T* =*T* (*r*, *z*) is

$$u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} = a\frac{\partial^2 T}{\partial z^2} + \frac{1}{\rho c\_p} \left[ a\left(\frac{\partial u}{\partial z}\right)^2 + b\left(-\frac{\partial u}{\partial z}\right)^{\ast \ast 1} \right]. \tag{26}$$

The corresponding thermal boundary conditions are

$$T = T\_w \quad \text{at} \ z = 0,\tag{27}$$

$$T \xrightarrow{} T\_{\bullet} \quad \text{as} \ z \xrightarrow{} \mathfrak{sp}. \tag{28}$$

Using the transformations (20) the above problem reduces to

$$\theta^{\mathfrak{p}} \star \text{Pr} \frac{\text{s} (2n - 1) \star (n + 2)}{n + 1} f \theta^{\mathfrak{p}} \star \text{Br} (f^{\mathfrak{p}})^2 \star \text{Pr} \, \text{E} c(-f^{\mathfrak{p}})^{\mathfrak{v} + 1} = 0,\tag{29}$$

$$\theta \begin{pmatrix} 0 \\ \end{pmatrix} = 1, \quad \text{and} \quad \theta \to 0 \text{ as } \eta \to \infty,\tag{30}$$

where *θ*(*η*)= *<sup>T</sup>* <sup>−</sup> *<sup>T</sup> <sup>∞</sup> <sup>T</sup> <sup>w</sup>* <sup>−</sup> *<sup>T</sup> <sup>∞</sup>* , *Br* <sup>=</sup> *aU* <sup>2</sup> *<sup>κ</sup>*(*<sup>T</sup> <sup>w</sup>* <sup>−</sup> *<sup>T</sup> <sup>∞</sup>*) the Brinkman number, *Ec* <sup>=</sup> *<sup>U</sup>* <sup>2</sup> *c <sup>p</sup>*(*<sup>T</sup> <sup>w</sup>* <sup>−</sup> *<sup>T</sup> <sup>∞</sup>*) the Eckert number and Pr= *rU R* −2 *n*+1 *<sup>α</sup>* the generalized Prandtl number.

The local Nusselt number *N ur* at the wall is defined as

$$N\mu\_r = \frac{rq\_w}{\kappa(T\_w - T\_\bullet)}\Big|\_{z=0} \tag{31}$$

where the wall heat flux at the wall is *qw* <sup>=</sup> <sup>−</sup>*κ*( <sup>∂</sup> *<sup>T</sup>* <sup>∂</sup> *<sup>z</sup>* ) | *<sup>z</sup>*=0, which by virtue of Eq. (31) reduces to

$$\operatorname{Re}\_{b}^{\bullet^{1/n}\bullet^{1}}\operatorname{Nu}\_{r}=-\theta'\begin{pmatrix}0\\0\end{pmatrix}.\tag{32}$$

#### **4. Solution procedure**

The two point boundary value problems comprising Eqs. (22) and (29) along with the associ‐ ated boundary conditions are solved by implicit finite difference scheme along with Keller box scheme. To implement the scheme, Eqs. (22) and (29) are written as a system of first-order differential equations in *η* as follows:

$$f' = p\_1 \tag{33}$$

$$p' = q\_{\prime} \tag{34}$$

$$Aq^{\prime} \star n(\! -q)^{\prime \bullet 1} q^{\prime} \star Dfq - sp^2 = 0,\tag{35}$$

$$
\theta' = \mathbf{t}\_{\prime} \tag{36}
$$

$$\text{tr}^{\prime} \star \text{Pr} \, E \alpha (\neg \eta)^{v \star 1} \star \text{Pr} \, D\dagger \star \text{Br} \eta^2 \equiv 0,\tag{37}$$

where *<sup>D</sup>* <sup>=</sup> *<sup>s</sup>*(2*<sup>n</sup>* <sup>−</sup> 1) + (*<sup>n</sup>* + 2) *<sup>n</sup>* + 1 .

The boundary conditions in terms of new variable are written as

$$\{f(0) = 0, \, p(0) = 1 \text{ and } \,\theta(0) = 1,\} \tag{38}$$

$$
\eta \xrightarrow{} 0 \text{ and } \theta \xrightarrow{} 0 \text{ as } \eta \xrightarrow{} \bullet. \tag{39}
$$

The functions and their derivatives are approximated by central difference at the midpoint *η j*− 1 2 of the segment *η <sup>j</sup>*−1*η<sup>j</sup>* , where *j* =1, 2, … *N* .

$$
\hbar \eta\_0 = 0, \ \eta\_j = \eta\_{j\pm 1} + h\_j, \ \eta\_N = \eta\_\mathbf{o}.\tag{40}
$$

Using the finite difference approximations equations (33) to (37) can be written as

$$(f\_{\rangle} - f\_{\left(\bullet\right)} - h\_{\left(P\right)\left(\bullet\right)} = 0,\tag{41}$$

$$(p\_{\rangle} - p\_{\rangle \bullet 1} - h\_{\rangle} q\_{\rangle \bullet \mathbf{4}^2} = 0,\tag{42}$$

$$\left(A(q\_{/} - q\_{/ \omega 1}) + nh\_{/}(-q\_{/ \omega 2})^{\nu \omega 1}(q\_{/} - q\_{/ \omega 1}) - sh\_{/}(p\_{/ \omega 1 2})^2 + Dh\_{/}f\_{/ \omega 1 2}q\_{/ \omega 2} = 0\right) \tag{43}$$

Convective Heat Transfer to Sisko Fluid over a Nonlinear Radially Stretching Sheet http://dx.doi.org/10.5772/60799 349

$$
\theta\_{\rangle} - \theta\_{\langle \bullet \mathbf{1}} - h\_{\langle \bullet \mathbf{1} \rangle} t\_{\langle \bullet \mathbf{1} \rangle^2} = 0,\tag{44}
$$

$$\mathbf{r}(t\_{\rangle} - t\_{|\boldsymbol{\omega}1}) \mathbf{+} \operatorname{Pr} \operatorname{Ech}\_{\rangle}(\mathbf{-}q\_{|\boldsymbol{\omega}\boldsymbol{\eta}2})^{\mathsf{u}\bullet 1}(q\_{|\boldsymbol{\cdot}} - q\_{|\boldsymbol{\omega}1}) \mathbf{+} \operatorname{Brh}\_{\rangle}(q\_{|\boldsymbol{\omega}\boldsymbol{\eta}2})^{2} \mathbf{+} \operatorname{DPr} h\_{|\boldsymbol{\cdot}\boldsymbol{\eta}2} t\_{|\boldsymbol{\omega}\boldsymbol{\eta}2} \mathbf{=} \mathbf{0},\tag{45}$$

where *<sup>j</sup>* =1, 2, 3, <sup>⋯</sup> *<sup>N</sup>* , *<sup>f</sup> <sup>j</sup>*−1/2 <sup>=</sup> *<sup>f</sup> <sup>j</sup>* <sup>+</sup> *<sup>f</sup> <sup>j</sup>*−<sup>1</sup> <sup>2</sup> , *<sup>p</sup> <sup>j</sup>*−1/2 <sup>=</sup> *pj* <sup>+</sup> *<sup>p</sup> <sup>j</sup>*−<sup>1</sup> <sup>2</sup> , *<sup>q</sup> <sup>j</sup>*−1/2 <sup>=</sup> *qj* <sup>+</sup> *<sup>q</sup> <sup>j</sup>*−<sup>1</sup> <sup>2</sup> , and *<sup>t</sup> <sup>j</sup>*−1/2 <sup>=</sup> *tj* <sup>+</sup> *<sup>t</sup> <sup>j</sup>*−<sup>1</sup> <sup>2</sup> .

Boundary conditions (38) and (39) are written as

$$f\_0 = 0, \ p\_0 = 1, \ \theta\_0 = 1,\tag{46}$$

$$p\_N = 0 \text{ and } \theta\_N = 0. \tag{47}$$

Eqs. (41) to (45) are system of nonlinear equations and these equations are linearized employing the Newton's method and using the expressions:

$$f\_{j}^{(\mathbf{k}\bullet\mathbf{1})} = f\_{j}^{(k)} \star \delta f\_{j}^{(k)},\ p\_{j}^{(k\bullet\mathbf{1})} = p\_{j}^{(k)} \star \delta p\_{j}^{(k)},\ q\_{j}^{(k\bullet\mathbf{1})} = q\_{j}^{(k)} \star \delta q\_{j}^{(k)},$$

$$\theta\_{\rangle}^{(\mathbf{k}\bullet\mathbf{1})} = \theta\_{\rangle}^{(k)} \star \delta \theta\_{\rangle}^{(k)},\ t\_{j}^{(k\bullet\mathbf{1})} = t\_{j}^{(k)} \star \delta t\_{j}^{(k)}.\tag{48}$$

where *k* =1, 2, 3, …

**4. Solution procedure**

348 Heat Transfer Studies and Applications

where *<sup>D</sup>* <sup>=</sup> *<sup>s</sup>*(2*<sup>n</sup>* <sup>−</sup> 1) + (*<sup>n</sup>* + 2)

of the segment *η <sup>j</sup>*−1*η<sup>j</sup>*

*η j*− 1 2 *<sup>n</sup>* + 1 .

differential equations in *η* as follows:

The two point boundary value problems comprising Eqs. (22) and (29) along with the associ‐ ated boundary conditions are solved by implicit finite difference scheme along with Keller box scheme. To implement the scheme, Eqs. (22) and (29) are written as a system of first-order

q

*f p* (0) 0, (0) 1 (0) 1, = **and** =

*p* ® 00. **and as** q

0 1 0, , . *j j jN*

1 2 <sup>1</sup> 1 2 <sup>1</sup> 1 2 12 12 ( ) ( ) ( )s( ) 0) *<sup>n</sup> A q q nh q q q h p Dh f q j j j j j j j j jj j*


Using the finite difference approximations equations (33) to (37) can be written as

q

 h

 h h

The functions and their derivatives are approximated by central difference at the midpoint

The boundary conditions in terms of new variable are written as

, where *j* =1, 2, … *N* .

 hh


h

*f p* ¢ = , (33)

*p q* ¢ = , (34)

¢ = *t*, (36)

= (38)

® ® ¥ (39)

*h* - ¥ = = + = (40)

1 12 0, *j j jj f f hp* - - - - = (41)

1 12 0, *j j jj p p hq* - - - - = (42)

1 2 ( ) 0, *<sup>n</sup> Aq n q q Dfq sp* - ¢ + - ¢ + - = (35)

1 2 Pr ( ) Pr 0, *<sup>n</sup> t Ec q Dft Brq* <sup>+</sup> ¢ + - + + = (37)

Putting the left hand side of the above expressions into Eqs. (41) to (45) and dropping the quadratic terms in *δ f* (*<sup>k</sup>* ) , *δ p* (*<sup>k</sup>* ) , *δq* (*<sup>k</sup>* ) , *δθ* (*<sup>k</sup>* ) and *δt* (*<sup>k</sup>* ) , the following linear equations are obtained:

$$
\delta \delta f\_{\rangle} - \delta f\_{|\bullet 1} - h\_{\rangle} \delta p\_{|\bullet 4|^2} = (r\_1)\_{|\bullet 4|^2 \prime} \tag{49}
$$

$$(\xi\_1)\_/\delta q\_/ \star (\xi\_2)\_/\delta q\_{/\Delta 1} \star (\xi\_3)\_/\delta f\_/ \star (\xi\_4)\_/\delta f\_{/\Delta 1} \star$$

$$
\left(\left(\xi\_{\sf s}\right)\_{/} \delta p\_{/} \star \left(\xi\_{\sf s}\right)\_{/} \delta p\_{/\sf r 1} \star \left(\xi\_{\sf r}\right)\_{/} \delta \theta\_{/} \star \left(\xi\_{\sf s}\right)\_{/} \delta \theta\_{/\sf r 1} = \left(r\_{2}\right)\_{/\sf r 1^{\sf r}}\tag{50}
$$

$$(\eta\_1)\_/\delta z\_/ \star (\eta\_2)\_/\delta z\_{/\bullet 1} \star (\eta\_3)\_/\delta f\_/ \star (\eta\_4)\_/\delta f\_{/\bullet 1} \star$$

$$
\sigma\_{\mathfrak{s}}(\eta\_{\mathfrak{s}})\_{/} \delta p\_{/} \star (\eta\_{\mathfrak{s}})\_{/} \delta p\_{/\mathfrak{s}1} \star (\eta\_{\mathfrak{r}})\_{/} \delta \theta\_{/} \star (\eta\_{\mathfrak{s}})\_{/} \delta \theta\_{/\mathfrak{s}1} = (r\_{3})\_{/ \mathfrak{s}1^{\prime}} \tag{51}
$$

$$
\delta \delta p\_{\rangle} - \delta p\_{\rangle\_{\bullet 1}} - h\_{\rangle} \delta q\_{|\bullet \vartheta 2} = (r\_4)\_{|\bullet \vartheta 2} \tag{52}
$$

$$(\delta\theta\_{\rangle} - \delta\theta\_{\rangle\omega\_1} - h\_{\rangle}\delta t\_{\rangle\omega\_{\uparrow}t\_2} = (r\_{\mathfrak{g}})\_{|\omega\rangle\omega\_{\uparrow}t\_2} \tag{53}$$

where

$$\zeta\_1^{\varepsilon} = \left[ A - n(n-1) \left\{ \frac{q\_{\mid} - q\_{\mid \sim 1}}{2} \right\} \left( -q\_{\mid \sim \ell^2} \right)^{n-2} + n \left\{ -q\_{\mid \sim \ell^2} \right\}^{n-1} + \frac{Dh\_{\mid}}{2} f\_{\mid \sim \ell^2} \right] \tag{54}$$

$$\zeta\_2 = \left[ -A - n(n-1) \left\{ \frac{q\_{\vert\_{\parallel}} - q\_{\vert\_{\perp 1}}}{2} \right\} \left( -q\_{\vert\_{\perp \neq \uparrow 2}} \right)^{n-2} + n \left\{ -q\_{\vert\_{\perp \neq \uparrow 2}} \right\}^{n-1} + \frac{Dh\_{\vert\_{\perp}}}{2} f\_{\vert\_{\perp \neq \uparrow 2}} \right]. \tag{55}$$

$$
\xi\_3^{\varepsilon} = \xi\_4^{\varepsilon} = \frac{Dh\_{\rangle}}{2} q\_{|\omega \cdot l| 2^{\varepsilon}}, \ \xi\_5^{\varepsilon} = \xi\_6^{\varepsilon} = -sh\_{\rangle} p\_{|\omega \cdot l| 2^{\varepsilon}}, \ \xi\_7^{\varepsilon} = \xi\_8^{\varepsilon} = 0,\tag{56}
$$

and

$$
\eta\_1 = 1 + \frac{D \mathrm{Pr}}{2} h\_{\rangle} t\_{\rangle \mathrm{-} \updownarrow \prime \prime} \ \eta\_2 = \eta\_1 - 2 \ \ \eta\_3 = \eta\_4 = \frac{D \mathrm{Pr}}{2} h\_{\rangle} t\_{\rangle \mathrm{-} \updownarrow \prime \prime} \tag{57}
$$

$$
\eta\_5 \equiv \eta\_6 \equiv \text{Brh}\_/\eta\_{\right] \omega \mu\_2} - \frac{\text{Pr} \, Ec}{2} (n+1) h\_j \left( -q\_{\right) \omega \mu\_2} \, \text{} \tag{58}
$$

The right hand sides of Eqs. (49) to (53) are given by

$$r\_1 = -\left(f\_{\rangle} - f\_{\right) + h\_{\rangle}} \mathbf{1} + h\_{\rangle} p\_{\perp \neq l^2 \mathbf{1}'} \tag{59}$$

$$r\_2 = -\left[A\left(q\_j - q\_{j-1}\right) + Dh\_j f\_{|\prec 4|} q\_{|\prec 4|} - sh\_j p\_{|\prec 4|}^2\right] - n\left(q\_j - q\_{|\prec 1}\right)\left(-q\_{|\prec 4|}\right)^{\nu = 1},\tag{60}$$

$$\Pr\_{3} = -\left[ \left( t\_{/} - t\_{/ \text{-} 1} \right) + D \Pr h\_{/} f\_{|\omega \downarrow \text{-} \downarrow \text{-} \downarrow \text{-} \downarrow} + Br h\_{/} g\_{|\omega \downarrow \text{-} \downarrow \text{-} \downarrow}^{2} + \Pr Ec \left( -q\_{/ \text{-} \downarrow \text{-} \downarrow} \right)^{\text{v} \star 1} \right] \tag{61}$$

<sup>4</sup> ( <sup>1</sup> ) 12 5 ( <sup>1</sup> ) 1 2 , . *j j j j j j jj r p p hq r t t ht* - - - - = - - + = - - + (62)

The boundary conditions (46) and (47) become

<sup>1</sup> 12 4 12 () , *j j jj j*

<sup>1</sup> 12 5 12 () , *j j jj j*

<sup>1</sup> 1 2 1 2 1 2 ( 1) , 2 2

<sup>2</sup> 1 2 1 2 1 2 ( 1) , 2 2

3 4 12 5 6 12 7 8 , , 0, <sup>2</sup>

1 12 2 1 3 4 1 2 Pr Pr <sup>1</sup> , 2, , 2 2 *j j j j D D*

5 6 1 2 ( 1 2 ) Pr ( 1) . <sup>2</sup>

*j j j j Ec*

( ) ( )( ) <sup>1</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> 12 12 1 2 1 12 , *<sup>n</sup> j j jj j jj jj j r A q q Dh f q sh p n q q q* -

( ) ( ) <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>1</sup> 12 12 1 2 1 2 Pr Pr , *<sup>n</sup> j j jj j j j <sup>j</sup> r t t D h f t Brh q Ec q* <sup>+</sup> - - - - -


<sup>é</sup> <sup>ù</sup> <sup>=</sup> - - <sup>+</sup> <sup>+</sup> <sup>+</sup> - êë úû (61)

<sup>4</sup> ( <sup>1</sup> ) 12 5 ( <sup>1</sup> ) 1 2 , . *j j j j j j jj r p p hq r t t ht* - - - - = - - + = - - + (62)

 hh*h t* - - = + = - = = (57)

hh

*j j j*

 xx

é æ - ö ù <sup>=</sup> ê- - - <sup>ç</sup> <sup>÷</sup> - <sup>+</sup> - <sup>+</sup> <sup>ú</sup> êë <sup>è</sup> <sup>ø</sup> úû

é æ - ö ù <sup>=</sup> <sup>ê</sup> - - <sup>ç</sup> <sup>÷</sup> - <sup>+</sup> - <sup>+</sup> <sup>ú</sup> êë <sup>è</sup> <sup>ø</sup> úû

*A nn q nq <sup>f</sup>* - - -

*A nn q nq <sup>f</sup>* - - -

( ) ( ) 2 1 <sup>1</sup>

*q q Dh*

*n n j j j*

( ) ( ) 2 1 <sup>1</sup>

*q q Dh*

*n n j j j*

*jj j*


*jj j*


 xx*q**sh p* - - = = = = - = = (56)

*n*

<sup>1</sup> ( <sup>1</sup> ) 1 2 , *j j jj r f f hp* - - = - - + (59)

*Brh q n hq* - - = = - + - (58)

*p p hq r* - - - - - = (52)

*ht r* - - - - - = (53)

(54)

(55)

 d

 d

dd

dq dq

*j*

*Dh*

*h f*

The right hand sides of Eqs. (49) to (53) are given by

xx

h h

h

where

and

x

350 Heat Transfer Studies and Applications

x

$$
\delta \delta f\_0 = 0, \ \delta p\_0 = 0, \ \delta q\_0 = 0, \ \delta \theta\_0 = 0, \ \delta q\_N = 0 \text{ and } \ \delta \theta\_N = 0. \tag{63}
$$

The linearized Eqs. (49) to (53) can be solved by using block elimination method as outlined by Na [23]. The iterative procedure is stopped when the difference in computing the velocity and temperature in the next iteration is less than 10<sup>−</sup><sup>5</sup> . The present method is unconditionally stable and has second-order accuracy.

#### **5. Exact solutions for particular cases**

It is pertinent to mention that Eq. (22) has simple exact solution to special cases, namely (i) *n* =0 and *s* =1 [22] and (*ii*) *n* =1 and *s* =3 [24]. For case (*i*), with *Br* =*Ec* =0, Eq. (29) reduces to

$$
\theta^\# \mathbf{+Pr} f \theta^\prime \mathbf{=0}.\tag{64}
$$

The exact solution to Eq. (64) in terms of the incomplete Gamma function, satisfying boundary conditions (30), is

$$\theta\left(\eta\right) = \frac{\Gamma\left(\frac{\text{Pr}}{\rho^{\text{2}}}, 0\right) - \Gamma\left(\frac{\text{Pr}}{\rho^{\text{2}}}, \frac{\text{Pr}}{\rho^{\text{2}}} e^{\mathsf{T}^{\beta\eta}}\right)}{\Gamma\left(\frac{\text{Pr}}{\rho^{\text{2}}}, 0\right) - \Gamma\left(\frac{\text{Pr}}{\rho^{\text{2}}}, \frac{\text{Pr}}{\rho^{\text{2}}}\right)},\tag{65}$$

where *β* = <sup>1</sup> *A* and Γ( ⋅ ) is the incomplete Gamma function. For case (*ii*), with *Br* =*Ec* =0, Eq. (29) reduces to

$$
\partial \!\!\/ \not\!\/ \mathbf{+} 3 \text{Pr} \, f \partial \!\!\/ \mathbf{'} \mathbf{0} = 0. \tag{66}
$$

Here the exact solution of Eq. (66) in terms of incomplete Gamma function, satisfying boundary conditions (30), is

$$\theta\left(\eta\right) = \frac{\Gamma\left(\frac{3\text{Pr}}{a^2}, 0\right) - \Gamma\left(\frac{3\text{Pr}}{a^2}, \frac{3\text{Pr}}{a^2} e^{\Delta\eta}\right)}{\Gamma\left(\frac{3\text{Pr}}{a^2}, 0\right) - \Gamma\left(\frac{3\text{Pr}}{a^2}, \frac{3\text{Pr}}{a^2}\right)}\text{,}\tag{67}$$

where *α* = <sup>3</sup> 1 + *<sup>A</sup>* .
