A tabulation of the local skin friction coefficient in terms of the comparison between the present results and the HAM results (ref. [22])
1. Introduction
Over the past several decades, the NavierStokes 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 nonNewtonian. That is, they might exhibit dramatic deviation from Newtonian behavior and display a range of nonNewtonian characteristics. A few points of nonNewtonian 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 nonNewtonian fluids are more appropriate than the Newtonian fluid. Consequently, the theory of nonNewtonian 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 nonNewtonian fluids as they possess very complex structure. However, there are some classes of fluids that cannot be classified as Newtonian or purely nonNewtonian such as waterborne coating etc. This situation demands some more general models which can be utilized for analysis of both Newtonian and nonNewtonian 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 nonNewtonian 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 nonNewtonian slurries [3]. The three parameters Sisko fluid model, which can be considered as a generalized powerlaw 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 [410].
Investigations of the boundary layer flow and heat transfer of nonNewtonian 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
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
2. Governing equations
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
In the above equations
where
The extra stress tensor
where
with
The quantity
represents an apparent or effective viscosity as a function of the shear rate. If
For the steady twodimensional axisymmetric flow, we assume the velocity, temperature and stress fields of the form
when
The steady twodimensional and incompressible equations of motion (2) including conservation of mass (1) and thermal energy (3) can be written as
where
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
where
3. Mathematical formulation
3.1. Flow analysis
Consider the steady, twodimensional and incompressible flow of Sisko fluid over a nonlinear radially stretching sheet. The fluid is confined in the region
The boundary conditions associated to flow field are
We define the following variables
where
On employing the above transformations, Eqs. (17) to (19) take the form [21]
where prime denotes differentiation with respect to
The physical quantity of major interest is the local skin friction coefficient and is given by [21]
3.2. Heat transfer analysis
In the assumption of boundary layer flow, the energy equation for the nonNewtonian Sisko fluid taking into account the viscous dissipation effects and neglecting the heat generation effects for the temperature field
The corresponding thermal boundary conditions are
Using the transformations (20) the above problem reduces to
where
The local Nusselt number
where the wall heat flux at the wall is
4. Solution procedure
The two point boundary value problems comprising Eqs. (22) and (29) along with the associated 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 firstorder differential equations in
where
The boundary conditions in terms of new variable are written as
The functions and their derivatives are approximated by central difference at the midpoint
Using the finite difference approximations equations (33) to (37) can be written as
where
Boundary conditions (38) and (39) are written as
Eqs. (41) to (45) are system of nonlinear equations and these equations are linearized employing the Newton’s method and using the expressions:
where
Putting the left hand side of the above expressions into Eqs. (41) to (45) and dropping the quadratic terms in
where
and
The right hand sides of Eqs. (49) to (53) are given by
The boundary conditions (46) and (47) become
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
5. Exact solutions for particular cases
It is pertinent to mention that Eq. (22) has simple exact solution to special cases, namely (i)
The exact solution to Eq. (64) in terms of the incomplete Gamma function, satisfying boundary conditions (30), is
where
For case
Here the exact solution of Eq. (66) in terms of incomplete Gamma function, satisfying boundary conditions (30), is
where
6. Validation of numerical results
The validation of present results is essential to check the credibility of the numerical solution methodology. The presently computed results are compared with the exact solutions obtained for some limiting cases of the problem. Figures 2 and 3 compare these results, and an excellent correspondence is seen to exist between the two sets of data. In addition, table 1 shows the comparison values of the local skin friction coefficient with those reported by of Khan and Shahzad [22]. It is seen that the comparison is in very good agreement, and thus gives us confidence to the accuracy of the numerical results.
7. Results and discussion
The main focus of the present chapter is to study the flow and heat transfer characteristics of a Sisko fluid over a nonlinear radially stretching sheet. To obtain physical insight of the flow and heat transfer, Eqs. (22) and (29) subject to boundary conditions (23) and (30) are solved numerically and the results are illustrated graphically. During the ensuing discussion, the assumption of incompressibility and isotropy of fluid is implicit. The influence of the flow behavior index
Figure 4 depicts the influence of the powerlaw index
The heat transfer aspects of the Sisko fluid over a constant surface temperature stretching sheet for shear thinning
The stretching parameter
The effect of the material parameter
The Prandtl number
Eckert number
Figures 10(a,b) compare the velocity and temperature profiles of the Bingham (
Adiabatic Eckert number
Table 2 summarizes the overall trends of the skin friction coefficient for shear thinning and thickening fluids when the material parameter
The stream function appearing in Eq. (20) is plotted in figure 12 for several values of











0  1  1.173721  1.173721  1.189598  1.189567 
1  1  1.659892  1.659891  1.605002  1.605010 
2  1  2.032953  2.032945  1.973087  1.973092 
3  1  2.347441  2.347451  2.297733  2.297713 
1  2  2.090755  2.090753  2.152145  2.152153 
1  3  2.449490  2.449491  2.621176  2.621182 










1.0  1.0  0.1  1.1738050  1.177395  1.62676  1.39104 
2.0  1.0  0.1  2.116299  1.233161  2.00146  1.45209 
3.0  1.0  0.1  2.427310  1.264815  2.33186  1.48665 
1.0  2.0  0.1  2.075148  1.125830  2.12359  1.60519 
2.0  2.0  0.1  2.544792  1.191590  2.58828  1.67792 
3.0  2.0  0.1  2.928524  1.228861  2.99557  1.72052 
1.5  1.5  0.1  2.140943  1.186123  2.107053  1.54061 
1.5  1.5  0.3  2.140943  1.043431  2.107053  1.44104 
1.5  1.5  0.5  2.140943  0.900738  2.107053  1.34148 





1  0.789183  1.019614 
2  1.186123  1.540612 
3  1.487571  1.935358 
4  1.739727  2.265692 
5  1.960520  2.555230 
8. Concluding remarks
In this chapter, a theoretical framework for analyzing the boundary layer flow and heat transfer with viscous dissipation to Sisko fluid over a nonlinear radially stretching sheet has been formulated. The governing partial differential equations were transformed into a system of nonlinear ordinary differential equations. The transformed ordinary differential equations were then solved numerically using implicit finite difference scheme along with Kellerbox scheme. The results were presented graphically and the effects of the powerlaw index
Our computations have indicated that the momentum and thermal boundary layers thickness were decreased by increasing the powerlaw index and the material parameter. Further it was noticed that the effects of the Prandtl and Eckert numbers on the temperature and thermal boundary layer were quite opposite. However, both the Prandtl and Eckert numbers were affected dominantly for shear thinning fluid as compared to that of the shear thickening fluid. Additionally, the Bingham fluid had the thickest momentum and thermal boundary layers as compared to those of the Sisko and Newtonian fluids.
Acknowledgments
This work has been supported by the Higher Education Commission (HEC) of Pakistan.
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