**8. Concluding remarks**

1 <sup>2</sup> Re*<sup>b</sup>* 1 *<sup>n</sup>*+1 *Cf n* =1 *n* =2

*A s Numerical results HAM results Numerical results HAM results* 1 -1.173721 -1.173721 -1.189598 -1.189567 1 -1.659892 -1.659891 -1.605002 -1.605010 1 -2.032953 -2.032945 -1.973087 -1.973092 1 -2.347441 -2.347451 -2.297733 -2.297713 2 -2.090755 -2.090753 -2.152145 -2.152153 3 -2.449490 -2.449491 -2.621176 -2.621182

**Table 1.** A tabulation of the local skin friction coefficient in terms of the comparison between the present results and

− 1 *<sup>n</sup>*+1 *N ur*

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

> Re*<sup>b</sup>* − 1 *<sup>n</sup>*+1 *N ur* Pr *n* =0.5 *n* =1.5 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

**Table 3.** A tabulation of the local Nusselt number for different values of Prandtl number when *s* =1.5 and *A*=1.5 are

*n* =0.5 *n* =1.5

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

*<sup>n</sup>*+1 *Cf* Re*<sup>b</sup>*

− 1 *<sup>n</sup>*+1 *N ur*

the HAM results (ref. [22])

360 Heat Transfer Studies and Applications

fixed.

*A s Ec* <sup>1</sup>

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

**Table 2.** A tabulation of the local skin friction and the local Nusselt number.

*<sup>n</sup>*+1 *Cf* Re*<sup>b</sup>*

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 Keller-box scheme. The results were presented graphically and the effects of the power-law index *n*, the material parameter *A*, the stretching parameter *s*, the Prandtl number Pr, and the Eckert number were discussed. It is pertinent to mention that the analysis in ref. [22] for velocity field was restricted to integer value of the power-law index *n*. However, the investigations in the present work were upgraded by adding the non-integral values of the power-law index *n* for the flow and temperature fields.

Our computations have indicated that the momentum and thermal boundary layers thickness were decreased by increasing the power-law 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.

### **Acknowledgements**

This work has been supported by the Higher Education Commission (HEC) of Pakistan.

#### **Author details**

Masood Khan1\*, Asif Munir1 and Azeem Shahzad2

\*Address all correspondence to: mkhan@qau.edu.pk; mkhan\_21@yahoo.com

1 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

2 Department of Basic Sciences, University of Engineering and Technology, Taxila, Pakistan
