Numerical Solutions for Some Problems of Flow of the non-Newtonian Fluids on Di erent Surfaces

Abstract

This thesis contains numerical and analytical solutions for systems of nonlinear equations governing uid ow, heat transfer and the concentration of some non􀀀Newtonian uids through di erent geometric shapes. Also presented a study to the error analysis in numerical methods by comparing it with the analytical methods and previously published work. It should be noted that the solution of the current results is obtained by designing Fortran, Matlab and Mathematica programmes and then the present graphics is drawn by designing Excel and Matlab programmes. This thesis consists of six chapters, which are followed by lists of references. Chapter(1) The introductory chapter is considered as a background for the material included in the thesis. The purpose of this chapter is to present a short introduction on numerical analysis and uid mechanics, a brief survey of famous numerical and analytical methods which using to solve uid mechanics problems, uid properties and the basic ow equations. Moreover, it contains a short survey of some needed concepts of the material used in this thesis. Chapter(2) The purpose of this chapter is to study the e ect of Papanastasiou viscosity on steady MHD ow and heat transfer between two parallel plates in the presence of dissipations and radiation. The dimensional quantities are applied to transform the governing equations into nonlinear partial di erential equations. In addition, the resulting equations solved numerically by using the nite di erence method (FDM). v SUMMARY Moreover, numerical results are presented for the distribution of velocity, temperature and local Nusselt number pro les for various parametric conditions. The e ects of varying the yield stress parameter D, the Hartman number Ha, Brinkman number Br and the radiation parameter are determined. In order to verify the e ciency of the proposed method in comparison with Di erential transform method (DTM), a comparison is presented in tables and gures for di erent values of di erent parameters. The tables and gures clearly show that the results by (FDM) are in good agreement with the results of analytical solution by using (DTM). Furthermore, at the end of this chapter the conclusions are summarized. Some results of this chapter was published in (Eleventh International Conference of Fluid Dynamics (ICFD11)), December 19􀀀21, 2013, Alexandria, Egypt. Chapter(3) The aim of this chapter is to study the e ect of chemical reaction and radiation on heat and mass transfer of stagnation point ow of micropolar uid through a porous medium. The governing equations are transformed into nonlinear ordinary di erential equations by applying the similarity transformation and then solved numerically by applying (FDM). The solutions are found to be governed by six parameters, the porosity parameter M, the material parameter K, the thermal radiation parameter Rd, the Prandtl number Pr, the Schmidt number Sc and the reaction􀀀rate parameter . Numerical results are presented the distribution of velocity, temperature and concentration pro les. Furthermore, comparisons of the present results with previously published work show that the present results have high accuracy and are found to be a good agreement. At the end of this chapter, the conclusions are summarized. Some results of this chapter are accepted for (INTERNATIONAL JOURNAL OF APPLIED MATHE- MATICS AND PHYSICS). Chapter(4) The main goal of this chapter is to study numerical and analytical treatment of MHD natural convection of an incompressible uid between two in nite parallel vertical plates through a porous medium vi SUMMARY using (FDM) and Multi􀀀step di erential transform method (MDTM). The governing equations are transformed into nonlinear partial di erential equations by applying the similarity variables and then solved numerically by applying (FDM) and analytically by using (MDTM). Figures illustrate the e ects of dimensionless non􀀀Newtonian viscosity , Prandtl number Pr, Eckert number E, porosity parameter Mp and magnetic parameter Mm on the nondimensional velocity and temperature. The gures and tables clearly show that the results by (FDM) and (MDTM) are in good agreement with the results of analytical solution of previously published works by using Homotopy perturbation method (HPM), Adomian decomposition method (ADM), Homotopy analysis method (HAM) and Di erential transform method (DTM). The work in this chapter is preparing to publication.