A Study on Numerical Solutions for Rotating Flows of a Third Grade Fluid with Partial Slip Conditions

Abstract

In this paper rotating flow of a third grade fluid past a porous plate with partial slip is studied .The nonlinear boundary value problem is solved using finite difference method. The variations of velocity components for various values of partial parameter �� 1are discussed the results are reported for conclusion. Keyword: third grade fluid, partial slip parameter, finite difference method, non - Newtonian fluid. I. Introduction Materials such as polymer solutions or melts, drilling mud, clastomers, certain oils and greases and many other emulsions are classified as non-Newtonian fluids. Due to complexity of fluids, there are many models describing the properties, but not all of non-Newtonian fluids. These models, however, cannot predict all the behaviours of non-Newtonian fluids, for example, normal stress differences, shear thinning or shear thickening, shear relaxation, elastic and memory effects etc. The pioneers of the field such as Coulomb, Navier, Girad, Poisson, Stokes, St. Venant and others recognised that boundary conditions are constitutive equations that should be determined by the material on either side of the boundary. The usual prescription of Dirichlet and Neumann conditions are often unsuitable for a realistic physical problem, for example the flow of polymers that stick-slip on the boundary. Recently non-standard boundary conditions have been considered from a rigorous mathematical perspective by Rao and Rajagopal. The earliest work S.Asgar,M.Mudassar Gulzar, M.Ayub (2006) are derived an analytical study of the rotating flow of a third grade fluid past a porous plate with partial slip effects. It serves as a flow model for the study of polymers. The analytic solution has been determined using homotopy analysis method (HAM). The work is extended to examine the effects of partial slip on the rotating flow past a uniformly porous plate. The fluid is incompressible and third grade .The numerical solution has been obtained using Finite difference method and finally focused on the effects of suction, blowing, third grade and slip parameter. II. A Study on rotating flow of a third grade fluid Consider the Cartesian coordinate system rotating uniformly with an angular velocity ٠about the z- axis ,taken positive in the vertically upward direction and the plate coinciding with the plane ��= 0 . The fluid flowing past a porous plate is third grade and incompressible .All the material parameters of the fluid are assumed to be constants . For the rotating frame, the equation of momentum