Analytical Solution of Modified Bingham Fluid Flow through Parallel Plates Channel Subjected to Forchheimer Medium and Hall Current Using Linearized Differential Transformation Method

Abstract

The study of a modified Bingham non-Newtonian fluid flow in Forchheimer porous medium subjected to Hall current is very important in many engineering applications such as petroleum engineering, discharge of groundwater in aquifers and MHD generators. The viscosity of modified Bingham fluid is expanded to powers of velocities and their derivatives using Taylor’s series to make the governing equations be familiar with the differential transformation method. Forchheimer model is needed at a high flow rate, where, flow will exhibit non–linearity with respect to velocity which make Darcy law inapplicable at these conditions. In this case, the momentum equations become highly non-linear. The classical differential transformation method transforms the non-linear differential equations to a non-linear algebraic system which gives more than one solution and may be unstable that leads to divergence of the required solution. The linearized differential transformation method (LMDTM) is a power series solution avoiding solution multiplicity and divergence by a linearization technique that is applied on non-linear governing equations to obtain the unique and convergent solution. The uniqueness, convergence and stability of the new technique are tested by comparisons with previously available results and it is also verified by the finite difference solution. The effects flow parameters on the velocity and friction factor are illustrated. The values of parameters in the present work are chosen according to: the available previous results, the power of the present method to compute over a large range of parameters and the distinctiveness between curves in figures and tables.