Impact of Inclined Magnetic Field on the Dynamics of Two Immiscible Viscous Fluids Flow in an Inclined Channel with Variable Permeability Porous Medium: A Finite Difference Approach

Abstract

This paper utilizes the finite difference method to numerically analyze the fluid flow in an inclined channel filled with a porous medium and containing two immiscible, viscous, incompressible, and electrically conducting fluids. These fluids have different viscosities and are arranged in two equal-width separate layers within the channel. The channel experiences a magnetic field that is inclined, and the permeability of the porous material changes across the width of the channel. The upper layer’s fluid has lower viscosity than the lower layer’s fluid. The Brinkman equation is employed to characterize the flow within the porous material. The paper presents numerical expressions for velocity and volumetric flow rate using the finite difference method, incorporating no-slip boundary conditions at the top and bottom plates and continuity of velocity with shear stress continuity at the interface. The effects of key parameters, such as the Hartmann number, Gravitational parameter and the permeability parameter, etc on the velocity distribution and volumetric flow rate are investigated. The results are graphically represented and thoroughly discussed.

Keywords: Brinkman equation; Variable Permeability; Hartmann number; viscosity ratio parameter; Finite difference method.

2020 Mathematics Subject Classification. 76S05; 76W05; 76M20