Green's Function for Laminar Flow in Channels with Porous Walls in the Presence of a Transverse Magnetic Field

Abstract

Despite the significant and ongoing interest in Green's functions from scientists, engineers and mathematicians, the area remains underdeveloped with respect to understanding problems from laminar fluid flow and magnetohydrodynamics in porous media. The purpose of this paper is to partially address this gap by constructing a new and explicit representation of the Green's function for a boundary value problem that is derived from laminar flow in channels with porous walls in the presence of a transverse magnetic field. We discuss some interesting consequences of our constructed Green's function, including: the establishment of an equivalent integral equation; and the generation of new information regarding solutions to our boundary value problem. We discover that, for any given transverse magnetic field, our laminar flow problem has a unique solution in a particular location provided the Reynolds number is sufficiently small, and that the solution may be approximated by Picard iterations.