When is the Porous, Laminar Flow Problem with Slip Condition Well Posed? And Where Does the Solution Lie?

Abstract

The aim of this article is to advance the current state of knowledge for steady, isothermal, incompressible, laminar flow within a channel featuring a non-zero tangential (or slip) velocity at the permeable walls. There has been significant interest in understanding the solutions to these problems. However, a firm mathematical understanding of the solutions to the slip problem and their properties is yet to be fully developed. For example, we still do not know: if the slip problem is well-posed; where the precise solution lies; if and how approximations converge to the solution; and what the estimates on approximation errors are. Herein we formulate a new mathematical foundation that includes existence; uniqueness; location; approximation; convergence and error estimates. Our strategy involves developing insight via new and interesting connections between the boundary value problem arising from modelling the laminar flow with slip velocity, and the theory of fixed points of operators.