Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

TL;DR

This paper investigates the existence, multiplicity, and asymptotic behavior of steady laminar flow solutions in a porous channel with asymmetric flow injection and suction, revealing multiple solution types and their high Reynolds number asymptotics.

Contribution

It identifies three types of solutions for asymmetric porous channel flow and constructs their asymptotic forms at high Reynolds numbers using boundary layer methods.

Findings

Three solution types (I, II, III) identified for asymmetric flow.

Unique solutions exist for Reynolds number R<14.10.

Multiple solutions emerge for R>14.10.

Abstract

The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. We show that there exist three solutions designated as type $I$, type $II$ and type $III$ for the asymmetric case. The numerical results suggest that a unique solution exists for the Reynolds number $0\leq R<14.10$ and two additional solutions appear for $R>14.10$. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. Asymptotic solutions are all verified by their corresponding numerical solutions.