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

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. The numerical results suggest that there exist three solutions designated as type I, type II and type III for the asymmetric case, type I solution exists for all non-negative Reynolds number and types II and III solutions appear simultaneously at a common Reynolds number that depends on the value of asymmetric parameter a and with the increase of a the common Reynolds numbers are decreasing. We also theoretically show that there exist three solutions. 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. These asymptotic solutions are all verified by their corresponding numerical solutions.