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.

中文翻译：

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层流通过具有不同渗透率的多孔通道的多重解及其渐近性

研究了具有两个可渗透壁的均匀多孔通道中稳定，不可压缩和充分展开的层流的相似解的存在性和多重性。我们将集中在所谓的非对称情况下，其中上壁具有一定的流量注入量，而下壁具有不同的吸力量。数值结果表明存在三种指定为| $ I $ |⁠，类型| $ II $ |的解。并输入| $ III $ | 对于非对称情况，键入| $ I $ | 存在所有非负雷诺数和类型| $ II $ |的解。和| $ III $ |解同时出现在公共雷诺数上，该雷诺数取决于非对称参数| $ a $ |的值。并随着| $ a $ |的增加 常见的雷诺数正在减少。我们还从理论上证明存在三种解决方案。对于最困难的高雷诺数情况，通过边界层校正或匹配渐近展开的方法构造了多个解中每个解的对应渐近解。这些渐近解都通过其相应的数值解来验证。