The Asymptotic Analysis And Computation For Laminar Flows Through A Porous Channel With Different Permeabilities

The laminar flow in a channel or pipe has wide applications in industrial engineering and biomedicine,researchers have carried out a lot of research in the above fields and obtained many meaningful results,however,there is no reasonable description method and flow mechanism for asymmetric flow.This paper extends the research of channel flow from special symmetrical flow to more general asymmetric flow and is devoted to the study of the asymmetric flow in a permeable channel and its asymptotic analysis and computation.First,we consider the the laminar flow problem in a two-dimensional porous channel of Navier Stokes equation with asymmetric boundary condition after a similarity transformation,and calculate the similarity multiple solutions(three solutions)in different parameter ranges,and classify these solutions according to Reynolds number and other parameters,and the behavior of solutions.These solutions appear in different Reynolds number ranges,especially when the Reynolds number increases,there are multiple solutions for laminar flow problems.Type ? solution exists for all non-negative Reynolds number and types II and III solutions appear simultaneously as Reynolds number is bigger than a bifurcation point,which epends on the asymmetric parameter.Furthermore,in order to verify the accuracy of the number of solutions obtained by numerical calculation,the existence of multiple solutions(three solutions)is proved theoretically through the analysis method,and the numerical results are consistent with them.Then for the high Reynolds number case we construct asymptotic solution for each of these three solutions using various singular perturbation methods(e.g.boundary layer correction and matched asymptotic expansion).Further these solutions are verified through numerical computations.Then,the mechanism of fluid transport in biological organisms is often closely related to the mechanism of fluid flow through retractable permeable channel.Hence,the paper considered an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field.The flow is driven by the expanding or contracting walls with different permeability.The asymptotic solutions are well constructed with singular perturbation method in two cases with large Reynolds numbers,i.e.,both walls of the channel are with suction,and one of the walls is with injection while the other one is with suction.For small Reynolds number cases,the regular perturbation method is used to construct the asymptotic solution.All the asymptotic results are finally verified by numerical results,meanwhile,the numerical solution is also verified by the asymptotic solution.Further more,the fluid flow in organism shows the characteristics of non-Newtonian fluid,which is similar to that of micropolar fluid.Hence,the laminar flow of a micropolar fluid in a uniformly porous channel with expanding or contracting walls under asymmetric boundary conditions is investigated.First,In order to overcome the difficulty of numerical calculation and make the parametric study more convenient,the singular perturbation method is used to construct the asymptotic solution for the case of high Reynolds number.Moreover,asymptotic solutions are generally required for the validation of numerical methods and computer softwares.Finally,the influence of Reynolds number,asymmetric parameter,expansion ratio and micropolar parameter on the flow is discussed numerically.