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A numerical study for multiple solutions of a singular boundary value problem arising from laminar flow in a porous pipe with moving wall. (English) Zbl 1354.35106
Summary: This paper is concerned with multiple solutions of a singular nonlinear boundary value problem (BVP) on the interval \([0, 1]\), which arises in a study of the laminar flow in a porous pipe with an expanding or contracting wall. For the singular nonlinear BVP, the correct boundary conditions are derived to guarantee that its linearization has a unique smooth solution. Then a numerical technique is proposed to find all possible multiple solutions. For the suction driven pipe flow with the expanding wall (e.g.  \(\alpha = 2\)), we find a new solution numerically and classify it as a type VI solution. The computed results agree well with what can be obtained by the bifurcation package AUTO. In addition, we also construct asymptotic solutions for a few cases of parameters, which agree well with numerical solutions. These serve as validations of our numerical results. Thus we believe that the numerical technique designed in the paper is reliable, and may be further applied to solve a variety of nonlinear equations that arise from other flow problems.

MSC:
35Q35 PDEs in connection with fluid mechanics
76S05 Flows in porous media; filtration; seepage
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
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