Applied Mathematics and Mechanics (English Edition) ›› 2014, Vol. 35 ›› Issue (2): 203-220.doi: https://doi.org/10.1007/s10483-014-1784-8

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Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

张燕1 林平1,2 司新辉1   

  1. 1. Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, P. R. China;
    2. Department of Mathematics, University of Dundee, Dundee DD1 4HN, U. K.
  • 收稿日期:2012-12-17 修回日期:2013-04-24 出版日期:2014-02-27 发布日期:2014-02-18

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

 ZHANG Yan1, LIN Ping1,2, SI Xin-Hui1   

  1. 1. Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, P. R. China;
    2. Department of Mathematics, University of Dundee, Dundee DD1 4HN, U. K.
  • Received:2012-12-17 Revised:2013-04-24 Online:2014-02-27 Published:2014-02-18

摘要: The cases of large Reynolds number and small expansion ratio for the asymmetric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Numerical methods are also designed to confirm the correctness of the present asymptotic solutions.

关键词: 泵, 缝隙高度可变, 压力分布模拟, 自适应有限单元法, singular perturbation method, regular perturbation method, porous expanding channel, expansion ratio

Abstract: The cases of large Reynolds number and small expansion ratio for the asymmetric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Numerical methods are also designed to confirm the correctness of the present asymptotic solutions.

Key words: pump, variable height gap, pressure distribution modeling, adaptive finite element method, singular perturbation method, regular perturbation method, porous expanding channel, expansion ratio

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