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Networks and Heterogeneous Media

2010, Volume 5, Issue 4: 783-812. doi: 10.3934/nhm.2010.5.783
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Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel

  • 1.
    Laboratory of Mathematics of the University of Saint-Etienne (LaMUSE), University Jean Monnet, 23, rue Dr Paul Michelon 42023 Saint-Etienne
  • 2.
    Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014 700 Bucharest
  • Received: 01 February 2010 Revised: 01 October 2010

  • Primary: 76M45; Secondary: 74F10.

  • The non-steady viscous flow in a thin channel with elastic wall is considered. The viscosity is constant everywhere except for some small neighborhood of the origin of the coordinate system, where it is a variable function. The problem contains two small parameters: ε, that is the ratio of the thickness of the channel and its length, and δ=εγ, γ≥3, that is the "softness of the wall", i.e. its inverse (rigidity) is great. An asymptotic expansion of the solution is constructed and, in particular, the leading term is described. An important new element of this paper is the procedure of construction of the boundary layer in the neighborhood of the origin of the coordinate system, generated by the variable viscosity. The error estimates for the difference of a truncated asymptotic ansatz and the exact solution are obtained. To this end, the existence and uniqueness of the solution are studied and some a priori estimates are proved.

    Citation: Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel[J]. Networks and Heterogeneous Media, 2010, 5(4): 783-812. doi: 10.3934/nhm.2010.5.783

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    [2] Grigory Panasenko, Ruxandra Stavre . Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks and Heterogeneous Media, 2010, 5(4): 783-812. doi: 10.3934/nhm.2010.5.783
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  • The non-steady viscous flow in a thin channel with elastic wall is considered. The viscosity is constant everywhere except for some small neighborhood of the origin of the coordinate system, where it is a variable function. The problem contains two small parameters: ε, that is the ratio of the thickness of the channel and its length, and δ=εγ, γ≥3, that is the "softness of the wall", i.e. its inverse (rigidity) is great. An asymptotic expansion of the solution is constructed and, in particular, the leading term is described. An important new element of this paper is the procedure of construction of the boundary layer in the neighborhood of the origin of the coordinate system, generated by the variable viscosity. The error estimates for the difference of a truncated asymptotic ansatz and the exact solution are obtained. To this end, the existence and uniqueness of the solution are studied and some a priori estimates are proved.


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    [13] G. P. Panasenko and R. Stavre, Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall, Networks and Heterogeneous Media, 3 (2008), 651-673.
    [14] G. P. Panasenko, Y. Sirakov and R. Stavre, Asymptotic and numerical modelling of a flow in a thin channel with visco-elastic wall, Int. J. Multiscale Comput. Engng., 5 (2007), 473-482. doi: 10.1615/IntJMultCompEng.v5.i6.40
    [15] D. Serre, Chute libre d'un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math., 4 (1987), 99-110.
    [16] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," 3rd edition, North-Holland, Amsterdam, 1984.

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