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The method of fundamental solution for the creeping flow around a sphere close to a membrane
Author(s) -
Debbech A.,
Elasmi L.,
Feuillebois F.
Publication year - 2010
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200900394
Subject(s) - stokes flow , gravitational singularity , mathematical analysis , mathematics , flow (mathematics) , particle (ecology) , boundary value problem , exact solutions in general relativity , mechanics , geometry , physics , oceanography , geology
The method of fundamental solution is used to calculate the creeping flow around a spherical solid particle close to a porous membrane. The equations for the flow in the porous medium and conditions at the interface are satisfied automatically with the use of a Green function calculated by Elasmi and Feuillebois [11]. Singularities, i.e. the Green function and some of its derivatives, are distributed inside the particle and their positions and intensities are optimized by minimizing the difference between the approximate velocity and the exact one on the particle surface in the sense of least squares. It is proved that this procedure also minimizes the dissipated energy of the error velocity, that is of the difference between the approximation and the exact value. For the example cases treated here of a sphere moving normal to a wall, the singularities are stokeslets and stokeslet quadrupoles (or source doublets). In the particular case of an impermeable wall, the method is the same as that of Zhou and Pozrikidis [19]. Their results are recovered and extended to a higher number of singularities. The method is then applied to the case of a solid sphere moving normal to a thin porous slab. By comparison with results obtained by Elasmi and Feuillebois [11] with the boundary integral method, it is found that a good approximation is obtained here with only a few singularities. A comparison is also made with Goren [13] who treated analytically a related problem in the case of a low porosity.