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On the eigenfunctions of the Stokes operator in a plane layer with a periodicity condition along it

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Abstract

In Rummler’s previous paper, formulas for the eigenfunctions of the Stokes operator were derived (in a rather concise form) in the case of a three-dimensional layer with a periodicity condition in orthogonal directions along the layer. In this paper, eigenfunctions and associated pressures are constructed and studied in a plane n-dimensional (specifically, two-dimensional) layer with a periodicity condition in orthogonal directions along the layer. A very simple and useful velocity representation in terms of the pressure gradient is used. As a result, the derivation of formulas is considerably simplified and reduced without applying cumbersome expressions and the eigenfunctions are expressed in terms of the associated pressures. Two-sided estimates are given, and the asymptotic behavior of nontrivial eigenvalues of the Stokes operator is analyzed.

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References

  1. B. Rummler, “The eigenfunction of the Stokes operator in special domains II,” Z. Angew. Math. Mech. 77, 669–675 (1997).

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Authors and Affiliations

  1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    B. V. Pal’tsev

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  1. B. V. Pal’tsev
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Correspondence to B. V. Pal’tsev.

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Original Russian Text © B.V. Pal’tsev, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 2, pp. 286–297.

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Pal’tsev, B.V. On the eigenfunctions of the Stokes operator in a plane layer with a periodicity condition along it. Comput. Math. and Math. Phys. 54, 303–314 (2014). https://doi.org/10.1134/S0965542514020109

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  • DOI: https://doi.org/10.1134/S0965542514020109

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