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Viscous Flows in Domains with a Multiply Connected Boundary

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New Directions in Mathematical Fluid Mechanics

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

In this paper we consider stationary Navier-Stokes equations in a bounded domain with a boundary, which has several connected components. The velocity vector is given on the boundary, where the fluxes differ from zero on its components. In the general case, the solvability of this problem has been an open question up to now. We provide a survey of previous results, which deal with partial versions of the problem. We construct an a priori estimate of the Dirichlet integral for a velocity vector in the case when the flow has an axis of symmetry and a plane of symmetry perpendicular to it, which also intersects each component of the boundary. Having available this estimate, we prove an existence theorem for the axially symmetric problem in a domain with a multiply connected boundary. We consider also the problem in a curvilinear ring and formulate a conditional result concerning its solvability.

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Author information

Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Lavrentyev Prospect 15, Novosibirsk, 630090, Russia

    V. V. Pukhnachev

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  1. V. V. Pukhnachev
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Editors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Vorob’evy Gory, 119991, Moscow, Russia

    Andrei V. Fursikov

  2. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, 630 Benedum Engineering Hall, Pittsburgh, PA, 15261, USA

    Giovanni P. Galdi

  3. Lavrentyev Institute of Hydrodynamics, Lavrentyev prospect 15, 630090, Novosibirsk, Russia

    Vladislav V. Pukhnachev

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Pukhnachev, V.V. (2009). Viscous Flows in Domains with a Multiply Connected Boundary. In: Fursikov, A.V., Galdi, G.P., Pukhnachev, V.V. (eds) New Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0152-8_17

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