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Steady Flow of a Navier–Stokes Liquid Past an Elastic Body

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Abstract

We perform a mathematical analysis of the steady flow of a viscous liquid, \({\mathcal{L}}\) , past a three-dimensional elastic body, \({\mathcal{B}}\) . We assume that \({\mathcal{L}}\) fills the whole space exterior to \({\mathcal{B}}\) , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v ∞. As for \({\mathcal{B}}\) , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady state fluid-structure problem with the surface of \({\mathcal{B}}\) as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small |v ∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.

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Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA

    Giovanni P. Galdi

  2. Institut für Mathematik, RWTH-Aachen, Aachen, Germany

    Mads Kyed

Authors
  1. Giovanni P. Galdi
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  2. Mads Kyed
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Correspondence to Giovanni P. Galdi.

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Communicated by S.S. Antman

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Galdi, G.P., Kyed, M. Steady Flow of a Navier–Stokes Liquid Past an Elastic Body. Arch Rational Mech Anal 194, 849–875 (2009). https://doi.org/10.1007/s00205-009-0224-y

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  • DOI: https://doi.org/10.1007/s00205-009-0224-y

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