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On the steady motion of a coupled system solid-liquid

About this Title

Josef Bemelmans, Institut für Mathematik, RWTH-Aachen, Germany, Giovanni P. Galdi, Department of Mechanical Engineering and Materials Science, University of Pittsburgh, U.S.A and Mads Kyed, Fachbereich Mathematik, Technische Universität Darmstadt, Germany

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 226, Number 1060
ISBNs: 978-0-8218-8773-8 (print); 978-1-4704-1060-5 (online)
Published electronically: March 1, 2013
Keywords:Navier-Stokes, nonlinear elasticity, fluid-structure interaction
MSC: Primary 35Q30, 76D05, 74B20, 74F10, 35R35

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Notation and Preliminaries
  • Chapter 3. Steady Free Motion: Definition and Formulation of the Problem
  • Chapter 4. Main Result
  • Chapter 5. Approximating Problem in Bounded Domains
  • Chapter 6. Proof of Main Theorem
  • Chapter 7. Bodies with Symmetry
  • Appendix A. Isolated Orientation


We study the unconstrained (free) motion of an elastic solid $\mathcal B$ in a Navier-Stokes liquid $\mathcal L$ occupying the whole space outside $\mathcal B$, under the assumption that a constant body force $\mathfrak{b}$ is acting on $\mathcal B$. More specifically, we are interested in the steady motion of the coupled system $\{\mathcal B,\mathcal L\}$, which means that there exists a frame with respect to which the relevant governing equations possess a time-independent solution. We prove the existence of such a frame, provided some smallness restrictions are imposed on the physical parameters, and the reference configuration of $\mathcal B$ satisfies suitable geometric properties.

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