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Steady weak solutions to an inflow/outflow driven compressible fluid-structure interaction problem

Boris Muha, Šárka Nečasová, Milan Pokorný, Srđan Trifunović, Justin T. Webster

Abstract

We study a stationary 3D/2D fluid-structure interaction problem between an elastic structure described by the linear plate equation and a fluid described by the compressible Navier-Stokes equations with hard-sphere pressure and inflow/outflow boundary data. This problem is motivated by wind-tunnel configuration and by the need for physically relevant steady states about which compressible flow-plate dynamics can be linearized. The main difficulty in the analysis is the lack of uniform estimates, both for approximate and weak solutions. In particular, the fixed-point construction for approximate solution yields a density estimate depending on approximate parameter, while the pressure estimate for the weak solution is only finite and non-quantifiable. As a result, large pressure loads can drive outward volume growth, while low pressure regions may lead to contact and therefore domain degeneration. This necessitates a novel approach based on a Lipschitz \emph{domain-correction} (barrier) mechanism that provides a framework in which solutions can be constructed without volume blow-up or degeneration of the domain. Constrained by the possibly very large fluid pressure load, our main result is the existence of a weak solution for a sufficiently large plate stiffness. Keywords: fluid-structure interaction, compressible Navier-Stokes, stationary weak solutions, hard-sphere pressure, inflow/outflow, linear plate, mathematical aeroelasticity

Steady weak solutions to an inflow/outflow driven compressible fluid-structure interaction problem

Abstract

We study a stationary 3D/2D fluid-structure interaction problem between an elastic structure described by the linear plate equation and a fluid described by the compressible Navier-Stokes equations with hard-sphere pressure and inflow/outflow boundary data. This problem is motivated by wind-tunnel configuration and by the need for physically relevant steady states about which compressible flow-plate dynamics can be linearized. The main difficulty in the analysis is the lack of uniform estimates, both for approximate and weak solutions. In particular, the fixed-point construction for approximate solution yields a density estimate depending on approximate parameter, while the pressure estimate for the weak solution is only finite and non-quantifiable. As a result, large pressure loads can drive outward volume growth, while low pressure regions may lead to contact and therefore domain degeneration. This necessitates a novel approach based on a Lipschitz \emph{domain-correction} (barrier) mechanism that provides a framework in which solutions can be constructed without volume blow-up or degeneration of the domain. Constrained by the possibly very large fluid pressure load, our main result is the existence of a weak solution for a sufficiently large plate stiffness. Keywords: fluid-structure interaction, compressible Navier-Stokes, stationary weak solutions, hard-sphere pressure, inflow/outflow, linear plate, mathematical aeroelasticity

Paper Structure

This paper contains 11 sections, 5 theorems, 121 equations, 1 figure.

Table of Contents

  1. The Model
  2. Weak solution and main result
  3. Literature review
  4. Construction of weak solutions on a corrected domain
  5. Preliminaries: minimal and maximal domain, fluid velocity extension
  6. The approximate problem
  7. Solving the approximate problem
  8. Limit $\varepsilon\to 0$
  9. Limit $\delta\to 0$
  10. Existence of a weak Solution: Proof of Theorem \ref{['main']}
  11. Acknowledgments

Key Result

Theorem 2.2

Let Assumption ass1 be in force. Let $\mathbf{u}_B\in C_0^2(\overline{\Sigma_{in}}\cup\overline{\Sigma_{out}})$ satisfying FluidBC:followup and and assume $\rho_B\in C(\Sigma_{in})$ with $0<\rho_B<\overline{\rho}$. Then, there is a lower threshold $\kappa_0>0$ such that for every $\kappa\geq \kappa_0$, there exists a weak solution $(\rho,\mathbf{u},w)$ in the sense of Definition weak:sol.

Figures (1)

  • Figure 1: Examples of a configuration domain $\mathscr{O}_0$ (left) and a section of a physical domain $\mathscr{O}(w)$ (right).

Theorems & Definitions (13)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.2: Main result
  • Remark 2.2: Generalizations
  • Definition 4.1
  • Theorem 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 3 more