We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. % Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality. 

/pdf/navier-stokes-fourier-fluids-interacting-with-elastic-shells-2v7gls9b.pdf

Navier-Stokes-Fourier fluids interacting with elastic shells

We give a short overview on new methods for non-linear dynamic problems in continuum mechanics developed recently in (Benesova, Kampschulte, Schwarzacher 2020, arXive:2008.04796). We describe the general idea of how to treat such problems starting from the energetic point of view and discuss some related recent and future results. Further we introduce an example that models the breathing through masks: An incompressible fluid governed by the Navier-Stokes equation is interacting with a porous, visco-elastic bulk solid. For that model we prove the existence of a weak solution via the new methods.

Variational methods for fluid-structure interaction and breathing through masks

Stability and error estimates of a linear numerical scheme approximating nonlinear fluid-structure interactions

In this paper we study the interaction problem between a nonlinear thermoelastic plate and a compressible viscous fluid with the adiabatic constant $\gamma>12/7$. The existence of a weak solution for this problem is obtained by constructing a time-continuous operator splitting scheme that decouples the fluid and the structure. The fluid sub-problem is given on a fixed reference domain in the arbitrary Lagrangian-Eulerian (ALE) formulation, and the continuity equation is damped on this domain as well. This allows the majority of the analysis to be performed on the fixed reference domain, while the convergence of the approximate pressure is obtained on the physical domain.

On the interaction problem between a compressible viscous fluid and a nonlinear thermoelastic plate

We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. % Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality.

In this paper we introduce a numerical scheme for fluid-structure interaction problems in two or three space dimensions: A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions.

pdf/on-numerical-approximations-to-fluid-structure-interactions-4nv5fqpy2q.pdf

On numerical approximations to fluid-structure interactions involving compressible fluids.

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On numerical approximations to fluid-structure interactions involving compressible fluids.