Navier-Stokes-Fourier fluids interacting with elastic shells

Abstract

We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a nonlinear 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. 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. Contents 1. Introduction 1.1. State of art 1.2. The model 1.3. Mathematical strategy 1.4. Overview of the paper 2. Preliminaries 2.1. Structural and constitutive assumptions 2.2. Function spaces on variable domains 2.3. Extensions on variable domains 2.4. Convergence in variable domains. 2.5. Weak solutions and main theorem 3. Equations for density and temperature in variable domains 3.1. The continuity equation 3.2. The internal energy equation 4. Construction of an approximate solution 4.1. Galerkin approximation 4.2. Total energy balance 4.3. Compactness of ∂ t η N 4.4. Compactness of the shell energy 5. Construction of a solution. 5.1. The limit system for ε → 0 5.2. Compactness of the shell energy 5.3. Proof of Theorem 2.16. References

Existence of a weak solution to a nonlinear fluid-structure interaction problem with heat exchange

“…We consider a model where both the fluid and the structure conduct heat and there is heat coupling given through temperature continuity and entropy flux given in (1.26) and (1.28). While, in the context of fluid-structure interaction, heat-conducting fluids have been studied in [6] and thermoelastic structures have been studied in [56], to the best of our knowledge this is the first work that takes into account heat conduction of both components, and heat exchange between the components. One nice consequence of this approach is that we were able to prove that the plate temperature is non-negative which does not hold if one considers just a linear plate model without coupling it to the fluid with heat exchange.…”

“…We consider general constitutive relations which are used in [15]. This is a generalization of the result from [6] where pressure is assumed to be of the following special form, p(ρ, ϑ) = ρ γ + a 3 ϑ 4 . We emphasize that these assumptions are somewhat restrictive from physical point of view.…”

“…We emphasize that these assumptions are somewhat restrictive from physical point of view. In particular, term ρϑ is not covered by analysis in [6] and that excludes a large number of physical cases (in particular perfect gases) where molecular pressure p M includes this coupling term. Moreover, the fluid entropy in [6] takes the form 4a 3 ϑ 3 ρ which allows the temperature to vanish on measurable sets, in contrast to our case where we have ϑ > 0 a.e.…”

“…To the best of our knowledge there are only a few very recent papers dealing with the mathematical analysis of FSI problems with a heat conducting fluid. In [42] the existence of a strong solution for small time or small data is proven in the case when there is an additional damping on the structure, while in [6] existence of a weak solution is obtained for an FSI problem with a nonlinear Koiter shell. In both of these papers the structure does not conduct heat.…”

Construction of a right inverse for the divergence in non-cylindrical time dependent domains

“…Higher integrability of the density, and consequently that of the pressure, is nowadays commonly obtained via Bogovskij operators [21,11]. The operator introduced here allows for a significant relaxation of the assumptions on the pressure in [11,12], where interaction of elastic shells and compressible fluids is studied. 1.5.…”