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Thermal effects in fluid structure interactions

Sourav Mitra, Sebastian Schwarzacher

TL;DR

The work addresses the global existence of bounded energy weak solutions for a thermo-fluid–structure system with two incompressible Navier–Stokes–Fourier fluids separated by a nonlinear Koiter shell, allowing heat transfer across a moving elastic interface. It advances a variational, two-time-scale framework that combines time-delayed minimization (τ-layer) and a regularization parameter κ, ensuring temperature positivity and deriving energy and entropy inequalities. Through careful compactness in moving domains (Aubin–Lions type results, solenoidal extensions) and multi-stage limit passages (τ→0, h→0, κ→0), the authors obtain weak solutions up to geometric degeneracy and provide treatment for insulating (λ=0) and superconductive (λ=∞) transmission limits. The results offer a robust mathematical foundation for heat-exchanging FSI with nonlinear shell energetics and pave the way for numerical schemes via discrete energy/entropy identities. Overall, the paper contributes a novel thermo-FSI theory with moving-boundary Koiter shells and rigorous convergence analysis in 3D.

Abstract

In this article we consider two different heat conducting fluids each modelled by the incompressible Navier-Stokes-Fourier system separated by a non-linear elastic Koiter shell. The motion of the shell changes the domain of definition of the two separated fluids. For this setting we show the existence of a weak solution. The heat capacity of the shell is given energetically. It allows to consider transmission laws ranging from insulation to superconductivity.We follow a variational approach for fluid-structure interactions. To include temperature a novel two step minimization scheme is used to produce an approximation. The weak solutions are energetically closed and include a strictly positive temperature.

Thermal effects in fluid structure interactions

TL;DR

The work addresses the global existence of bounded energy weak solutions for a thermo-fluid–structure system with two incompressible Navier–Stokes–Fourier fluids separated by a nonlinear Koiter shell, allowing heat transfer across a moving elastic interface. It advances a variational, two-time-scale framework that combines time-delayed minimization (τ-layer) and a regularization parameter κ, ensuring temperature positivity and deriving energy and entropy inequalities. Through careful compactness in moving domains (Aubin–Lions type results, solenoidal extensions) and multi-stage limit passages (τ→0, h→0, κ→0), the authors obtain weak solutions up to geometric degeneracy and provide treatment for insulating (λ=0) and superconductive (λ=∞) transmission limits. The results offer a robust mathematical foundation for heat-exchanging FSI with nonlinear shell energetics and pave the way for numerical schemes via discrete energy/entropy identities. Overall, the paper contributes a novel thermo-FSI theory with moving-boundary Koiter shells and rigorous convergence analysis in 3D.

Abstract

In this article we consider two different heat conducting fluids each modelled by the incompressible Navier-Stokes-Fourier system separated by a non-linear elastic Koiter shell. The motion of the shell changes the domain of definition of the two separated fluids. For this setting we show the existence of a weak solution. The heat capacity of the shell is given energetically. It allows to consider transmission laws ranging from insulation to superconductivity.We follow a variational approach for fluid-structure interactions. To include temperature a novel two step minimization scheme is used to produce an approximation. The weak solutions are energetically closed and include a strictly positive temperature.

Paper Structure

This paper contains 47 sections, 23 theorems, 319 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definition of a bounded energy weak solution and main result
  3. Novelty and significance
  4. Ideas, strategy and some further comments
  5. Bibliographical remarks.
  6. The structure of the paper.
  7. Non-linear Koiter energy, geometry near the interface and some key lemmas
  8. Non-linear Koiter energy
  9. Geometry of the moving interface, trace and embeddings
  10. Solenoidal extension
  11. Notion of convergence and an Aubin-Lions type theorem
  12. A time delayed approximation layer
  13. Definition of solution to a time delayed problem and existence result
  14. $\tau$ layer, the itirative approximations
  15. Solving \ref{['min1']} and some a priori estimates
  16. ...and 32 more sections

Key Result

Theorem 1.1

Assume that: $(i)$$\Omega\subset\mathbb{R}^{3}$ is a given bounded domain with $C^{\infty}$ boundary $\partial\Omega.$ Further suppose $\Omega^{2}_{o}\Subset \Omega$ (as in Figure $1.$) such that $\operatorname{dist}(\partial\Omega^{2}_{o},\partial\Omega)>0$ and $\partial\Omega^{2}_{o}$ (the boundar for some $y\in\Gamma$ or the Koiter energy degenerates, i.e., for some $y\in\Gamma.$

Figures (1)

  • Figure 1: Sketch of the domain $\Omega$ where $\Omega=\overline{{\Omega^{1}(t)}}\cup{\Omega^{2}(t)}$ and the boundary of ${\Omega^{2}(t)}$ is $\Sigma_{\eta}.$

Theorems & Definitions (42)

  • Definition 1
  • Remark 1: Compatibility to strong solutions
  • Remark 2
  • Remark 3: Pressure from temperature
  • Remark 4
  • Remark 5: Entropy-inequality
  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 1
  • ...and 32 more