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A collision result for both non-Newtonian and heat conducting Newtonian compressible fluids

Šárka Nečasová, Florian Oschmann

TL;DR

The paper investigates finite-time collision between a rigid body and the boundary in a compressible, possibly non-Newtonian, fluid within a bounded domain. It develops a test-function framework based on a divergence-free field that aligns with the solid’s motion to derive a weak-form energy balance, yielding explicit collision criteria. The main contributions extend known Newtonian collision results to non-Newtonian and heat-conducting fluids by detailing parameter-dependent bounds on the cusp exponent and viscosity-growth effects, and by adapting the analysis to a temperature-dependent viscosity setting with updated collision conditions. These results enhance understanding of fluid-structure interaction in compressible regimes and underline how viscosity behavior and thermal effects influence collision outcomes in practical applications.

Abstract

We generalize the known collision results for a solid in a 3D compressible Newtonian fluid to compressible non-Newtonian ones, and to Newtonian fluids with temperature depending viscosities.

A collision result for both non-Newtonian and heat conducting Newtonian compressible fluids

TL;DR

The paper investigates finite-time collision between a rigid body and the boundary in a compressible, possibly non-Newtonian, fluid within a bounded domain. It develops a test-function framework based on a divergence-free field that aligns with the solid’s motion to derive a weak-form energy balance, yielding explicit collision criteria. The main contributions extend known Newtonian collision results to non-Newtonian and heat-conducting fluids by detailing parameter-dependent bounds on the cusp exponent and viscosity-growth effects, and by adapting the analysis to a temperature-dependent viscosity setting with updated collision conditions. These results enhance understanding of fluid-structure interaction in compressible regimes and underline how viscosity behavior and thermal effects influence collision outcomes in practical applications.

Abstract

We generalize the known collision results for a solid in a 3D compressible Newtonian fluid to compressible non-Newtonian ones, and to Newtonian fluids with temperature depending viscosities.
Paper Structure (9 sections, 3 theorems, 67 equations, 1 figure)

This paper contains 9 sections, 3 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations:
  3. General assumptions
  4. The stress tensor and uniform bounds
  5. The solid's shape and main result
  6. Construction of test function and proof of main result
  7. Test function
  8. Estimates near the collision -- Proof of Theorem \ref{['theo1']}
  9. Newtonian flow with temperature-growing viscosity

Key Result

Theorem 2.2

Let $\gamma>\frac{3}{2}$, $2 \leq p < 3$, $0<\alpha\leq 1$, and $\Omega,\ \mathcal{B}\subset\mathbb R^3$ be bounded domains of class $C^{1,\alpha}$. Let $(\rho, \mathbf u, {\mathbf G})$ be a weak solution to NSE enjoying the bounds UnifBds, let $\mathbb S$ comply with S1--S3, and assume that a1--a6

Figures (1)

  • Figure 1: The body $\mathcal{B}$ and fluid $\mathcal{F}$ in the container $\Omega$

Theorems & Definitions (10)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • Remark 4.2