Collision/No-collision results of a solid body with its container in a 3D compressible viscous fluid

TL;DR

This work analyzes collision versus no-collision of a rigid body moving inside a bounded 3D domain filled with a viscous compressible fluid. It introduces a cusp-sensitive test function adapted to the vanishing gap and derives rigorous energy bounds to show that boundary roughness, quantified by $C^{1,\alpha}$ regularity, can induce finite-time contact with the container when $0<\alpha\le 1$, $\gamma>3$, and $\alpha<\min\{\tfrac{1}{3},\tfrac{3(\gamma-3)}{4\gamma+3}\}$, provided the solid mass is large and the initial vertical velocity is small; the proof hinges on careful estimation of drag-like terms near the cusp. The paper also establishes no-collision results in a separate setting: a smooth body with a PD feedback controller guarantees a positive clearance from the boundary under small-data energy bounds, and higher regularity data yield no-collision without control. Overall, the results extend incompressible collision theories to the compressible Navier–Stokes regime and quantify how boundary roughness and compressibility shape contact dynamics in fluid–structure interactions.

Abstract

We consider a bounded domain $Ω\subset\mathbb R^3$ and a rigid body $\mathcal{S}(t)\subsetΩ$ moving inside a viscous compressible Newtonian fluid. We exploit the roughness of the body to show that the solid collides its container in finite time. We investigate the case when the boundary of the body is of $C^{1,α}$-regularity and show that collision can happen for some suitable range of $α$. We also discuss some no-collision results for the smooth body case.

Figures (1)

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

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Proposition 2.1
• Remark 2.2
• proof : Proof of Proposition \ref{['prop:EI']}