Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid

TL;DR

This paper analyzes the asymptotic behavior of a Neumann problem related to a rigid body's collision with a cavity filled with perfect fluid, revealing conditions for finite-time contact and velocity behavior at contact.

Contribution

It introduces a novel asymptotic analysis of the Dirichlet energy in a cusp domain, linking geometric contact properties to collision dynamics in fluid-structure interaction.

Findings

Solid always reaches the cavity in finite time.

Velocity at contact depends on the tangency exponent α.

The analysis transforms the problem into a domain with a horizontal strip.

Abstract

We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated to the solution of a Laplace Neumann problem as the distance $\varepsilon>0$ between the solid and the cavity's bottom tends to zero. Denoting by $\alpha>0$ the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a non zero velocity for $\alpha <2$ (real shock case), and with null velocity for $\alpha \geqslant 2$ (smooth landing case). Our proof is based on a suitable change of variables sending to infinity the cusp singularity at the contact. More precisely, for every $\varepsilon\geqslant 0$, we transform the Laplace Neumann problem into a generalized Neumann problem set on a domain containing a horizontal strip $]0,\ell_\varepsilon[\times ]0,1[$, where $\ell_\varepsilon\to +\infty$.