Interface dynamics in Hele-Shaw flows with centrifugal forces. Preventing cusp singularities with rotation

TL;DR

This paper presents exact solutions for Hele-Shaw flows in a rotating cell, demonstrating how rotation can prevent finite-time cusp singularities and providing conditions for stability across initial conditions.

Contribution

It introduces a class of exact solutions showing how centrifugal forces influence cusp formation, offering criteria to prevent singularities in Hele-Shaw flows with rotation.

Findings

Rotation can prevent cusp formation in Hele-Shaw flows

Existence of a critical rotation rate for stability

A simple stability condition related to linear analysis

Abstract

A class of exact solutions of Hele-Shaw flows without surface tension in a rotating cell is reported. We show that the interplay between injection and rotation modifies drastically the scenario of formation of finite-time cusp singularities. For a subclass of solutions, we show that, for any given initial condition, there exists a critical rotation rate above which cusp formation is prevented. We also find an exact sufficient condition to avoid cusps simultaneously for all initial conditions. This condition admits a simple interpretation related to the linear stability problem.