Multi-parameter Hopf bifurcations of rimming flows

TL;DR

This paper analyzes the stability of steady states in rimming flows, demonstrating how parameter variations can lead to Hopf bifurcations and periodic oscillations in a thin fluid film within a rotating cylinder.

Contribution

It extends multi-parameter Hopf bifurcation theory to infinite-dimensional systems and rigorously proves the emergence of periodic solutions in rimming flows.

Findings

Eigenvalues transition from stable to unstable with parameter changes

Periodic solutions bifurcate from critical equilibria

Theoretical framework applies to thin film flow dynamics

Abstract

In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of (Benilov, Kopteva, O'Brien, 2005), which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fr\'echet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems.