Multi-parameter Hopf bifurcations of rimming flows

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 Fréchet 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.

39 pages

References in corpus (6)

• On the nature of ill-posedness of the forward-backward heat equation
• Convergence to equilibrium for a thin film equation on a cylindrical surface
• Pareto optimal structures producing resonances of minimal decay under $L^1$-type constraints
• Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions
• On computing the instability index of a non-selfadjoint differential operator associated with coating and rimming flows
• Convergence of eigenvalues for a highly non-self-adjoint differential operator