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Hyperbolic & Dispersive PDE Seminar

Michael Weinstein -- Free boundary problems arising in the dynamics of a gas bubble in an unbounded liquid

Michael Weinstein (Columbia University)

Location:  Hill 705
Date & time: Thursday, 18 April 2024 at 3:50AM - 4:50PM

Abstract: Consider a deforming gas bubble immersed in an unbounded liquid with surface tension. For an incompressible and inviscid liquid Rayleigh found (in the linearized approximation) that a spherically symmetric equilbrium bubble is neutrally stable; an infinitesimally small perturbation excites undamped oscillations in all spherical harmonic components of the perturbation. However, an actual fluid-bubble system comes with dissipation mechanisms: thermal diffusion, viscosity, and in the case where the liquid is compressible, acoustic radiation. 

We discuss recent results on the thermal decay of radial bubble oscillations in an incompressible liquid. We focus on the approximate (isobaric) model of A. Prosperetti [J. Fluid Mech. 1991]; see also Biro-Velazquez [SIMA 2000]. 

i) We show that if the liquid has non-zero viscosity and surface tension, then all equilibrium bubbles are spherically symmetric by an application of Alexandrov’s theorem on closed constant-mean-curvature surfaces.

 ii) The model exhibits a one-parameter manifold of spherically symmetric equilibria (steady states), which is parametrized by the bubble mass (encompassing all regular spherical equilibria).  We prove that the manifold of spherical equilibria is an attracting centre manifold relative to small spherically symmetric perturbations, and that solutions approach this manifold at an exponential rate as time advances. 

iii) We also study the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic external sound field. We prove that this periodically forced system admits a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations.

 For the general asymmetric dynamics about a spherical bubble equilibria, we expect the deforming bubble to eventually relax to a sphere. Iwill discuss work in progress on asymmetric dynamics of these models and future directions.