Mathematical Sciences Department Colloquium
Speaker: Chen-Chih Lai
Wednesday, January 22, 2025
11:00 am - 12:00 pm
Stratton 201
Title: Free boundary problems of PDEs concerning thermal effects in bubble dynamics
Abstract: Bubble dynamics plays a crucial role not only in fundamental and applied physics but also in various engineering and industrial applications. In this talk, we will discuss mathematical models describing the deformation of a gas bubble in a liquid. These models fall under the category of fluid interface problems, a subclass of free boundary problems.
To provide a comprehensive understanding, we will begin with a brief overview of fundamental PDEs governing fluid dynamics, the associated boundary conditions, and the broader context of bubble dynamics. Subsequently, attention will be directed towards the thermal decay of bubble oscillation, particularly examining the approximate isobaric model proposed by A. Prosperetti in [J. Fluid Mech. 1991], under which the gas pressure within the bubble is spatially uniform and follows the ideal gas law. This model exhibits a one-parameter family of spherical equilibria, parametrized by the bubble mass. We prove that this family forms an attracting centre manifold for small spherically symmetric perturbations, with solutions approaching the manifold at an exponential rate over time. Furthermore, we show that under either liquid viscosity or irrotational flow assumptions, any equilibrium bubble must be spherical. Additionally, the manifold of spherically symmetric equilibria captures all regular spherically symmetric equilibrium.
We also explore the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic, spherically symmetric external sound field. For this periodically forced system, we establish the existence of a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations.
In the latter part of the talk, I will discuss some limitations of the isobaric model in a more general (nonspherically symmetric) irrotational setting. Specifically, I will address issues such as (1) the undamped oscillations of shape modes due to spatial uniformity of the gas pressure, and (2) the incompatibility between viscosity and irrotationality assumptions. Our results suggest that to accurately capture the effect of thermal damping on the dynamics of general deformations of a gas bubble, the model should be considered within a framework that includes either non-zero vorticity, corrections to the isobaric approximation, or both.
If time permits, I will present ongoing work on the existence of nonspherically symmetric equilibrium bubbles in a rotational framework.
This talk is based on joint work with Michael I. Weinstein ([Arch. Ration. Mech. Anal. 2023], [Nonlinear Anal. 2024], [arXiv:2408.03787], and work in progress).