Thermal relaxation toward equilibrium and periodically pulsating spherical bubbles in an incompressible liquid

Abstract

We study the radial relaxation dynamics toward equilibrium and time-periodic pulsating spherically symmetric gas bubbles in an incompressible liquid due to thermal effects. The asymptotic model (Prosperetti, 1991; Biro and Velazquez, 2000) is one where the pressure within the gas bubble is spatially uniform and satisfies an ideal gas law, relating the pressure, density and temperature of the gas. The temperature of the surrounding liquid is taken to be constant and the behavior of the liquid pressure at infinity is prescribed to be constant or periodic in time. In (Lai and Weinstein, 2023), for the case where the liquid pressure at infinity is a positive constant, we proved the existence of a one-parameter manifold of spherical equilibria, parametrized by the bubble mass, and further proved that it is a nonlinearly and exponentially asymptotically stable center manifold. In the present article, we first refine the exponential time-decay estimates, via a study of the linearized dynamics subject to the constraint of fixed mass. We then study the nonlinear radial dynamics of the bubble–fluid system subject to a small-amplitude, time-periodic far-field pressure. We prove that nonlinearly and exponentially asymptotically stable time-periodically pulsating solutions of the nonlinear model exist for all sufficiently small forcing amplitudes. The existence of such states is formulated as a fixed point problem for the Poincaré return map, and the existence of a fixed point makes use of our exponential time-decay estimates of the linearized problem.