Asymmetric deformations of a perturbed spherical bubble in an incompressible fluid

Abstract: We study the dynamics of a gas bubble in a fluid with surface tension, initially near a spherical equilibrium. While there are many studies and applications of radial bubble dynamics, the theory of general deformations from a spherical equilibrium is less developed. We aim to understand how asymmetrically perturbed equilibrium bubbles evolve toward spherical equilibrium due to thermal or viscous dissipation in an incompressible liquid. We focus on the isobaric approximation [Prosperetti, JFM, 1991], under which the gas pressure within the bubble is spatially uniform and obeys the ideal gas law. The liquid outside the bubble is incompressible, irrotational, and has surface tension. We prove that any equilibrium gas bubble must be spherical by showing that the bubble boundary is a closed surface of constant mean curvature. We then study the initial value problem (IVP) for the coupled PDEs, constitutive laws and interface conditions of the isobaric approximation for general (asymmetric) small initial perturbations of the spherical bubble in the linearized approximation. Our first result, considering thermal damping without viscosity, proves that the linearized IVP is globally well-posed. The monopole (radial) component of the perturbation decays exponentially over time, while the multipole (non-radial) components undergo undamped oscillations. This indicates a limitation of the isobaric model for non-spherical dynamics. Our second result, incorporating viscous dissipation, shows that the IVP is linear and nonlinearly ill-posed due to an incompatibility of normal stress boundary conditions, for non-spherical solutions, and the irrotationality assumption. Our study concludes that to accurately capture the dynamics of general deformations of a gas bubble, the model must account for either vorticity generated at the bubble-fluid boundary, spatial non-uniformities in the gas pressure, or both.