In this dissertation, we investigate the dynamics of a gas bubble in an inviscid, compressible liquid, including the effect of surface tension. Kinematic and dynamic boundary conditions couple the bubble surface deformation dynamics with the dynamics of waves in the fluid.
This system has a spherical equilibrium state, resulting from the balance of the pressure at infinity and the gas pressure within the bubble. First, we study the linearized dynamics about this equilibrium state in a center of mass frame: (1) We prove that the velocity potential and bubble surface perturbation satisfy pointwise in space exponential time-decay estimates. (2) The time-decay rate is governed by scattering resonances, eigenvalues of a non-self-adjoint spectral problem. These are pole singularities in the lower half plane of the analytic continuation of a resolvent operator from the upper half plane, across the real axis into the lower half plane. (3) The time-decay estimates are a consequence of resonance mode expansions for the velocity potential and bubble surface perturbations. (4) For small compressibility (Mach number, a ratio of bubble wall velocity to sound speed, ε), this is a singular perturbation of the incompressible limit. The scattering resonances which govern the anomalously slow time-decay, are called Rayleigh resonances. Asymptotics, supported by high-precision numerical computations, indicate that the Rayleigh resonances which are closest to the real axis satisfy the estimate [special characters omitted](exp(−κε−2We)), κ > 0. Here, We denotes the Weber number, a dimensionless ratio comparing inertia and surface tension. (5) To obtain the above results we prove a general result, of independent interest, estimating the Neumann to Dirichlet map for the wave equation, exterior to a sphere.
Next, the linearized dynamics are generalized to include scattering by incident waves: (1) To obtain the bubble perturbation and velocity potential, we study the wave equation outside of a sphere with Neumann boundary conditions and nontrivial initial wave conditions. (2) The time-dynamics are controlled by the scattering resonances from previous setting with no incident waves.
Then the linear analysis is extended to quadratic order in perturbations: (1) The second order system is forced by quadratic products of first order solutions. Thus, the resonance modes are coupled at second order. An individual asymmetric bubble mode can induce symmetric oscillations and two consecutive asymmetric nodes can cause bubble translation. Translation excites all modes. (2) An explicit solution for the velocity of the center of mass is given, along with the corresponding fluid velocity potential.
Finally, we present preliminary asymptotic results for viscous perturbations to the linearized compressible problem: (1) There are modes of very large index for which radiation damping is stronger than viscous damping.