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SIAM J. Math. Anal. 43, pp. 828-876 (49 pages)
Radiative Decay of Bubble Oscillations in a Compressible Fluid
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Add to FacebookConsider the dynamics of a gas bubble in an inviscid, compressible liquid with 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. We study the linearized dynamics about this equilibrium state in a center-of-mass frame. We prove that the velocity potential and bubble surface perturbation satisfy pointwise-in-space exponential time-decay estimates. The time-decay rate is governed by the imaginary parts of scattering resonances. These are characterized by a non–self-adjoint spectral problem or as 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. The time-decay estimates are a consequence of resonance mode expansions for the velocity potential and bubble surface perturbations. The weakly compressible case (small Mach number, $\epsilon$) is a singular perturbation of the incompressible limit. The scattering resonances which govern the remarkably slow time-decay are Rayleigh resonances, associated with capillary waves, due to surface tension, on the bubble surface, which impart their energy slowly to the unbounded fluid. Rigorous results, asymptotics, and high-precision numerical studies indicate that the Rayleigh resonances which are closest to the real axis satisfy $\bigl|\frac{\Im\lambda_\star(\epsilon)}{\Re\lambda_\star(\epsilon)}\bigr|=\mathcal{O}\bigl(\exp(-\kappa~\mathrm{We}~\epsilon^{-2})\bigr), \kappa>0$. Here, $\mathrm{We}$ denotes the Weber number, a dimensionless ratio comparing inertia and surface tension. To obtain the above results we prove a general result estimating the Neumann to Dirichlet map for the wave equation exterior to a sphere.
© 2011 Society for Industrial and Applied Mathematics
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Received July 26, 2010
Accepted December 06, 2010
Published online March 31, 2011
Accepted December 06, 2010
Published online March 31, 2011
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