Overview
A time-domain pressure-interface evolution model has been developed and analyzed for a compressible gas bubble within a compressible liquid, incorporating surface tension effects. The model originates from a linearization of the nonlinear two-phase Euler free-boundary problem around a spherical Young-Laplace equilibrium. The outcome is a coupled bulk-surface hyperbolic system that describes the liquid pressure, the gas pressure, and the normal displacement of the interface.
Research Context
The analysis of gas bubble dynamics in compressible liquids, particularly with surface tension, presents complexities due to the nature of the governing equations. The surface-tension quadratic form, central to the model, is indefinite, which complicates time-domain analysis. This indefiniteness arises because various components of the spherical harmonics behave differently: the $Y_0^0$ component represents the volume-changing breathing mode, the $Y_1^m$ ($m = \{-1,0,1\}$) components correspond to neutral translations, and only the higher spherical harmonics yield a coercive shape-mode energy. This inherent complexity necessitates a distinct analytical approach for different components of the system.
Approach
The research strategy involved a decomposition of the system based on the behavior of the surface-tension quadratic form. The coercive sector, encompassing the higher spherical harmonics, was addressed using a $C^0$-semigroup argument. In parallel, the breathing mode ($Y_0^0$ component), which lacks coercivity, was analyzed separately through Fourier-Laplace methods. This dual approach allowed for the demonstration of well-posedness for admissible finite-energy data and the existence of classical solutions, provided natural compatibility conditions are met.
Beyond the primary formulation, the researchers justified two specific limiting descriptions of the model:
- In an acoustic quasi-static regime, the developed model was demonstrated to reduce to the linearized Rayleigh-Plesset equation for the breathing mode. Simultaneously, it reduced to the linearized Rayleigh-Lamb equations for the shape modes.
- In a separate regime, the model was shown to simplify to the frozen-interface acoustic transmission model.
A frequency-domain analysis was also conducted to identify corresponding resonance mechanisms. This analysis linked the Minnaert, Rayleigh-Lamb, and Fabry-Pérot-type resonances as different components and limits stemming from the same pressure-interface formulation.
Findings
- A time-domain pressure-interface evolution model was derived for a compressible gas bubble in a compressible liquid with surface tension, based on linearization of the nonlinear two-phase Euler free-boundary problem.
- The model, expressed as a coupled bulk-surface hyperbolic system, describes liquid pressure, gas pressure, and normal interface displacement.
- Well-posedness was established for admissible finite-energy data and classical solutions under natural compatibility conditions, utilizing a decomposition that addressed the indefinite nature of the surface-tension quadratic form. \(C^0\)-semigroup arguments were applied to the coercive sector, while Fourier-Laplace methods were used for the breathing mode.
- The model quantifiably reduces to the linearized Rayleigh-Plesset equation for the breathing mode and the linearized Rayleigh-Lamb equations for the shape modes in an acoustic quasi-static regime.
- In a different regime, the model reduces to the frozen-interface acoustic transmission model.
- Frequency-domain analysis identified Minnaert, Rayleigh-Lamb, and Fabry-Pérot-type resonance mechanisms as integral components and limits of the pressure-interface formulation.