Finite time blowup of incompressible flows surrounding compressible bubbles evolving under soft equations of state

Abstract

We explore the dynamics of a compressible fluid bubble surrounded by an incompressible fluid of infinite extent in three dimensions, constructing bubble solutions with finite time blowup under this framework when the equation of state relating pressure and volume is soft (e.g. with volume singularities that are locally weaker than that in the Boyle–Mariotte law), resulting in a finite time blowup of the surrounding incompressible fluid as well. We focus on two families of solutions, corresponding to a soft polytropic process (with the bubble decreasing in size until eventual collapse, resulting in velocity and pressure blowup) and a cavitation equation of state (with the bubble expanding until it reaches a critical cavitation volume, at which pressure blows up to negative infinity, indicating a vacuum). Interestingly, the kinetic energy of these solutions remains bounded up to the finite blowup time, making these solutions more physically plausible than those developing infinite energy. For all cases considered, we construct exact solutions for specific parameter sets, as well as analytical and numerical solutions that show the robustness of the qualitative blowup behaviours for more generic parameter sets. Our approach suggests novel—and perhaps physical—routes to the finite time blowup of fluid equations.