Nonspherically symmetric equilibrium bubbles in a steadily rotating incompressible fluid

TL;DR

This paper introduces new non-spherical equilibrium bubble solutions in a rotating incompressible fluid, extending previous models and demonstrating their shapes through numerical PINN simulations.

Contribution

It establishes the existence of rotational equilibrium solutions and constructs a nonspherical, horn-torus-shaped bubble, extending prior spherical results to axisymmetric, azimuthal flows.

Findings

Existence of rotational equilibrium solutions.

Construction of horn-torus-shaped bubble.

Numerical PINN simulation of bubble shape.

Abstract

This note presents two nontrivial, rotational equilibrium solutions to the spatial uniform gas pressure (isobaric) approximate model of Prosperetti in the inviscid case. Building on Gavrilov's work [GAFA 2019], we first establish the existence of equilibrium solutions with nontrivial (rotational) liquid flow. Second, we construct a nonspherically symmetric, horn-torus-shaped equilibrium bubble under mild spatial decay conditions of the liquid flow. In addition, we extend earlier results on the characterization of spherical equilibrium bubbles to the axisymmetric, purely azimuthal setting. Finally, we implement a numerical simulation of the equilibrium bubble shape using the Physics-Informed Neural Network (PINN) approximation.