Steady bubbles and drops in inviscid fluids

TL;DR

This work rigorously constructs steady, near-spherical bubbles and drops as traveling-wave solutions to the axisymmetric two-phase Euler equations with surface tension. It centers on perturbations of Hill's spherical vortex enclosed by a spherical vortex sheet, with a two-parameter Weber-number framework ($We$ and $\gamma$); the spherical reference state exists when $We=\gamma$, and the analysis identifies a sequence of critical values $\gamma_k$ where nontrivial bifurcations occur via Crandall--Rabinowitz, complemented by an implicit-function regime away from these points. The authors develop a robust functional-analytic setup on Sobolev spaces on the sphere, express the interfacial condition as a functional $\mathcal{F}(\gamma,We,\eta)$ decomposed into a jump term and curvature term, and compute the linearization to reveal a Dirichlet-to-Neumann structure that governs bifurcation. The main contributions include the rigorous existence and local uniqueness of near-spherical traveling-wave solutions, explicit bifurcation analysis at $\gamma_k$, quantitative asymptotics for small perturbations, and a demonstration that surface tension enriches the two-phase problem beyond the one-fluid Hill vortex, where uniqueness (mod translations) holds for fixed circulation. These results provide a rigorous mathematical grounding for shape bifurcations of bubbles and drops in inviscid, high-Reynolds-number flows and illuminate the role of surface tension in stabilizing interacting vortex structures.

Abstract

We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill's vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall-Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill's spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation.

Figures (3)

• Figure 1: Schematic plots of the spherical vortex ($\operatorname{We} = \gamma$) and the perturbations for some $\gamma \notin \Gamma$ depending on the proportion of $\operatorname{We}$ and $\gamma$.
• Figure 2: Schematic plots of the spherical vortex ($\operatorname{We} = \gamma$) and of the bifurcations at $\operatorname{We}_1$ and $\operatorname{We}_2$.
• Figure 3: Values of $\gamma_k$, $k = 1,\dots,15$ approximated numerically.

Theorems & Definitions (25)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Theorem 2.1
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4