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Non-Existence of thick bubble rings at low Weber numbers

Yuanjiang Han, Christian Seis

Abstract

We examine the existence of thick bubble rings within the framework of the free-boundary capillary Euler equations, focusing on the regime of low Weber numbers. Although spheroidal bubbles are known to approach a spherical shape in this limit, the possibility of thick bubble rings persisting at low Weber numbers has remained uncertain. In contrast to the ordinary Euler equations, which admit thick vortex ring solutions, our analysis reveals that the free-boundary capillary Euler equations do not support thick bubble rings at low Weber numbers. This distinction highlights the significant impact of surface tension on the behavior of vortex rings in the capillary regime.

Non-Existence of thick bubble rings at low Weber numbers

Abstract

We examine the existence of thick bubble rings within the framework of the free-boundary capillary Euler equations, focusing on the regime of low Weber numbers. Although spheroidal bubbles are known to approach a spherical shape in this limit, the possibility of thick bubble rings persisting at low Weber numbers has remained uncertain. In contrast to the ordinary Euler equations, which admit thick vortex ring solutions, our analysis reveals that the free-boundary capillary Euler equations do not support thick bubble rings at low Weber numbers. This distinction highlights the significant impact of surface tension on the behavior of vortex rings in the capillary regime.
Paper Structure (3 sections, 7 theorems, 66 equations, 2 figures)

This paper contains 3 sections, 7 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical setting and results
  3. Proofs

Key Result

Theorem 1

There exists a positive constant $C$ such that for any solution $(E, \psi, W, \gamma, \lambda)$ to the equations 4, 3, 5, 6, 7 that satisfies the curvature condition 1 and the maximum principle 47, it follows that where $\mu = R/\sqrt{|E|}$.

Figures (2)

  • Figure 1: A thick vortex inside of the a semi-circle of the same geometric center $(R,0)$.
  • Figure 2: Construction of the triangle contained in $E_+$ and the trapezoid containing $E_-$ in the case of a symmetric convex set $E$.

Theorems & Definitions (16)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Proposition 1
  • ...and 6 more