A stable self-similar singularity of evaporating drops: ellipsoidal collapse to a point

TL;DR

This paper provides a rigorous mathematical proof that small perturbations of spherical evaporating drops follow the classical $D^{2}$ law and evolve into self-similar ellipsoids during collapse.

Contribution

It offers the first complete mathematical validation of the classical $D^{2}$ law and describes the asymptotic ellipsoidal shape of collapsing drops.

Findings

Small perturbations of spheres follow the $D^{2}$ law.

Drops converge to a self-similar ellipsoid during collapse.

Shape parameters depend on initial conditions.

Abstract

We study the problem of evaporating drops contracting to a point. Going back to Maxwell and Langmuir, the existence of a spherical solution for which evaporating drops collapse to a point in a self-similar manner is well established in the physical literature. The diameter of the drop follows the so-called $D^{2}$ law: the second power of the drop-diameter decays linearly in time. In this study we provide a complete mathematical proof of this classical law. We prove that evaporating drops which are initially small perturbations of a sphere collapse to a point and the shape of the drop converges to a self-similar ellipsoid whose center, orientation, and semi-axes are determined by the initial shape.