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

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.

29 pages, To appear in Archive for Rational Mechanics and Analysis, minor corrections

References in corpus (3)

• Stokes flow near the contact line of an evaporating drop
• Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension
• Orthogonality conditions and asymptotic stability in the Stefan problem with surface tension