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.

On melting and freezing for the 2D radial Stefan problem

“…The first main open problem following this work is the understanding of the full non radial stability of the free boundary problem in the stable melting regime k = 0 which should be amenable to our approach. Let us mention that a related problem in the context of evaporating drops was recently studied and solved in the setting of a self-similar collapse in the very nice work [15]. The second main open problem is to give a complete description of the flow for small initial data, and here we expect that the constructions and underlying functional framework of Theorem 1.1 and Theorem 1.2 are essential steps.…”

The contact line of an evaporating droplet over a solid wedge and the pinned–unpinned transition

“…The vapour concentration ϕ outside the droplet satisfies a diffusion equation. Assuming that the evaporation process is slow, the diffusion equation is reduced to Laplace's equation (Deegan et al 1997;Eggers & Pismen 2010;Gelderblom et al 2012;Fontelos, Hong & Hwang 2015). As boundary conditions, we impose no vapour flux across the mound wall and a vapour saturation value ϕ s at the droplet surface ϕ = ϕ s .…”