Coalescence and capillary breakup of liquid volumes

Abstract

The problem of the mathematical modeling of coalescence and breakup of liquid volumes surrounded by an inviscid gas is considered. As is shown, an unphysical singularity in the known self-similar solutions of the Navier–Stokes equations intended to describe the topological transition of the flow domain arises as a consequence of the assumption that the free surface becomes smooth immediately after the onset of coalescence or remains so up to the very moment of breakup. Then the standard kinematic boundary condition prescribes that fluid particles belonging to the free surface remain there at all times and thus couples the scales for lengths and velocities in a self-similar solution leading to the singularity. An alternative approach allowing one to remove the singularity at a macroscopic level is formulated. Its key idea is that the topological transition, being a particular case of an interface formation/disappearance process, is associated with a free-surface cusp either propagating away from the point of initial contact of two volumes leading to their coalescence or “severing” a liquid thread connecting them in the case of breakup. The interface becomes (or, in a reverse flow, ceases to be) smooth at a finite distance from the point where, in the standard approach, a singularity would have taken place. An earlier developed macroscopic theory of interface formation/disappearance is applied without any ad hoc changes.