Topological Transitions in Liquid Liquid Interfaces

A set of fundamental yet ill understood phenomena in uid dynamics involves changes in the topology of interfaces between partially miscible or nominally immiscible uids Such changes occur for example when continuous jets pinch o into droplets when sheared interfaces atomize and when droplets of one uid reconnect with one another These topological transitions occur in many practical applications involving transport mixing and separation of petroleum chemical and food products as well as contaminated waste streams The dynamics of topological transitions are di cult to understand and model for several reasons For one the uids in which these transitions occur are complex A second problem associated with topological transitions is caused by the short time scales over which they occur In practical ows the transition time scales are much shorter than the local ow time scales making the transitions di cult to characterize experimentally or compute numerically A third problem associated with transitions is purely numerical how does one handle the change in interface topology in a physically justi ed way In this paper we will address the last problem in the context of incompressible uid ows Many researchers see for example have tried using ad hoc methods to change the topology of interfaces While this approach often referred to as contour surgery allows topological transitions to be overcome it is di cult to justify the reconnection conditions based on physical principles In a few special cases involving uid gas interfaces it is possible to develop physically based reconnection conditions by using special similarity solutions of the Navier Stokes equations see For ows involving liquid liquid interfaces however the dynamics are more complicated and no such similarity solutions have been constructed