Michael Siegel, associate professor of mathematical science, focuses his research on the analysis and numerical computation of moving boundary problems that arise in fluid mechanics, materials science, and physiology. With grant support from the NSF, he is investigating moving boundary problems that are important in technological applications, ncluding analytical studies of the evolution of slender axisymmetric bubbles with surfactant, pinch off (topological singularities) in slender bubbles, and the singular effects of surface tension in the dynamics of two-finger competition in Hele-Shaw flow.
A Hele-Shaw cell consists of two closely spaced glass plates, with a viscous fluid occupying the gap between the plates. When air is injected into the viscous fluid (say, through a hole in one of the plates) an interesting fingering pattern develops on the interface. G. I. Taylor realized in 1956 that the equations governing flow in a Hele-Shaw cell are analogous to those governing flow in porous media, and therefore the cell could serve as a simple apparatus to study flow instabilities arising during oil recovery
The figure above depicts some unexpected results obtained in a study of finger competition in a Hele-Shaw cell (joint work with E. Paune and J. Casademunt, University of Barcelona). The figure represents the effects of small surface tension during the `competition of two fingers of air propagating into the viscous liquid. The solid lines show the interfacial trajectories for surface tension values (a) .01 (b) .005 (c) .001 (d) .0005 (with (a) being the lower left figure box and (d) being the upper right). The dashed line corresponds to the zero surface tension evolution. The surprising feature here is that the presence of small surface tension leads to a dramatically different outcome in finger competition when compared to the zero surface tension evolution.