HELE-SHAW FLOW AND PATTERN-FORMATION IN A TIME-DEPENDENT GAP

We consider Bow in a Hele-Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid und er a lateral straining Bow, sucking in the interface and causing it to buckle. The resulting short-lived patterns can resemble a network of connections with triple junctions. The basic instability-a variant of the Saffman-Taylor instability-is found in a version of the two-dimens ional Darcy's law, where the divergence condition is modified to accou nt for the lifting of the plate. For analytic data, we establish the e xistence, uniqueness and regularity of solutions when the surface tens ion is zero. We also construct some exact analytic solutions, both wit h and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble f ission. Further, we present the results of numerical simulations of th e bubble motion, examining in particular the distinctive pattern forma tion resulting from the Saffman-Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface t ension would fission into two bubbles.