Hele–Shaw flow and pattern formation in a time-dependent gap

We consider flow in a Hele–Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid under a lateral straining flow, 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-dimensional Darcy’s law, where the divergence condition is modified to account for the lifting of the plate. For analytic data, we establish the existence, uniqueness and regularity of solutions when the surface tension is zero. We also construct some exact analytic solutions, both with and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble fission. Further, we present the results of numerical simulations of the bubble motion, examining in particular the distinctive pattern formation resulting from the Saffman–Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface tension would fission into two bubbles. PACS numbers: 4720D, 4720M, 4755D, 8385P AMS classification scheme numbers: 76E30, 76D45