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We study a model for the evolution of an axially symmetric bubble of inviscidfluid in a homogeneous porous medium otherwise saturated with a viscous fluid.The model is a moving boundary problem that is a higher-dimensional analogue ofHele-Shaw flow. Here we are concerned with the development of pinch-offsingularities characterised by a blow-up of the interface curvature and thebubble subsequently breaking up into two; these singularities do not occur inthe corresponding two-dimensional Hele-Shaw problem. By applying a novelnumerical scheme based on the level set method, we show that solutions to ourproblem can undergo pinch-off in various geometries. A similarity analysissuggests that the minimum radius behaves as a power law in time with exponent$\alpha = 1/3$ just before and after pinch-off has occurred, regardless of theinitial conditions; our numerical results support this prediction. Further, weapply our numerical scheme to simulate the time-dependent development andtranslation of axially symmetric Saffman-Taylor fingers and Taylor-Saffmanbubbles in a cylindrical tube, highlighting key similarities and differenceswith the well-studied two-dimensional cases.

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow