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Numerical and analytical studies of Hele-Shaw flows

In this talk I shall discuss two studies of Hele-Shaw flow. In the first, I shall consider numerical simulations of viscous fingering experiments in a radial Hele-Shaw cell whose geometry and/or flow-rate is in some way different from the traditional configuration. In this context the motivation is to devise strategies for controlling the viscous fingering patterns and/or reducing the growth of the interfacial fingers. For example, I report on how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. In another example, I illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. In the second Hele-Shaw study in this talk, I will summarise how techniques in exponential asymptotics can be used to understand the formation of double-, triple- and multiply-tipped bubbles propagating steadily in a Hele-Shaw channel.