On the selection of Saffman–Taylor viscous fingers for divergent flow in a wedge

Abstract

We study self-similar viscous fingering for the case of divergent flow within a wedge-shaped Hele-Shaw cell. Previous authors have conjectured the existence of a countably infinite number of selected solutions, each distinguished by a different value of the relative finger angle. Interestingly, the associated solution branches have been posited to merge and disappear in pairs as the surface tension decreases. For the first time, we demonstrate how the selection mechanism can be derived based on exponential asymptotics. Asymptotic predictions of the finger-to-wedge angle are additionally given for different sized wedges and surface-tension values. The merging of solution branches is explained; this feature is qualitatively different to the case of classic Saffman–Taylor viscous fingering in a parallel channel configuration. Moreover, because the asymptotic framework does not highly depend on specifics of the wedge geometry, the proposed theory for branch merging in our self-similar problem likely relates much more widely to tip-splitting instabilities in time-dependent flows in circular and other geometries, where the viscous fingers destabilise and divide in two.