Exponential asymptotics for the Saffman-Taylor problem in a wedge #

Cecilie Andersen, Chris Lustri, Scott McCue, Phil Trinh

15:30 Monday in 2Q42.

Part of the Advances in water waves and free-surface flows session.

Abstract #

Saffman Taylor viscous fingering is a classic problem in potential flow theory and exponential asymptotics. In this problem an inviscid fluid is injected into a Hele Shaw channel displacing a viscous fluid and in the steady state a single finger occupies some proportion of the width of the channel. As surface tension approaches zero a countably infinite set of permissible fingers will select a unique zero surface tension solution. Exponential asymptotics have proved essential in deriving this selection mechanism. In this talk I discuss a generalisation of the Saffman Taylor problem to a wedge geometry. This is motivated by desires to understand the more physically relevant Saffman Taylor instability in a circular geometry with injection of the inviscid fluid outwards from a central source. The wedge geometry already shows much greater complexity than the classic problem in a channel. In particular, the bifurcation diagram changes qualitatively and now the countably infinite set of permissible fingers all disappear in pairs before the zero surface tension limit can be reached. Notably, we are still able to capture this behaviour with exponential asymptotics techniques and I will present the key ideas from this analysis.