Abstract

In a traditional Hele-Shaw configuration, the governing
equation for the pressure is Laplace's equation; thus, mathematical models for
Hele-Shaw flows are amenable to complex analysis. We consider here one such problem, where a
bubble is moving steadily in a Hele-Shaw cell.
This is like the classical Taylor-Saffman bubble, except we suppose the
domain extends out infinitely far in all directions. By applying a conformal mapping, we produce
numerical evidence to suggest that solutions to this problem behave in an
analogous way to well-studied finger and bubble problems in a Hele-Shaw
channel. However, the selection of the
ratio of bubble speeds to background velocity for our problem appears to follow
a very different surface tension scaling to the channel cases. We apply techniques in exponential
asymptotics to solve the selection problem analytically, confirming the
numerical results, including the predicted surface tension scaling laws.
Further, our analysis sheds light on the multiple tips in the shape of the
bubbles along solution branches, which appear to be caused by switching on and
off exponentially small wavelike contributions across Stokes lines in a
conformally mapped plane. These results
are likely to provide insight into other well-known selection problems in
Hele-Shaw flows.