Bifurcations of drops and bubbles propagating in variable-depth Hele-Shaw channels

The steady propagation of air bubbles through a Hele-Shaw channel with either a rectangular or partially occluded cross section is known to exhibit solution multiplicity for steadily propagating bubbles, along with complicated transient behaviour where the bubble may visit several edge states or even change topology several times, before typically reaching its final propagation mode. Many of these phenomena can be observed both in experimental realisations and in numerical simulations based on simple Darcy models of flow and bubble propagation in a Hele-Shaw cell. In this paper, we investigate the corresponding problem for the propagation of a viscous drop (with viscosity $$\nu $$ ν relative to the surrounding fluid) using a Darcy model. We explore the effect of drop viscosity on the steady solution structure for drops in rectangular channels or with imposed height variations. Under the Darcy model in a uniform channel, steady solutions for bubbles map directly on to those for drops with any internal viscosity $$\nu \ne 1$$ ν ≠ 1 . Hence, the solution multiplicity predicted for bubbles also occurs for drops, although for $$\nu >1$$ ν > 1 , the interface shape is reversed with inflection points appearing at the rear rather than the front of the drop. The equivalence between bubbles and drops breaks down for transient behaviour, at the introduction of any height variation, for multiple bodies of different viscosity ratios and for more detailed models which produce a more complicated flow in the interior of the drop. We show that the introduction of topography variations affects bubbles and drops differently, with very viscous drops preferentially moving towards more constricted regions of the channel. Both bubbles and drops can undergo transient behaviour which involves breakup into two almost equal bodies, which then symmetry break before either recombining or separating indefinitely.