Biphasic co-flow through a sudden expansion or contraction of a Hele-Shaw channel

In the present paper we study the biphasic co-flow through a Hele-Shaw cell with an abrupt change, either expansion or contraction, in channel width. Assuming gentle variation of the flow in the streamwise direction (valid at small capillary number), we derive the third-order nonlinear differential equation governing the shape of the interface separating the two fluids. The interfacial profiles obtained by solving this nonlinear equation are further used as the initial guess for the interface shape in the rigorous numerical solution of the two-dimensional free-boundary problem by the finite element method. The theoretical results are then compared with the experimental findings showing a good agreement. The amplitude of the capillary ridge emerging upstream from a sudden expansion (for moderate expansion ratios up to 1:3) and leading to narrowing of the thread of the disperse phase is significant, however not large enough to trigger its instability and breakup.