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

We study the biphasic co-flow through an expansion (or contraction) of the shallow (Hele-Shaw) microchannel. Assuming gentle streamwise variation of the flow, we derive the 3rd-order nonlinear differential equation governing the steady-state shape of the interface separating the two fluids. The interfacial profiles obtained by integrating this nonlinear equation are further compared to numerical solution of the two-dimensional free-boundary problem and experimental results. The amplitude of the capillary ridge emerging upstream from a sudden expansion and leading to narrowing of the thread of the inner phase is substantial, however, not large enough to trigger instability and breakup.