We consider the displacement of an inviscid fluid (air) by a viscous fluid (oil) in a narrow channel with gap-thickness modulations. The interfacial dynamics of this problem is strongly nonlocal and exhibits competing effects from capillarity and permeability. We derive analytical predictions of steady-state front morphologies, which are exact at linear level in the case of a persistent modulation in the direction of front advancement. The theoretical predictions are in good agreement with experimental measurements of steady-state front morphologies obtained in a Hele-Shaw cell with modulations of the channel depth, consisting on three parallel tracks of reduced depth, for small gap modulations. The relative average distance between theoretical and experimental fronts in the region around the central track is smaller than about $4\%$, provided that the height of the tracks is less than $13\%$ of the total channel depth and the local distortion of the front height $h$ is small enough $\left(\left|\mathbf{\nabla }h\right|<0.8\right)$ for the linear approximation to hold.

• Received 26 May 2017

DOI:https://doi.org/10.1103/PhysRevFluids.2.094006