Horizontal capillary flow of a Newtonian liquid in a narrow gap between a plane wall and a sinusoidal wall

An experimental and theoretical study of the capillary flow of a Newtonian liquid (mineral oil) in a Hele-Shaw cell in which the gap varies sinusoidally in one coordinate direction, and flow takes place in the direction of constant channel cross-sectional area is reported. The geometric non-uniformity of the gap is observed to produce interface fingering. Finger length is observed to increase with decreasing spacing between plates of fixed shape, and with increasing gross penetration distance. In the regime of interest, finger length is observed to increase slowly with increasing interface advancement, motivating a quasi-steady model in which gross interface advancement is predicted by a Lucas-Washburn model and interface fingering is predicted by a Hele-Shaw model of steady flow. The steady interface velocity in the Hele-Shaw model is set equal to the instantaneous interface velocity predicted by the Lucas-Washburn model. Fingering predicted by the quasi-steady model matches the experimentally observed trends with regards to plate spacing and gross penetration distance.