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The stable-downward vertical displacement of an air–liquid meniscus in a two-dimensional Hele-Shaw cell is studied in the limit of small capillary ( Ca ⁎ ) and Bond ( Bo H ⁎ ) numbers. Subsequently, a thin film of thickness h is deposited on the substrate walls for Ca ⁎ ⪯i Bo H ⁎ . Using a reverse Washburn analysis as it applies to two-dimensional geometries in the Stokes flow limit, we develop mathematical expressions for the speed of the average location of the meniscus x ¯ N spanning the smallest gap dimension a . Furthermore, we show that the other spanwise meniscus is stable using classic linear stability analysis for immiscible fluid displacements. Asymptotic analysis of the average meniscus speed yields an expression for the film thickness as a function of several measurable parameters. Experiments are performed using silicone oil (kinematic viscosities 1 × 10 − 5 , 1 × 10 − 4 and 1 × 10 − 3 m 2 /s) while varying the initial displacement height of the meniscus before descent x ¯ N 0 , and the Hele-Shaw cell gap spacing. Analysis of the transient film thickness h , estimated from the average meniscus position, speed and physical properties, is in good agreement and suggests self-similarity. |
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