Gravity and capillary pressure-driven drainage in a vertical Hele-Shaw cell: Thin film deposition

• Thin film deposition in a vertical Hele-Shaw parallel plate capacitor cell is analyzed.
• The model is developed using a using reverse Washburn analysis in two-dimensions.
• The meniscus interface is displaced from equilibrium using electro-capillary pressure.
• The stability of the receding interface is shown using Saffman–Taylor linear stability analysis.
• Final film thickness values are estimated to be 5–10% of the parallel-plate capacitor spacing.

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⁎)(Ca⁎) and Bond (BoH⁎) numbers. Subsequently, a thin film of thickness hh is deposited on the substrate walls for Ca⁎⪯¡BoH⁎. 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 aa. 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 1×10−51×10−5, 1×10−41×10−4 and 1×10−31×10−3 m2/s) while varying the initial displacement height of the meniscus before descent x¯N0, and the Hele-Shaw cell gap spacing. Analysis of the transient film thickness hh, estimated from the average meniscus position, speed and physical properties, is in good agreement and suggests self-similarity.

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