Wetting of a slit in an inclined plane by a falling liquid film

What is it about?

Liquid film flow along an inclined plane featuring a slit, normal to the main direction of flow, creates a second gas-liquid interface connecting the two side-walls of the slit. This inner interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. The exact liquid configuration is determined by employing the Galerkin/finite element method to solve the two-dimensional Navier-Stokes equations at steady-state. The interplay of inertia, viscous, gravity and capillary forces along with the substrate wettability and orientation with respect to gravity and the width of the slit determine the extent of liquid penetration and free surface deformation. Finite wetting lengths are predicted in hydrophilic and hydrophobic substrates for inclination angles more or less than the vertical, respectively. Multiple steady solutions, connected by turning points forming a hysteresis loop, are revealed by pseudo arc-length continuation. Under these conditions, small changes in certain parameter values leads to an abrupt change in the wetting length and the deformation amplitude of the outer film surface. In hydrophilic substrates the wetting lengths exhibit a local minimum for small values of the Reynolds number and a very small range of Bond numbers; when inertia increases, they exhibit the hysteresis loop with the second limit point in a very short range of Weber numbers. Simple force balances determine the proper rescaling in each case, so that critical points in families of solutions for different liquids or contact angles collapse. The flow inside the slit is quite slower, in general, because of viscous dissipation and includes counter rotating vortices often resembling those reported by (Moffatt, 1964). In hydrophobic substrates the wetting lengths decrease monotonically until the first limit point of the hysteresis loop, which occurs in a limited range of Bond numbers when the Kapitza number is less than 300 and in a limited range of Weber numbers otherwise. Here additional solution families are possible as well, where one or both contact points (Cassie state) coincide with the slit corners.

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Why is it important?

Liquid film flow over an inclined plane has been studied extensively. Until now, no published results exist about the effect on the flow of a narrow slit in the solid surface. The extent to which liquids enter the slit depends on the slit size, hydrophobicity, material properties and flow rate. All these have been examined thoroughly. Hysteresis has been found to arise in certain parameter ranges. The results have important implications for our understanding of superhydrophobic surfaces and how they function.