On the degree of wetting of a slit by a liquid filmflowing along an inclined plane


D.Pettas, G.Karapetsas, Y.Dimakopoulos, J.Tsamopoulos


Liquid film flow along an inclined plane featuring a slit, normal to the main directionof flow, creates a second gas–liquid interface connecting the two side walls of theslit. This inner interface forms two three-phase contact lines and supports a widelyvarying amount of liquid under different physical and geometrical conditions. Theexact liquid configuration is determined by employing the Galerkin/finite elementmethod to solve the two-dimensional Navier–Stokes equations at steady state. Theinterplay of inertia, viscous, gravity and capillary forces along with the substratewettability and orientation with respect to gravity and the width of the slit determinethe extent of liquid penetration and free-surface deformation. Finite wetting lengthsare predicted in hydrophilic and hydrophobic substrates for inclination angles moreor less than the vertical, respectively. Multiple steady solutions, connected by turningpoints forming a hysteresis loop, are revealed by pseudo-arclength continuation. Underthese conditions, small changes in certain parameter values leads to an abrupt changein the wetting length and the deformation amplitude of the outer film surface. Inhydrophilic substrates the wetting lengths exhibit a local minimum for small valuesof the Reynolds number and a very small range of Bond numbers; when inertiaincreases, they exhibit the hysteresis loop with the second limit point in a very shortrange of Weber numbers. Simple force balances determine the proper rescaling ineach case, so that critical points in families of solutions for different liquids or contactangles collapse. The flow inside the slit is quite slow in general because of viscousdissipation and includes counter-rotating vortices often resembling those reported byMoffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18). In hydrophobic substrates, thewetting lengths decrease monotonically until the first limit point of the hysteresisloop, which occurs in a limited range of Bond numbers when the Kapitza number isless than 300 and in a limited range of Weber numbers otherwise. Here additionalsolution families are possible as well, where one or both contact points (Cassie state)coincide with the slit corners.


coating, microfluidics, thin films

DOI: https://doi.org/10.1017/jfm.2017.190