Self-similar regimes of liquid-layer spreading along a superhydrophobic surface

Abstract

Within the Stokes film approximation, unsteady spreading of a thin layer of a heavy viscous fluid along a horizontal superhydrophobic surface is studied in the presence of a given localized mass supply in the film. The forced (induced by the mass supply) spreading regimes are considered, for which the surface tension effects are insignificant. Plane and axisymmetric flows along the principal direction of the slip tensor of the superhydrophobic surface are studied, when the corresponding slip tensor component is either a constant or a power function of the spatial coordinate, measured in the direction of spreading. An evolution equation for the film thickness is derived. It is shown that this equation has self-similar solutions of a source type. The examples of self-similar solutions are constructed for power and exponential time dependences of mass supply. In the final part of the paper, some of the solutions constructed are generalized to the case of a weak dependence of the flow on the second spatial coordinate, caused by a slight variability of the slip coefficient in the direction normal to that of spreading. The constructed self-similar solutions can be used for experimental determination of the parameters important for hydrodynamics, e.g. the slip tensor components of commercial superhydrophobic surfaces.