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.In the last decade, in the literature an interest sharply increased in the so-called superhydrophobic surfaces [1], which possess the self-cleaning property of a lotus leaf, along which liquid droplets roll off with a minimal friction. Typically, the superhydrophobicity is achieved through a combination of chemical hydrophobicity and a mechanical surface texturing. The texturing here means the creation of surface roughness in the form of microholes or microprotrusions (with the dimensions of micrometers), in which or between which stable air microbubbles are present. In the literature, this state of the hydrophobic surface is called the Cassie state [2]. On the superhydrophobic surface, the static contact angles attain 120 or more degrees. When a fluid flows along such surface, a noticeable macroscopic slip is observed. This is attributable to the fact that the fluid moving along a superhydrophobic surface contacts on the microlevel with both the segments of a solid surface (with no-slip condition) and the surfaces of bubbles, on which the friction almost vanishes. In the averaged description of the fluid flow near a superhydrophobic surface on the scales much greater than the dimensions of the microinhomogeneities and microbubbles, as the effective boundary condition, the Navier condition [3, 4] is usually formulated:

Lava spreading during volcanic eruptions on the condition of partial slip along the underlying surface

“…In the present study we solve a problem similar to [3,14,24,25] but on the condition of partial slip along the underlying surface with the velocity being a power function of friction. As in [3,14,24,25], the problem is reduced to a nonlinear second-order partial differential equation.…”

“…As in [3,14,24,25], the problem is reduced to a nonlinear second-order partial differential equation. If the exponent in the function that determines the total extruded magma volume and the exponent in the slip law are specifically linked, then the solution of this equation can be reduced by the variable separation method to solving a single nonlinear second-order ordinary differential equation.…”

“…Here, the use of the slip condition is predetermined by the properties of the surface itself and the coefficient of proportionality between the velocity and friction at the surface is generally not constant but a given function of coordinates. In [24,25], for this slip law various self-similar solutions, similar to [3,14] are constructed.…”

“…In [24,25], the spreading of a liquid along a superhydrophobic surface was investigated. Here, the use of the slip condition is predetermined by the properties of the surface itself and the coefficient of proportionality between the velocity and friction at the surface is generally not constant but a given function of coordinates.…”

“…Nevertheless, for some problems the slip along the rigid surface is more motivated (see, for example, review [16]). In problems of liquid spreading the slipcondition was used in the thin-layer approximation in [17][18][19][20][21][22][23][24][25] for Newtonian and in [4] for non-Newtonian liquids.…”