A slow steady flow of a viscous fluid over a superhydrophobic surface with a periodic striped system of 2D rectangular microcavities is considered. The microcavities contain small gas bubbles on the curved surface of which the shear stress vanishes. The general case is analyzed when the bubble occupies only a part of the cavity, and the flow velocity far from the surface is directed at an arbitrary angle to the cavity edge. Due to the linearity of the Stokes flow problem, the solution is split into two parts, corresponding to the flows perpendicular and along the cavities. Two variants of a boundary element method are developed and used to construct numerical solutions on the scale of a single cavity with periodic boundary conditions. By averaging these solutions, the average slip velocity and the slip length tensor components are calculated over a wide range of variation of governing parameters for the cases of a shear-driven flow and a pressure-driven channel flow. For a sufficiently high pressure drop in a microchannel of finite length, the variation of the bubble surface shift into the cavities induced by the streamwise pressure variation is estimated from numerical calculations.

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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:

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