Slow viscous flow near a superhydrophobic surface with trapped gas bubbles

Abstract

The Boundary Element Method (BEM) is used to solve the problem of Stokes flow of a viscous fluid over a periodic striped texture of a superhydrophobic surface (SHS), partially filled with frictionless gas bubbles. The shape of the bubble surfaces and the position of the meniscus pinning points relative to the cavity walls are taken into account in the study. Two kinds of flows important for practical applications are considered: a pressure-driven flow in a thin channel with a bottom superhydrophobic wall and a shear-driven flow over a periodic texture. We study the flow pattern in the fluid over a single cavity containing a bubble with a curved phase interface shifted into the cavity. A parametric numerical study of the averaged slip length of the SHS is performed as a function of the geometric parameters of the texture. It is shown that the curvature of the phase interface and/or its shift into the cavity both result in the decrease in the average slip length. It is demonstrated that the BEM can be an efficient tool for studying Stokes flows over textured superhydrophobic surfaces with different geometries of microcavities and phase interfaces. We consider a steady-state Stokes flow of a viscous fluid over a periodic striped texture (with period length L) of a superhydrophobic surface (SHS) containing rectangular microcavities (of width d) partially filled with gas bubbles on the surface of which the shear stress is neglected. The gas bubbles are stably trapped in the cavities by the surface tension force. For a Stokes flow regime, the shape of the gas bubbles is the same as in statics and can be approximated by a circle arc with the curvature radius determined by the static angle of wettability at the pinning points of the meniscus on the cavity walls. The dimensional position of the pinning points δ is assumed to be known. We consider the most general case when the fluid velocity vector is directed at an arbitrary angle to the stripes of the SHS. Due to the linearity of the Stokes equations for the 3D flow, the problem of the flow over a striped SHS is split in two problems corresponding to the flows perpendicular to the stripes (2D problem) and along the stripes (1D problem). These flow directions correspond to the principal directions of the slip-length tensor of the striped SHS [1]. Let L be the length scale of the problem, and U and µU/L be the velocity and pressure scales, respectively. In dimensionless form, the equations of fluid motion in the principal directions over a single cavity of a periodic striped SHS are written as follows:Here, u p = (u p , v p ) is the velocity field for the flow normal to the stripes, u l is the fluid velocity along the stripes, p is the dimensionless pressure in the fluid; j = 1 and 0 in the cases of a pressure-driven flow and a shear-driven flow, respectively. We specify the following boundary conditions at all points of the boundary of the 2D flow domain containing phase interface. On the solid walls, the no-slip condition for the velocity is specified; the upper bounda...