Capillary channels in a gravitational field

Abstract

We investigate the shape of the liquid–air free interface in the next two settings. First, we insert two parallel vertical plates sufficiently close in an infinite reservoir of liquid. Due to capillary and gravity forces and when the equilibrium is achieved, the liquid rises to a certain height. Then the liquid–air interface meets the vertical walls at prescribed angles and its mean curvature is a linear function of the height, that is, of the coordinate function that defines the gravity field. We study the shapes of these interfaces and their qualitative properties assuming natural hypothesis on symmetry. One matter of interest is to obtain estimates of the size of the capillary meniscus, such as its height, in terms of the boundary data. In the second setting, we consider a horizontal hydrophilic strip surrounded by a solid region of hydrophobic character. We spread the liquid over the strip driven by wettability in such a way that the liquid remains confined up to the boundary of the strip. In a state of equilibrium and assuming that the liquid is invariant in the direction of the non-bounded coordinate of the strip, we prove results on the existence and the uniqueness and we analyse the behaviour of the interface, specially related to the estimates of volume enclosed by the surface. Both settings are particular situations of the one-dimensional case of the capillarity problem, which has been studied in the literature to describe the shape of a liquid that faces a vertical plate.