Prescribed Contact Angle Research Articles

On the existence of a capillary surface with prescribed contact angle

In this note we consider the problem of the existence and regularity of an equilibrium-free surface S of a liquid that partially fills a cylindrical container, as determined by surface forces, gravitational forces and boundary adhesion. Let D denote 'a right cross section of the cylinder and aD its boundary. We represent the capillary surface S by a height function u(x, y), (x, y)~ D. The form of the capillary surface S as determined by the principle of virtual work (cf. [l]) leads to an equation of the form

ON THE BEHAVIOR OF A CAPILLARY SURFACE IN A WEDGE

Estimates above and below are obtained for the height of the equilibrium-free surface of a liquid when the liquid partially fills a cylindrical container whose cross section contains a corner with interior angle 2alpha. The surface is characterized by the condition that its mean curvature be proportional to its height above a reference plane (or, in the case of zero gravity, that the mean curvature be constant), and by the requirement that it meet the container wall with prescribed contact angle gamma. It turns out that the qualitative behavior of such a surface near the vertex changes markedly, according as alpha + gamma < (1/2)pi, or alpha + gamma >/= (1/2)pi. In the former case, the surface is either unbounded or fails to exist, while in the latter case every such surface is bounded. Some experimental comparisons are indicated, and an application to the problem of describing the mechanism of water rise in trees is discussed.The above results describe a limiting case among corresponding properties that hold for surfaces defined over domains with smooth boundaries. This extension is indicated, as well as a formal extension to n-dimensional surfaces; here the interest centers on the fact that it is the mean curvature of an (n-1)-dimensional boundary element that controls the local behavior of the n-dimensional solution surface.