Capillary Surfaces

When two fluids (at least one a liquid) are adjacent, the free surface of their interface is called a capillary surface. I study stationary capillary problems, in which neither fluid is flowing. The only unknown, then, in stationary capillary problems is the position of the capillary surface. The study of capillary surfaces goes back to antiquity (see Finn's book ``Equilibrium Capillary Surfaces'' for some historical comments) but is still far from settled. The main impetus to study capillary surfaces today comes from dealing with liquids in spacecraft, where the surface's shape will be determined by surface tension, although capillary problems arise in other contexts as well.

Capillary surfaces are closely related to minimal surfaces. In fact, capillary surfaces minimize a more general functional than do minimal surfaces. The standard mathematical model for volume constrained capillary problems is as follows: a set in is the region occupied by one of the fluids, which contacts a fixed surface . In micro-gravity (i.e., gravity is negligible) the energy functional is

where is the free surface of (i.e., that portion of which is not contained in ), is the portion of which is wetted by (i.e., ), and is a constant depending on the materials involved. We seek local minima of this energy functional if is required to satisfy a volume constraint, that is, we only consider volume conserving perturbations. The first variation yields that the mean curvature of is a constant H, and that the angle between the normal to and the normal to along the curve of contact is , at least when is smooth enough to have a normal. A surface satisfying this first order condition is called stationary. If gravity is included, the mean curvature is an affine function of height, but the contact angle condition remains the same.

For anyone interested in looking at these surfaces, Finn's book is a good place to start.

Tom Vogel
Fri Jan 12 16:24:29 CST 1996