Cylindrically singular solutions of the capillary problem

This work was done jointly with Robert Neel. We consider the problem of determining the surface interface of fluid partly filling a semi-infinite capillary tube of general section D and closed at one end, in the absence of gravity. It is known that the formal equations for a surface interface as a graph covering the base do not in general admit regular solutions. In the present work it is shown that whenever smooth solutions fail to exist there will nevertheless exist a solution in a singular sense suggested by physical intuition. These surfaces are smooth graphs with constant mean curvature H, over subsets of the base domain D that are bounded within D by subarcs of semicircles C of radius 1/2H and meeting the boundary of D in the prescribed contact angle. As C is approached within the subdomain of regularity, the solution surface is asymptotic to the vertical cylinder over C. Surfaces with that behavior have been observed physically.

In some configurations of particular interest, the procedure leads to unique determination of such a "C-singular" solution. However, uniqueness cannot in general be expected, as is shown by example. Further examples show a) that C-singular solutions may appear also when smooth solutions exist, and b) they may fail to occur in that case, depending on the particular geometry.