Liquid bridges, edge blobs, and Scherk-type capillary surfaces

Paul Concus, Robert Finn, and John McCuan

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Submission date: 20. Dec. 1999
Pages: 41
published in: Indiana University mathematics journal, 50 (2001) 1, p. 411-441 
DOI number (of the published article): 10.1512/iumj.2001.50.1849
Bibtex
MSC-Numbers: 76B45, 76D45, 53A10, 49Q05, 53C42, 76E10
Keywords and phrases: capillarity surface, stability, contact angle, liquid bridge, mean curvature
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Abstract:
It is shown that, with the exception of very particular cases, any tubular liquid bridge configuration joining parallel plates in the absence of gravity must change discontinuously with tilting of the plates, thereby proving a conjecture of Concus and Finn [Phys. Fluids 10 (1998) 39--43]. Thus the stability criteria that have appeared previously in the literature, which take no account of such tilting, are to some extent misleading. Conceivable configurations of the liquid mass following a plate tilting are characterized, and conditions are presented under which stable drops in wedges, with disk-type or tubular free bounding surfaces, can be expected. As a corollary of the study, a new existence theorem for H-graphs over a square with discontinuous data is obtained. The resulting surfaces can be interpreted as generalizations of the Scherk minimal surface in two senses:
(a) the requirement of zero mean curvature is weakened to constant mean curvature, and
(b) the boundary data of the Scherk surface, which alternate between the constants +infinity and -infinity on adjacent sides of a square, are replaced by capillary data alternating between two constant values, restricted by a geometrical criterion.