Liquid bridges, edge blobs, and Scherk-type capillary surfaces (Q2758108)

The authors show that, with the exception of particular cases, any tubular liquid bridge configuration joining parallel plates in the absence of gravity must change discontinuously with tilding of the plates. This surprising result is caused by the strong nonlinearity of the problem. The proof is based on the integration of equation for the position vector, and on appropriate integrations by parts. Moreover, possible configurations in a wedge of given opening angle \(2\alpha\) are discussed if the contact angles on the plates are \(\gamma_1\) and \(\gamma_2\), respectively. It turns out, for example, that only a spherical edge blob can sit in the edge if \((\gamma_1,\gamma_2)\) is in the interior of the rectangle: \(\pi-2\alpha\leq\gamma_1+\gamma_2\leq\pi +2\alpha\) and \(2\alpha-\pi\leq\gamma_1-\gamma_2\leq\pi-2\alpha\). The proof of this result is based on regularity results for second-order elliptic equations over domains with corners, and by using Codazzi equations for a surface of constant mean curvature which imply that \((L-M)-2iM\) is analytic on the surface. Then the behaviour of the surface near the corners implies that the surface is totally umbilic, and hence it is spherical. As a corollary, a new existence theorem for H-graphs over a square with discontinuous data is obtained which can be interpreted as a generalization of Scherk minimal surface.