Capillarity and Archimedes’ principle

John McCuan and Ray Treinen

Vol. 265 (2013), No. 1, 123–150

DOI: 10.2140/pjm.2013.265.123

Abstract

We consider some of the complications that arise in attempting to generalize a version of Archimedes’ principle concerning floating bodies to account for capillary effects. The main result provides a means to relate the floating position (depth in the liquid) of a symmetrically floating sphere in terms of other observable geometric quantities.

A similar result is obtained for an idealized case corresponding to a symmetrically floating infinite cylinder.

These results depend on a definition of equilibrium for capillary systems with floating objects which to our knowledge has not formally appeared in the literature. The definition, in turn, depends on a variational formula for floating bodies which was derived in a special case earlier (Pacific J. Math. 231:1 (2007), 167–191) and is here generalized to account for gravitational forces.

A formal application of our results is made to the problem of a ball floating in an infinite bath asymptotic to a prescribed level. We obtain existence and nonuniqueness results.

Mathematical Subject Classification 2010

Primary: 76B45

Milestones

Received: 15 January 2009

Revised: 25 April 2013

Accepted: 29 April 2013

Published: 28 August 2013