Capillarity and archimedes' principle

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. © 2013 Mathematical Sciences Publishers.