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.

Extended Annular Capillary Surfaces

“…The natural extension would be to include inclination angles ψa, ψ b ∈ [−π, π]. This leads to the possibility of an inflection point on the curve, in which case (6). (7), (8) are needed.…”

“…We call any solution of (6). (7), (8) with these properties the profile of an extended capillary surface.…”

“…In Figure 2 Additional discussion of floating drops and applications of extended capillary surfaces may be found in [2], [3], and [8]. A discussion of a floating sphere in a bounded container may be found in [6]. …”

Examples of Non-Uniqueness of the Equilibrium States for a Floating Ball

“…Then we will follow this by a measurement of the mathematical energy, specifically of interest for the cases of non-uniqueness. For a theoretical treatment of this setting in general, see first McCuan and Treinen [1], and also McCuan [2] for further details. We will use the results of those variational arguments in what follows.…”

“…In [1], we derived an additional necessary condition that does not appear in the classical literature consisting of fluid interactions with rigid solid objects. A manuscript by Finn [3] is the standard reference for the classical literature.…”