During the 17th century, the French priest and chemist Mariotte observed that two floating bodies can attract or repel each other, and he attempted (without success!) to find physical laws describing the behavior. Two centuries later, Laplace studied the question in the context of then novel surface tension formulation, and as an initial step sought the horizontal force acting on two semi-infinite parallel plates of possibly differing materials, held rigidly and partly immersed in an infinite liquid bath. He discovered that even for this idealized configuration, remarkably varying behavior occurs, depending on physical parameters. As one example, repelling force can change to (unbounded) attracting force as the plates approach each other.
Despite this and other such discoveries, there was apparently no further activity on the topic for over two centuries, until one of the present authors wrote on it from a new perspective, emphasizing geometric content. In the present continuing work, we show that the behavior can be (strikingly) still more varied than Laplace has indicated, that the different forms of behavior can be delineated by explicit geometric criteria, and that the force can be predicted to arbitrary accuracy. We provide asymptotic growth and decay estimates in limiting configurations, and we point out a discontinuous dependence on the data. We show that the transformation from repelling to attracting can occur in an exotic way that we could not have anticipated.
The work described is joint, by Robert Finn and Devin Lu.