The Laplace Parallel Plates Problem in Capillarity Theory

Abstract

When two parallel plates of perhaps different materials are immersed vertically into a fluid bath in a vertically downward gravity field, they may experience an attracting or repelling force. Following a framework introduced by Laplace, we seek explication based on surface tension theory, which we use to correlate the forces with estimates for the height of the liquid surface. In this context, we provide (as special cases of more general estimates), an exact formula for the mean rise height in the channel formed by the plates, together with a strict bound for the variation of rise height within the channel. We base our procedures directly on the nonlinear formulation of the governing equations as formulated by Young and by Laplace, and introduce no further structural hypotheses. As support for this decision we present an explicit example based on our Theorem 5, displaying errors of unlimited magnitude that can arise from customary linearization procedures, and which can also obscure physical phenomena of central interest. As corollary of the development, we obtain also a new characterization of the classes of attracting and repelling solutions. We establish further an asymptotically exact form of the Laplace discovery, that aside from an isolated exceptional case of repelling forces bounded from infinity above (and, as we show, bounded also from zero below), the force between the plates always becomes attracting and grows as the inverse square of the separation distance, as the plates are brought together with fixed contact angles. We continue our efforts, initiated by us in earlier publications, toward clarifying this exotic behavior in quantitative terms.