Force Profiles for Parallel Plates Partially Immersed in a Liquid Bath

Abstract

For two parallel plates held fixed and dipped in an idealized infinite liquid bath in a gravity field, we define the force profile to be the function ${F:(0,\infty)\to \mathbb{R}}$ which assigns to each plate separation distance ${d\in (0,\infty)}$ the signed (horizontal) force density of mutual attraction between the plates (with repulsive forces corresponding to negative values). We show that precisely three nontrivial qualitative profiles are possible depending on the adhesion properties of the plates, namely the contact angles. Results of Finn, Lu, and Bhatnagar in three recent papers which treat this problem originally suggested by Laplace, show that the profiles are well-defined and depend, in a qualitative sense, only on the contact angle pair ${(\gamma_1,\gamma_2)}$ associated with the adhesion properties of the inner facing surfaces of the plates. We also describe symmetry relations satisfied by the quantitative force profiles with respect to ${(\gamma_1,\gamma_2)\in [0,\pi]\times[0,\pi]}$. Taken together, our discussion provides a unified overview of the recent progress on the problem and an alternative approach to several of the main results.