On Young’s Paradox, and the attractions of immersed parallel plates

Abstract

A seemingly paradoxical prediction for behavior of objects with nonconstant contact angles when dipped into fluids is clarified in a new way. The method is then applied to the problem of determining the attraction (or repulsion) of parallel vertical plates dipped into an infinite liquid bath. A criterion is given for determining whether the plates attract or repel each other, and estimates for the forces are obtained, which are asymptotically exact for small plate separations. It is shown that the attracting force is asymptotically proportional inversely to the square of the distance between the plates; however the repelling force remains under a fixed bound in magnitude. In all cases the net forces are independent of the contact angles of the exterior fluid with the plates, although that is not the case for the individual forces ascribable to pressure differences and to surface tensions. It is shown that regardless of the data, each of the plates experiences the same net force as does the other. Finally a new and more precise and inclusive clarification is given for a phenomenon described in 1806 by Laplace, who noted conditions under which repelling forces can change abruptly into (much larger) attracting forces.