Equilibrium and mutual attraction or repulsion of objects supported by surface tension

Abstract

A theory is presented to explain known phenomena associated with lightweight objects on the surface of a liquid, and to predict other phenomena. Such objects are supported partly by hydrostatic pressure and partly by surface tension, and this latter component causes the surface of the liquid to deflect in a manner that decays rapidly with distance from each object. The forces between such objects, which may be of attraction or repulsion, stem from the interaction of these surface deflexions and are here determined by reference to the equal and opposite forces required to maintain static equilibrium. As an essential preliminary to this, and in the context of a linear theory, the equilibrium equations are derived for a floating object of arbitrary shape where the deflexions and slopes of the adjoining free surface of the liquid are also arbitrary but small. These equilibrium equations provide the boundary conditions that determine the deflexion of the free surface of the liquid based on Laplace's surface tension equation. The forces of mutual attraction or repulsion are shown to be given by certain contour integrals involving the squares of the surface deflexion and slopes surrounding each object.