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

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.One–dimensional cases of infinite strips supported on an infinite expanse of liquid are considered in detail because, first, they admit of exact, nonlinear solutions so that the range of validity of the linear theory may be estimated and, second, they demonstrate in a simple manner many of the phenomena associated with objects supported by surface tension, including: mutual attraction leading to coalesence, characteristic deflexion patterns in rafts dependent on the individual strip width, localized mutual repulsion between objects of different weights, and extensive mutual repulsion between an object and a boundary. It is also shown that a phenomenon of mutual alignment occurs with certain strips with pie–crust edge undulations.An analysis is given of the toppling instability of an upright circular cylinder, and its equilibrium state if its centre of gravity is radially offset; we also outline an inverse method of analysis for determining the forces of mutual attraction between discs of arbitrary shape, and the method is demonstrated for two touching oval discs. Finally, attention is given to the forces of mutual attraction within and between coalescences of numerous objects whose individual linear dimensions are small in comparison with the capillary length.Various results are supported by experiment.