Effect of the vessel wall on the stability of floating bodies

Abstract

The restoring force acting on a vertical cylinder floating or suspended at the liquid interface in a vertical cylindrical vessel is determined when the cylinder is displaced horizontally from the symmetrical position. When both cylinder and vessel walls are wetted, or unwetted, the force urges the inner cylinder towards the wall, but when one wall is wetted and the other not the force acts to recentralize the inner cylinder when the cylinder radii are small. This latter force, although smaller than the former, may be used to centralize a body floating in a cylindrical vessel. When both walls are wetted, or unwetted, an approximate expression for the restoring force f acting on a cylinder of radius r i displaced by a distance δ from the axis of the vessel of radius r o is f = — 4 πσ 2 r i δ / ∆ ρ g ∆ r 3 [1 - ( δ 2 / ∆ r 2 )] 3/2 where σ is the tension of the fluid-fluid interface, ∆ ρ is the density difference between the phases and ∆ r the difference between the radii r o and r i . When one wall is wetted and the other is not, ∆ r may be replaced by r o + r i in the above equation when r o and r i are small, and f then becomes positive. The expression for wetted walls has been experimentally confirmed for different liquids and cylinder geometries. Limiting solutions have been derived for small gap widths and small displacements of the inner cylinder. The latter solution agrees well with the more exact analytical expressions which are also derived.