A surprising bubble catastrophe

Abstract

Consider a tube with several orifices out of which fluid of a given volume protrudes. In gravity free conditions the equilibrium configuration will be one which minimizes total surface area subject to the volume constraint. The surface of each liquid drop will have the same constant mean curvature. Suppose that the orifices are cirlces with radii ri where each exposed drop is a spherical cap. We analyze this problem from the viewpoint of catastrophe theory. For a tube with two circular openings the interesting situation occurs when the configuration supports a double hemisphere (h, h) equilibrium. This gives a cusp catastrophe with the radii r1 , r2 as universal unfolding parameters. For the case of three openings with a triple hemisphere equilibrium (h, h, h) we obtain an elliptic umbilic with the radii r1, r2, r3 as unfolding parameters. Further surprising phenomena occur along the cusp lines emanating from the elliptic umbilic.