A surprising bubble catastrophe
The author considers 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. The orifices are circles with radii \(r_i\) where each exposed drop is a spherical cap. The author investigates 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 radii \(r_1\) and \(r_2\) as universal unfolding parameters. For the case of three openings with a triple hemisphere equilibrium \((h,h,h)\), an elliptic umbilic with the radii \(r_i\) \((i=1,2, 3)\) as unfolding parameters is obtained. Surprising phenomena occur along the cusp lines emanating from the elliptic umbilic. Section 1 is an introduction. Section 2 contains the basic formulas needed for the analysis. In Section 3, the author gives a quick review of the key definitions and tools of catastrophe theory. In Section 4, he carries out the analysis for the two bubble experiments for the sake of completeness. In Section 5, he analyzes the three bubble configuration.
scientific article; zbMATH DE number 1356083