The shapes and stability of captive rotating drops

Abstract

Computer-aided means are presented for constructing entire families of solutions to Young and Laplace’s nonlinear partial differential equation of capillarity, with the enclosed volume prescribed, and of determining the stability of the solutions and bifurcations between families having different three-dimensional symmetry properties; equivalently, these are means for surveying the topography of corresponding energy surfaces in especially convenient finite-dimensional function spaces spanned by socalled finite element bases in which both the solutions and variations of them are represented. The means are a finite element algorithm employing Newton iteration and, for the stability and bifurcation eigenproblem, a block-Lanczos method. The algorithm is applied to gyrostatic liquid drops of fixed volume held captive between two co-rotating, parallel, concentric faces or contact circles and acted on by surface tension and centrifugal force. The results for the special case of captive cylindrical drops compare well with published and new results of conventional stability and bifurcation analysis. Axisymmetric drop shapes that evolve from rest shapes of constant mean curvature are found to form a one-parameter family in rotational Bond number £ = Q ² R ³ Ap/8cr. Bifurcating axisymmetric and three-dimensional families are calculated. The limit of stability is found to lie in the family of simplest axisymmetric drops, except in the case of very fat ones, which exchange stability with G-shaped drops, a remarkable fact. Implications for the experiments of Plateau, Carruthers & Grasso, and others are discussed.