EQUILIBRIA, STABILITY AND BIFURCATIONS OF ROTATING COLUMNS OF FLUID SUBJECTED TO PLANAR DISTURBANCES

Long gyrostatically rotating cylindrical drops held together by surface tension are amenable to conventional bifurcation analysis as well as newer, computer-aided methods of analysis, and so are useful prototypes of three-dimensional drops. Instability to Rayleigh's axisymmetric mode is set aside and effects of translationally symmetric (planar) disturbances are investigated. The shapes and stability of drops on and near the family of perfect cylinders are determined by means of the power series method of Millman & Keller. Nonlinearly distorted shapes and their stability are calculated by Galerkin or finite-element analysis, using either (a) a polar or (b) a composite polar and cartesian representation of drop shape. The disadvantage of using single-coordinate representation of drop shapes near break-up is brought out. The new results show that a family of symmetric two-lobed shapes bifurcates from the main family of perfectly cylindrical shapes when the rotation rate attains a critical value, in accord with the linearized hydrodynamic analysis of Hocking. Moreover, a new family of asymmetric two-lobed shapes is uncovered that bifurcates from and rejoins the family of symmetric two-lobed shapes, using a polar representation of drop shape. Plainly, the new shape family is the two-dimensional analog of the family of three-dimensional capillary peanuts discovered by Brown & Scriven, who used a spherical polar representation of drop shape. By way of contrast, the results obtained using a composite polar and cartesian representation of drop shape show that the gyrostatic family of symmetric two-lobed shapes does not exchange stability with a family of asymmetric shapes and eventually breaks up by becoming a family of self-intersecting shapes.