The dependence of the shape and stability of captive rotating drops on multiple parameters

Abstract
The dependence of the shape and stability of rigidly rotating captive drops on multiple parameters is analysed by applying asymptotic and computer-aided techniques from bifurcation theory to the Young-Laplace equation which governs meniscus shape. In accordance with Brown and Scriven, equilibrium shapes for drops with the volume of a cylinder and without gravity are grouped into families of like symmetry that branch from the cylindrical shape at specific values of rotation rate, measured by the rotational Bond number E. Here, the evolution of these families with changes in drop volume Y , drop length B , and gravitational Bond number G is presented. Criteria are laid out for predicting drop stability from the evolution of shape families in this parameter-space and they circumvent much of the extensive solution of eigenproblems used previously. Asymptotic analysis describes drops slightly different from the cylindrical ones and shows that some shape bifurcations from cylinders to wavy, axisymmetric menisci are ruptured by small changes in drop volume or gravity. Near these points at least one of the shape families singularly develops a fold or limit point. Numerical methods couple finite-element representation of drop shape which is valid for a wide range of parameters with computer-implemented techniques for tracking shape families. An algorithm is presented that calculates, in two parameters, the loci of bifurcation or limit points; this is used to map drop stability for the four-dimensional parameter space (E, / B , G ). The numerical and asymptotic results compare well in the small region of parameters where the latter are valid. An exchange of axisymmetric mode for instability is predicted numerically for drop volumes much smaller than that of a cylinder.

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