Free surface shapes in rigid body rotation: Exact solutions, asymptotics and approximants
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We analyze steady interface shapes in zero gravity in rotating right circular cylindrical containers under rigid body rotation. Predictions are made near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation. We examine geometries where the container is axially infinite and derive properties of their solutions. We then examine in detail two special cases of menisci in a cylindrical container: a meniscus spanning the cross section; and a meniscus forming a bubble. In each case we develop exact solutions for the meniscus height and the bubble length as infinite series in powers of appropriate rotation parameters; and we find the respective asymptotic behaviors as the shapes approach their critical configuration. Finally we apply the method of asymptotic approximants to yield analytical expressions for the height of the meniscus and the length of a spinning bubble over the whole range of rotation speeds. As the spinning bubble method is commonly used to measure surface tension, the latter result has practical relevance.
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