Free surface shapes in rigid body rotation: Exact solutions, asymptotics and approximants

We analyze steady interface shapes in rotating right-circular cylindrical containers under rigid body rotation in zero gravity. 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 respective axial lengths 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 axial lengths of menisci over the whole range of rotation speeds. In this application, the analytical solution is employed to examine errors in the assumption that the interface is a right circular cylinder; this assumption is key to the spinning bubble method used to measure surface tension.