On the shape of air-liquid interfaces with surface tension that bound rigidly-rotating liquids in partially filled containers

The interface shape of a fluid in rigid body rotation about its axis and partially filling the container is often the subject of a homework problem in the first graduate fluids class. In that problem, surface tension is neglected, the interface shape is parabolic and the contact angle boundary condition is not satisfied in general. When surface tension is accounted for, the shapes exhibit much richer dependencies as a function of rotation velocity. We analyze steady interface shapes in rotating right-circular cylindrical containers under rigid body rotation in zero gravity. We pay especial attention to shapes near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation at certain specific rotational speeds. 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 introduced by the assumption that the interface is a right circular cylinder; this assumption is key to the spinning bubble method used to measure surface tension.