Transient dynamics of a rotating spherical liquid drop
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The transient rotation of a liquid drop in an infinite gaseous medium is analyzed for two cases: viscous retardation from constant-speed rotation, and steady-torque start-up with development to the steady state. This situation arises when a levitated liquid drop is rotated with an acoustically applied torque. Subsequent changes in the torque cause transient effects. To understand such a system, the two basic problems of start-up and retardation are studied. The rotational Reynolds number is considered to be low enough so that nonlinear inertial effects may be neglected. The Laplace-transform method is used to deal with the time dependence. Since the fluid velocity has only the azimuthal direction (and the profile is independent of it), and the other angular dependence is factorable as sin θ, the solution turns out to be effectively a two-variable problem in r and t. Nevertheless, the finite mass of the spherical drop and its finite viscosity make it a mathematically challenging problem. Besides the full analytical solution, results are obtained in the limits of a solid sphere, and small time for the liquid drop. In all liquid-drop cases, the results are limited to the drop viscosity being higher than the surrounding region. For several common liquids in a gas, the flow field indicates nearly a solid-sphere like behavior, except at small times in the region near the interface. The deviations from the asymptotic center-point velocity are amplified for illustration.
KeywordsLaplace transform Levitation Rotating drop Stokes flow Transient fluid dynamics
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