Energy stability theory of decelerating swirl flows

An infinite circular cylinder, filled with viscous, incompressible fluid, is rotating as a solid body. At time t = 0 the angular velocity ω(t) of the cylinder is decreased in a prescribed fashion. The resulting swirl flow is susceptible to centrifugal instabilities. The method of energy is used to determine sufficient conditions, R < R_G, where R is a Reynolds number, such that rotationally symmetric disturbances of arbitrary amplitude decay to zero with time. Both impulsive and smooth angular velocity histories of the container are considered. The analysis further provides a lower bound t ⁺ on the onset time before which all disturbances must decay to zero, for R > R_G. If the final state (as t→∝) also corresponds to solid body rotation (e.g., when the cylinder is brought to rest), then the analysis simultaneously provides an upper bound t ‡ on the decay time after which all disturbances decay zero, for R > R_G. In this case centrifugal instabilities are confined to times t in the interval t ⁺ < t < ‡.