Spinning viscous sheets

We study the extensional flow of a circular viscous sheet driven by centrifugal forces. For a liquid of constant extensional viscosity, we show the existence of a similarity solution for the thickness of the sheet and the radial speed of the liquid. The radius of the circular sheet is found to increase with time $t$ as $(1-t)^{-1/2}$, and hence becomes singular over a timescale sets by the kinematic viscosity of the liquid and the angular speed. We then investigate the case of a liquid that has the extensional viscosity that increases with increasing extension rate and investigate how the dynamics is affected by such ``extensional thickening.''