Dynamics of instabilities in a viscous fluid rotating with an acceleration

The flow of a viscous incompressible fluid between coaxial cylinders rotating with a constant rigid-body acceleration about the axis is studied numerically. A one-dimensional time-dependent solution of the Navier-Stokes equations is constructed analytically for the motion starting from the state of rest. On the initial time interval the one-dimensional unsteady fluid motion is unstable. Small perturbations introduced into the flow initiate the formation of secondary vortex flows with a velocity component along the axis. The dynamics of the developing instabilities and their dissipation are studied numerically. A condition determining the dimensions of the unsteady secondary flow zone is formulated. The unsteady regime is transient and starting from a certain instant of time the flow becomes stable.