Sink flow of a rotating stratified fluid from a horizontal circular opening is treated as a time-dependent problem that begins from a quiescent state. This problem is considered on an assumption that both the deviation of the density distribution and the Mach number are small. This flow is then governed by the Navier-Stokes set of equations for an incompressible fluid, which is solved numerically by the finite-difference method. It is clarified that, as time goes on, the withdrawal thickness increases owing to the effects of Coriolis and centrifugal forces when the viscous effect is negligible. The case of dominant viscous effect is also examined and interesting feature is found.