On the nonlinear stability of slowly varying time-dependent viscous flows

In this paper we investigate the manner in which finite-amplitude disturbances are set up in viscous fluid flows that are changing slowly in time. It is shown that, when the appropriate Reynolds or Rayleigh number is slowly increased, then, no matter how slowly this change takes place, there is always a short time interval where a quasi-steady approach breaks down. In this time interval a finite-amplitude solution is set up which ultimately approaches that predicted by a quasi-steady theory. In order to demonstrate our ideas we discuss the Taylor-vortex problem in a situation in which the speed of the inner cylinder changes slowly in time. In particular we discuss the case when the speed of the inner cylinder is modulated slowly in time and it is found that at low frequencies the disturbances of most physical relevance are not periodic solutions of the equations of motion.