  The effect of rotation on the classical gravity-driven Rayleigh-Taylor
instability has been shown to influence the scale of the perturbations that
develop at the unstable interface and consequently alter the speed of
propagation of the front. The present authors argued that this is as a result
of a competition between the destabilizing effect of gravity and the
stabilizing effect of the rotation. In the present paper we consider the
extreme limit of high rotation rates in which rotational forces dominate and
gravitational forces may be ignored. The two liquid layers initially form
concentric cylinders, centred on the axis of rotation. The configuration may be
thought of as a fluid-fluid centrifuge. There are two types of perturbation to
the interface that may be considered, an azimuthal perturbation around the
circumference of the interface and a varicose perturbation in the axial
direction along the length of the interface. It is the first of these types of
perturbation that we consider here, and so the flow may be considered
essentially two-dimensional, taking place in a circular domain.
  We carry out a linear stability analysis on a perturbation to the hydrostatic
background state and derive a fourth order Orr-Sommerfeld-like equation that
governs the system. We consider the dynamics of systems of stable and unstable
configurations, inviscid and viscous fluids, immiscible fluid layers with
surface tension, and miscible fluid layers that may have some initial diffusion
of density. Theoretical predictions are compared with numerical experiments and
the agreement is shown to be good. We do not restrict our analysis to equal
volume fluid layers and so our results also have applications in coating and
lubrication problems in rapidly rotating systems and machinery.
