We consider the effect of high rotation rates on two liquid layers that 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. A linear stability analysis is carried out on a perturbation to the hydrostatic background state and a fourth-order Orr–Sommerfeld-like equation that governs the system is derived. 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. In the most simple case of two layers of inviscid fluid separated by a sharp interface with no surface tension acting, we show that the effects of the curvature of the interface and the confinement of the system may be characterised by a modified Atwood number. The classical Atwood number is recovered in the limit of high azimuthal wavenumber, or the outer fluid layer being unconfined. 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.