Families of reversing and non-reversing Taylor vortex flows between two co-oscillating cylinders with different amplitudes

This paper deals with the centrifugal instability of time-modulated Taylor-Couette flow for the case in which the inner and outer cylinders are co-oscillating around zero mean with the angular velocities Ωin = Ω0 cos(ωt) and Ωout = εΩ0 cos(ωt), respectively (Ω0, ω, and ε denote, respectively, the amplitude, the frequency of the modulated rotation, and the amplitudes ratio). The small-gap equations for the stability of this flow with respect to axisymmetric disturbances are derived and solved on the basis of Floquet theory. We recover in the case ε = 0 where the outer cylinder is stationary while the inner is modulated the two well-known reversing and non-reversing Taylor vortex flows. Attention is focused on the evolution of these time-dependent flows when one allows the oscillation of the outer cylinder. It turns out that an increase in the parameter ε leads to the discovery of families of reversing and non-reversing flows and other interesting bifurcation phenomena including codimension-two bifurcation points. In addition, a proper tuning of this parameter ε provides a control of the onset of instability as well as the nature of the primary bifurcation. Moreover, it is shown that when ε > 1, the instability is suppressed in low frequencies and the flow is always stable in good agreement to what is obtained by a quasi-steady approach where transient instability is detected. This latter is attributed to the fluid inertia taking place when the cylinders are reversing their rotation’s direction. However, no effect of the parameter ε is observed in high frequencies where the instability develops in thin boundary Stokes layers near the oscillating cylinders.