Bifurcations and instabilities in sliding Couette flow

We carry out linear and nonlinear analyses on a flow between two infinitely long concentric cylinders with the radii a and b subject to a sliding motion of the inner cylinder in the axial direction. We confirm the linear stability result of Gittler (Acta Mechanica, vol. 101, 1993, p. 1) for the axisymmetric case, namely the flow is linearly stable against axisymmetric perturbations when the radius ratio eta = a/b is greater than 0.1415. We extend his analysis to the non-axisymmetric case and find that the stability of the flow is still determined by axisymmetric perturbations. Our nonlinear analysis exhibits that (i) finite-amplitude axisymmetric solutions exist far below the linear critical Reynolds number for eta < 0.1415 and (ii) non-axisymmetric travelling wave solutions appear abruptly at a finite Reynolds number even for eta > 0.1415 where the linear critical state is absent.