The sliding Couette flow problem
Seminar Room 1, Newton Institute
The sliding Couette flow, categorised by Joseph (1976), is a flow between concentric cylinders of radii, a and b ( > a), where the inner cylinder is pulled with an axial speed, U , relative to the stationary outer cylinder. It is known that the linear critical Reynolds number based on the speed U and the gap width, b-a, is infinite, at least when the radius ratio is not very small, so that secondary flows, if they exist, must bifurcate abruptly from the laminar state. The absence of linear instabilities occurs similarly in the problems, such as plane Couette flow, pipe Poiseuille flow and flow in a square duct, which have been extensively explored with success in recent years. As far as the author knows finite amplitude solutions in the sliding Couette flow have not yet been found.
In this short paper we analyse both linear and nonlinear instabilities of the sliding Couette flow in the limit of narrow gap. Following Masuda, Fukuda & Nagata (2008) we apply a uniform rotation.O, in the streamwise direction in order to provoke rotational instabilities. The idea is to see whether bifurcated flows developed with increasing O may be sustained in the subcritical region and even exist as O is reduced back to zero.
We show numerically that the critical Reynolds number approaches the global stability limit determined by energy theory in the limit of large rotation rate. A nonlinear analysis indicates that secondary flows bifurcating at a moderate rotation rate are characterized by three-dimensional spiral vortex structures. Attempted continuation of the secondary flow branch to the zero rotational rate will be discussed.
 Joseph,D. D. (1976) Stability of Fluid Motions I, Springer-Verlag.
 Masuda, S., Fukuda, S. & Nagata, M. (2008) 'Instabilities of plane Poiseuille flow with a streamwise system rotation', J. Fluid Mech., 603, 189-206.