J. Fluid Mech. (2007), vol. 588, pp. 59–74. c  2007 Cambridge University Press doi:10.1017/S0022112007007422 Printed in the United Kingdom 59 On the compressible Taylor–Couette problem A. MANELA 1 AND I. FRANKEL 2 1 Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 2 Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel (Received 11 October 2006 and in revised form 13 May 2007) We consider the linear temporal stability of a Couette flow of a Maxwell gas within the gap between a rotating inner cylinder and a concentric stationary outer cylinder both maintained at the same temperature. The neutral curve is obtained for arbitrary Mach (Ma) and arbitrarily small Knudsen (Kn) numbers by use of a ‘slip-flow’ continuum model and is verified via comparison to direct simulation Monte Carlo results. At subsonic rotation speeds we find, for the radial ratios considered here, that the neutral curve nearly coincides with the constant-Reynolds-number curve pertaining to the critical value for the onset of instability in the corresponding incompressible-flow problem. With increasing Mach number, transition is deferred to larger Reynolds numbers. It is remarkable that for a fixed Reynolds number, instability is always eventually suppressed beyond some supersonic rotation speed. To clarify this we examine the variation with increasing Ma of the reference Couette flow and analyse the narrow-gap limit of the compressible TC problem. The results of these suggest that, as in the incompressible problem, the onset of instability at supersonic speeds is still essentially determined through the balance of inertial and viscous-dissipative effects. Suppression of instability is brought about by increased rates of dissipation associated with the elevated bulk-fluid temperatures occurring at supersonic speeds. A useful approximation is obtained for the neutral curve throughout the entire range of Mach numbers by an adaptation of the familiar incompressible stability criteria with the critical Reynolds (or Taylor) numbers now based on average fluid properties. The narrow-gap analysis further indicates that the resulting approximate neutral curve obtained in the (Ma, Kn) plane consists of two branches: (i) the subsonic part corresponding to a constant ratio Ma/Kn (i.e. a constant critical Reynolds number) and (ii) a supersonic branch which at large Ma values corresponds to a constant product Ma Kn. Finally, our analysis helps to resolve some conflicting views in the literature regarding apparently destabilizing compressibility effects. 1. Introduction The Taylor–Couette (TC) instability in a fluid between rotating concentric cylinders giving rise to a secondary vortical flow (‘Taylor vortices’) is a classical problem in hydrodynamic stability theory (Chandrasekhar 1961; Drazin & Reid 1981; Koschmieder 1993). The problem has been investigated extensively for incompressible fluids. Rayleigh (1916) formulated a stability criterion for the mean viscous flow which is based on the inviscid perturbation equations. Subsequent analyses have considered the viscous perturbation equations where transition to instability is governed by Re c , a critical value of the Reynolds (Re) number, depending on the ratios of cylinders’ radii and rotation speeds. Taylor (1923) studied the narrow-gap approximation. He