Axisymmetric stability of the flow between two exactly counter-rotating disks with large aspect ratio

We study the first bifurcation in the axisymmetric flow between two exactly counter-rotating disks with very large aspect ratio $ \Gamma\,{\equiv}\, R/H$, where $R$ is the disk radius and $2 H$ is the inter-disk spacing. The scaling law for the critical Reynolds number is found to be $\Rey_c \propto \Gamma^{-1/2}$, with $\Rey \,{\equiv}\, \Omega H^2/\nu$, $\Omega$ being the magnitude of the angular velocity and $\nu$ the kinematic viscosity. An asymptotic analysis for large $\Gamma$ is developed, in which curvature is neglected, but the centrifugal acceleration term is retained. The Navier–Stokes equations then reduce to leading order to those in a Cartesian frame, and the axisymmetric base flow to a parallel flow. This allows us locally to use a Fourier decomposition along the radial direction. In this framework, we explain the physical mechanism of the instability invoking the linear azimuthal velocity profile and the effect of centrifugal acceleration.