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 Γ ≡ R/H, where R is the disk radius and 2H is the inter-disk spacing. The scaling law for the critical Reynolds number is found to be R ec Γ- 1/2 , with Re ≡ ΩH 2 /ν, Ω being the magnitude of the angular velocity and ν the kinematic viscosity. An asymptotic analysis for large Γ 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.