Linear stability of circular Couette flow in the limit of small radius ratio

In the context of a detailed study of the linear stability of spiral Poiseuille flow at small radius ratio (Cotrell and Pearlstein, $J. Fluid Mech.$, in press), we have shown that in the limiting case of no rotation, annular Poiseuille flow is linearly stable at all $Re$, provided that the radius ratio lies below a critical value. Here, we consider the other limiting case, of no axial flow, and report a numerical investigation of the stability of circular Couette flow for small radius ratio. The results are compared to experimental work of Theodorsen for a whirling shaft in an unbounded, otherwise quiescent fluid.