The linear stability of parallel flow in a concentric annulus to infinitesimal, axially symmetric disturbances is considered. First, the Poiseuille flow in annular cylinders is studied with the ratiokof the outer to inner cylinder as a parameter. The critical Reynolds number is a monotone function of this radius ratio, increasing without bound ask → ∞(Hagen‐Poiseuille flow) from the plane Poiseuille flow limit(k = 1). Second, a one‐parameter family of skewed (variable viscosity) flows in a fixed annulus is studied. The neutral curves for many of these skewed profiles have a second minimum, which for sufficiently skewed profiles, gives the lowest value of the Reynolds number. These two minima are associated with two very different distributions of Reynolds stress. Both distributions are such that on part of the channel the energy is transferred from the disturbance motion to the basic motion and both can be explained by analysis of the Reynolds stress jump condition and the known structure of Reynolds stress near a rigid wall.