| The general stability criteria of inviscid Couette flow between two coaxial cylinders
with angular velocity Ω(r) are obtained analytically. First, a necessary instability
criterion for centrifugal flows is derived as ξ′(Ω - Ωs) < 0 (or ξ′-(Ω - Ωs) < 0)
somewhere in the flow field, where ξ is the vorticitiy of profile and Ωs is the angular
velocity at the inflection point ξ′ = 0. Second, a criterion for stability is found as
-(μ1 + 1-r2) < f(r) = –ξ′–
Ω- Ωs < 0, where μ1 is the smallest eigenvalue. The new
criteria are the analogues of the criteria for parallel flows, which are special cases of
Arnol’d’s nonlinear criteria. Specifically, Pedley’s cirterion is proved to be an special
case of Rayleigh’s criterion. Moreover, the criteria for parallel flows can also be
derived from those for the rotating flows. These results extend the previous theorems
and would intrigue future research on the mechanism of hydrodynamic instability. |