On the nonlinear stability of the circular Couette flow to viscous axisymmetric perturbations

An axisymmetric viscous nonlinear stability analysis of the circular Couette flow to any finite amplitude perturbation is developed. The analysis is based on investigating the reduced Arnol'd energy-Casimir function $A_{rd}$, which consists of the sum of the total kinetic energy of the flow $E$ and the Casimir circulation dependent function $C_S$, i.e. $A_{rd}=E+C_S$. In this case, $\Delta A_{rd}$ is used as a Lyapunov function, which represents the difference between the reduced Arnol'd function at a later time $t$ and the corresponding base flow value. The requirement for the temporal decay of $\Delta A_{rd}$ leads to two novel conditions for the nonlinear stability of this steady flow against axisymmetric viscous perturbations of any finite amplitude. We also establish for the very first time a definite nonlinear stability region in terms of the operational parameters for the circular Couette flow. Once the flow is nonlinearly stable and stays axisymmetric, it always decays asymptotically to a unique steady state defined by the rotating cylinders. The results from this research shed a new fundamental physical insight into a classical flow problem that was studied for many decades.