Stability of Spatially Periodic Supercritical Flows in Hydrodynamics

Recently, Eckhaus developed a theory for a class of nonlinear stability problems which can be formulated in terms of a scalar partial differential equation with quadratic nonlinearities. It is demonstrated that Eckhaus' work on the development and stability of periodic solutions can be extended to a class of nonlinear matrix partial differential equations. The equations governing axisymmetric viscous flow between concentric rotating cylinders belong to the class of equations considered. When the Taylor number $T$ is slightly above the minimum critical value $T_c$ there exists an interval of possible equilibrium flows (Taylor‐vortex flows) growing out of the instability of Couette flow. It is shown that within the interval of possible Taylor‐vortex flows, there exists a subinterval of stable vortex flows.