Initiation of Self-Oscillations at Loss of Stability of Spatially-Periodic, Three-Dimensional Viscous Flows with Respect to Long-Wave Perturbations

We study the long-wave asymptotics of a secondary regime formed at loss of stability of a stationary spatially-periodic flow, as one of the periods grows without bound (the wavenumber vanishes). It is shown that if the main flow, aligned with the long period, is non-zero and certain nondegeneracy conditions are satisfied, then a self-oscillation regime branches off the main solution as the viscosity decreases, both soft and rigid modes of loss of stability being possible. Explicit formulas for the main terms of the asymptotics are derived. Examples of the self-oscillations on particular flows are calculated and the behavior of the fluid particle paths in the self-oscillation traveling-wave' regime branching off the shear flow is studied.