Instability of long-wave viscous flows (Q806465)

The asymptotics of the problem of stability of spatially periodic three- dimensional viscous incompressible flows are considered in the case in which one of the periods increases without bound, while the other two are fixed. Long-wave modes, for which asymptotic expansions in powers of the wave number \(\alpha\) are constructed, are considered. The conditions of solvability of the higher approximations are used for determining the decay rate expansion coefficients. When the main longitudinal (along the long period) flow is nonzero, the principal approximation of the long- wave mode is a traveling wave of the transverse flow with a superimposed complex longitudinal flow. The latter is generated by the interaction between the traveling wave and the main logitudinal flow.