On the problem of shear flow stability with respect to long-wave perturbations

To find secondary flow branching to the steady flow it is necessary to consider linear spectral problem and linear adjoint problem. Long-wave asymptotics of linear adjoint problem in two-dimensional case is under consideration. We assume the periodicity with spatial variables when one of the periods tends to infinity. Recurrence formulas are obtained for the $k$th term of the velocity and pressure asymptotics. If the deviation of the velocity from its period-average value is an odd function of spatial variable, the velocity coefficients are odd for odd $k$ and even for even $k$. The relations between coefficients of linear adjoint problem and linear spectral problem are obtained.