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 kth 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 btained.