On the stability of two-dimensional flows close to the shear

We consider the stability problem for two-dimensional spatially periodic flows of general form, close to the shear, assuming that the ratio of the periods tends to zero, and the average of the velocity component corresponding to the “long” period is non-zero. The first terms of the long-wavelength asymptotics are found. The coefficients of the asymptotic expansions are explicitly expressed in terms of some Wronskians and integral operators of Volterra type, as in the case of shear basic flow. The structure of eigenvalues and eigenfunctions for the first terms of asymptotics is identified, a comparison with the case of shear flow is made. We study subclasses of the considered class of flows in which the general properties of the qualitative behavior of eigenvalues and eigenfunctions are found. Plots of neutral curves are constructed. The most dangerous disturbances are numerically found. Fluid particle trajectories in the self-oscillatory regime in the linear approximation are given.