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This study provides sufficient conditions for the temporal monotonic decay of enstrophy for two-dimensional perturbations traveling in the incompressible, viscous, plane Poiseuille, and Couette flows. Extension of Synge's procedure [J. L. Synge, Proc. Fifth Int. Congress Appl. Mech. 2, 326 (1938); Semicentenn. Publ. Am. Math. Soc. 2, 227 (1938)] to the initial-value problem allow us to find the region of the wave-number-Reynolds-number map where the enstrophy of any initial disturbance cannot grow. This region is wider than that of the kinetic energy. We also show that the parameter space is split into two regions with clearly distinct propagation and dispersion properties.

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows: the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and nondispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum behave in a dispersive manner. By contrast, if the wave number is above a specific threshold, a sharp dispersive-to-nondispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocity can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.