Multiscale three-dimensional vortex pairs of the Poiseuille flow

In three dimensions, the Boussinesq-Rayleigh series solutions of the unsteady Navier-Stokes equations are computed by symbolic programming and evaluated by parallel computing. For generating functions, which are bounded together with their derivatives, the absolute convergence of the series solutions is shown by converting the differential recurrent relations into tensor recurrent relations and using the comparison and ratio tests. A pyramidal structure of four-dimensional tensors of derivatives, which is employed in the tensor recurrent relations, is obtained by induction. It is shown that the general solutions away from boundaries are nonlinear superpositions of the Stokes flow, the Bernoulli flow, the Couette flow, and the Poiseuille flow, which are unsteady, three-dimensional continuations of the classical flows at high Reynolds numbers. The general solution for the Poiseuille flow is specified by periodic generating functions, which model mixing in one-, two-, and three- dimensional flows away from boundaries. The emergence and interaction of multiscale vortex pairs are treated mathematically due to the existence of multi-valued general solutions for streamlines of the Poiseuille flow at high Reynolds numbers.