The present theoretical article considers the nonlinear interaction of oblique three dimensional Tollmien-Schlichting waves and induced or input longitudinal vortex motion, mainly for channel flow at large Reynolds numbers. Both the waves and the vortices are controlled by viscous-inviscid balancing but their respective flow structures are rather different because of the different typical timescales involved. This leads to the vortex-wave interaction being governed by nonlinear evolution equations on the vortex timescale, even though the wave amplitudes are notably small. The analogue in boundary-layer transition, addressed in a previous paper, is also re-considered here. Computational and analytical properties of the interaction equations for both channel flows and boundary layers are investigated, along with certain connections with companion studies of other vortex-wave interactions in channel flow. The nonlinear interactions in channel flow are found to lead to finitetime blow-up in amplitudes or to sustained vortex flow at large scaled times, depending on the input conditions. In particular, increasing the input amplitudes of the vortex or the wave can readily provoke blow-up even in the linearly stable regime; whereas in the case of sustained vortex flow new physical effects come into play on slightly longer timescales. Again, a very interesting feature is that the blowup response is found to be confined to a small range of wave angles near 45° relative to the original flow direction.
Study of the Blow Up of the Maximal Solution to the Three-Dimensional Magnetohydrodynamic System in Lei-Lin-Gevrey Spaces
