We derive necessary conditions that traveling wave and other solutions of the Navier–Stokes equations must satisfy in the pipe, Couette, and channel flow geometries. Some conditions are exact and must hold for any traveling wave solution or periodic solution irrespective of the Reynolds number (Re). Other conditions are asymptotic in the limit Re. For the pipe flow geometry, we give computations up to Re=100000 showing the connection of our asymptotic conditions to critical layers that accompany vortex structures at high Re.

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