On flow of a Navier–Stokes fluid in curved pipes. Part I: Steady flow Original Research Article

We consider the steady, fully developed motion of a Navier–Stokes fluid in a curved pipe of cross-section DD under a given axial pressure gradient GG. We show that, if GG is constant, this problem has a smooth steady solution, for arbitrary values of the Dean’s number κκ, for DD of arbitrary shape and for any curvature ratio δδ of the pipe. This solution is also unique for κκ sufficiently small. Moreover, we prove that the solution is unidirectional (no secondary motion) if and only if κ=0κ=0. Finally, we show the same properties for the approximations to the Navier–Stokes equations called “Dean’s equations” and provide a rigorous way in which solutions to the full Navier–Stokes equations approach those to this approximation in the limit of δ→0δ→0.