On flow of a Navier-Stokes fluid in curved pipes. Part I: Steady flow

We consider the steady, fully developed motion of a Navier-Stokes fluid in a curved pipe of cross-section D under a given axial pressure gradient G. We show that, if G is constant, this problem has a smooth steady solution, for arbitrary values of the Dean's number κ, for D of arbitrary shape and for any curvature ratio δ of the pipe. This solution is also unique for κ sufficiently small. Moreover, we prove that the solution is unidirectional (no secondary motion) if and only if κ=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→0.