Existence and uniqueness of steady, fully developed flows of second order fluids in curved pipes

Steady, fully developed flows of second order fluids in curved pipes of circular cross-section have previously been studied using regular perturbation methods.(2, 3,12,20) These perturbation solutions are applicable for pipes with small curvature ratio: The cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. It was shown by Jitchote and Robertson(12) that perturbation equations could be ill-posed when the second normal stress coefficient is nonzero. Motivated by the singular nature of the perturbation equations, here, we study the full governing equations without introducing assumptions inherent in perturbation methods. In particular, we examine the existence and uniqueness of solutions to the full governing equations for second order fluids. We show rigorously that a solution to the full problem exists and is locally unique for small non-dimensional pressure drop, in agreement with earlier results obtained using a formal expansion in the curvature ratio.(12) The results obtained here axe valid for arbitrarily shaped cross-section (sufficiently smooth) and for all curvature ratios. An operator splitting method has been employed which may be useful for numerical studies of steady and unsteady flows of second order fluids in curved pipes.