Dean flow of a Bingham fluid in a curved rectangular duct

In this paper we study the flow of a Bingham fluid on a curved (rectangular) pipe. Flows in this kind of geometries present secondary flows due to presence of centripetal forces in the radial direction. This so called Dean Flow has been extensively studied for Newtonian fluids. In considering a yield stress fluid in similar geometries the picture is less clear. Unfortunately, there is not an analytical solution, so the flow has been barely studied. Thus the purpose of this work. We consider only steady flow and develop a solution as a perturbation series in terms of the Dean number, Dn. The leading order solution is an axisymmetric azimuthal flow, driven by the pressure gradient. We compute this flow numerically using an augmented Lagrangian method on a regular rectangular mesh. We have also studied the limiting case of zero flow and we give a general expression for the critical Bingham number, B-c, for this to happen. The first order velocity perturbation produces 2 recirculating vortices, at top and bottom walls, analogous to those of the Newtonian flow. The size of the vortices appears to decrease slightly as B -> B-c(-), and the magnitude of the secondary flow decays to zero significantly faster than that of the leading order flow. Finally we discuss the short comings of the perturbation series approach and propose how to overcome them.