Flow of second order fluids in curved pipes

Beginning with the work of Dean in 1927, regular perturbation methods have been used to study flows of incompressible Newtonian, generalized Newtonian and viscoelastic fluids in curved pipes of circular cross section. In thes e studies, the perturbation parameter is the curvature ratio: the cross sec tional radius of the pipe divided by the radius of curvature of the pipe ce nterline. Closed form perturbation solutions for the equations of motion fo r second order fluids have previously been obtained by other authors for th e special case when the second normal stress coefficient is zero, H.G. Shar ma, A. Prakash, Ind. J. Pure Appl. Math 8 (1977) 546-557; P.J. Bowen, Ph.D. Thesis, University of Wales, (1990); P.J. Bowen, A.R. Davies, K. Waiters, J. Non-Newtonian Fluid Mech. 38 (1991) 113-126. Here, we obtain closed form solutions for the perturbation equations even when the second normal stres s difference is non-zero. We show that for a countable number of combinatio ns of non-dimensional parameters a perturbation solution exists but is not unique. For other combinations of parameters a perturbation solution does n ot even exist. This latter result implies that for these parameter values t here does not exist a steady, fully developed solution for flow in curved p ipes which is a perturbation of the straight pipe solution, regardless of t he magnitude of the curvature ratio. We emphasize that these singular point s do not arise when the second normal stress coefficient is zero. A solutio n to the perturbation equations exists and is unique for values of the mate rial constants which correspond to real polymeric fluids. For these values of the material constants, the secondary motion at zero Reynolds number is qualitatively similar to that arising in Newtonian fluids due to inertial e ffects. (C) 2000 Elsevier Science B.V. All rights reserved.