On steady flows of viscoelastic fluids in curved pipes

In this talk, we will discuss results for steady, fully developed flows of viscoelastic fluids in curved pipes and contrast this behavior with flows of Newtonian fluids. Following the approach of W. R. Dean and other authors, we have used regular perturbation methods to study flows of viscoelastic fluids in curved pipes. The perturbation parameter is the curvature ratio: the cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. We have obtained explicit solutions to the perturbation equations at first order for second order fluids and a modified Oldroyd-B fluid. In the absence of inertial effects, flows of Newtonian fluids in curved pipes display a secondary flow, rather a uniaxial flow exists which differs only slightly from the straight pipe solution. In contrast, even in the absence of inertial effects, the class of viscoelastic fluids studied display a secondary motion (see, e.g. Thomas 1963, Bowen et al. 1991, Robertson and Muller 1996). Significantly, for a countable number of combinations of material parameters and Reynolds numbers, there is a loss of uniqueness of the solution to the perturbation equations. For other values of material parameters and Reynolds number, a solution does not even exist. There is a region in parameter space which is free of such singularities. It is interesting that these singularities do not arise when the second normal stress coefficient is zero. This lack of existence to the perturbation equations regardless of the magnitude of the curvature ratio, implies a lack of existence of a solution which is a steady, fully developed perturbation of the straight pipe solution. The implications of this result are under investigation.