Flow of Oldroyd-B fluids in curved pipes of circular and annular cross-section

Abstract

Perturbation methods are used to study steady, fully developed flow of Oldroyd-B fluids through curved pipes for both pipes of circular and annular cross-section. The perturbation parameter is the curvature ratio, given by the cross-sectional radius of the pipe divided by the constant radius of the pipe centerline. We compare results for creeping and non-creeping flows for both viscoelastic and Newtonian fluids. In pipes of circular cross-section, the velocity field for creeping flows of Oldroyd-B fluids is qualitatively similar to that found for non-creeping flows of Newtonian fluids. Namely, in addition to the primary flow, there is a secondary motion consisting of counter-rotating vortices. In curved annular pipes, two pairs of counter-rotating vortices are generated by either inertial or elastic effects. In this geometry, the differences between creeping flow of viscoelastic fluids and non-creeping flow of Newtonian fluids are dramatically accentuated at small values of inner to outer pipe radius, r i r o . For Newtonian fluids, as r i r o approaches zero, the magnitude and size of the vortices adjacent to the inner cylinder shrink to zero. However, for creeping flow of Oldroyd-B fluids, the inner vortex pair is comparable to the outer vortex pair in both size and strength, even for values of r i r o as small as 0.01. For pipes of circular cross-section, the effect of elasticity on the drag is considered and earlier predictions by Bowen et al. for the upper convected Maxwell fluid are extended to the Oldroyd-B fluid for non-zero Reynolds number.