Fully developed viscous and viscoelastic flows in curved pipes

Some h-p finite element computations have been carried out to obtain soluti ons for fully developed laminar flows in curved pipes with curvature ratios from 0.001 to 0.5. An Oldroyd-3-constant model is used to represent the vi scoelastic fluid, which includes the upper-convected Maxwell (UCM) model an d the Oldroyd-B model as special cases. With this model we can examine sepa rately the effects of the fluid inertia, and the first and second normal-st ress differences. From analysis of the global torque and force balances, th ree criteria are proposed for this problem to estimate the errors in the co mputations. Moreover, the finite element solutions are accurately confirmed by the perturbation solutions of Robertson & Muller (1996) in the cases of small Reynolds/Deborah numbers. Our numerical solutions and an order-of-magnitude analysis of the governing equations elucidate the mechanism of the secondary flow in the absence of second normal-stress difference. For Newtonian flow, the pressure gradient near the wall region is the driving force for the secondary flow; for creep ing viscoelastic flow, the combination of large axial normal stress with st reamline curvature, the so-called hoop stress near the wall, promotes a sec ondary flow in the same direction as the inertial secondary flow, despite t he adverse pressure gradient there; in the case of inertial viscoelastic fl ow, both the larger axial normal stress and the smaller inertia near the wa ll promote the secondary flow. For both Newtonian and viscoelastic fluids the secondary volumetric fluxes per unit of work consumption and per unit of axial volumetric flux first in crease then decrease as the Reynolds/Deborah number increases; this behavio ur should be of interest in engineering applications. Typical negative values of second normal-stress difference can drastically suppress the secondary flow and in the case of small curvature ratios, make the flow approximate the corresponding Poiseuille flow in a straight pipe. The viscoelasticity of Oldroyd-B fluid causes drag enhancement compared to Newtonian flow. Adding a typical negative second normal-stress difference produces large drag reductions for a small curvature ratio delta = 0.01; ho wever, for a large curvature ratio delta = 0.2, although the secondary flow s are also drastically attenuated by the second normal-stress difference, t he flow resistance remains considerably higher than in Newtonian flow. It was observed that for the UCM and Oldroyd-B models, the limiting Deborah numbers met in our steady solution calculations obey the same scaling crit erion as proposed by McKinley et al. (1996) for elastic instabilities; we p resent an intriguing problem on the relation between the Newton iteration f or steady solutions and the linear stability analyses.