Fully developed viscous and viscoelastic flows in curved pipes. (Q2744299)

The authors have carried out some \(h\)-\(p\) finite element computations to obtain solutions for fully developed laminar flows in curved pipe with curvature ratios from 0.001 to 0.5. An Oldroyd-3-constant model is used to represent the viscoelastic fluid, which includes the upper-convected Maxwell (UCM) model and the Oldroyd-B model as special cases. The numerical solutions and an order-of-magnitude analysis of the governing equations elucidate the mechanism of secondary flow in the absence of second normal-stress difference. For Newtonian flow, the pressure gradient in the wall region is the driving force for the secondary flow; for creeping viscoelastic flow, the combination of large axial normal stress with streamline curvature, the so-called hoop stress near the wall, promotes a secondary flow in the same direction as the inertial secondary flow, despite the adverse pressure gradient; in the case of inertial viscoelastic flow, both the larger axial normal stress and the smaller inertia near the wall promote the secondary flow. It was observed that for the UCM and Oldroyd-B models, the limiting Deborah numbers met in the steady solution calculations obey the same scaling criterion as proposed by \textit{G. H. McKinley}, \textit{P. Pakdel} and \textit{A. Öztekin} [J. Non-Newtonian Fluid Mech. 67, 19--47 (1996)] for elastic instabilities.