Pengtao Yue, Joseph Dooley & James J. Feng

*J. Rheol.* **52**, 315-332 (2008)

**Abstract** - This paper investigates the mechanism for secondary recirculations in non-Newtonian flows in a non-circular pipe, and develops a general criterion on the direction of the secondary flow based on the fluid rheology and the cross-sectional geometry of the pipe. Although the secondary flow is usually attributed to the second normal stress difference N_{2}, the relationship between the two turns out to be more involved than previously assumed. By theoretical analysis and numerical computations using the Giesekus model, we show that N_{2} produces an effective body force that, if non-conservative, gives rise to secondary flows in the transverse direction. From this understanding, we propose a criterion for the direction of the secondary flow based on the second normal stress coefficient &Psi_{2} and the shear viscosity &eta_{s}: if &Psi_{2}/&eta_{s} is an increasing function of the strain rate, the fluid flows from high shear regions to low shear regions along the walls, and vice versa. This criterion accounts for all the prior computational work and resolves some inconsistencies in the literature. It is also consistent with all experimental observations to date.