Flow of second-order fluid in a curved duct with square cross-section

This paper investigates the inertial and creeping flow of a second-order fluid in a curved duct with a square cross-section. Numerical modeling is employed to analyze fluid flow, and the governing equations are discretized using the finite difference method on a staggered mesh. The marker-and-cell method is employed to allocate the parameters on the staggered mesh, and static pressure is calculated using the artificial compressibility approach. The effect of centrifugal force due to the curvature of the duct and the opposing effects of the first and second normal stress difference on the flow field are investigated. In addition, the order-of-magnitude technique is used to derive the force balance relations for the core region of flow. Based on these relations, the performance mechanism of centrifugal force and normal stress differences on the generation of secondary flows is considered. We also present an analytical relation for the axial velocity profile and flow resistance ratio of creeping flow. For this kind of flow, previous studies have investigated the effect of the first normal stress difference on the transition from one pair to two pairs of vortices while we show that the negative second normal stress difference has the opposite effect on this transition and can stabilize the flow.