# The Role of Elasticity in the Vortex Formation in Polymeric Flow around a Sharp Bend
## 1 Introduction  
Many modern-day products, such as rubber tires and plastic bags, are made from polymers. These items are made by specialized processing machines that handle polymer solutions in pipes, conduits, and accessories such as bends and elbows. These polymeric solutions, as with any other liquid, are subjected to many fluid dynamic effects. One such effect is the formation of recirculation zones, or vortices in abrupt, changes of flow direction as in contractions and bends. These vortices are a feature where some amount of the fluid becomes trapped in a cyclone like structure near or around corners, justifying those authors that refer to them as separation bubbles [1]. Recirculation zones can have detrimental effects on the flow of polymers and polymer solutions, affecting their manufacturing processes. As a result, it is important to know the conditions under which vortices form. The presence of vortices near or around corners is one of the most significant alterations in channel flow for both non-Newtonian and Newtonian fluids. Vortices are well known to form in Newtonian fluids due to the effect of inertia [2]. However, for polymers and polymer solutions not all the physics involved are fully understood [3, 4]. Understanding the flow behavior of polymers and polymer solutions is essential to design and optimize fluid flow systems in many practical applications. For example, in microfluidics the polymer concentration has a significant effect on viscoelastic behavior by altering the base flow or result in flow instabilities. In that regard, [PERSON] et al. [5] studiedthe flows of dilute and semi-dilute polymer solutions in sharp 90\({}^{\circ}\) micro-bends in channels of rectangular cross-section. Their flow visualizations show that a vortex is present in the inner, upstream corner of the bend and grows with increasing Reynolds and Weissenberg numbers for flows of shear-thinning, semi-dilute polymeric solutions. They reported that secondary flows were not present for Newtonian flows under similar conditions and that a vortex is absent for flow of a dilute, non-shear thinning PEO solution. They concluded that shear-thinning appears to be central to the presence of an elastic secondary flow in this geometry. Their experiments were carried at very low Reynolds number (\(10^{-6}<\text{Re}<0.03\)), and Weissenberg numbers ranging from 0.42 to 126. More recently, [PERSON] et al. [6] reported instabilities in viscoelastic flow in a 90\({}^{\circ}\) bent channel. They observed that the flow instability in an aqueous PEO solution occurs when the concentration of PEO is as low as 50 ppm. Investigating the effects of the polymer concentration, flow rate, and elasticity number, they found that the flow is stabilized in shear-thinning fluids, whereas the flow instability is amplified when both elastic and inertial effects are pronounced. Their experiments were carried out at Reynolds numbers ranging from 0.3 to 3.0 and Weissenberg numbers of 0 (Newtonian flow) to 40. Both studies [5; 6] coincide in pointing to the shear thinning properties of the solution, a decrease in viscosity under shear strain, as the main reason for the formation of vortices and secondary flows. At the macroscopic level, it is also well known that polymers, as well as surfactants, are frequently added to Newtonian fluids with the purpose of reducing friction losses in straight pipe turbulent flow. In some cases, the drag-reducing rate is as high as 75% [7] making them attractive for using in complex industrial pipe flow systems. Friction losses in industrial piping systems are mostly due to accessories such as bends, tees, and valves, rather than in the straight pipes. Understanding how polymers affect the Newtonian flow in bends and accessories would help in evaluating their drag reduction potential and application in intricate piping systems. [PERSON] et al. [8] studied the bend flow characteristics of two surfactant solutions, experimentally and numerically.