# The Role of Elasticity in the Vortex Formation in Polymeric Flow around a Sharp Bend
## 2 Materials and Methods  
### Governing Equations  
Two types of fluids were studied in this research, a Newtonian fluid and a polymeric fluid. Newtonian flows are characterized by Navier-Stokes equations. Polymer flows require additional equations to characterize the elastic behavior of the polymer chains. In this work, the polymer chains are characterized using the FENE-P model (finitely extensible nonlinear elastic model with Peterlin closure [9]). The FENE-P model was chosen due to its versatility and simplicity in characterizing polymer behavior. In the FENE-P model, polymers are represented as dumbbells, two masses connected by a spring, as in many other polymer models [10]. Nevertheless, in the FENE-P model the spring has a finite stretching limit. For this work it is assumed that the fluid is incompressible for both Newtonian and polymer-based flows, for which the continuity equation becomes:  
\[\
abla.\overset{\rightarrow}{v}\ =\ 0, \tag{1}\]  
where \(\overset{\rightarrow}{v}\) is the velocity of the fluid. The general momentum equation of motion is given by:  
\[\rho\left(\frac{\partial\overset{\rightarrow}{v}}{\partial t}+\overset{ \rightarrow}{v}.\
abla\overset{\rightarrow}{v}\right)\ =\ -\
abla.\overset{ \rightarrow}{\underline{\Pi}}+\rho\overset{\rightarrow}{\delta}, \tag{2}\]  
where \(\underline{\Pi}\) is the total stress tensor:  
\[\underline{\Pi}\ =\ \underline{\tau}+p\underline{l}, \tag{3}\]  
with \(\underline{\tau}\) as the extra stress tensor, \(p\) the hydrostatic pressure, and \(\underline{l}\) the identity tensor. For an incompressible Newtonian fluid, the stress tensor \(\underline{\tau}\) is given by:  
\[\underline{\tau}\ =\ -\eta_{s}\overset{\rightarrow}{\underline{\gamma}}, \tag{4}\]  
\[\overset{\rightarrow}{\underline{\gamma}}\ =\ \left[\
abla\overset{ \rightarrow}{v}+\left(\
abla\overset{\rightarrow}{v}\right)^{T}\right], \tag{5}\]  
where \(\eta_{s}\) is the Newtonian shear viscosity, and \(\dfrac{\gamma}{\underline{\tau}}\) is the strain rate tensor. Combining (3)-(5), substituting into (2), and neglecting the effects of gravity yields the well-known Navier-Stokes equation of fluid dynamics, see below. For a detailed derivation of (6) see ref. [11]. \[\rho\left(\frac{\partial\overset{\rightarrow}{v}}{\partial t}+\overset{ \rightarrow}{v}.\
abla\overset{\rightarrow}{v}\right)\ =\ -\
abla p+\eta_{s}\
abla^{2}\overset{ \rightarrow}{v}. \tag{6}\]  
For complex fluids such as polymers and polymer solutions, Equation (4) is not sufficient to describe the dynamics of the flows. An additional stress term is added to the Newtonian constitutive equation. For complex fluids:  
\[\underline{\tau}\ =\ \underline{\tau}_{N}+\underline{\tau}_{p}, \tag{7}\]where \(\underline{\underline{\gamma}}_{N}\) denotes the Newtonian stress component, and \(\underline{\underline{\gamma}}_{P}\) denotes the stress component characterizing the polymer behavior. The equation used to estimate \(\underline{\underline{\gamma}}_{P}\) depends on the particular model of polymer physics being employed. For the purposes of this research, the FENE-P constitutive equation is given by [10]:  
\[\underline{\underline{\tau}}_{P}\ =\ G_{O}f\left(\underline{\underline{I}}- \underline{\underline{A}}\right), \tag{8}\]  
\[\lambda\underline{\underline{A}}_{(1)}+f(\underline{\underline{A}}-\underline {\underline{I}})\ =\ 0, \tag{9}\]  
where \(G_{O}\) is the elastic modulus, and \(\lambda\) is the relaxation time of the polymers. The conformation tensor \(\underline{\underline{A}}\) measures the stretch and orientation of the polymers.