# A Primer on CFD-DEM for Polymer-Filled Suspensions
## 3 Particle-Laden Flows
### Governing Equations for the Continuum Phase (CFD)  
In this work, set II (or form A in [10]) of the incompressible volume-averaged Navier-Stokes equation is considered for the fluid phase [47]. The volume-averaged continuity equation reads as  
\[\frac{\partial\underline{e}_{f}}{\partial t}+\
abla\cdot(\epsilon_{f} \underline{z})=0, \tag{26}\]and the volume-averaged linear momentum equations are given by  
\[\frac{\partial(\epsilon_{f}\underline{\varphi})}{\partial t}+\
abla\cdot( \epsilon_{f}\underline{\varphi}\underline{\upsilon})=-\epsilon_{f}\
abla P- \underline{E}^{pf}+\epsilon_{f}\
abla\cdot\underline{\underline{\tau}}+ \epsilon_{f}\underline{\wp}, \tag{27}\]  
where \(\epsilon_{f}\) is the fluid volume fraction (also denoted as the void fraction), \(P\) is the modified pressure \((p/\rho)\) and the fluid-phase viscous stress tensor \(\underline{\underline{\tau}}\) is given by:  
\[\underline{\underline{\tau}}=\frac{\eta}{\rho}\bigg{[}(\
abla\underline{ \underline{\upsilon}})+(\
abla\underline{\underline{\upsilon}})^{T}-\frac{2} {3}(\
abla\cdot\underline{\upsilon})\delta_{K}\bigg{]}, \tag{28}\]  
where \(\delta_{K}\) is the Kronecker delta. Recall that for a power law fluid the viscosity is a function of the magnitude of the rate of deformation tensor \(\eta(|\underline{D}|)=k|\underline{D}|^{n-1}\). During the CFD-DEM calculations several particle-fluid momentum exchange correlations can be applied [47]. In this work, the momentum exchange term from the particles to the fluid is enforced via the source term \(\underline{E}^{pf}\) in the volume-averaged momentum equations (see the second term on the right-hand side of Equation (27)). Again, following the formulation of set II in [47], the momentum exchange term \(\underline{E}^{pf}\) reads as follows  
\[\underline{E}^{pf}=\frac{1}{\Delta V_{i}}\sum_{i=1}^{n_{p}}\Bigl{(}\underline{ f}_{i}^{d}+\underline{f}_{i}^{vm}+\underline{f}_{i}^{Bass}+\underline{f}_{i}^{ Saff}+\underline{f}_{i}^{Mag}\Bigr{)}, \tag{29}\]  
where \(\Delta V_{i}\) is the volume of the fluid cell where particle \(i\) is located and \(n_{p}\) is the number of particles in that cell. Taking into account Equations (25) and (29) we would need to find expressions for each force term. However, depending on the physics of the problem under consideration, it can be possible to neglect some of the force terms. In this work, we only consider the drag, pressure, and viscous forces. The drag force \(\underline{f}_{i}^{d}\) on a particle in the fluid is calculated here by the Di Felice [48] correlation, which is valid for a wide variety of both fixed-bed and suspended-particle systems, being given by  
\[\underline{f}_{i}^{d}=\frac{1}{8}C_{d}\rho\pi d_{p}^{2}\Bigl{(}\underline{ \underline{\upsilon}}-\underline{\wp}_{i}^{p}\Bigr{)}|\underline{\underline{ \upsilon}}-\underline{\wp}_{i}^{p}|\epsilon_{f}^{-\chi}, \tag{30}\]  
where \(d_{p}\) is the particle diameter, \(\underline{\wp}_{i}^{p}\) is the translational velocity of particle \(i\) and \(C_{d}\) is the drag coefficient expressed as  
\[C_{d}=\left(0.63+\frac{4.8}{\sqrt{Re_{p}}}\right)^{2}, \tag{31}\]  
with the particle Reynolds number, \(Re_{p}\), defined by  
\[Re_{p}=\frac{\epsilon_{f}\rho d_{p}|\underline{\underline{\upsilon}}- \underline{\wp}_{i}^{p}|}{\eta}, \tag{32}\]  
and the empirical expression for the exponent \(\chi\) is given by  
\[\chi=3.7-0.65\text{exp}\Biggl{[}-\frac{\left(1.5-\log(Re_{p})\right)^{2}}{2} \Biggr{]}. \tag{33}\]  
The pressure and viscous forces are used because, with the _unresolved_ CFD-DEM approach, the particles are not discretized explicitly in the CFD part [49; 50]. The pressure gradient force, \(\underline{f}_{i}^{\
abla p}\), which results from the difference in pressure across the fluid and particle's surface, reads as follows [47]  
\[\underline{f}_{i}^{\
abla p}=-V_{i}\
abla p, \tag{34}\]  
where \(V_{i}\) is the volume of particle \(i\).