# A Primer on CFD-DEM for Polymer-Filled Suspensions
## 3 Particle-Laden Flows
### Governing Equations for the Dispersed Phase (DEM)  
To model the dispersed phase, the discrete element method (DEM) is employed. Following [42], the DEM is used to solve [PERSON]'s second law equations for the motion of a particle \(i\) in the suspension  
\[m_{i}\frac{dg_{i}^{p}}{dt}=\sum_{j=1}^{n_{i}^{c}}\underline{f}_{ij}^{c}+\sum _{k=1}^{n_{i}^{mc}}\underline{f}_{ik}^{nc}+\underline{f}_{i}^{pf}+\underline{ f}_{i}^{S}, \tag{10}\]  
\[\underline{I}_{i}\frac{d\underline{\omega}_{i}^{p}}{dt}=\sum_{j=1}^{n_{i}^{c }}\underline{M}_{ij}, \tag{11}\]  
where \(m_{i}\) and \(I_{i}\) are particle's \(i\)'s mass and moment of inertia, respectively, \(\underline{v}_{i}^{p}\) and \(\underline{\omega}_{i}^{p}\) are particle \(i\)'s linear and angular velocities, respectively, \(\underline{f}_{ij}^{c}\) and \(\underline{M}_{ij}\) are the contact force and torque promoted by the \(j\) contact (caused by a particle or a wall) with particle \(i\), respectively; \(n_{i}^{c}\) is the total number of contacts experienced by particle \(i\), \(\underline{f}_{ik}^{nc}\) are the non-contact (long-range) forces promoted by particle \(k\) or other sources on particle \(i\); \(n_{i}^{nc}\) is the total number  
Figure 6: Analytical and numerical results for power law torque calculations on parallel-plate rheometer with a gap of 0.5 mm and \(n=0.3\), \(n=0.6\) and \(n=0.9\). of non-contacts experienced by particle \(i\), \(\underline{f}_{i}^{pf}\) are the particle-fluid interaction forces, and \(\underline{F}_{i}^{g}\) is the force due to the gravitational acceleration vector. In this work, the contact force due to the interaction between two rigid particles, \(\underline{f}_{ij}^{c}\), was computed by the Hertzian spring dashpot model [43]. The normal component contribution, \(\underline{f}_{nij}^{c}\), resulting from the contact of particle \(j\) with particle \(i\), is given by  
\[\underline{f}_{nij}^{c}=(-k_{n}|\delta_{n}|-\eta_{n}\underline{x}_{ij}^{p} \cdot\underline{u})\underline{u}, \tag{12}\]  
and, if the contact is between particle \(i\) and a wall, \(\underline{f}_{niiv}^{c}\), then it is calculated as  
\[\underline{f}_{niiv}^{c}=(-k_{nw}|\delta_{nw}|-\eta_{nw}\underline{v}_{nw}^{ p}\cdot\underline{u}_{w})\underline{u}_{w}. \tag{13}\]  
where \(\underline{u}\) is the unit vector pointing from the center of particle \(i\) to that of particle \(j\) and \(\underline{u}_{w}\) is the unit vector of the surface normal pointing outwards. Here, we should know beforehand the stiffness and the damping coefficients of particles in the normal directions, \(k_{n}\) and \(\eta_{n}\), respectively, and also the ones due to the contact of a particle with the walls, \(k_{nw}\) and \(\eta_{nw}\). In addition, for the spring-dashpot model, the particles are allowed to overlap an amount of \(\delta_{n}\) and \(\delta_{nw}\) normal displacements when in contact with another particle or with a wall, respectively. The magnitude of these displacements read as follows  
\[|\delta_{n}|=r_{i}+r_{j}-|\underline{p}_{j}-\underline{p}_{i}|, \tag{14}\]  
and  
\[|\delta_{nw}|=r_{i}-|\underline{p}_{i}-\underline{p}_{c}|, \tag{15}\]  
where \(r_{i}\) and \(r_{j}\) are the radii of particles \(i\) and \(j\), respectively, \(\underline{p}_{i}\), \(\underline{p}_{j}\), and \(\underline{p}_{c}\) are the vectors with the coordinates that give the locations of particle \(i\), particle \(j\), and the contact point on the wall, respectively. Subsequently, due to the particle-particle interaction, we need to define the velocity of particle \(i\) relative to particle \(j\), \(\underline{v}_{ij}^{p}\),  
\[\underline{v}_{ij}^{p}=\underline{v}_{i}^{p}-\underline{v}_{j}^{p}, \tag{16}\]  
and, if the contact is between particle \(i\) and a wall,  
\[\underline{v}_{iw}^{p}=\underline{v}_{i}^{p}-\underline{v}_{w}^{p}, \tag{17}\]  
where \(\underline{v}_{i}^{p}\) and \(\underline{v}_{j}^{p}\) are particle \(i\) and particle \(j\) linear velocities, respectively, and \(\underline{v}_{w}^{p}\) is the slip velocity of the sphere-wall contact point.