# A Primer on CFD-DEM for Polymer-Filled Suspensions
## 4 Rheology of Filled Systems  
After validating the methodology used for torque calculation with a single-phase flow of the Newtonian and power law fluids, a CFD-DEM numerical model for multiphase flows of suspensions was assessed. The main coupled model parameters used were: spherical particles of radius \(r=100\)\(\eta\)m and density \(\rho_{p}=1000\) kg/m\({}^{3}\), a Newtonian fluid with the same rheological parameters of Section 2.3, DEM timestep \(=1\times 10^{-7}\) s, CFD timestep \(=1\times 10^{-5}\) s, coupling interval \(=100\), form A for the coupled flow [10,47], [PERSON]'s drag model, pressure gradient force model, and viscous force model. The Young's modulus used was equal to \(3\times 10^{9}\) Pa and the Poisson ratio was equal to 0.33. The damping and friction coefficient values were 0.9 and 0.3, respectively. It should be remarked that we obtained the steady-state results by considering both the evolution of the velocity and pressure residuals, and, the evolution of the torque. Figure 7 shows the comparison between numerical torque results for suspensions without particles and those with 10% and 30% of particle volume fractions for a Newtonian fluid. The numerical results shown in Figure 7 were fitted by a first-order polynomial. We can see that the ordinate of the intersection point of those polynomials with the \(y\)-axis increases with the increase of the particles volume fraction, which indicates that yield stress phenomena occur when the fluid is in the presence of particles, in accordance with the results published in the literature [8,15]. When increasing the shear rate, the fluid seems to show solid-like behavior at first (at really low shear rates), i.e., it does not deform any further, but at higher shear rates the fluid deforms. This could be due to the fact that the presence of solid particles gives the suspension a more solid-like behavior at low deformations. At higher deformations, the liquid-like behavior prevails and the suspension flows. Figure 7 also shows that the torque increases with the increase in the number of solid particles in suspension. This result was to be expected as the presence of particles leads to more _resistance_ within the suspension. This resistance is further enhanced by the interaction between the particles, the drag of the particles, and gravity. Note also that at higher shear rates, the influences of different particle percentages on the torque are more pronounced, as expected. Figure 8 shows the particle distribution of a suspension inside a parallel-plate rheometer with a gap of 0.5 mm and 10% particle volume fraction. In Figure 8a we see the view from below (fixed plate) and in Figure 8b the view from above (moving plate). It is interesting to see the transient dynamics of the suspension flow. For \(t=1\), the number of particles seen from above is small, but for \(t=4\) the number of particles is much higher, giving the impression that the suspended particles are moving toward the upper wall. However, looking at the view from below, the number of particles appears to be more or less the same over time. This means that the particles that migrate to the upper wall are far away from the lower and upper plates. The reason for this behavior can be attributed to the fact that the mixed interaction between particles, particles and walls, the rotation of the upper plate, and gravity lead to a pulling force that can attract the middle particles (in the \(z\)-direction). Note also the particle formation in the view from below (Figure 8a). Due to the particle-particle interaction and the resistance that the fluid exerts on the particles, a radial distribution of particles is observed for \(t=1\). As time increases, the particles become more random and dispersed, as expected. This result is not seen in the view from above and could therefore also be related to gravity. Figure 7: Numerical torque calculations on a parallel-plate rheometer with a gap of 0.5 mm and 0%, 10%, and 30% of particle volume fractions for a Newtonian fluid.