# A Primer on CFD-DEM for Polymer-Filled Suspensions
## 2 Single-Phase Fluid Flows  
### Governing Equations  
First, we will consider that an incompressible fluid is confined between two circular plates, as shown in the schematic diagram of Figure 1. To take advantage of the symmetry of this flow, we can write the governing equations for the motion of a laminar, incompressible, and isothermal fluid in cylindrical coordinates. The equation describing the conservation of mass is as follows,  
\[\frac{1}{r}\frac{\partial(rv_{r})}{\partial r}+\frac{1}{r}\frac{\partial v_{ \theta}}{\partial\theta}+\frac{\partial v_{z}}{\partial z}=0, \tag{1}\]  
and the linear momentum equations are given by  
\[\rho\bigg{(}\frac{\partial v_{r}}{\partial t}+v_{r}\frac{\partial v _{r}}{\partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{r}}{\partial\theta}- \frac{v_{\theta}^{2}}{r}+v_{z}\frac{\partial v_{r}}{\partial z}\bigg{)}=- \frac{\partial p}{\partial r}+\bigg{(}\frac{1}{r}\frac{\partial(r\tau_{rr})}{ \partial r}+\frac{1}{r}\frac{\partial\tau_{\theta r}}{\partial\theta}-\frac{ \tau_{\theta\theta}}{r}+\frac{\partial\tau_{rz}}{\partial z}\bigg{)}+\rho g_{r}, \tag{2a}\] \[\rho\bigg{(}\frac{\partial v_{\theta}}{\partial t}+v_{r}\frac{ \partial v_{\theta}}{\partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{\theta }}{\partial\theta}+\frac{v_{\theta}v_{r}}{r}+v_{z}\frac{\partial v_{\theta}}{ \partial z}\bigg{)}=-\frac{1}{r}\frac{\partial p}{\partial\theta}+\bigg{(} \frac{1}{r^{2}}\frac{\partial(r^{2}\tau_{\theta\theta})}{\partial r}+\frac{1 }{r}\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\frac{\partial\tau_{ \theta z}}{\partial z}\bigg{)}+\rho g_{\theta},\] (2b) \[\rho\bigg{(}\frac{\partial v_{z}}{\partial t}+v_{r}\frac{ \partial v_{z}}{\partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{z}}{\partial \theta}+v_{z}\frac{\partial v_{z}}{\partial z}\bigg{)}=-\frac{\partial p}{ \partial z}+\bigg{(}\frac{1}{r}\frac{\partial(r\tau_{rz})}{\partial r}+\frac{ 1}{r}\frac{\partial\tau_{\theta z}}{\partial\theta}+\frac{\partial\tau_{ zz}}{\partial z}\bigg{)}+\rho g_{z}. \tag{2c}\]  
Here, \(t\) is time, \(\underline{r}=(v_{r},v_{\theta},v_{z})\) are the three components of fluid velocity vector \(\underline{v}\) in cylindrical coordinates \((r,\theta,z)\), \(\tau_{rr}\), \(\tau_{\theta\theta}\), \(\tau_{zz}\), are the normal stress components, \(\tau_{r\theta}\), \(\tau_{rz}\), \(\tau_{z\theta}\) are the shear stress components of the stress tensor \(\underline{r}\), \(\rho\) is the fluid density and \(\underline{g}=(g_{r},g_{\theta},g_{z})\) are the three components of the gravity vector \(\underline{g}\).  
For a Newtonian fluid, the stress tensor is given by,  
\[\underline{\tau}=2\eta\underline{\underline{\underline{D}}}=\eta\left(\
abla \underline{v}+(\
abla\underline{v})^{T}\right), \tag{3}\]  
with \(\eta\) being the fluid dynamic viscosity, \(\underline{\underline{D}}\) is the rate of the deformation tensor and \(\
abla\underline{v}\) is the gradient of velocity (both given in cylindrical coordinates).  
For the power law fluid, the stress tensor is given by,  
\[\underline{\underline{\tau}}=2\eta(|\underline{\underline{D}}|)\underline{ \underline{D}}, \tag{4}\]where the viscosity is now a function of the magnitude of the rate of the deformation tensor (\(|\underline{\underline{D}}|\)),  
\[\eta(|\underline{\underline{D}}|)=k|\underline{\underline{D}}|^{n-1}, \tag{5}\]  
with \(n\) being the power law constant and \(k\) being the flow consistency index.