## 2 Model Development
### Governing Equations  
Because of the Euler-Euler formulation separate equations are used for the particle and fluid phase for both momentum and mass. A phase momentum source term describes SID for the main channel in these equations. To make use of the principle of SID, the flow needs to be laminar (turbulence will destroy the concentration profiles). Furthermore, non-reactive hard and inert particles are used that do not dissolve, and do not stick to the walls. The above-mentioned conditions allow for us to assume that there are no internal forces and phase mass source terms. Also, a constant density and an absence of mass transfer between phases can be assumed. This results in the equations given next.  
Continuity equation for the fluid:  
\[\frac{\partial}{\partial t}\int\limits_{V}\big{(}1-\alpha_{p}\big{)}\rho_{f} dV+\oint\limits_{A}\big{(}1-\alpha_{p}\big{)}\rho_{f}v_{f}da=0 \tag{1}\]  
Continuity equation for the particles:  
\[\frac{\partial}{\partial t}\int\limits_{V}\alpha_{p}dV+\oint\limits_{A}\alpha_ {p}v_{p}da=0 \tag{2}\]  
Momentum equation for the fluid:  
\[\begin{array}{c}\frac{\partial}{\partial t}\int\limits_{V}\big{(}1-\alpha_{p }\big{)}\rho_{f}v_{f}dV+\oint\limits_{A}(1-\alpha_{p})\rho_{f}v_{f}\otimes \Big{(}v_{f}-v_{g}\Big{)}da=-\int\limits_{V}\big{(}1-\alpha_{p}\big{)}\
abla p \ dV\\ +\int\limits_{V}(1-\alpha_{p})\rho_{f}\ g\ dV+\oint\limits_{A}(1-\alpha_{p}) \tau_{f}da-\int\limits_{V}F_{p}^{D}dV-\int\limits_{V}F^{SID}dV\end{array} \tag{3}\]  
Momentum equation for the particles:  
\[\begin{array}{c}\frac{\partial}{\partial t}\int\limits_{V}\alpha_{p}\rho_{p} v_{p}dV+\oint\limits_{A}\alpha_{p}\rho_{p}v_{p}\otimes\big{(}v_{p}-v_{g} \big{)}da=-\int\limits_{V}\alpha_{p}\
abla p\ dV+\int\limits_{V}\alpha_{p}\rho _{p}\ g\ dV\\ +\oint\limits_{A}\alpha_{p}\tau_{p}da+\int\limits_{V}F_{p}^{D}dV+\int\limits_ {V}F^{SID}dV\end{array} \tag{4}\]  
Here, \(\alpha_{i}\) is the volume fraction of phase \(i\) (-) \(\left(\sum\limits_{i}\alpha_{i}=1\right)\), \(\rho_{i}\) the density of phase \(i\) (kg/m\({}^{3}\)), \(v_{i}\) the velocity of phase \(i\) (m/s), \(v_{g}\) the grid velocity (m/s), \(p\) the pressure (Pa), \(g\) the gravity vector (m/s\({}^{2}\)), \(\tau_{i}\) the molecular stresses of phase \(i\) (Pa), \(F^{D}\) the drag force (N/m\({}^{3}\)), and \(F^{SID}\) the shear induced diffusion force (N/m\({}^{3}\)). The shear induced diffusion force is given below in Equation 5 for which the closure relations are given in Table 1.  
\[F^{SID}=\frac{\Big{(}D_{a_{p}}\
abla\alpha_{p}+D_{\gamma}\
abla\dot{\gamma} \Big{)}}{M} \tag{5}\]  
With \(\dot{\gamma}\) the shear rate (1/s), \(D_{a_{p}}\) the volume fraction dependent diffusion coefficient (m\({}^{2}\)/s), \(D_{\dot{\gamma}}\) the shear rate dependent diffusion coefficient (m\({}^{2}\)), and \(M\) the mobility (m\({}^{2}\)/(Pa\(\cdot\)s)). The drag force is described by the following generic relation:  
\[F_{ij}^{D}=A_{ij}^{D}\big{(}v_{j}-v_{i}\big{)} \tag{6}\]  
Here, \(A_{ij}^{D}\) is the linearized drag coefficient (Pa\(\cdot\)s/m\({}^{2}\)). For more details about the drag force see Table 2.  
To make the key points of our numerical model clearer, a flow chart is presented in Figure 4 that gives the most important points used in our model.