## 2 Materials and Methods
### Formulation of the Discrete Element Model
#### 2.1.1 Governing Relations of the Discrete Element Method  
The DEM governs a system, which can be considered an assembly of discrete particles interacting with one another in the frame of Newtonian mechanics. An arbitrary particle \(i\) in the system of \(N\) particles undergoes translational and rotational motion as follows:  
\[m_{i}\frac{d^{2}\mathbf{x}_{i}}{dt^{2}}=\mathbf{F}_{i},\;\;I_{i}\frac{d\mathbf{\omega}_{i} }{dt}=\mathbf{M}_{i}, \tag{1}\]  
where \(t\) is time, and \(m_{i}\) and \(I_{i}\) are the mass and the moment of inertia of the particle, respectively, while the vectors \(\mathbf{x}_{i}\) and \(\mathbf{\omega}_{i}\) determine the position of the center of the particle \(i\) and the rotational velocity around the particle's center of mass. The vectors \(\mathbf{F}_{i}\) and \(\mathbf{M}_{i}\) represent the resultant force and the resultant moment, acting in the center of the particle \(i\). Equation (1), along with the specified initial conditions, presents a universal mathematical model, capturing a large amount of problems encountered by the DEM and various extensions.  
The forces acting on the particle may be classified into the forces induced by the external fields, the contact forces between the contacting particles, and the bond forces between the bonded particles. Thus, the resultant force can be expressed by the external forces, the sum of the contact forces \(\mathbf{F}_{ij}^{c}\) between the contacting particles that are indicated by subscript \(j=1\), \(N_{c}\), and by the sum of the bond forces \(\mathbf{F}_{ik}^{b}\) between the bonded particles that are indicated by subscript \(k=1\), \(N_{b}\):  
\[\mathbf{F}_{i}=m_{i}\mathbf{g}+\sum_{j=1,j\
eq i}^{N_{c}}\mathbf{F}_{ij}^{c}+\sum_{k=1,k,j\
eq i}^{N_{b}}\mathbf{F}_{ik}^{b}, \tag{2}\]  
where \(\mathbf{g}\) is the acceleration due to gravity. Thus, in the present work, the electromagnetic force, the aerodynamic force, and other external forces [29], except for the gravity force, are not considered.  
The rotational motion is governed by the respective moments. The moment of the particle \(i\) can be restricted by the contribution of moments \(M^{\rm c}_{ij}\) of the contacting particles and by the contribution of moments \(M^{b}_{ik}\) of the bonded particles:  
\[M_{i}=\sum_{j=1,j\
eq i}^{N_{i}}M^{\rm c}_{ij}+\sum_{k=1,k\
eq i}^{N_{b}}M^{b}_{ ik}. \tag{3}\]  
In the case of granular flows, there are no bonded particles, therefore, \(N_{b}\), \(F^{b}_{ik}\), and \(M^{b}_{ik}\) are always equal to zero. In the case of the media discretized by the bonded particles, both counterparts can be not equal to zero. The moment and the force acting between two particles \(i\) and \(j\) can be expressed as the sum of the normal and tangential components:  
\[F^{o}_{ij}=F^{o}_{ij,n}+F^{o}_{ij,t}, \tag{4}\]  
\[M^{o}_{ij}=M^{o}_{ij,n}+M^{o}_{ij,t}, \tag{5}\]  
where \(F^{o}_{ij,n}\) and \(F^{o}_{ij,t}\) are the normal and tangential vector components of the force acting between the particles \(i\) and \(j\), and \(M^{o}_{ij,n}\) and \(M^{o}_{ij,t}\) are the normal and tangential components of the moment, respectively. The superscript \(o\) can be equal to \(c\) and \(b\) in the case of the granular flows and the bonded particles, respectively.