## 4 Lbm-Dem  
The application of coupled LBM-DEM requires the assumption that particles, whether in fluid lattices or solids, are rigid. The fluid phase is considered incompressible and Newtonian. Particle deformation is limited to the local surface area of the contact point. The interaction between particles primarily occurs through surface contact collisions while disregarding other weak forces. When considering issues pertaining to particle transport, solid particles are primarily propelled by the combined action of hydrodynamic forces exerted by the fluid, as well as body forces. In the context of LBM, fluid particles collide with the boundaries of solid particles exchanging momentum and altering their motion. The first step in establishing a model of fluid and solid particle interaction involes representing particles using lattice nodes. Figure 3 categorizes solid and fluid nodes into three distinct groups: (1) Fluid boundary nodes, comprising fluid nodes that are directly adjacent to at least one solid node; (2) solid boundary nodes, representing solid nodes that are connected to at least one fluid node; and (3) internal solid nodes, which refer to solid nodes that are not linked to any fluid nodes. The connection between fluid boundary nodes and solid boundary nodes is referred to as a boundary link which constantly changes in an on-off manner when particles move, significantly impacting calculated interaction forces. By incorporating collision operators into the immersed moving boundary (IMB) algorithm, boundary nodes undergo updates at every time step [24], as follows:  
\[f_{i}(\mathbf{x}+\mathbf{c}_{i}\Delta t,t+\Delta t)=f_{i}(\mathbf{x},t)-\frac{ \Delta t}{\tau}(1-B)(f_{i}(\mathbf{x},t)-f_{i}^{\text{eq}}(\mathbf{x}t))+(1-B) F_{i}\Delta t+B\mathbf{\Omega}_{i}^{\mathbf{s}} \tag{27}\]  
where parameter \(B\) is defined as  
\[B=\frac{\varepsilon(\tau-1/2)}{(1-\varepsilon)+(\tau-1/2)} \tag{28}\]  
and where \(\mathbf{\Omega}_{i}^{\mathbf{s}}\) is a novel collision operator for nodes that are partially or fully encompassed by a solid. \(\varepsilon\) represents the volume fraction of solids that overlaps with a lattice node. If \(\varepsilon=1\), the lattice node effectively assumes the properties of a solid node, causing fluid particles to rebound in their original incoming directions. If \(\varepsilon=0\), the collision equation reverts to the standard LBM collision equation. The new collision operator \(\mathbf{\Omega}_{i}^{\mathbf{s}}\) applies to nodes that have overlap with solids, as follows:  
\[\mathbf{\Omega}_{i}^{\mathbf{s}}=f_{-i}(x_{t},t)-f_{i}(x_{t},t)+f_{i}^{\text{ eq}}(\rho,u_{\text{s}})-f_{-i}^{\text{eq}}(\rho,u) \tag{29}\]  
where \(u_{\text{s}}\) denotes the velocity of the solid and the subscript \(-i\) is the opposite direction to \(i\)  
Figure 3: Internal boundary model of coupled LBM-DEM. The LBM fluid adds an additional hydrodynamic force (\(F_{l}\)), which is given as follows:  
\[F_{f}=\frac{h^{2}}{\Delta t}\sum B_{n}\sum_{i}\mathbf{\Omega_{i}^{s}e}_{i} \tag{30}\]  
and the torque is given by  
\[M_{f}=\frac{h^{2}}{\Delta t}\sum_{n}\left(x_{i}-x_{s}\right)\times\left(B_{n} \sum_{i}\mathbf{\Omega_{i}^{s}e}_{i}\right) \tag{31}\]  
where \(x_{s}\) represents the center location of the solid particle and \(B_{n}\) denotes the new collision operator \(B\) at node \(n\). The second summation in both equations is over the lattice directions. The force and torque will be applied to coupled LBM-DEM simulations. In this calculation, both the interparticle contact force and the hydrodynamic force acting on the particle under analysis are taken into consideration, as follows:  
\[ma=F_{f}+mg \tag{32}\] \[J\ddot{\theta}=M_{f} \tag{33}\]  
where \(m\) represents the mass, \(J\) represents the moment of inertia, \(\ddot{\theta}\) represents the angular acceleration, and \(g\) represents the acceleration due to gravity.