It is considered a system of spherical free particles in a rectangular volume. Particles have random initial location and velocity distributions in the closed rectangular volume.
Particle force balance is expressed by the right side of equation (1)
, (1)
where v is the single particle velocity, Fg is gravity force, M.B magnetic force due to outer magnetic field interaction, η air drag friction coefficient. In that terms (1) includes gravity, magnetic, static and dynamic drag forces (Fig.1). Magnetic flux density B in the particle domain is calculated by secondary Biot-Savart static field model.
, (2)
where J is the source current density and r is distance vector from the source and local particle coordinates.
Particle dynamics method is performed by nodal solver, based on FDM method, with step-wise transient collision estimator and static outer magnetic field gradient intensity force correction.
, (3)
Particle velocity field is the primary field. As a second correction field is used magnetic field, independently computed by Biot-Savart formulation.
Force balance schematic is represented in Fig.1. It shows particle interactions in the modeling domain, according to equation (1).
Fig.1 Particle dynamic force balance.
Particle velocity field is the primary field. As a second correction field is used magnetic field, independently computed by Biot-Savart formulation. Step-wise modeling scheme is presented in Fig. 2. Magnetic field distribution is static and it is not influenced by particle kinetics. For each particle reposition, local magnetic field values are taken from the magnetic field map.
Fig.2 Step-wise modeling scheme.
Direct one-side coupling of partial kinetics model and magnetic field model is used, as it is represented on Fig.2.
Modeling algorithm is presented in Fig. 3. It is a step-wise sequence, where for each time-step local velocities for each particle are calculated with (1), next collisions with other particles and domain walls are found and corrected and final particle distribution is obtained. Process is repeated for all time-steps.
Fig.3
3D Cartesian coordinate system is used. Modeling domain is considered as a rectangular box. 3 DoFs per particle are calculated, which defines local coordinates (x, y, z). First order FDM integration is applied for each individual particle. Correction scheme for collisions is also added, assuming elastic collision with full impulse transfer. Here is modeled 1000 particles system moving in a 3D box [10 × 10 × 10] mm rectangular domain. More details on FDM implementation are presented in the next paper section.
Fig.4