At each time step, drag and torque acting on the particle are computed using explicit expressions involving the velocity, pressure, and shear stress distribution in the fluid cells surrounding the particle [9]. The total fluid forces and torques acting on a macroscopic particle in the direction consist of virtual mass, pressure, and viscous fluid components:

(13–1)

The ^th virtual mass component of the fluid force and torque experienced by a particle, , is calculated as the integral of the rate of change of momentum for all fluid cells within a particle volume:

(13–2)

where is the cell fluid mass; and are the fluid and particle velocities in the direction , respectively; and is the flow time step.

The ^th pressure component of the fluid force and torque acting on the particle surface, , is calculated based on the pressure distribution around the particle:

(13–3)

where is the pressure, is the approximated area of a particle surface in a fluid cell touching the particle, is the radius vector from the fluid cell center to the particle center, and is the Cartesian component of vector in the ^th direction.

The ^th viscous component of the fluid force and torque acting on a particle surface, , is calculated based on the shear stress distribution around the particle:

(13–4)

where is the shear stress in the positive ^th direction acting on a plane perpendicular to the ^th direction, and is the Cartesian component of vector in the ^th direction.

Based on fluid forces and torques, the new particle position, velocities, and accelerations are calculated at each flow time step.