2.1.2. Particle Motion

Particles are tracked in the Lagrangian frame; particle position and velocity are obtained by integration of the governing ordinary differential equations along the particle trajectory. The motion of a single particle is governed by the conservation of mass equation and Newton's second law.

$$\frac{dm\_p}{dt} = 0,\tag{9}$$

$$m\_p \frac{d\sigma}{dt} = F\_\prime \tag{10}$$

$$I\_p \frac{d\omega}{dt} = \mathbf{M}\_\prime \tag{11}$$

where *mp* is the particle mass, *Ip* is the particle moment of inertia, *v* is the particle velocity, *ω* is the particle angular velocity, *F* is the resultant force on the particle and *M* is the resultant torque on the particle. The forces acting on the particle are the drag (*Fd*), the buoyancy force (*Fb*), the gravitational force (*Fg*), the pressure gradient force (*Fpg*), the virtual mass force (*Fvm*) and the Magnus lift due to particle rotation (*Fml*):

$$m\_p \frac{d\upsilon}{dt} = F\_d + F\_b + F\_{\mathcal{S}} + F\_{p\mathcal{S}} + F\_{\upsilon m} + F\_{ml\prime} \tag{12}$$

These forces are expressed as:

$$F\_d = m\_p \frac{18\mu\_c}{\rho\_p d\_p^2} \frac{C\_D Re\_p}{24} (\mu - v),\tag{13}$$

$$F\_{\mathbf{b}} + F\_{\mathbf{g}} = m\_p \frac{\mathbf{g} (\rho\_{\mathcal{P}} - \rho\_{\mathcal{E}})}{\rho\_{\mathcal{P}}},\tag{14}$$

$$F\_{\rm p\mathcal{g}} = m\_p \frac{\rho\_c}{\rho\_p} \frac{Du}{Dt} \,\tag{15}$$

$$F\_{\rm vm} = \frac{1}{2} m\_p \frac{\rho\_c}{\rho\_p} \left( \frac{D\mathbf{u}}{Dt} - \frac{d\mathbf{v}}{dt} \right),\tag{16}$$

$$F\_{\rm ml} = \frac{1}{2} A\_p \mathbb{C}\_{RL} \rho\_c \frac{|\mathfrak{u} - \mathfrak{v}|}{|\Omega|} ((\mathfrak{u} - \mathfrak{v}) \times \Omega), \tag{17}$$

where *ρ<sup>p</sup>* is the particle density, *dp* is the particle diameter, *CD* is the drag coefficient, *Rep* is the particle Reynolds number, *A<sup>p</sup>* is the projected particle surface area, *CRL* is the rotational lift coefficient and **Ω** is the relative angular velocity of the particle. For the drag coefficient (*CD*), a correlation by Morsi and Alexander [24] is adopted, as it is applicable over a wide range of Reynolds numbers.

For the rotational lift coefficient (*CRL*), an approach proposed by Tsuji et al. [25] is adopted, as it is applicable to high particle Reynolds numbers (Rep < 1600). The torque applied to the particle (*M*) can be expressed in terms of rotational drag acting on the particle. Equation (11) is rewritten as

$$I\_p \frac{d\omega}{dt} = \frac{\rho\_c}{2} \left(\frac{d\_p}{2}\right)^5 \mathbb{C}\_{\omega} |\Omega| \cdot \Omega,\tag{18}$$

where *Cω* is the rotational drag coefficient. The correlation proposed by Dennis et al. [26] is used.
