*2.4. Momentum Conservation Equation*

The momentum conservation equation of the q phase is

$$\begin{aligned} \frac{\partial}{\partial t} \Big( \mathbf{a}\_{\mathrm{q}} \rho\_{\mathrm{q}} \stackrel{\rightarrow}{\boldsymbol{\nabla}}\_{\mathrm{q}} \Big) + \nabla \cdot \Big( \mathbf{a}\_{\mathrm{q}} \rho\_{\mathrm{q}} \stackrel{\rightarrow}{\boldsymbol{\nabla}}\_{\mathrm{q}} \stackrel{\rightarrow}{\boldsymbol{\nabla}}\_{\mathrm{q}} \Big) &= \ -\mathbf{a}\_{\mathrm{q}} \nabla p + \nabla \cdot \overline{\overline{\boldsymbol{\nabla}}}\_{\mathrm{q}} + \mathbf{a}\_{\mathrm{q}} \rho\_{\mathrm{q}} \stackrel{\rightarrow}{\boldsymbol{g}} \\ &+ \sum\_{\mathbf{P}=1}^{\mathrm{n}} \left( \overrightarrow{R}\_{\mathrm{P}\mathrm{q}} + \dot{m}\_{\mathrm{P}\mathrm{q}} \overrightarrow{\boldsymbol{v}}\_{\mathrm{P}\mathrm{q}} - \dot{m}\_{\mathrm{qp}} \overrightarrow{\boldsymbol{v}}\_{\mathrm{qp}} \right) + \left( \overrightarrow{F}\_{\mathrm{q}} + \overrightarrow{F}\_{\mathrm{lift},\mathrm{q}} + \overrightarrow{F}\_{\mathrm{vm},\mathrm{q}} \right) \end{aligned} \tag{4}$$

where <sup>τ</sup><sup>q</sup> is the pressure strain tensor of the q phase; <sup>→</sup> *F* q, → *<sup>F</sup>*lift,q, and <sup>→</sup> *F*vm,q are the volume force, lift force, and virtual mass force of the q phase, respectively; and <sup>→</sup> *R*pq is an interaction term.
