# Fluid Variables

> Variable | Definition
> --- | ---
> \(\rho_g\) | Fluid density
> \(\varepsilon_g\) | Volume fraction of fluid (= 1 if no particles)
> \(U_g\) | Fluid velocity
> \(\tau\) | Viscous stress tensor
> \(g\) | Gravitational acceleration

# Fluid Equations

Conservation of fluid mass:

\[\frac{\partial (\varepsilon_g \rho_g)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g) = 0\]

Unlike two-fluid modeling, \(\varepsilon_g = 1 - \varepsilon_s\), is not a solution variable. Rather, \(\varepsilon_s\) is evaluated from the particle field through volume filtering,

\[(1 - \varepsilon_g) A (\boldsymbol{x},t) \approx \sum_{i=1}^{N_p} A_i(\boldsymbol{X}_i,t) {\mathcal G} (\left| \boldsymbol{x} - \boldsymbol{X}_i \right|) {\mathcal V}_i\]

where \(A_i\) is a general particle property, \({\mathcal V}_i\) is the particle volume and \(\mathcal G\) is a transfer kernel with compact support–here linear hat. Setting \(A_i = 1\) gives the local void fraction.

Assuming the fluid phase is incompressible, \(\frac{D \rho_g}{Dt} = 0\), the conservation of fluid mass is equivalent to the conservation of fluid volume:

\[\frac{\partial \varepsilon_g}{\partial t} + \nabla \cdot (\varepsilon_g U_g) = 0\]

The conservation of fluid momentum is:

\[\frac{ \partial (\varepsilon_g \rho_g U_g)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g U_g) + \varepsilon_g \nabla p_g = \nabla \cdot \tau + M_{sg} + \varepsilon_g \rho_g g\]

where \(M_{sg} = - M_{gs}\) is the generalized interfacial momentum transfer from the solid particles to the fluid-phase. Like \(\varepsilon_s\), \(M_{gs}\) is determined from the L-E transfer kernel by setting \(A_i = F_{gi}\), where \(F_{gi}\) is the force due to the fluid-phase on the ith particle. Following MFiX classic (and many other CFD-DEM codes designed for high density ratio gas-solids flows), only buoyancy (pressure gradient) and steady drag are considered:

\[F_{gi} = - \mathcal{V}_i \nabla p_g - \frac{1}{2} C_D \rho_g \boldsymbol{V}_{ig} \left|\boldsymbol{V}_{ig}\right| A_i^{(proj)}\]

where \(\boldsymbol{V}_{ig} = \boldsymbol{V}_i - \boldsymbol{U}_g ( \boldsymbol{X}_i )\) is the velocity of ith particle relative to the fluid-phase (at the particle position \(\boldsymbol{X}_i\)). \(F_{gi}\) is closed by the specification of a drag coefficient, \(C_D\). Currently, MFIX-Exa includes Wen-Yu, Gidaspow and BVK2 drag laws.

In chemical engineering literature, it is common to lump all drag-related terms of \(M_{gs}\) into \(\beta\). With this simplification and some re-arrangement, the fluid momentum takes the more convenient form:

\[\frac{ \partial (\varepsilon_g \rho_g U_g)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g U_g) + \varepsilon_g \nabla p_g = \nabla \cdot \tau + \sum_p \beta_p (V_p - U_g) + \varepsilon_g \rho_g g\]