Fluid Variables

Variable	Definition
${\rho }_g$	Fluid density
${\epsilon }_g$	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:

$$\frac{\partial ({\epsilon }_g{\rho }_g)}{\partial t}+\mathrm{\nabla }\cdot ({\epsilon }_g{\rho }_g{U}_g)=0$$

Unlike two-fluid modeling, ${\epsilon }_g=1-{\epsilon }_s$, is not a solution variable. Rather, ${\epsilon }_s$ is evaluated from the particle field through volume filtering,

$$(1-{\epsilon }_g)A(\mathit{x},t)\approx \sum _{i=1}^{N_p}{A}_i({\mathit{X}}_i,t)\mathcal{G}(|\mathit{x}-{\mathit{X}}_i|){\mathcal{V}}_i$$

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:

$$\frac{\partial {\epsilon }_g}{\partial t}+\mathrm{\nabla }\cdot ({\epsilon }_g{U}_g)=0$$

The conservation of fluid momentum is:

$$\frac{\partial ({\epsilon }_g{\rho }_g{U}_g)}{\partial t}+\mathrm{\nabla }\cdot ({\epsilon }_g{\rho }_g{U}_g{U}_g)+{\epsilon }_g\mathrm{\nabla }{p}_g=\mathrm{\nabla }\cdot \tau +M_{sg}+{\epsilon }_g{\rho }_{g}g$$

where $M_{sg}=-M_{gs}$ is the generalized interfacial momentum transfer from the solid particles to the fluid-phase. Like ${\epsilon }_s$, $M_{gs}$ 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:

$$F_{gi}=-{\mathcal{V}}_i\mathrm{\nabla }{p}_g-\frac{1}{2}{C}_D{\rho }_g{\mathit{V}}_{ig}\left|{\mathit{V}}_{ig}\right|A_i^{(proj)}$$

where ${\mathit{V}}_{ig}={\mathit{V}}_i-{\mathit{U}}_g({\mathit{X}}_i)$ is the velocity of ith particle relative to the fluid-phase (at the particle position ${\mathit{X}}_i$). $F_{gi}$ 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:

$$\frac{\partial ({\epsilon }_g{\rho }_g{U}_g)}{\partial t}+\mathrm{\nabla }\cdot ({\epsilon }_g{\rho }_g{U}_g{U}_g)+{\epsilon }_g\mathrm{\nabla }{p}_g=\mathrm{\nabla }\cdot \tau +\sum _p{\beta }_p(V_p-U_g)+{\epsilon }_g{\rho }_{g}g$$