Fluid Variables

Variable	Definition
\(\rho\)	Fluid density
\(U\)	Fluid velocity
\(\tau\)	Viscous stress tensor
\(\mu_s\)	scalar diffusivity
\(\mu_T\)	thermal conductivity
\({\bf g}\)	Gravitational acceleration
\({\bf H}_U\)	\(= (H_x , H_y , H_z )\), External Forces
\(H_s\)	External sources
\(H_T\)	External heat sources

Fluid Equations

Conservation of fluid mass:

\[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho U) = 0\]

Conservation of fluid momentum:

\[\frac{ \partial (\rho U)}{\partial t} + \nabla \cdot (\rho U U) + \nabla p = \nabla \cdot \tau + {\bf H}_U\]

Incompressibility constraint:

\[\nabla \cdot U = 0\]

Tracer(s): for conservative,

\[\frac{\partial \rho s}{\partial t} + \nabla \cdot (\rho U s) = \nabla \cdot \mu_s \nabla s + \rho H_s\]

or, for non-conservative,

\[\frac{\partial s}{\partial t} + U \cdot \nabla s = \nabla \cdot \mu_s \nabla s + H_s\]

Optional temperature:

\[\rho c_p \left( \frac{\partial T}{\partial t} + U \cdot \nabla T \right) = \nabla \cdot \mu_T \nabla T + H_T\]

By default, \(H_s = 0\), \(H_T = 0\), and \({\bf H}_U = {\bf 0}\). If gravity is set during runtime, then \({\bf H}_U\) defaults to \(\rho {\bf g}\).