# Fluid Variables

Variable

Definition

\(\rho\)

Fluid density

\(U\)

Fluid velocity

\(\tau\)

Viscous stress tensor

\({\bf H}_U\)

\(= (H_x , H_y , H_z )\), External Forces

\(H_s\)

External sources

# Fluid Equations

Conservation of fluid mass:

Conservation of fluid momentum:

Velocity constraint:

where \(S\) is zero by default, to model incompressible flow. The \(S \ne 0\) case is discussed below.

Tracer(s):

for conservatively advected scalars and

for passively advected scalars. In general, one could advect an arbitrary number of scalars.

IAMR has the ability to incorporate general, user-defined external forcing and source terms. The default behaviour is that
\(H_s=0\), and \({\bf H}_U\) represents gravitational forces, with \({\bf H}_U= (0 , 0 , -\rho g )\) in 3d and
\({\bf H}_U= (0 , -\rho g )\) in 2d, where \(g\) is the magnitude of the gravitational acceleration. However, since
by default, \(g=0\), \({\bf H}_U = 0\) unless `ns.gravity`

is set (for more info see Physics Parameters).

By default, IAMR solves the momentum equation in convective form. The inputs parameter `ns.do_mom_diff = 1`

is used to
switch to conservation form. Tracers are passively advected by default. The inputs parameter `ns.do_cons_trac = 1`

switches the first tracer to conservative. A second tracer can be included with `ns.do_trac2 = 1`

, and it can be
conservatively advected with `ns.do_cons_trac2 = 1`

.

IAMR also has the option to solve for temperature, along with a modified divergence constraint on the velocity field:

Here, the divergence constraint captures compressibily effects due to thermal diffusion.
To enable the temperature solve, use `ns.do_temp = 1`

and set `ns.temp_cond_coef`

to represent \(\lambda / c_p\),
which is taken to be constant. More sophiticated treatments are possible; if interested, please open an issue on github:
https://github.com/AMReX-Codes/IAMR/issues