/**
# Mixing an uniformly stratified fluid

Disk-shaped fans of size $V = \pi R^2 \times w$ at a height $h$ above
the surface ($h>R$) are located on a $L \times L$ infinite-grid layout
($L>R$ is the periodicity length scale). Each fan accelerates the
hosting (Bounsinesq) atmosphere thet is initially at rest and has a
reference density $\rho_0$ according to a kinetic-energy input rate
$P$ (i.e. effective power). The resulting fan's jets are affected by
the presence of a constant vertical density gradient. Due to gravity
it is so-called `stable' and can be associated with a the
Brunt-Vaisala frequency ($N$).

We *can* define a kinematic-power scale (i.e. $\mathcal{P}$ with
units $\mathrm{m}^2 / \mathrm{s}^3$),

$$\mathcal{P}=\frac{P}{\rho_0V}.$$

The six system parameters $R, w, h, L, \mathcal{P}$ and $N$, with
units of length and/or time give rise to four
dimensionless groups. E.g.:

Three geometrical groups,

$$\Pi_1 = w/R$$
$$\Pi_2 = L/R$$
$$\Pi_3 = h/R$$

Being the dimensioness width, periodicity length scale and
height. Furthermore, there is a group that concerns the mixing capacity
of a fan: 

$$\Pi_4 = \frac{\mathcal{P}}{R^2N^3}$$

Noting that the `mixing capacity' should be interpreted with respect
to the geometrical parameters.

## Additional complexity

The orientation of the fan is described by it's angles with respect to
some reference coordinate system.  In general a fan can be rotated
over three angles (say $\theta, \phi, \psi$). If,

$$ \{\theta, \phi, \psi\} = \overrightarrow{f}(t)$$

Then the formulation of $\overrightarrow{f}(t)$ is an important system
specification that affects mixing aswell. However, addional complexity
arises as the functional form of $f$ may not live in an ordered
parameter space.
*/