Editing Euler equations (fluid dynamics) (section)

===Nondimensionalisation===
{{See also|Cauchy momentum equation#Nondimensionalisation}}
{{Unreferenced section|date=April 2021}}
In order to make the equations dimensionless, a characteristic length <math>r_0</math>, and a characteristic velocity <math>u_0</math>, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
<math display="block">\begin{align}
       u^* & \equiv \frac{u}{u_0}, &
       r^* & \equiv \frac{r}{r_0}, \\[5pt]
       t^* & \equiv \frac{u_0}{r_0} t, &
       p^* & \equiv \frac{w}{u_0^2}, \\[5pt]
  \nabla^* & \equiv r_0 \nabla.
\end{align}</math>
and of the field [[unit vector]]:
<math display="block">\hat{\mathbf g}\equiv \frac {\mathbf g} g.</math>

Substitution of these inversed relations in Euler equations, defining the [[Froude number]], yields (omitting the * at apix):
{{Equation box 1
|indent=:
|title='''Incompressible Euler equations with constant and uniform density'''<br/>(''nondimensional form'')
|equation=<math>\begin{align} 
   {D\mathbf{u} \over Dt} &= -\nabla w + \frac{1}{\mathrm{Fr}} \hat{\mathbf{g}}\\
  \nabla \cdot \mathbf{u} &= 0
\end{align}</math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA
}}

Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with [[perturbation theory]].