In order to make the equations dimensionless, a characteristic length r, and a characteristic velocity u, 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:
Substitution of these inverse relations in the Euler momentum equations, and definition of the Froude number:
and the Euler number:
the equations are finally expressed (with the material derivative and now omitting the indexes):
Cauchy-type equations in the high Froude limit  (corresponding to negligible external field) are named free equations. On the other hand, in the low Euler limit  (corresponding to negligible stress) general Cauchy momentum equation becomes an inhomogeneous Burgers equation (here we make explicit the material derivative):
This is an inhomogeneous pure advection equation, as much as the Stokes equation is a pure diffusion equation.