  We examine how stationary solutions to Galerkin approximations of the
Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to
$\infty$. An appropriate scaling is used to place the Grashof number as a new
coefficient of the nonlinear term, while the body force is fixed. A new type of
asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced.
Relations among the terms in the expansion are obtained by following a
procedure that compares and totally orders positive sequences generated by the
expansion. The same methodology applies to the case of perturbed body forces
and similar results are obtained. We demonstrate with a class of forces and
solutions that have convergent asymptotic expansions in $G$. All the results
hold in both two and three dimensions, as well as for both no-slip and periodic
boundary conditions.
