  In this paper, we examine a finite element approximation of the steady
$p(\cdot)$-Navier-Stokes equations ($p(\cdot)$ is variable dependent) and prove
orders of convergence by assuming natural fractional regularity assumptions on
the velocity vector field and the kinematic pressure. Compared to previous
results, we treat the convective term and employ a more practicable
discretization of the power-law index $p(\cdot)$. Numerical experiments confirm
the quasi-optimality of the a priori error estimates (for the velocity) with
respect to fractional regularity assumptions on the velocity vector field and
the kinematic pressure.
