  The Bhatnagar-Gross-Krook kinetic model of the Boltzmann equation is solved
for the steady cylindrical Poiseuille flow fed by a constant gravity field. The
solution is obtained as a perturbation expansion in powers of the field
(through fourth order) and for a general class of repulsive potentials. The
results, which are hardly sensitive to the interaction potential, suggest that
the expansion is only asymptotic. A critical comparison with the profiles
predicted by the Navier-Stokes equations shows that the latter fail over
distances comparable to the mean free path. In particular, while the
Navier-Stokes description predicts a monotonically decreasing temperature as
one moves apart from the cylinder axis, the kinetic theory description shows
that the temperature has a local minimum at the axis and reaches a maximum
value at a distance of the order of the mean free path. Within that distance,
the radial heat flows from the colder to the hotter points, in contrast to what
is expected from the Fourier law. Furthermore, a longitudinal component of the
heat flux exists in the absence of gradients along the longitudinal direction.
Non-Newtonian effects, such as a non uniform hydrostatic pressure and normal
stress differences, are also present.
