  The steady state of a dilute gas enclosed between two infinite parallel
plates in relative motion and under the action of a uniform body force parallel
to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation
is analytically solved for this Couette-Poiseuille flow to first order in the
force and for arbitrary values of the Knudsen number associated with the shear
rate. This allows us to investigate the influence of the external force on the
non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille
flow is analyzed when the shear-rate Knudsen number and the scaled force are of
the same order and terms up to second order are retained. In this way, the
transition from the bimodal temperature profile characteristic of the pure
force-driven Poiseuille flow to the parabolic profile characteristic of the
pure Couette flow through several intermediate stages in the Couette-Poiseuille
flow are described. A critical comparison with the Navier-Stokes solution of
the problem is carried out.
