  Slow flows of a slightly rarefied gas under high thermal stresses are
considered. The correct fluid-dynamic description of this class of flows is
based on the Kogan--Galkin--Friedlander equations, containing some
non-Navier--Stokes terms in the momentum equation. Appropriate boundary
conditions are determined from the asymptotic analysis of the Knudsen layer on
the basis of the Boltzmann equation. Boundary conditions up to the second order
of the Knudsen number are studied. Some two-dimensional examples are examined
for their comparative analysis. The fluid-dynamic results are supported by
numerical solution of the Boltzmann equation obtained by the Tcheremissine's
projection-interpolation discrete-velocity method extended for nonuniform
grids. The competition pattern between the first- and the second-order
nonlinear thermal-stress flows has been obtained for the first time.
