  The state of a single-species monatomic gas from near-equilibrium to highly
nonequilibrium conditions is investigated using analytical and numerical
methods. Normal solutions of the Boltzmann equation for Fourier flow (uniform
heat flux) and Couette flow (uniform shear stress) are found in terms of the
heat-flux and shear-stress Knudsen numbers. Analytical solutions are found for
inverse-power-law molecules from hard-sphere through Maxwell at small Knudsen
numbers using Chapman-Enskog (CE) theory and for Maxwell molecules at finite
Knudsen numbers using a moment-hierarchy (MH) method. Corresponding numerical
solutions are obtained using the Direct Simulation Monte Carlo (DSMC) method of
Bird. The thermal conductivity, the viscosity, and the Sonine-polynomial
coefficients of the velocity distribution function from DSMC agree with CE
results at small Knudsen numbers and with MH results at finite Knudsen numbers.
Subtle differences between inverse-power-law, variable-soft-sphere, and
variable-hard-sphere representations of Maxwell molecules are observed. The MH
and DSMC results both indicate that the effective thermal conductivity and the
effective viscosity for Maxwell molecules are independent of the heat-flux
Knudsen number, and additional DSMC simulations indicate that these transport
properties for hard-sphere molecules decrease slightly as the heat-flux Knudsen
number is increased. Similarly, the MH and DSMC results indicate that the
effective thermal conductivity and the effective viscosity for Maxwell
molecules decrease as the shear-stress Knudsen number is increased, and
additional DSMC simulations indicate the same behavior for hard-sphere
molecules. These results provide strong evidence that the DSMC method can be
used to determine the state of a gas under highly nonequilibrium conditions
