Normal solutions of the Boltzmann equation for highly nonequilibrium Fourier flow and Couette flow

摘要

The state of a single-species monatomic gas from near-equilibrium to highlynonequilibrium conditions is investigated using analytical and numericalmethods. Normal solutions of the Boltzmann equation for Fourier flow (uniformheat flux) and Couette flow (uniform shear stress) are found in terms of theheat-flux and shear-stress Knudsen numbers. Analytical solutions are found forinverse-power-law molecules from hard-sphere through Maxwell at small Knudsennumbers using Chapman-Enskog (CE) theory and for Maxwell molecules at finiteKnudsen numbers using a moment-hierarchy (MH) method. Corresponding numericalsolutions are obtained using the Direct Simulation Monte Carlo (DSMC) method ofBird. The thermal conductivity, the viscosity, and the Sonine-polynomialcoefficients of the velocity distribution function from DSMC agree with CEresults at small Knudsen numbers and with MH results at finite Knudsen numbers.Subtle differences between inverse-power-law, variable-soft-sphere, andvariable-hard-sphere representations of Maxwell molecules are observed. The MHand DSMC results both indicate that the effective thermal conductivity and theeffective viscosity for Maxwell molecules are independent of the heat-fluxKnudsen number, and additional DSMC simulations indicate that these transportproperties for hard-sphere molecules decrease slightly as the heat-flux Knudsennumber is increased. Similarly, the MH and DSMC results indicate that theeffective thermal conductivity and the effective viscosity for Maxwellmolecules decrease as the shear-stress Knudsen number is increased, andadditional DSMC simulations indicate the same behavior for hard-spheremolecules. These results provide strong evidence that the DSMC method can beused to determine the state of a gas under highly nonequilibrium conditions