The planar Fourier Bow for a dilute gas of hard spheres is studied by means of the directsimulation Monte Carlo method to solve the Boltzmann equation. Two different types of boundary conditions are considered. In the conventional conditions, the gas can be seen as enclosed between two baths at equilibrium at wall temperatures. In the alternative conditions, both baths are out of equilibrium in states close to the one of the actual gas. It is shown that these alternative conditions are more appropriate to analyze bulk transport properties, as they reduce the boundary efFects. The deviation of the heat Bux from the Fourier law is small, even for large thermal gradients. In addition, the velocity distribution function is obtained and compared with the exact solution of the Bhatnagar-Gross-Krook model.
The steady planar Poiseuille flow generated by a constant external force is analyzed in the context of the nonlinear Bhatnagar–Gross–Krook kinetic equation for a gas of Maxwell molecules. An exact solution is found for a particular value of the force parameter. At a hydrodynamic level, the solution is characterized by a parabolic profile of the flow velocity with respect to a space variable scaled with the local collision frequency, a parabolic profile of the temperature with respect to the same variable, and a constant pressure. The (dimensionless) ratios between the quadratic coefficients and the external force are equal to 146 for the flow velocity and 65 for the temperature, as compared with the values 1/2 and 0, respectively, in the Navier–Stokes order. The fluxes of momentum and energy are explicitly evaluated. The anisotropy of the velocity distribution is made evident by the diagonal elements of the pressure tensor: Pyy/Pxx=0.031, Pzz/Pxx=0.081. Finally, the velocity distribution function is obtained in terms of quadratures.
The behaviour of rarefied monatomic gas of Maxwell particles within a rectangular enclosure is investigated, with the Navier–Stokes and Fourier field of equations with first- (NSF) and second-order boundary conditions (NSF2) of the velocity slip and temperature jump, and the regularized 13 moments approach (R13). The enclosure considered has a heated bottom with lateral walls that have specular reflection. The effect of the three dimensionless parameters characterizing the simulated problem, the cavity aspect ratio, the Knudsen number, and the temperature ratio of the hot over the cold walls, on the flow and bulk quantities is examined. For the small Knudsen numbers the flow presents one type of streamlines from the cold to hot plate in both NSF and R13 theories, while by increasing the Knudsen number the flow becomes more complex and presents hot to cold flow streamlines in the extended approach of R13. These rarefaction effects cannot be predicted by the classical continuum approach of NSF. The increase of the temperature ratio in R13 affects the hot to cold flow, which begins to vanish, while this type of streamline does not appear by decreasing the aspect ratio.
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