Abstract
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.
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