Generalized hydrodynamics for a Poiseuille flow: Theory and simulations

From the complete Boltzmann's equation we obtain general hydrodynamic equations for the laminar stationary Poiseuille flow driven by an acceleration of gravity g. This theoretical framework implies highly nonlinear transport equations. The hydrodynamic equations are perturbatively solved up to sixth order using a small adimensional parameter F proportional to g. The predictions are compared with our own simulational results obtaining very good agreement. A second and small adimensional parameter that naturally enters the formalism is a Knudsen number Kn proportional to the ratio between the mean free path and the width of the Poiseuille channel and it serves to understand the role of the finite size effects. It will be seen in particular that there is a heat flux with a normal component qy and a heat flux qₓ parallel to the isotherms and that their ratio is inversely proportional to the Reynolds number: qₓ/qy~F/Kn~1/Re.