Generalized hydrodynamics approach to the Knudsen problem

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Abstract
It is shown that the Knudsen problem (‘‘paradox’’) can be resolved by the flow-rate formula for a non-Newtonian fluid flowing under a pressure gradient in a circular tube, as reported elsewhere. The flow-rate formula exhibits a minimum in the low-pressure regime but gives the Hagen-Poiseuille linear dependence on pressure in the high-pressure regime. Since the generalized hydrodynamic equations used for the derivation of the flow-rate formula are derived from the Boltzmann equation, the present solution of the Knudsen problem is well founded on the kinetic theory of gases. The nonlinear transport processes, which increasingly manifest themselves as the gas density decreases, are the cause for the emergence of a minimum in the flow rate for rarefied gases in circular tube flow. The entropy production and also the drag coefficient are calculated as a function of pressure difference, Reynolds number, and other parameters characteristic of the system.