# Synopsis: In thin air

#### Thermodynamically Admissible 13 Moment Equations from the Boltzmann Equation

Hans Christian Öttinger

Published March 22, 2010

In a sufficiently rarefied gas, the tools of standard hydrodynamics, namely, the Navier-Stokes-Fourier equations (based on Newton’s second law applied to fluid motion), no longer apply. On the other hand, tracking the statistics of individual molecules, as is done in Boltzmann’s kinetic equation approach, is computationally prohibitive.

Writing in Physical Review Letters, Hans Christian Öttinger at the ETH in Zurich, Switzerland, strengthens a description of rarefied gas flow that is intermediate between these two extremes. Previous work showed that Boltzmann’s kinetic equation can yield a set of $13$ moment equations that describe the momentum of the gas and heat flux within it. In his work, Öttinger looks to the laws of nonequilibrium thermodynamics to provide the necessary constraints on the equations. For example, entropy is conserved in reversible processes and standard hydrodynamics is recovered in the limiting case.

The results could advance efforts—particularly those based on numerical calculations—to describe the kinetics of nonequilibrium gases. These calculations are important for understanding gas flow in the smallest of channels—as in microfluidics—and aerodynamics of satellites and space stations in the outer limits of our atmosphere. – Jessica Thomas