Keywords: extended hydrodynamics, extended thermodynamics, rarefied gas flow, method of moments, non-equilibrium flow, Chapman-Enskog expansion.

Over many years, the Navier-Stokes-Fourier (NSF) equations have proved to be able to provide an accurate description of many flow phenomena. However, despite their great success at representing continuum flow problems, there are several areas where the NSF equations fail to predict the flow properties correctly. In particular, the NSF equations are known to be poor at predicting the thickness of a strong shock wave and at capturing the flow physics at high altitudes. The reason for their failure is the breakdown in the assumption that the flow is in thermodynamic equilibrium. Another area where the NSF equations fail is in vacuum technology. However, with the realization of micro-technology, the reliability of the NSF equations is again under question. Micro-devices have characteristic length scales that range from 0.1 microns through to 1 mm. In a one micron device, the number of molecules colliding with the walls becomes comparable to the number of intermolecular collisions, even at atmospheric pressures. Under such conditions, the system exhibits a strong degree of non-equilibrium and, because the NSF equations are only suitable for flows close to equilibrium, they are not reliable at predicting flow rates, velocities etc.

For gaseous flows, it is possible to define a number that provides insight about the degree of non-equilibrium in a gas. This is known as the Knudsen number, Kn, and it relates the mean free path of the gas molecules, lambda, to a characteristic dimension of the device, L, i.e. Kn=lambda/L. The gas is considered to be in thermal equilibrium when Kn<=0.001. It is possible to extend the range of applicability of the NSF equations to the so-called slip regime, where Kn<=0.1, by modifying the wall boundary conditions to account for velocity-slip and temperature-jump effects. Although this approach is well accepted for engineering problems, it can only provide information about the mean flow quantities. To extend the validity of the NSF equations into flows where non-equilibrium effects begin to dominate, i.e. beyond Kn=0.1, entails considerable mathematical complexity. In our work, we have been developing extended hydrodynamic techniques based on a Hermite polynomial expansion of the molecular distribution function combined with a Chapman-Enskog expansion to regularize the set of equations. This approach was first introduced by Harold Grad in 1949 who developed the thirteen-moment equations. With the advent of micro-technology, there has been renewed interest in developing models that can accurately capture gaseous transport at the micro-scale. We have been pioneering the development of the twenty-six moment equations with a focus on capturing low-speed flows in the range 0.1<=Kn~1.0 and will present recent results in this challenging numerical field.

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