Hypersonic flows about space vehicles in low earth orbits and flows in microchannels of microelectromechanical devices produce local Knudsen numbers which lie in the continuum-transition regime. The Navier-Stokes equations cannot model these flows adequately since they are based on the assumption of small deviation from local thermodynamic equilibrium. A number of extended hydrodynamics (E-H) or generalized hydrodynamics (G-H) models as well as the Direct Simulation Monte Carlo (DSMC) approach have been proposed to model the flows in the continuum-transition regime over the past 50 years. One of these models is the Burnett equations which are obtained from the Chapman-Enskog expansion of the Boltzmann equation (with Knudsen number (Kn) as a small parameter) to $OKn2.$ With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with this model. Several improvements to the conventional Burnett equations have been proposed in recent years to address both the physical and numerical issues; two of the most well known are the Augmented Burnett Equations and the BGK-Burnett Equations. This review article traces the history of the Burnett model and describes some of the recent developments. The relationship between the Burnett equations and Grad’s 13 moment equations as shown by Struchtrup by employing the Maxwell-Truesdell-Green iteration is also presented. Also, the recent work of Jin and Slemrod on regularization of the Burnett equations via viscoelastic relaxation that ensures positive entropy production and eliminates the instability paradox is discussed. Numerical solutions in 1D, 2D, and 3D are provided to assess the accuracy and applicability of Burnett equations for modeling flows in the continuum-transition regime. The important issue of surface boundary conditions is addressed. Computations are compared with the available experimental data, Navier-Stokes calculations, Burnett solutions of other investigators, and DSMC solutions wherever possible. This review article cites 56 references.

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