Burnett equations

During the derivation of the Fokker—Planck equation and telegrapher s equation, the derivatives with respect to distance were dropped at the second order. However, there is a case for including the third- and fourth-order spatial derivatives, though the third-order derivatives W2 will average to zero. This procedure leads to the Burnett equation... 

Of more concern are the comments by De Schepper et al. [528] and Resibois and De Leener [490]. They have discussed whether such a fourth-order derivative can have meaning. A mode-coupling theory and a kinetic theory of hard spheres both indicate that the Burnett coefficient diverges at tin. There seems little or no reason for the continued use of the Burnett equation in discussing chemical reaction rates in solution. Other effects are clearly more important and far more reasonable from a theoretical point of view. 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

It has been argued that in the higher Knudsen number regime, the Burnett equations will allow continued application of the continuum approach. In practice, many problems have been encountered in the numerical solution and physical properties of the Burnett equations. In particular, it has been demonstrated that these equations violate the second law of thermodynamics. Work on use of the Burnett equations continues, but it appears to be unlikely that this approach will extend our computational capabilities much further into the high Knudsen number regime than that offered by the Navier-Stokes equations. 

In addition to the limitations of the continuum approaches in being able to accurately represent transport processes under strongly nonequilibrium conditions, the formulation of physically meaningful boundary conditions may also be problematic. For the Euler equations, the boundary conditions at the vehicle surface must be adiabatic for energy and no slip for momentum. Use of the Navier-Stokes equations allows stipulation of isothermal temperature and slip velocity conditions. However, under strongly nonequilibrium conditions, these boundary conditions will fail to reproduce the physical behavior accurately. The situation for the Burnett equations is even worse since the required boundary conditions must include second order effects. 

The Boltzmann equation is solved by the particulate methods, the Molecular Dynamics (MD), the Direct Simulation Monte Carlo (DSMC) method, or by deriving higher order fluid dynamics approximations beyond Navier-Stokes, which are the Burnett Equations. The Burnett equation... 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

When the second order approximations to the pressure tensor and the heat flux vector are inserted into the general conservation equation, one obtains the set of PDEs for the density, velocity and temperature which are called the Burnett equations. In principle, these equations are regarded as valid for non-equilibrium flows. However, the use of these equations never led to any noticeable success (e.g., [28], pp. 150-151) [39], p. 464), merely due to the severe problem of providing additional boundary conditions for the higher order derivatives of the gas properties. Thus the second order approximation will not be considered in further details in this book. 

From the DSMC results and solutions of the linearized Boltzmann equation, it is evident that the velocity profiles in pipes, channels and ducts remain approximately parabolic for a large range of Knudsen number. This is also consistent with the analysis of the Navier-Stokes and Burnett equations in long channels, as documented in Ref [1]. Based on this observation, we model the velocity profile as parabolic in the entire Knudsen regime, with a consistent slip condition. We write the dimensional form for velocity distribution in a channel of height h. 

We developed a unified flow model that can accurately predict the volumetric flowrate, velocity profile, and pressure distribution in the entire Knudsen regime for rectangular ducts. The new model is based on the hypothesis that the velocity distribution remains parabolic in the transition flow regime, which is supported by the asymptotic analysis of the Burnett equations [1]. The general velocity slip boundary condition and the rarefaction correction factor are the two primary components of this unified model. 

If there are large concentration gradients in the system, Fick s law must be modified to include higher spatial gradients. We then obtain Burnett equations. 

The results themselves have a subtlety associated with their interpretation owing to the presence of the volume-ratio parameter and, optionally, the initial density parameter. The Burnett equations have more flexibility to fit Burnett data than only a density series to PVT data. The statistical uncertainties reflect the quality of the experimental data relative to the particular model used to describe the experiment. The estimation of accuracy for Burnett results is necessarily somewhat subjective since the effect of systematic errors on parameter values is not explicit in nonlinear equations, such as the Burnett equations. Accuracy, however, can be estimated from a study of the effects of systematic errors in computer model calculations and from the magnitude of the change in the volume-ratio value determined with nonideal and nearly ideal gases. For these reasons, we include such information along with our virial coefficient results for ethylene. 

Bao F, Yu X, Lin J (2012) Simulation of gas flows in micro/nano systems using the Burnett equations. JPhys Conf Ser 362 1-12... 

Burnett equations—have been worked out in detail, " but as we shall discuss below the applications of these higher-order equations are limited. 

The boundary conditions to be used with the Burnett equations have also been determined for a BGK model by Sone, and for more general models by de Wit using variational methods, but in fact this set of boundary conditions is not complete. The Burnett and higher-order hydrodynamic equations have nonphysical solutions showing spatial variations on the length scale of the mean free path. One would like to have boundary conditions that could be used to reject these unphysical solutions. However, the available set of boundary conditions is not sufficient for that. Instead one must postulate that the rapidly varying solutions are absent and then use the available boundary conditions to determine the remaining hydrodynamic solution. 

Once one has decided to formulate a dynamical theory for Fourier components conserved variables at long times and small k, extra simplifications occur. Note that the right-hand side of Mori s equation (10) may be a quite complicated function of k for arbitrary k. However, for small A , it is common to expand K t) /ft, and in a power series in k. For the conserved-fluid variables, schemes that expand the equations of motion to order A , k, k j and k, respectively, are the Euler, Navier-Stokes, Burnett, and super-Burnett equations. Since = /k j, and since ift contains A once, K contains A twice, while x does not contain A at all [Eqs. (5), (6), and (11)], the leading term in / > = is 0 ik), while the leading term in Kx Ms 0(k ). The leading term in i(o is imaginary while the leading term in F is real. The Euler equations are reversible, i.e., show no dissipation, while the introduction of the k term in K at the Navier-Stokes level causes irreversibility. 

Illustrated in Figure 4.22, the Burnett equations have been used to predict the pressure distribution across the gap between the inner (rotating) and outer cylinders. These data are compared with the DSMC data of Nanbu (1983) and that obtained using the Navier-Stokes equation. Regardless of the value of Kn, the conventional solution shows no pressure variation across the channel, while the Burnett and DSMC solutions are essentially in agreement. 

The coefficients tu, depend on the gas model and R is the specified gas constant. Since the Burnett equations are obtained by a second-order Chapman-Enskog expansion in Kn, they require second-order slip boundary condition. However, it may be noted that it has been observed that the second-order slip b.c. are inaccurate for Kn > 0.2. The Burnett equation can be used to obtain analytical/numerical solutions for at least a portion of the transition regime for a monoatomic gas. 

The term Kn M iPf /Pf- is relatively small for low Mach number flows in the early transition regime (i.e., Kn< 1). Hence, Burnett equation reduces to... 

Hence, the streamwise Burnett equation is reduced to the form obtained in the N-S limit. The cross flow momentum equation shows that the pressure gradient in that direction is balanced by the Burnett normal stress, which in the case of continuum is identically zero on the flat surface. 

The Burnett equation can be used to obtain numerical/analytical solution for at least a portion of t/ie regime (0.1 [Pg.96]

---