# Interdisciplinary Applied Mathematics

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B

дщ u?

= —2MW1 + —

dxj

P

duk дщ    ( D дщ дщ duk

wi-——h UJ2 yrv—2:

dxk dxj

Dt dx

j

dxk dx^

д2Т    1 dp dT    R дТ dT

j

3 dxjdxj    4 pT dxj dxj    5T dxj dx

(2.25)

dui duk dxk dxj

where a bar over a tensor designates a nondivergent symmetric tensor, i.e., fij = (fij + fji)/2 — \$ij/3fmm.

Similar expressions are valid for the heat flux qB (Zhong, 1993). The coefficients шi depend on the gas model and have been tabulated for hard spheres and Maxwellian gas models (Schamberg, 1947; Zhong, 1993). Since the Burnett equations are of second order in Kn, they are valid in the early transition flow regime. However, fine-grid numerical solutions of certain

X

n

##### s

FIGURE 2.5. Control surface for tangential momentum flux near an isothermal wall moving at velocity Uw.

versions of the Burnett equations result in small wavelength instabilities. The cause of this instability has been traced to violation of the second law of thermodynamics (Balakrishnan, 2004). Using the Chapman-Enskog expansion and the Bhatnagar-Gross-Krook model of the collision integral, Balakrishnan (2004) derived the BGK-Burnett equations, and reported that the entropy-consistent behavior of the BGK-Burnett equations depends on the moment closure coefficients and approximations of the total derivative terms (-^) in equation (2.25). In the following we use the exact definition of the total derivative instead of the Euler approximation most commonly used in hypersonic rarefied flows (Zhong, 1993). Numerical solutions of the Burnett equations for sevearal gas microflows can be found in (Agarwal et al., 2001; Agarwal and Yun, 2002; Xu, 2003; Lockerby and Reese, 2003; Xue et al., 2003).

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