# Interdisciplinary Applied Mathematics

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Cartesian coordinate system (x1,x2) = (x,y).

Remark: The conservation equations (2.16) are valid for continuum as well as for rarefied flows. However, the viscous stresses (aij) and the heat flux (qi) have to be determined differently for different flow regimes (see ection 15.4.2). Specifically, the thermal stresses

d2T    1 d2T

dxidxj    3 dx 12

in the momentum equation (derived from the Boltzmann equation) are not included in the Newtonian law for fluids. Similarly, the term in the energy equation

d2Uj

dx2

is not present in the Fourier law. These terms are derived in the asymptotic analysis of the Boltzmann equation in the limit of small deviation from equilibrium (Sone, 2002). For small Knudsen number flows and with O(M) ~ O(Kn), the thermal stress in the momentum equation can be absorbed in the pressure term. However, if the Reynolds number of the system is large or the temperature variation is not small, then the thermal stress cannot be included in the pressure term. In this case, these extra terms have to be included explicitly in the governing equations, which are different from the above compressible Navier-Stokes equations (Sone, 2002). To this end, also the work of (Myong, 1998) may be consulted. He derived thermodynamically consistent hydrodynamic models for high Knudsen number gas flows, valid uniformly for all Mach number flows and satisfying the second law of thermodynamics.

###### 2.2.1 First-Order Models

By first-order models we refer to the approximation of the Boltzmann equa tion up to O(Kn), i.e., the compressible Navier-Stokes equations. The con-

stitutive laws from equations (2.2a) and (2.2b) are

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