Equilibrium Flow The Euler Equations

Applying the local Maxwellian distribution function (2.225), explicit expressions can be obtained for the heat flux, q and the pressure tensor P. 

In the zero order solution, there is no heat flow and the pressure tensor is diagonal. The diagonal elements in the pressure tensor denote the normal stresses that are identical to the thermodynamic pressure, as deflned by (2.68). 

These conclusions are not obvious at first sight, so a brief mathematical verification is provided. In Cartesian tensor notation, the pressure tensor (2.64) is given by  

Furthermore, as the integration is over the whole velocity space, it can be proved that any integral vanishes when the integrand is odd. Inspection of the pressure tensor shows that, since Q and Cj themselves are odd functions of Cj and Cj respectively, whereas the exponential is an even function, the integrand as a whole is odd when i j and even when i = j. 

The derivation of the shear and normal stress terms is examined for two representative cases, for which we pick out the odd term i = 1, j = 2) and the even term i = j = 1). The two analytical integral solutions required are taken from the literature . 

In the following mathematical manipulation we are solving even integrals on the form  

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