Gradient Perturbations Navier Stokes Equations

Appl3ung the Enskog perturbation method we intend to describe the prop -erties of gases which are only slightly different from equilibrium. Only under these conditions will the flux vectors be about linear in the derivatives so that the formal deflnitions of the transport coefficients apply. In this limit the distribution function is still nearly Maxwellian, and the Boltzmann equation can be solved by a perturbation method. The resulting solutions are then used to obtain expressions for the heat and momentum fluxes and for the corresponding transport coefficients. 

The flrst step in the Enskog expansion is to introduce a perturbation parameter e into the Boltzmann equation to enforce a state of equilibrium flow as the gas is dominated by a large collision term  

If these series expansions are introduced into the modified Boltzmann equation (2.241), and the coefficients of equal powers of equated, the distribution function can be uniquely determined to a specified order provided that / still satisfies the moment relations defining the physical properties like density, gas velocity and temperature. 

The solution of (2.244) gives rise to the absolute Maxwellian, f, as discussed in sect 2.7.2. 

Considering the first order approximation, (2.245) is supposedly solved for /. It is then required that the physical properties (p,v,T) are determined by the zero order approximation, f, whereas the higher order terms in the expansion contribute to q and o only. 

in the first order perturbation solution f is written in terms of the perturbation function P. Hence, 

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Step 1 is purely hydrodynamic and relates the perturbation Q to the velocity near the wall which is the only relevant quantity for the mass transfer response. at are either wall velocity gradients or coefficients involved in the velocity expansion near the wall. This step requires the use of Navier-Stokes equations and will be treated in Chapter 2. 

The viscous stresses only come into play for systems containing significant velocity gradients within the fluid. Nevertheless, very large gradients are not required as the Navier-Stokes equations can be derived from the Chapman-Enskog perturbation theory. On the other hand, for the non-equilibrium boundary layer and shock wave systems, i.e., systems which deviates considerably from equilibrium, higher order expansions are apparently needed [28]. Actually, at least for shock waves the validity of the Maxwell-Boltzmann equation becomes questionable. 

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. 

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