In N-S the  velocity is the average mixture velocity, 

v =sumiN (rhoi*vi)/sumiN(rhoi)  , a mass weighted velocity over all component i, 

where i is the velocity of a component of the mixture formed by N chemical species. ( the same approach is given in molar base)

v is the velocity "we can see", to measure with a PIV or other any apparatus. 

The Diffusion velocity of a i component is  the relative velocity (v-vi), and the duffusive flux J"i =rhoi*(v-vi).

The 1st Ficks law establish that  J"i = - rho*Di, effGrad(Xi), where Xi, is the mass fraction of i in the mixture with mena density equal to rho. Hence,  in a flow with velocity v - the convective or advective velocity, we have  components with velocity  vi v, due the diffusive fluxes generating by  each gradient mass concentration or chemical potentials. 

This relations are valid in a macroscopic scale, where the flow velocity  that we "see" is v. In a meso scale we need accounting for the statistical fluctuations and the Lattice- Boltzmann approach is  more suitable. 

The Knudsen number is a good parameter to evaluate, if the mean path is very smaller than the  characteristic flow length Kn1 a stochastic approach must be considered as LB.

In the derivation the NS equations we consider the bulk velocity, or as aforementioned, the mixture average velocity   to accounting for the momentum transport  due convection. Considering for simplicity, a bicomponent mixture, The momentum transport due diffusive flux is conservative because J"1 =J"2 is also conservative and any mass flux com velocity v brings a momentum quantity m"diff*v,  and this part of momentum transport is mutually canceled.

In the energy transport equation this diffusive flux can carry enthalpy, and this term is present in the energy balance equation. We can have heat transfer against the thermal gradient due a  big concentration gradient which transports this heat. So this term does not vanish. Fluent is one code that considers this effect.

Dear Ricardo, Your rewritten correlation from usually used textbooks.  But it is not correct. The average vi (of component  i) must have 2 parts (branches) : W-as in mechanics of reversibly part, and irreversibly part – diffusion term. These 2 parts (branch)  have different property of symmetry.  In these case there can not be introduced (defined) a Sum of two vectors which have different  properties symmetry .

It is may be more clear for pure gas. In these case will be clear that usually correlation a priory assumed that diffusion process in pure gas absent and in finally these assumption «provide” (tautology).

Lev  

Interesting Lev, I've downloaded two PDF from you. ( the others written in Cyrillic, I can't  read, sorry). I'll read to learn more about this. Have you more references where this is explained. I found a interesting information,  and I'd like to learn more about it.

Tks

Dear Ricardo Vicente (from Brazil?),

I have 3 monograph +aprox. 100 articles in Russian (sorry) and it is difficult translate in English (may be in Latin ?) but in its are lot of mathematics (2 vectors have sum only if its both Belong to one class: excuse me).  Lev Ivanovich.

yes, I'm   from Brasil. Latin is better to me. I understand a little bit, Portuguese is so far away fro latin nowadays.... But I'll try use the google translation service.  I think it will help. Do not worry. Tks again!

Let me rephrase the question to check  I understand correctly:

Why is there no diffusion of particles/molecules in the macroscopic NS equation?

The answer to this is a result of the averaging process that is used to move from a particle/molecular scale to a continuum scale.  At the molecular scale there are indeed two components to the particle motion, a coherent, convective part and a stochastic diffusive part.  The NS equation is the volume/short time average of the microsystem, and hence it preserves the coherent part, but the stochastic part averages to zero and hence doesn't appear in the bulk NS equation. 

Generally speaking microscale fluctuations dont translate up to the continuum scale, but one well known exception for NS is the Reynolds Stress model of microscale turbulence.  In this case the averages of the cross-correlations of the microscale velocity fluctuations are preserved in the form of a Reynolds stress tensor.  

http://en.wikipedia.org/wiki/Reynolds_stress_equation_model

You might also find Beyond Equilibrium Thermodynamics by Hans-Christian Oettinger answers some of your questions. 

Your answer is based on the model in which disappearance of diffusion processes on macro level by averaging is initially simply postulated, and then represents it as "proof" (it is tautology).

Lev

The formal answer to this question will be that particle diffusion velocity in NS equation is neglected. This is called "fluid approach" that is good for the description of collective effects but is well known to fail for the description of individual particles, or other mesoscopic phenomena.

Still, there are methods to describe mesoscopic processes within the fluid approach.

One, is to introduce the stochastic force in the NS equation. In such case, fluctuation-dissipation theory is employed.

Another method is to follow two-fluid description. In this case two components are described individually and friction properties between these two flows define the dissipative properties of two component medium. This approach is farther developed for the description of plasma, where two, or even three fluid descriptions are known.

Of cause probably most rigorous way is to not use the fluid approach when dealing with mesoscopic phenomena and proceed with particle distribution function in kinetic theory. I believe this is the approach you persue in your publications. Still, results seem similar to those that can be obtained via two fluid approach (e.g. Eq. 20 from JTP 1986). I am not expert in this area, but I think it can be helpful to make direct comparison and estimate improvements offered by your method, especially in the case of non-neutral fluids.

Sincerely,

A.T.

Dear Alexander. Your answer demands more careful studying.

It seems to me you can read in Russian. In Russian editions such questions for a long time were considered  also I shall try to pick up the appropriate references.

Lev Ivanovich

Dear Lev, I didn't understand completely what you mean by "velocity of the flow mass diffusional nature". Looking at your papers it seems that you used a sort of "mixture formulation". If one applies "rigorously" this approach, it actually appears a "small scale velocity" in the Navier-Stokes equations. This velocity correlated by itself in the diadic product give rise to the so called "Korteweg stresses". Korteweg stresses are important only at the interface of two fluids and in many applications are just neglected.

I don't know if this is the velocity you are talking about. 

Regards,

Andrea

It has been suggested by Howard Brenner that NS should include diffusion of mass. He has published a few papers where he develops his theory.

In order not to complicate the issue will not consider a mesoscopic system. Consider the simplest system in the form of an ideal simple gas, where is the temperature gradient.    see attach    Lev 

Modeling of diffusion velocity is standard in the extended NS equations, when you are solving more than one mass conservation equation and have other driving terms present (such as chemical reactions) that could create a concentration gradient to act as the driving force.

Let me try again.

To paraphrase, it seems that you are noting that the macro transport coefficients ought to be cross-coupled e.g. a temperature gradient will drive an associated  mass flux down the temperature gradient?

In which case - I agree this is neglected in the NS equation.  I think I have seen some formulations of coupled sets of transport equations that take this coupling into account, but it's  long time ago and I can't remember any specific reference (sorry).

As someone mentioned above, Brenner wrote some good discussions of 'issues with the NS equations', which I think also addressed this point.