Dear Anatoly,

you are indeed right, continuum model has its drawbacks. Regarding the models, you are right, there is kinetic theories for gases, which use velocity distribution functions and essentially operates in 6-dimensional space. Nevertheless, it is still a continuum model and can not be used for dense liquids.

However, for dense liquids there are also good models around. The class of simulations you would need to look for is "molecular dynamics (MD) simulations". It is a discrete model. Specific model for water (bear in mind, that MD models has to be specific to different chemical constituents of the liquid) has been developed for quite long time, here is one well respected publication of initial improvements (see attached link).

One has to note though, that these simulations are extremely expensive, however they are more accurate than any continuum models. I have seen some very recent works, in which attempts are made to couple both continuum and MD approaches, but I don't have it at my fingertips currently.

http://scitation.aip.org/content/aip/journal/jcp/60/4/10.1063/1.1681229

Dear Anatoly,

you are indeed right, continuum model has its drawbacks. Regarding the models, you are right, there is kinetic theories for gases, which use velocity distribution functions and essentially operates in 6-dimensional space. Nevertheless, it is still a continuum model and can not be used for dense liquids.

However, for dense liquids there are also good models around. The class of simulations you would need to look for is "molecular dynamics (MD) simulations". It is a discrete model. Specific model for water (bear in mind, that MD models has to be specific to different chemical constituents of the liquid) has been developed for quite long time, here is one well respected publication of initial improvements (see attached link).

One has to note though, that these simulations are extremely expensive, however they are more accurate than any continuum models. I have seen some very recent works, in which attempts are made to couple both continuum and MD approaches, but I don't have it at my fingertips currently.

http://scitation.aip.org/content/aip/journal/jcp/60/4/10.1063/1.1681229

Dear Anatoly,

The model named "regularized 13-moment (R13) equations" is an improved continuum model for molecular gases and has been employed to study several classical problems of the fluid mechanics. The model contains balance equations for stress and heat flux in addition to usual mass, momentum and energy balance equations, and the system is closed in a sophisticated way by considering the "order of magnitude" of moments.

However, to the best of my knowledge, turbulence has never been studied yet with this model. Please have a look in the attached papers and also refer to the book "Macroscopic Transport Equations for Rarefied Gas Flows" by "Prof. Henning Struchtrup". You may find more information on the homepages of Prof. Henning Struchtrup (http://www.engr.uvic.ca/~struchtr/) and Prof. Manuel Torrilhon (http://www.mathcces.rwth-aachen.de/5people/torrilhon/start). Both the authors have researchgate accounts as well.

I am sure that both of them will be happy if you need any help/suggestion in employing their model for studying the turbulence.

I hope that this helps.

Thanks and regards,

Vinay

http://www.engr.uvic.ca/~struchtr/

http://www.mathcces.rwth-aachen.de/5people/torrilhon/start

Dear Anatoly,

You could read the attached review paper, and then follow up on my book referenced in it. In them, I argue the thesis that a proper fluid mechanics has to start from the bottom up, ie by employing molecular dynamics. On the QJRMetS website, there is a correction to the journal article, it is 780_ftp.pdf.

Good luck,

Adrian Tuck

The continuum model is correct when its assumptions are valid (obviously). The problem is that even modern computing power is insufficient to exactly solve the continuum equations for practically all turbulent flows, hence the wide range of simplified models.

Dear Prof. Andrew John Yule,

But why is it hard to introduce the self diffusion of a liquid particle directly into the liquid volume notion? Maybe here some random work theory can  be applied? The liquid volume notion seems to me very unrealistic.

Dear Anatoly,

If you find DNS too inaccurate, and molecular dynamics too time-consuming, you could also try Lattice Boltzmann. 

Good luck,

Sita Drost

Dear Sita Drost!

Sorry, I am from Russia :), I did not understand your words excepting molecular dynamics (MD).

I say that for MD you need to choose right force potentials which has in turn many parameters. It is also a problem! And at the end you will have picoseconds of life of physical system! How do we even decide to use these results in physics...;)?

What is DNS and Lattice Boltzman? DNS is numerical simulations? If so it is the continuum model still:)!

But why is it hard to introduce the self diffusion of a liquid particle directly into the liquid volume notion? Maybe here some random work theory can  be applied? The liquid volume notion seems to me very unrealistic. One can choose a probability function as a continuous differentiable function?

Dear Anatoly, the turbulence problem is solved for infinitely high Reynolds numbers, i.e. for Eularian fluids. One nice example is Karman's constant as 1/SQRT(2*Pi) = 0.399. The theory is based on the idea that the vortices in the turbulent vortex ensemble can be treated as colliding particles and some earlier work by Fokker, Planck, Landau, Kolmogorov and Hans Herrmann. Meanwhile also versions for Newtonian and non-Newtonian fluids are avalable. Mathematically it is written as continuum equations for rms turbulent kinetic energy and rms vorticity. So it fits nicely in traditional code enviroments. All details on my RG website.

Dear Anatoly,

Sorry about that. What I meant to say is that Lattice Boltzmann might be a nice compromise between Molecular Dynamics (very accurate, but also extremely time-consuming and therefore not very suitable for macro-scale problems) and DNS (= Direct Numerical Simulation; continuum model, accuracy depends on e.g. time step & grid size). So, if you DO NOT want to use the continuum model, or Molecular Dynamics, maybe you DO want to use the Lattice Boltzmann method.

Lattice Boltzmann is a meso-scale method, that is, it originates from a molecular description of a fluid (Boltzmann equation), but it uses fictive "particles" that are much larger than molecules (and yet much smaller than the macro-scale of the flow, hence the term meso-scale). It's computationally efficient and hence, suitable for the simulation of macro-scale flows. I'm not quite sure what you mean by "introduce self diffusion of a liquid particle... etc.", but I think the Lattice Boltzmann method does something like that.

If you'd like to know more details, you can check Google/Wikipedia. An example of an introductory text about Lattice Boltzmann is https://www.ndsu.edu/fileadmin/physics.ndsu.edu/Wagner/LBbook.pdf

All the best,

Sita

https://www.ndsu.edu/fileadmin/physics.ndsu.edu/Wagner/LBbook.pdf

Anatoly, you are completely right that there are many turbulence theories and much more semi-empirical models. However, there is only ONE theory which is not only free of empirical params but also predicts the universal constants of turbulent motions, among them the von-Karman constant. Lev Landau wrote in his textbook series (Landau-Lifshitz) that one can only measure this constant! But here he failed. Enjoy. If you have questions contact me directly.

Article Universal equations and constants of turbulent motion

Dobry vetsher, here we are, see attachment. There is more on my website and another paper for physica scripta is just in print.

Article Universal equations and constants of turbulent motion

Conference Paper Eulerian turbulence and non-Newtonian fluids.

I think that the key would be if turbulence has intrinsically inside some issue that makes the continuum model no longer valid. But the Kolmogorov scale (in the fluids we are used to work on) is much greater than the free mean path, therefore I cannot see a reason to immagine that a particle-based model can really work better or can provide more theoretical results than a solution based on the continuum. Maybe, I am wrong...

However, I know about turbulence being studied in quantum theory field.

Dear Anatoly,

Turbulence can be studied properly by Navier-Stokes equations (DNS). Therefore turbulent flow simulation is not an unsolved problem and indeed DNS can reproduce experiments very accurately (see e.g. "Turbulent Flows" by S.B.Pope). Certainly the question of computational cost is a different story.

Also note that Knudsen and Reynolds numbers are inversely proportional which means that in a typical turbulent flow, molecular scales are irrelevant. Hence in majority of cases, continuum theory is sufficient to address turbulence.

Molecular behaviour is not irrelevant in the atmosphere, or any other gas where the observed scale invariance is different than what emerges from a molecular dynamics treatment. Current computer capacity can handle about 10**13 molecules/cc, enough to simulate three orders of magnitude in time, out to microseconds at STP. There is still a gap of 5 orders of magnitude to be spanned out to macroscopic experimental observations, but if the scalings are the same the scaling exponents can be used to cover economically subgrid scale parametrizations. The continuum assumption has to be justified from the bottom up rather than from the top down; gases are composed of molecules and in non-equilibrium conditions use of the gas constant R to express all molecular behaviour may not be adequate in principle, and in practice in the atmosphere.

Dear Sirs,

Is anybody familiar with nuclear magnetic resonance (NMR) experiments which allow to measure as macroscopic velocity of fluid particle as the self-diffusion of a molecule in the liquid particle completely noninvasively and nondestructively (no any powder in water or the like)? I know that about 20 years ago prof. John Gore (he was in Yale University then have gone to Vanderbilt university) made NMR experiments to measure velocity distribution in sudden expansion tubes for turbulent flows. You can find the articles in the journal of magnetic resonance. 

Does anybody know or give references about NMR or other studies which explore the relationship of a molecule self-diffusion in/from a liquid particle and the macroscopic velocity of this particle?

Yes, there are the next direction to  develop FM models in approximation of continuous media taking into account the compatibility condition of the fundamental set. The differential fluid mechanics occupies intermediate position between classical macroscopic fluid mechanics and atomic-molecular  physics.

The next constructive  step is more detailed inspecting the thermodynamic potential for non-homogeneous fluids with interfaces and contact boundaries.

Yuli D. Chashechkin

Anatoli, are you really sure that you correctly understand the notion "self diffusion"? In my view it has to do with chem potential and less with fluid mech. If so then you should better talk about MIXING. Here the first etap in a turbulent fluid is done by fluid-mechanics down to a "graininess" of the Kolmotorov scale. What happens next in the course of mixing down to molecular and chemically significant scales is driven by molecular diffusion and/or your diffusion WITHIN quasi-laminar patches or ever shrinking "graines". Priyatnogo dnya ...

Research Turbulent mixing and a generalized phase transition in shear...

Conference Paper Turbulent Mixing in Non-Newtonian Dispersions

Article Turbulent mixing and a generalized phase transition in shear...

Dear Helmut Ziegfeld Baumert,

1. The self-diffusion of molecule is a random movement of molecule in a media which only contains of this type molecules. For example we have a pure water, then we can talk about the self-diffusion of water molecules. Einstein formula for the self-diffusion (see for example physical kinetics textbooks or Landau`s hydrodynamics volume) is as follows. 

=sqrt(6Dt),

where -rms displacement of a molecule, D-the molecule self-diffusion coefficient, t - time of the displacement.

So say t=1 ms, D=2.3*10^(-5) cm^2/s (it is for water at 20 0C) then

=sqrt(6*2.3*0.00001*0.001)=3.7*10^(-4)=3.7 microns.

The conclusion is that during the time of 1 millisecond the water molecule has moved away at the distance of 3.7 microns.

2. The reason why the self diffusion of a molecule crucially relates to fluid mechanics (FM) is as follows. In FM the continuum is a mathematically geometric 3D body which consists of THE SAME PARTICLES (OR POINTS OR MOLECULES as you wish to imagine) all the time ;). But according to the item 1 it is NOT CORRECT.

Could you better elaborate your conclusion at point #2?

I consider the continuum model as the intepretation of a continuous function f(x) as an integral function of a probability distribution over a small (but greater than the f.m.p) volume around x...In other words, the function is continuous by means of the volume V(x). I do not remember that the theory forces the same particles to be in the volume...

In the last 10 years or so, there were a number of papers whose authors claim that classical hydrodynamics is not correct, but must be augmented by considering two distinct velocities, one to describe mass transfer, the other to describe momentum. Sometimes people speak of volume velocities as well. These velocities differ by a term which is typically related to self-diffusion and density gradient, which seems to be what you are looking for. Main author on this was Howard Brenner, but others chipped in as well (Durst, Dadzie,...). There are quite a number of variants of "augmented hydrodynamics".

Some references on this are in our paper [Inconsistency of a dissipative contribution to the mass flux in hydrodynamics, Hans Christian Öttinger, Henning Struchtrup, and Mario Liu, Phys. Rev. E 80, 056303 – Published 10 November 2009]. As the title shows, we are highly critical of the various models. Note that some later attempts to "augment" hydrodynamics try to address the various points of the paper.

I have not seen any convincing argument in favor of this. The two velocities create quite some problems, such as the question which one has to vanish for a fluid at rest. A gas under temperature gradient in a closed system clearly has no mass flux, but would then have momentum, although the box is at rest... and so on.

Kinetic theory yields that both velocities (for mass flux and momentum) agree, which is a reflection of the fact that there is only one velocity for a particle.

Note also that kinetic theory describes the gas as particles entering and leaving any volume element you chose in your flow domain, hence there are always different particles in a volume element. Think of a gas in equilibrium: each particle has a more or less random path (described by the self-diffusion coefficient, by the way), but in average the number of particles in a volume element does not change. As many particles leave as enter (not really, there are small fluctuations).

It was said before: Within the range of applicability, hydrodynamics works very well, in full agreement to solutions of kinetic theory, or, I believe, molecular dynamics--there is no need for additional terms.

TL;DR: Attempts were made to add self-diffusion terms to hydrodynamics, but none of these is convincing.

Dear Anatoly,

I only refer to the turbulent aspect in your question and here my answer remains correct: I intruduced in 2005 a new picture of turbulence as an ENSEMBLE ("gas") of interacting VORTEX DIPOLES and applied standard techniques (Fokker-Planck) to derive a two-eq turbulence model for the r.m.s. mean values, known in the literature as k-omega, but with a clean mechanistic background and: prediction of the universal constants, Karman and Kolmogorov constants in almost perfect agreement with observations.

Your self-diffusion problem is somehow trivial compared with the above - at least as long as you not better describe where the crucial point really sits in your perspective. As a practice-oriented theoretician I would like to know where and how you touch the limits of the existing theories, equation systems etc.

Regarding turbulence, there was such a clear point as Lev Landau in his book series with Lifshitz about Theoretical Physics wrote that the von-Karman constant can ONLY BE MEASURED. This was driving me to calculate it and found 1/SQRT(2*Pi) = 0.399 where Princeton superpipe gave 0.40 +/- 0.02. As a good sports Landau would have survived that, I am sure, and his picture decorates (together with pictures of Taylor, Flo Ziegfeld, Kolmogorov and Einstein family) one of my office walls.

Please be more specific.

Dear Sirs,

All equations in fluid mechanics are written for the liquid volume (the continuum) which consists of the SAME PARTICLES (POINTS, MOLECULES as you suppose it is). It is  Lagrange description of a fluid, see e.g. https://en.wikipedia.org/wiki/Continuum_mechanics.

After this usually one does a transformation from Lagrange variables to Euler`s one and you come to the Navier–Stokes equation.

Dear Helmut Ziegfeld Baumert,

I just see that the continuum model is very different from nature, a modern physical reality. Why do you and others use it even when you can not model/predict the stress distribution along the wall in 2D step-wise channel for Re up to 1000000??!! If you read Russian language you can read the enclosed textbook made by the one of the best aerodynamic departments of Russia (and USSR) which was formerly headed by famous prof. Loizjanskij.

Many of today's problems in fluid mechanics have distant origins due to the lack of clear axiomatic and universally accepted definitions of basic concepts (in fact the concept of fluid mechanics in different models, flow, wave, swirl, etc.). As a result, the same symbols used in different theories acquire different meanings.

Moreover the basic criterion of observability of physical quantities, which provides the ability to monitor the adequacy of the theory does not controlled as well as condition of the identity transformation in procedure of simplifying the equations.

The basic ideology of fluid mechanics based on the Euler’s equations and continuity (ideal fluid) or continuity and the Navier-Stokes equations for viscous fluid of a uniform density. The velocity of a fluid is selected as basic flow parameter in these systems.

However, the boundaries of an arbitrarily chosen element of the homogeneous continuous medium (a liquid particle) cannot be physically defined, which eliminates the possibility of controlling its position and determination of the velocity magnitude, respectively. The hypothesis of "passive marker" is not confirmed by a careful experiment. Liquid miscible markers break down on fiber in the flow; a solid body revolves and disturbs the environment. (Stepanova E. V., Chashechkin Yu. D. Marker transport in a composite vortex // Fluid Dyn. 2010. V. 45 (6). P. 843 – 858). With decreasing size a particle begins to participate in Brownian motion.

Equations of modern fluid mechanics are postulated on the basis of conservation laws or their analogues that are balance equations for non-equilibrium media. Conserved quantities (invariants) determine the choice of the basic observable physical parameters of the medium (the thermodynamic potential, density, temperature, pressure, concentration of components) and a fluid flow (momentum, energy). The momentum value is determined by the force action of the flow on an obstacle and volumetric or mass flow rate.

The fundamental system of classical fluid mechanics equations (continuity, the Navier-Stokes, Fourier and Flick equations), which expresses the laws of conservation, defines a fluid flow as dynamics of the fields of independent variables. The system solutions define the patterns of the fields and the dynamics of their changes. The every field is characterized by its own spatial and temporal scales (to be determined), their relationships is characteristic velocity, which generally will be individual for each field.

Systems need to be analyzed, taking into account the condition of compatibility determining the rank of a complete system, order of its linearized version and the degree of the characteristic (dispersion) equation. Classification of component flows is given in Chashechkin Yu. D. Differential fluid mechanics – harmonization of analytical, numerical and laboratory models of flows. P. 61-91. // Mathematical Modeling and Optimization of Complex Structures. Springer Series “Computational Methods in Applied Sciences” V. 40. 2016. DOI: 10.1007/978-3-319-23564-6-5.

The approximation of a homogeneous fluid is a special degenerate case, which cannot serve as the basis for the analysis of the general situation.

The heterogeneity of the liquid composition requires a detailed account of the thermodynamic potentials including chemical potential impact to the actual dynamics of fields, especially in the fine structure of the flow, typical for the real environments and technology.

The requirement of identity transformations in theoretical mechanics of fluids is violated almost everywhere. On the first pages of textbooks and books the fundamental system is printed and then the constitutive (the theory of turbulence) or reduced models (linear model approximation of the boundary layer, the acoustic approximation) are analyzed. New equations have new symmetries, respectively; they define new physical quantities, which are not correspond to the old quantities (Baidulov V. G. and Chashechkin Yu. D. Invariant properties of systems of equations of the mechanics of inhomogeneous fluids // J. Appl. Maths Mech. 2011. V. 75(4). P. 390 – 397.). That is the same symbols have a different meaning in different models and must be given different symbols. All this reflects the confusion of modern theoretical and "almost killed" experimental fluid mechanics.

Separately should be said about the boundary conditions. Near the surface of the liquid (free, or at the contact boundary with another medium) change the thermodynamic potentials. It is always manifest individual atomic-molecular properties inherent in each medium in the values of the surface tension, the composition of the medium and the component thermodynamic potentials.

Atomic-molecular properties also appear in the values of the characteristic times of transformation of some types of energy into another, which can transfer with a slow characteristic time of macroscopic processes (as ratio of scales and velocities) and the fastest time of atomic and molecular processes (rapid release of available surface potential energy at the confluence of two fluids and destruction of the free surfaces).

Summary.

Classical mechanics of fluids has a constructive potential for further development to describe flows dynamics and structure on the macroscopic scales (up to molecular cluster size ~10^(- 6) cm).

Realization of fluid mechanics potential requires careful description of the medium (determining thermodynamic potentials, physical quantities, state equations, constants and kinetic coefficients).

The basis of the flows description is the fundamental system of equations (continuity, transport of momentum, energy substances) with physically reasonable boundary conditions. Analysis of the system should take into account the control of the criteria of compatibility (defined the rank of the system) as well as the identity of transformation.

Fluid flows are transport of momentum and energy, accompanied by consistent changes in the physical parameters of the medium. Fluid flow dynamics determines the changes in the structure of independent fields of physical quantities, each of which has its own geometry (pattern, structure, scales).

Hydrodynamics needs to develop methods for measuring the all observable physical quantities (momentum, density, energy, the concentration of components) allowing to resolve a fine field structure and visualizing large-scale processes that ensure the completeness of the experiment.

In this context, the concept of self-diffusion is non-physical as the process is not characterized by the observed physical quantities. Rational mechanics of fluids does not require the introduction of additional unobserved variables.

Dear Yuli D. Chashechkin,

Why the self-diffusion of a molecule is unobservable variable?? You measure  a dynamics viscosity in fluid mechanics (FM) experiment!! The dynamic viscosity is included in Navier–Stokes equation (I do not account now the 2nd viscosity coefficient). The every (!!) FM textbook teaches the student that the viscosity of fluid, e.g. gases (see, e.g. classical Russian Emzev`s book) is described by a diffusion of molecules between two neighboring layers of fluid. This diffusion is a reason of the transfer of momentum then the one layer decelerates and the other accelerates. Then the textbook says that in FM we do not study this molecular process and phenomenologically describes this process by the friction (or viscosity) coefficient. But again the dynamic viscosity is measurable quantity in FM! Do you agree with the my last sentence?

Dear Anatoly A Khripov,

You have no right to mix observable physical quantities and products of mathematical calculations based on specific theoretical model and have to think how  they can be distinguished.

Observable are invariants that are basic properties of space-time (distance or interval)  or conservative value in the sets of equations (mass in mechanics, momentum, energy, concentrations). Diffusion coefficient  is defined from measurements of concentrations.

Theories prescribe connections between observable and non-observable parameters but cannot replace foundation of sciences.

YuDC

Anatoly,

I think Yuli's last answer is correct. His remarks also apply to entropy, which is a measure - ie a mathematical construct - and not a dynamical variable, especially in non-equilibrium systems, which macroscopic fluid flows are of necessity. See ISBN 9780199236534, "Atmospheric Turbulence: A Molecular Dynamics Perspective", OUP, 2008.

Adrian

Dear Anatoly,

We need to distinguish observable and measurable quantities (as Yuli wrote above). The self-diffusion is a fact, but the diffusion coefficient (as well as the viscosity coefficient, heat conduction coefficient, etc.) is not an observable quantity but a measurable one. This coefficient is the result of the prescribed mathematical model (invented by a human being, subjective), i.e. it is the coefficient in the mathematical expression, but it does not exist in reality. If one would suggest historically another mathematical expression for all these transport phenomena we would now talk about not a diffusion coefficient but about a different quantity. This means that such coefficients are not universal.

Another basic example is the pressure and density. The pressure is measurable but not observable while the mass density is observable (it is just a number of molecules). The fact is that it is easy to measure pressure and much more difficult to measure density. We feel pressure and we think that this is a real entity, but it is not.

In particular, my friends and I are developing a continuum theory for viscous fluids where there is no diffusion, viscosity and other coefficients et all, please see

http://www.sciencedirect.com/science/article/pii/S0021999116000693 (open access)

If you take a look at our model, you will not recognize here any of the famous transport laws such as Newton’s law of viscosity, Fourier law of heat conduction etc. But they are here anyway, they are intrinsic properties of the model. We are trying to employ in the model only basic objective  (as Yuli Chashechkin wrote above) space-time properties such as times, length (deformation), velocity, etc.

The continuum approach is not so bad as you think, but certainly it is not perfect yet, and even after more than 100 years, there is plenty of room to improve it. And please don’t think that the Navier-Stokes is the only possible way to describe viscous Newtonian flows in the framework of continuum mechanics. Our model is another one and extremely different, maybe there is a third one, etc. 

http://www.sciencedirect.com/science/article/pii/S0021999116000693

Dear Sirs,

I just repeat my understanding of FM, maybe I am wrong.

1. In the FM theory the continuum volume consists of the same particles (points molecules). It is called be Lagrange description of motion (see my previous messages). But such a volume does not exist in real liquid or gas due to the self-diffusion (e.g. during 1 millisecond a water molecule moves at the distance of about 4 microns, see my previous messages). 

2. All the equations of FM are deduced from the following method. One applies the 2nd Newton`s law to the continuum volume. But such a volume is simply a fantasy!

3. Finally we rewrite the FM equations (which is written in Lagrange description) in terms of the Euler`s description of motion. In order to do this we use the fact the continuum volume consists of the same points. In other words there is one-to-one correspondence between the initial (in time) position R0=R0(a,b,c,t0) and the later position R=R(x,y,z,t) of every point of the continuum volume. It is equivalent to the fact that Jacobian D(x,y,z)/D(a,b,c) is not equal to zero (https://ru.wikipedia.org/wiki/%D0%AF%D0%BA%D0%BE%D0%B1%D0%B8%D0%B0%D0%BD). Consequently we can write the functions a=a(x,y,z,t), b=b(x,y,z,t), c=c(x,y,z,t). Then we can substitute these ones to the Lagrange description which is Vx=Vx(a,b,c,t),  Vy=Vy(a,b,c,t), Vz=Vz(a,b,c,t). And we come to Euler`s description which is Vx= Vx(x,y,z,t), Vy= Vy(x,y,z,t), Vz=Vz(x,y,z,t). The all FM equations including Navier-Stokes one are written in Euler`s description of motion. And the FM experiments measure Euler velocities.

4. Based on the above I ask you for the following. Why does physicists use this continuum model for about 140 years? This model is unphysical! Maybe the problems with the existence of hundreds of the turbulence theories and their sickness to describe even the turbulent flow in the simplest channels (see my above messages) are in the application of this old fantastic model???

Dear Anatoly,

Let me slightly correct your understanding of continuum mechanics. For sure, for any length scale L, there is a time during which all the continuum particles (finite volumes) of size L consist of the same molecules. Otherwise, this violates the causality principle (molecules move at finite speeds). So, if we fix a time \tau, there is always a length L that all volumes ~L^3 consist of the same molecules for t < \tau. During this time, such a volume (ensemble of molecules) behaves like a single “particle”. We call this new entity by the continuum particle or fluid particle, or material element, or whatever.

Such particles don’t have to exist forever but only during this very short time \tau (lifetime). Something like 10^{-9} sec. After that time, we again decompose the fluid on a new set of fluid particles.

I don’t think that the Lagrangian approach is applicable to the above picture, only Eulerian, because the particles exist only during time \tau, and thus, the notion of the Lagrangian labels and trajectories is not applicable. I would like to emphasize that there is no need to require that the particles exist forever.

One of the misleading things in the continuum theories, like the Navier-Stokes for example, is that it is always said that the fluids are represented by a set of finite(!) volumes (continuum particles) but then the size of this volumes is ignored in the model, they become mathematical points of zero size, which is, of course, unphysical.

I think that a proper continuum theory should contain this non-zero length scale. Our model is an attempt to build such a theory. It contains a time scale which explicitly defines the scale of the fluid particles. Below this scale, the continuum description is not applicable.

Dear Ilya,

1. I can not understand your words "Lagrangian approach is not applicable in FM". FM can not exist without Lagrangian approach!! It is impossible.

2. Your time 10^-9 is so slow that you come to ideal fluid without viscosity, so you have Euler`s equations of FM. These equations seems to me are much more simple to solve. Let`s remember a fluid in rotating cylinder :))

Dear Anatoly,

If I understand it correctly, the problem that you are pointing out with respect to "continuum volumes consisting same molecules", can be treated through fluctuating hydrodynamics formulation (see Landau & Lifshitz fluid mechanics book). In this formalism, one can introduce the notion of self-diffusion through density fluctuations as well (see Hydrodynamic Fluctuations in Fluids and Fluid Mixtures by de Zarat Leira & Sengers).

Yet let me emphasize that I think the departure of fluctuating hydrodynamics from classical Navier-Stokes is only relevant as molecular scales become important. Therefore, I cant imagine that at high Reynolds number flow (where energy is contained in very large scales), you observe any deviation between Navier-Stokes and fluctuating hydrodynamics. Also, I dont agree with your statement "you can not model/predict the stress distribution along the wall in 2D step-wise channel for Re up to 1000000". It seems to me that you are mixing turbulence models and direct Navier-Stokes description.

Dear Anatoly,

1. "I can not understand your words "Lagrangian approach is not applicable in FM". FM can not exist without Lagrangian approach!! It is impossible."

In my last comment, I meant a different point. Of course, the Lagrangian description is widely used in the classical FM. But the Lagrangian approach implies that the continuum particles do not rearrange, i.e. they don’t change their neighbors because otherwise the Lagrangian coordinates are destroyed. While in my view, macroscopic flow in the context of finite-scale particles is the process of these particles rearrangements(!), i.e. particles do change neighbors. And this is the key, this is the essence of any flows. So, in my comment, I meant that I have no idea how to introduce Lagrangian coordinates in the situation when the continuum particles rearrange. But the good thing is that we don’t need the Lagrangian description for this. The Eulerian description works pretty well :)

2. "Your time 10^-9 is so slow that you come to ideal fluid without viscosity, so you have Euler`s equations of FM. These equations seems to me are much more simple to solve. Let`s remember a fluid in rotating cylinder :))" 

No-no, a short time doesn’t mean that the resistance of a fluid to shearing vanishes. The resistance is defined not only by the time of the particle interaction but also by the strength of this interaction. So, the fluid is indeed viscous. And also if you think that this time is small, you then need to have a reference time to compare with.

Dear Sirs,

Let us talk about experiments in FM. For example, the lagrangian particle, I mean a molecule, its macroscopic ordered and chaotic motion can be simultaneously seen by nuclear magnetic resonance (NMR, MRI) method. Did somebody CHECKED :) the FM solutions in lagrangian approach and NMR experiments. I know it is really possible.

Did anyone provide the references?

Dear Anatoly,

I still do not clearly understand what you are looking for.... concerning experimental visualization of molecule motion, it would be on the scale of 1-100 A, I doubt you can rigorously define a lagrangian law as dx/dt = u(x(t),t). On the other hane, the observation does not define yet a "model" for the fluid motion. What is more, we already addressed that the large part of the FM problems show characteristic scales below the fmp. Hypersonic flows, rarefied flows can be some example of the limits you can reach.

Dear Filippo,

It depends on the time scale and average macroscopic velocity. If it is more then 0.01 second and the velocity is more that 10 cm/s then a  molecule will ride at the distance:

1. More then r= v*t= 10*0.01=0.1 cm, due to macroscopic motion

2. More 0.01 cm , due to the self-diffusion.

Both processes can be seen, say in any modern NMR spectrometer or MR scanner for animals which equipped by self-diffusion and velocity measuring blocks, see NMR pioneering german (now not a german :)) company: bruker.com

You can solve numerically FM equations in lagrangian approach and compare with NMR experiments. It opens a possibility to modify FM equations. I just think..