...consequently Navier-Stokes equations are e.g. not applicable to gaseous systems under extremely low pressure or where gas is as much diluted that the mean free path between molecules starts to get large (near vacuum conditions, high stratosphere conditions, plasma). So e.g. the atmospheric flow around a space shuttle during the re-entry from space into the atmosphere cannot be predicted by solving the N-S-equations.

Regards,

Dr. Th. Frank.

This also happens in very small pores (smaller than mean free path) at ambient pressure and can be noticed when measuring gas permeability in shale (pore throat size 100 nm or less). This is called the Klikenberg effect.

I think that the key to focus on the lack of application of the NSE is to understand the limits starting from the integral form of the conservation equation (mass, momentum, total energy). We get all first spatial derivatives disappear and the variables that still require differentiation (and hence require the continuum model) are due to the adoption of Newton and Fourier models (velocity gradient and temperature gradient). Therefore we can conclude that the lack of applicability is due not specifically to the general form of the NSE but to the constitutive laws we introduce to close the equations.

There are analytic solutions to compressible Navier-Stokes equations for shocks in ideal gases. These go back to Becker (1922) and Morduchow-Libby (1949) and predict the continuous shock structure. There are also experimental data for shock structure (Alsmeyer, Schmidt, many others). NS predicts shocks that are only half as wide as the measurements and differ significantly in shape. The disagreement is not related to Knudsen number, but rather to the failure of local thermodynamic equilibrium. Note, direct solution of the Boltzmann equation by DSMC agrees well with the experimental data.

Dear/Len G Margolin

Analytical solution of Compressible Navier-Stokes Equations gives inaccurate prediction compared with experimental measurements , however the approximate solution of N.S.(analytical) will lead to obtain the trend of experimental data. On the other hand, using Boltzman equation (DSMC) is more realistic and gives satisfactory agreement with measurements.

Regards

The Klikenberg Effect is also important when replacing oil by gas in lab permeability tests for reservoir characterisation. Apparently, liquids adhere to the pore walls while gas slips through them without much difficulty. Usually, lab permeability tests are completed at pressures well above the atmospheric.

Achmed, there's virtuyll a countable \infty of cases to which N-S is NOT applicable. One is actually not turbulence as such any more because I believe that we have today - in the footsteps of Kelvin, Fokker, Planck, Kolmogorov, Herrmann and myself - a complete statistical theory: AS LONG AS turbulence is in STEADY STATE, i.e. as long as the energy flow through the spectra is constant in time and neither accelerated or de-celerated. In my view this would be the next step.

Here is another area of science which seems to lack of accuracy in definitions.

Indeed, when fluid is a continuum? Very good question and should be grabbed through it's very fundamentals.

Strictly speaking I consider that if Kinetic gas theory is valid, then we cant have viscosity (resistance against deformation), and therefore we also can't have any continuity. -> NSE is not valid!

So maybe that what we consider as a viscosity in the gases, is actually only inertia.

If this is accepted, then we could ask the question where then lies the exact transition from viscous interactions to inertial interactions? If laminar flow is possible, then the viscosity is obviously present. And who have lived in cold climate must have experienced when air is flowing undoubtedly laminar.

This means that the viscosity is not lost in the phase-transition from liquid to gas.

Ok, I think this is as far as we can go with classical physics. Actually we can't even exactly define the difference of solids and liquids in classical physics. And because of critical point, this is actually true also for the difference between liquid and gas.

Here we need to take the step to modern physics; QED. Viscosity is obviously nothing else than electromagnetic interactions of atoms. If we think real solids and Newtonian fluid; their behavior is theoretically considered exactly the same. Just the value of the shear stress is different. This mean that in the modern physics point of view, they might be seen as the same, because it's only chrystal-structure (or the lack of it) which makes this value different. But it's obviously impossible for the single atom to know, if it's part of some structure or not. It's just doing it's own QED interactions with it's neighbor atoms and that's it.

When we think these interactions, we immediately notice that there is easy definition to viscosity; If the atoms are attracted (pull) to each other, there is viscosity. But if there is just propulsion (push) difference between the atoms, they belong to the kinetic gas theory.

So if we could find an exact definition to this change point from pull to push, that would be similarly the exact limit for NSE equations.

I once got an idea, I guessed a new physical law; Maybe it's the speed of light, which defines this limit. And I know this is difficult to think with all these Planck-constants etc. as we actually don't (officially) even truly know what they are and mean. What is gravity and all these black-holes mambo-jambo of "modern physics". So here i wan't to strictly stay on the "classical physics" and not start any conversation about some "string theory" on n-dimensions.

So let's just take this Idea with the speed of light an calculate some consequences. And then let's just compare these calculations to what we can observe in nature.

Working Paper STATES OF MATTER defined from The Speed of Light

And if the observations fits the theory, (as they do) then I am satisfied. The states of matter can be defined through the speed of light. And even though i don't know any person who could peer review this to me. It doesn't really matter. I peer reviewed it my self by looking another planets. And another changes, like that from gas to plasma.

and it was

BINGO

BINGO

BINGO

BINGO

... so I am pretty sure that in next decades this will get recognized, and I only need to wait for that to happen. Maybe someone tries to stole this idea, or maybe some founds my papers badly written. Well this doesn't really influence to anything which atoms do. We can't invent "physics", we can only try to understand it.

And this understanding is a private issue. Everyone is responsible for their own understanding.

I wish to you all an nice Sunday.

I will be busy in my everyday life in next weeks.

I would like to introduce the work of Yoshio Sone (Annual Review of Fluid Mechanics32 (2000) 779-81), which showed that even in the limit of vanishing Knudsen number, the Navier-Stokes equations are not complete in describing the gas dynamics in some cases.

Dear Dr. Ahmed Abdelhameed,

To my knowledge, the Navier-stokes (NS) and Fourier equation it not applicable in the nanoscale regime. To solve heat transfer problems in nanostructures, we usually refer to the Boltzmann transport equation and its derived models. The NS equation is available only for Kn=0,01, however for Kn=1 (quasi-ballistic regime) we cannot describe the nature of heat transport due to fluctuations and such perturbations inside the domain. As Knudsen increase (Kn=10), a ballistic regime predominate the heat conduction and a more discontinuities appears in the temperature profiles. In fact, we can introduce such enhancements in models transport like (phonon hydrodynamics model, Ballistic-Diffusive equation, Guyer-Krumhansl equation ….). Actually, those models can predict the heat transfer in micro and nanoscale regime. In addition, they are able to describe the scattering mechanism occurs in surfaces, interfaces and walls. As an efficient boundary condition, I think that the temperature-jump boundary condition can solve the problems of the non-continuous heat conduction. When the Mean Free Path (MFP) becomes comparable to the characteristic length, the gradient temperature decrease due to non-local effects that probably leads to this discontinuity in the heat flux and temperature behavior. For rarefied gas flow, I think that we can apply the Thermon gas model.

Best regards,

Houssem Rezgui

Dear/Houssem Rezgui

Thank you so much for valuable and constructive reply , but for some nano-fluid cases inside pipes the predicted results from N-S equations are very close to numerical results based on the Boltzmann transport equation.

Regards

Dear Dr. Ahmed Abdelhameed,

Maybe, in the case of Kn=0.1 NS and BTE can give the same results. Kn= 0.1 refer to a hydrodynamic level and we can apply both the BTE and NS equation also the Guyer-Krumhansl equation (GKE) can resolve the problem. Actually, we can use NS in this case of small Kn due to the dominance of normal (N) processes scattering. However, we cannot apply NS in the case of Kn>1 due to resistive (R) process. We should resolve the BTE under additional relaxation time to describe the perturbation, which must be influence the nanoscale heat transfer.

Houssem Rezgui It is a common mistake to think that small knudsen number is all that is required to derive NS from Boltzmann. In fact, the derivation also requires local thermodynamic equilibrium, which is an independent assumption.

Dear Prof. Len G Margolin,

Of course, we should refer to thermodynamics theories to study gas flow. Classical thermodynamics has been used for modeling a local equilibrium. Navier-stokes formalism cannot describe the heat transfer in nanoscale regime because NS predict only a local thermal transport. However, for Kn >1 we should include non-local effects occurs in nanosystems. For this reason, NS deemed to be unavailable for medium Knudsen number. As I denoted in previous comments, we refer to a derived transport models from the BTE. To discuss any transport model we should introduce such approximation because it is a difficult task to solve the BTE due to collision terms. Hence, solving the BTE need some boundary conditions like (velocity slip boundary, Temperature-jump,…). Knudsen number has a fundamental description in solving multiscale heat transfer problems:

Kn=0.001 : Continuum flows ( Classical Fourier equation)

0.01

It would be helpful if we can discern between the critique to the use of the general form of NS equations as deduced from the Reynolds theorem or to the specific models that are introduced to close the set of equations, that is Newton, Fourier, ideal gas and so on. ;-)

Generally if you have a flow with Kn number less than 0.001, you can consider it as continuum and you can use NS equations to predict the flow characteristics. If your Kn number is much larger than 0.001 (but still smaller than 0.1), NS equations are still valid but you need to modify your boundary condition, specifically you need to define a slip boundary conditions on the wall. We can say this theory works for transition regime (0.1

I agree well with your explanation Dr. Reza Rasooli . We, should introduce an efficient boundary condition. The important task is how we can relate such computational model with slip boundary condition. For this reason, I insist to consider the Knudsen scale.

Hi Tapan -- The Chapman-Enskog approximation is a perturbation expansion of the equilibrium particle velocity distribution. So the essential starting point is that the zero order solution is the Maxwell-Boltzmann (Gaussian) distribution. This is independent of Knudsen number, in fact is an assumption about the size of the gradients. One way to think about this is to consider the next order of expansion beyond NS (i.e., the Burnett equations) and ask what assumptions are required to rederive NS from Burnett. In the Burnett regime, quantities like heat flux obey their own PDEs. To get back to NS, one must set the time derivatives of those higher moment equations to zero which is, more precisely speaking, the definition of local thermodynamic equilibrium. Kremer's book has a very nice discussion of this point (GM Kremer, Introduction to the Boltzmann Equation and Transport Processes in Gases).

An even more fundamental viewpoint is to recall that the solution of the Boltzmann equation lives in a 6-dimensional space. Chapman Enskog only does its averaging in the 3-dimensional velocity space. When one averages also in the coordinate space, you derive new sources of dissipation, what the turbulence people have called anomalous dissipation, which are independent of viscosity and heat conduction.

Physicists who are interested in the "failures of Navier--Stokes" should read some of the fundamental papers of Becker (1922), Onsager, and Kolmogorov. In general, NS fails for high Reynolds number flows for reasons other than Knudsen number. I have also written several papers on this subject which I would be pleased to send to you.

Dear Ahmed: I would prefer to turn the question around. I.e. which flow conditions are - more or less in a precise sense - correctly described by NSE?

Or do you have a very specific reason to pose your question the way you did it initially? Teaching, or problem solving?

I kind of agree with prof @Len G Margolin. The assumption of local thermodynamic equilibrium is a key assumption if one wants to derive the Navier-Stokes and Fourier's (NSF) laws. For example, we assume that the thermodynamic variables (energy, entropy, enthalpy, Gibbs equation, etc) depends 'only' on the equilibrium variables.

For micro/nano flows this assumption is usually not valid hence the NSF model looses its validity. Other physical situations can also lead to the violation of local equilibrium assumptions--such as strong shocks (high Mach number flows) --where very strong gradients appears. Now the question arises, where the Knudsen number entres in such problems? It is a number which relates microscopic (length or may be time?) scales to the macroscopic physical length scales. If one comes from a statistical mechanics point of view (Boltzmann equation for example), in particular for gases, the choice for microscopic scales is clear (mean free path or time) and one can do a formal asymptomatic expansion (Chapman Enskog type) to drive the NSF equations. Many people has shown how local equilibrium assumption relates back to the Knudsen number, Chapman, Truesdell, Muller, to name the few.

My concern (hence a question) is: as we know now (if Molecular simulations to be believed) that for nano-confined liquids, the NSF equations are not valid. But there is no equivalence of mean free path in liquids (even if one insists defining it, it would be in angstroms). Hence there is no clear scale separation, so why does NSF model fails? And moreover, is there any master equation (equivalent to the Boltzmann equation) for liquids. Does Enskog-Vlasov model a good model for (all) liquids?

Dear/Helmut

Thank you so much for your concern, as a matter of fact my question aims to focus on the ability of NSE to specify most of fluid problems based on that NSE comes from real theory so it is able to deal with flow applications.

Regards

Sergey Popov

Could you elaborate further what do you mean posting this plot? I suppose is an experimental measurement of velocity and static pressure (thermodinamic) in time. But I suppose that this is still a flow problem that has a statistical energy equilibrium.

@Anirudh Rana I am curious about your references to Chapman, Truesdell and Muller. The Boltzmann equation has an advective term as well as the collision integral. The Knudsen number is meant to guarantee that the "system" has sufficient collisions to tend to equilibrium and so is based on the collision term. However it is the advective term that drives the system away from equilibrium. The ratio of these terms at the kinetic level is very similar to the Reynolds number in continuum flow. On length scales where the advective term dominates, equilibrium cannot be reached.

So to summarize, collisions put a lower limit on system size while advection puts an upper limit on system size. In some cases, there is no realizable system size that obeys both constraints. That is when LTE fails (cannot get close to Maxwell Boltzmann statistics).

To my knowledge, neither Chapman, Truesdell nor Muller are concerned with the effects of advection. Perhaps you know of references that I am unaware of? (Note that I refer to system size rather than the undefined "macroscopic length.")

Privyet, Sergey Popov:

Possibly you may have a look at my "Universal constants and ... " (at ReseachGate). There is a point on Karman at plane boudaries, a parameter-free theory gives 1/sqrt(2*Pi) ~ 0.399. This is for PLANE surfaces. But what about curved surfaces like in "your" NACA0012, or in Princeton's superpipe? It is worth discussing.

Rarefied gas dynamics is one of the field in fluid mechanics where continuum approximation breaks and we cannot use naiver stokes equation.

There is no good NS solutions for transition regime flow....

Dear/Nazaruddin

Transition regime flow solution using NSE is based on how to deal with grid generation and specification of the boundary conditions precisely. However, Transition zone Reynolds number is between (2000 -3000) it means you need to determine the right Reynolds number using some experimental measurements as a guide.

Warm Regards

Dear Ahmed Abdelhameed ,

The Navier-Stokes equation, in its original form, is unsolvable analytically. So, there is no problem can be solved by this equation. We can solve it by using several assumptions including the numerical approaches. The problem when we use the modified/simple N-S equation is its accuracy. We can use this equation for any problem in fluid dynamics, but the accuracy might be very low.

Thank you.

Have a nice time.

Best regards.

Dear Nazaruddin Sinaga -- there is indeed one very famous analytic solution to the Navier--Stokes equations. In 1922, Becker found an exact solution of NSE for the finite structure of a shock wave in a perfect gas. It was only 5 years after the publication of the Chapman--Enskog approximation, which derived the relation between the (macroscopic) viscosity and the microscopic mean free path. Becker realized that his solution showed the shock width predicted by NSE would be only about one mean free path, implying that the Knudsen number was of order 1. Thus would of course invalidate the validity of the Chapman--Enskog perturbation expansion.

In the 1960s, experiments showed that shocks in argon were in fact nearly twice as wide as NSE predicts. The Knudsen number in these shocks is less than 1/3, which should allow a valid expansion. Furthermore, calculations with higher order rarefied gas models (Burnett, super-Burnett) show little improvement. The issue is not the accuracy of numerical approaches, nor of any simplification of NSE. The problem is one of physics, in particular that the Chapman-Enskog approximation supposes local thermodynamic equilibrium (LTE) which is unjustified in a shock. And BTW, numerical solutions of the Boltzmann equation very accurately predict the width and shape of measured shocks.

In the transition regions between the laminar and turbulent flow region, the predicted of the flow behaviour is very difficult.

Since the direct solution of NSE is out of hand, it is very hard to say precisely where it cannot work, because still the solution for a myriad of applications is not known completely such as transition regions with high pressure or velocity gradients, separation of flow, turbulent layer in boundary regions and etc... as well as that the basic assumption is held for continuum state, so if it does not the case NSE might not end up with the solution.