The Knudsen number is defined via the mean free path l_fp. In liquids the mean free path hardly can be introduced. Though liquids differ greatly from the solids, the marked molecule in liquid actually is moving under the impact of the self-consistent-field caused by the other molecules. Introducing the concept of the quasi-particles we can introduce the mean free path for those quasi-particles. The last is of the order of the mean distance between the molecules. Knudsen number as Kn_p=l_fp/D, where D is the nanoparticle diameter formally can be introduced, but how can it be used for the continuous model justification? The Knudsen number arises as a dimensionless parameter in the kinetic equation. Small Knudsen numbers correspond to the continuous model, even taking into account, that for the dense gas, all the more for the liquids, much more complicated kinetic equation compared to the Boltzmann equation should be used. That parameter has no practical use without considering of any type of the kinetic equation. For the nanoparticles Kn_p may not be small, so I see no application field for such parameter, specially taking into consideration the long-range interaction of the particles via the liquid.

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Dear Omar Z. Sharaf

The mean free path can be defined for any flowing medium.

The density of liquids and solids, in other words the mean molecular distances at these two phases do not differ much from each other.

The mean free path of gases can be defined from the Kinetic theory of gases, while the mean free path of liquids mainly depends on their structure factor.

The answer of your question is Yes, you can use Knudsen number for evaluating the continuum assumption, however, I have not seen any reference to assess the validity range of continuum assumption for liquid flows.

As you know, the criterion of Kn

I think it depends on your nanoparticles size range, thus if your Knudsen number is relatively small then the assumption of continuum is valid but if it's close to or greater than 1, you can't assume continuum for your study.

Omar Z. Sharaf , I am not a theoretician, in our joint paper, Dr. Baumert is the one who can theoretically formulate what has been experimentally found by me and my coworkers first. Anyway, I try to send some light and hope it will be somewhat enlighting:

In nanoparticle dispersions (you wrote "suspended" which is not the same as a real colloidal dispersion, but I assume you are referring to the same type of systems), you have 2 continuous phases,

- one is the dispersion medium, a liquid

- the other one are the dispersed nanoparticles which are NOT completely separated from each other, these are NOT statistically evenly distributed in the liquid, but they are forming 3-dimensionally structured elongated and branched filaments and ultra-thin layers,

while these dispersed particles have at least one strongly adsorbed layer of the dispersion liquid molecules, which layer is connecting 2 neighbouring dispersed particles so that they are forming a chain (let's say: a pearly chain.

Although the following paper is dealing with dispersions in thermoplastic polymers, the basics are the same for dispersions which are liquids at ambient temperature conditions:

Article Dispersion hypothesis and non-equilibrium thermodynamics: ke...

I hope this may help.

Dear Omar,

From physical point of view, in a dense liquid the mean free path is nonsense. the reason is that every molecule is under a force field from another molecules all time of its existence.

Therefore to my taste the Knudcen number is useless.

You can use the average distance between nanoparticles D and the diameter nanoparticles d , then some criterion D/d appears.

Good luck

Sergey

Tapan K. Sengupta

You cannot define mean free path for solids, however, it can be defined for liquids where collisions also occur.

I agree with you that as the medium gets denser the definition of mean free path become nonsense.

https://en.wikiversity.org/wiki/Microfluid_Mechanics/Models_for_Gases_and_Liquids

Dear Sergey Fisenko

I think the nanoparticles diameters should be compared with the mean molecular spacing of the liquid, not the average distance between the nanoparticles.

What do you think?

Thank you Dr. Tapan K. Sengupta

But I do agree with Dr. Morteza Anbarsooz--the situation for liquids and solids is not the same.

Although the dynamics of a liquid molecule will be dominated by its potential interactions with neighboring molecules, there is a MFP between molecular collisions (on the order of 0.2 Angstroms for liquid water).

In fact, you can also define a MFP for photons and charge carriers. Basically, any moving particle can have a MFP, as far as I know. Solids do not have moving particles, whereas liquids do.

Hi Omar,

this is a really interesting topic. Generally, a system of nanoparticles suspended in a fluid forms a nanofluid system. In this case, the whole system can be treated as a two-phase model (mixture granular model). I have read some numerical applications in this field, specifically of a liquid (water-Al2O3-Alumina moving at Re = 1050) and solid dispersed particles having 100nm in diameter. The model takes into account the collision liquid-solid and solid-solid: the granular viscosity accounts the shear viscosity arising from particle momentum exchange due to translation and collision. Another model you could use is the Eulerian multiphase model. I agree with Morteza Anbarsooz about the possibity to use the Knudsen number in your case. If you isolate the solid-solid interaction you could estimate the free path of the particles using the Knudsen number and the mean molecular spacing of the liquid as reference length.

Regards

Dear Prof. Tapan K. Sengupta

I see your point.

However, for liquid molecule collisions with each other, you can find that the MFP is indeed defined. This, for example, could be found in any textbook on microfluidics (e.g., Micro- and Nanoscale Fluid Mechanics by Kirby).

I recently also found the definition of Knudsen number for particulate flows in Crowe's classical two-phase flow handbook and in Adhesive Particle Flow by Marshall, defined as the ratio between MFP of the fluid molecules and the solid particles diameter as Dr. Morteza Anbarsooz has mentioned. It is stated in these references that the fluid (whether gas or liquid) around particles can indeed be treated as a continuum provided that Knudsen number is

Under laminar conditions, Knudsen is generally a first guess. A higher-order guess would be to look at the length and duration of nano-chain formation etc. This is MUCH different under turbulent flow conds. Then we may get very complex reaction-diffusion systems, WITHIN and BELOW the level of the macroscopic turbulent scale. This real stuff. Nor all, but VERY MUCH depends upon details of your problem.

The Knudsen number is defined via the mean free path l_fp. In liquids the mean free path hardly can be introduced. Though liquids differ greatly from the solids, the marked molecule in liquid actually is moving under the impact of the self-consistent-field caused by the other molecules. Introducing the concept of the quasi-particles we can introduce the mean free path for those quasi-particles. The last is of the order of the mean distance between the molecules. Knudsen number as Kn_p=l_fp/D, where D is the nanoparticle diameter formally can be introduced, but how can it be used for the continuous model justification? The Knudsen number arises as a dimensionless parameter in the kinetic equation. Small Knudsen numbers correspond to the continuous model, even taking into account, that for the dense gas, all the more for the liquids, much more complicated kinetic equation compared to the Boltzmann equation should be used. That parameter has no practical use without considering of any type of the kinetic equation. For the nanoparticles Kn_p may not be small, so I see no application field for such parameter, specially taking into consideration the long-range interaction of the particles via the liquid.

Thank you very much Prof. Yuriy Gorbachev , Prof. Tapan K. Sengupta

I came out of this post with much more insight and appreciation for the subject matter.

' In the laminar flow of nanoparticles suspended in a liquid, can we use Knudsen number to show that the liquid phase can be treated as a continuum? '

Yes, if the total amount of nanoparticles is significantly less than the liquid phase.

The question ' If so, how can we define the Knudsen number for solid nanoparticles suspended in a liquid carrier? ' is incorrect. You do not take the laminar nanoparticle flow . If you used the laminar nanoparticle flow .... or suspension liquid in laminar nanoparticle flow. Then your question would be correct.

Some of your theoretical considerations (and words) I do not really understand, but partially, I get the feeling that you do not really (not practically) *know* how such a system behaves and is structured.

Dmitry Tsyganov wrote "... if the total amount of nanoparticles is significantly less than the liquid phase" - if you are really dealing with *NANO*particles in a liquid, then you will not be able to *disperse* more than 0.5 to 1.5% of such small particles in a liquid.

Why? because every particles becomes *dispersed* by formation of adsorbed layers of the dispersion liquid which takes liquid molecules out of the liquid and they become definitely fixed, immobile.

So, the a.m. "if" consideration is practically not to be considered.

tbc (2)

(2) Omar Zharaf asked "... flow of nanoparticles suspended in a liquid ..."

I already wrote in my first answer "suspended" is a wrong term (unless you have aggregated nanoparticles which would anyway settle, that's not what you are looking at).

Suspension is a system like sand particles in water.

Nanoparticles in a liquid is a colloidal system, the nanoparticles are *dispersed*, and that is accompanied by adsorption of liquid molecules, a huge amount of those molecules become immobile.

The wording "laminar flow of nanoparticles in a liquid" is also probably a wording which indicates a poor understanding of what is really happening in a liquid: the particles as such do not flow, neither in laminar nor in turbulent mode. The total liquid system, the colloidal dispersion as a whole is flowing, and the dispersed nanoparticles with it.

tbc (3)

Dear prof. Bernhard Wessling , I used to think more widely. For me, there may be a liquid (may be gas ) from ONLY nanoparticles.

(3) The nanoparticles - after dispersion is completed - will form complex 3-dimensional structures, like branched pearl chains, seams, membranes ... These structures are governing the flow properties, the colloidal liquid's viscosity.

Above, Vincenzo Sammaranto mentioned " collision liquid-solid and solid-solid ": the solid nanoparticles do not expose their surface any more! their surface is covered by an AT LEAST monomolecular liquid molecules layer which has a kind of glassy status (not liquid, not solid). So there is no "liquid-solid" and especially not "solid-solid" collision to be considered.

Also, it needs to be taken into account, that the pearl chain structures are quite rigid, the cohesion forces are really strong, otherwise we would always observe a break-down of the viscosity under high shear which is not the case.

Dmitry L. Tsyganov frankly spoken, I do not understand in your last comment what a liquid (maybe gas) consisting only of nanoparticles could be and why this has something to do with the starting question about "nanoparticles in a liquid".

Dear prof. Bernhard Wessling, Have you ever stimulated gas as the hard-sphere spheres? Replace hard-sphere spheres on nanoparticles and you get gas from the nanoparticles.

I think the concept of quasi-PARICLES, as mentioned by Yuriy Gorbachev, points into the right direction. However, things are often much more complex as explained by Bernhard Wessling. I.e. even the concept of a PARTICLE needs possibly to be replaced here. Besides its practical aspects, it would be a nice and challenging play field for theoreticians.

As long as the flow is always and everywhere laminar, then FINE, but if turbulence cannot be excluded withint the process chain, then real work is ahead ...

Summarizing: An problem-specific analogy of Knudsen cannot be excluded, but there are deeper questions to be possibly answered, in particular if the media are reactive.

Dmitry L. Tsyganov Sure, I can look at a gas as if the molecules were hard, but

a) a molecule is NOT a PARTICLE, a particle behaves much more dfferent than a single molecule; you can not compare a single gas molecule (atom) with a particle which has a surface, surface tension etc - all of this a gas atom / molecule does not have

b) a gas is not a dispersion, and not liquid in which nanoparticles had been dispersed.

Thank you very much, Prof. Bernhard Wessling for the clarifications.

I changed the wording of the question according to your corrections.

I did mean that nanoparticles are "dispersed" in the liquid host forming a colloidal dispersion. To avoid any ambiguity, I rephrased the question as "flow of a nanofluid".

I also meant the flow of the liquid carrying the colloidal nanoparticles, not flow of the nanoparticles alone. So my wording was poor, indeed.

It might be worth mentioning; however, that in some other two-phase flow modeling approaches, such as the Eulerian-Eulerian two-fluid model, nanoparticles are themselves treated from a continuum perspective. So saying something like "flow of the nanoparticle phase" is common, which raises even more questions in my opinion.

Omar Z. Sharaf Thanks for considering my input, however according to my knowledge, the term "nanofluid" is not appropriate; I would reword it like this:

" In the laminar flow of a colloidal dispersion (i.e., nanoparticles dispersed in a liquid) ..."

I think no. It is more logical to consider the ratio of the minimum thickness of the liquid layer (for example, the minimum diameter of the pore cross section in a porous medium) to the diameter of the liquid molecule.

Best Regards,

Mikhail Markov

Perhaps the criterion is associated with the possibility of neglecting molecular forces, for example, the disjoining pressure. That is, if the thickness of the fluid layer is greater than the characteristic distance at which the molecular forces act. Maybe it is an equivalent of small Knudsen numbers flow for gases?

I agree with other answers that a Knudsen number for molecules in a liquid, has no physical meaning. However if you consider a nano-particle suspended in a liquid, the question could be: "When can you consider the liquid surrounding a nano-particle to be a continuum?". If this is the question I guess that the ratio of the liquid molecule diameter to nano-particle diameter can be considered as a kind of "Kundsen" number. This ratio can be considered as a measure of the thermal fluctuations acting on the nano-particle. Thermal fluctuations should scale with the square-root of the number of collisions between liquid molecules and the nano-particle. If this what you are interested in, you should read the literature on Brownian motion.