Transition to the Fluid Dynamic Limit: Mathematical Models and Simulation Results

Round 1

Reviewer 1 Report

In this manuscript, an improved mathematical framework (trace theory) is developed to illustrate the differences in the effects of linearized and non linearized Boltzmann operators, and BGK models on simulation results under fluid mechanics limits. However, there is no clear explanation of the advantages and disadvantages of this framework compared to other methods, such as Chapman-Enskog expansion method. It is recommended to summarize and explain the significance and results of this work at the end of the article.

Minor questions and suggestions:

1) In the Abstract, what is the BKG system?

2) For Figure 1, it is recommended to add variable descriptions corresponding to the coordinate axis to make the information clearer.

Author Response

I thank the referee for the careful reading, the hints and suggestions. In the new version, I have included the following modifications.

In order to resume the advantages and disadvantages of the various methods I have included Conclusions at the end of the paper.

I have included two new references explaining the BGK (Bhatnagar-Gross-Krook) system and its various versions.

I have changed the subtext of Fig.1.

Reviewer 2 Report

The current work is well preseneted ana the results and conclusion is also very well presentted. I have some remarkt to the work:

1.The references are not in correct order.

2. Please give the dimension of each parameter also explain all the parameter in the equations.

3. The figures have to be more clear for example fig. 5.

Author Response

I thank the referee for the careful reading, the hints and suggestions. In the new version, I have included the following modifications.

In order to resume the advantages and disadvantages of the various methods I have included Conclusions at the end of the paper.

I have included two new references explaining the BGK (Bhatnagar-Gross-Krook) system and its various versions.

I have reordered the reference list.

I have changed the subtext of Fig.1.

Concerning your further comments:

The parameters are all in dimensionless form as is standard in theoretical papers. To explain all their physical meanings would make the essential arguments much more intransparent. Therefore I hesitate to make these changes.

Unfortunately, the graphic software produced the figures in the given quality. I see no way to change this. However, all the results concern qualitative rather than quantitative aspects. These should be recognizable. E.g. Fig. 5 indicates the relaxation of the moment differences from the left (perturbation) to the right (equilibrium zone). However, the differences do not vanish. That is what should be demonstrated.

Reviewer 3 Report

Comments for author File: Comments.pdf

Author Response

I thank the referee for the careful reading, the hints and suggestions.

Concerning your major concerns:

-- Coincidence with reference [5]: The first version of the present work (written down in [5]) had the aim to develop a mathematical theory about the transition from gas kinetics to fluid mechanics, and to find acceptance among mathematicians and theoretically oriented physicists and engineers. As a confirmation of the underlying ideas we included the result presented in Fig. 2 which very well illuminates the difference to the classical theory. It proves the existence of two different solutions to the balance equation, one of which corresponds to standard theory but is unstable, while the stable one represents the new approach. Lea Bold, at that time one of my students, found out this result numerically. That's why she was invited as a coauthor. Meanwhile her scientific interest has changed and she is doing research in a different field.

The present paper addresses in first line people doing applications. To this aim the heat layer and the evaporation condensation problem for mixtures have been worked out. My first intent was to shorten the theoretical part (indeed sections 2 to 4 could be shortened). But in its present form the paper seems to me more self-contained and accessible for people from different scientific fields. If the referees agree, I would like to keep in as it is.

-- The main scope of the paper was to find a theoretical derivation of closure relations which avoid the mysteries (``ghost effect'') and some of the deficiencies of the standard theory. A comparison e.g. with experimental data is beyond the scope, since it requires a careful adjustment of the collision models to the experimental setting. A comparison with other fluid models (standard Navier-Stokes, BGK) is contained in the paper. Of course the present approach has higher computational costs than standard ones, since it requires the local evaluations of the balance equations which are formulated in phase space. However, this procedure becomes necessary only in exceptional situations e.g. close to interfaces (see Conclusions).

-- Equation (49) is indeed time-dependent. However, even for time dependent macroscopic moment equations the steady version of (49) makes sense, since the microscopic time scale $\tau$ is faster than the macroscopic time $t$.

-- I corrected all minor errors including wrong references to formulas as far as I found them. The plots are the output of my graphics software. At present I see no chance to change them.

Concerning your minor concerns:

-- We only use scalar products with weight 1. Terms of the form $e^{-v^2}$ enter through one of the terms within the brackets, e.g. terms in $e{\cal M}$. I did not understand your question about orthogonality in case of non-zero bulk velocity.

--The elements of ${\cal E}$ are given by the parameters $\rho$, $\overline{v}$ and $T$. You get the elements of the tangent space if you derive $e\in{\cal E}$ with respect to $\overline{v}$ and $T$.

-- Tangent space ``in $e$'' is correct. To each $e\in{\cal E}$ is attached a tangent space (given by the derivatives at point $e$) which goes through $e$.

-- The term $()_{\perp}$ is related to the decomposition at the beginning of the proof of Lemma (4.6). Even if ${\bf f}$ is an element of $e{\cal M}$, the derivative $(v\cdot \nabla_x {\bf f})$ contains terms orthogonal to $e{\cal M}$. These are the reasons, why closure relations have to be formulated.

-- $(0)$, and $(0,\overline{T})$ correspond to the parameters of $\hat{\pi}_x$ and $\hat{\sigma}_x$ like in formulas (27) and (30).

-- In (41) the time derivative vanishes, since all parts of $\partial_te$ are in $e\cdot{\cal M}$ and do not have an orthogonal part.

-- In line 222 I avoided the term manifold since this would introduce another mathematical technical term.

-- In (58), (59) I cannot find any change of notation.

-- Changing $\overline{v}_x$ only represents a shift in velocity space. For the calculation of the closure moments, this is of no relevance.

-- The formulas (79) and (80) concern equilibrium functions $e$. Formula (74) concerns an element of trace space (which is a small perturbation of $e$). Thus the right hand side does not cancel.

-- Formula below line 338: The $x$-derivative was a typo. As far as I see the definition of $\hat{\sigma}_x$ is as before.

-- I have done all the corrections you mentioned in your list.

In the new version, I have included the following modifications.

-- In order to meet some of the main concerns of the referees, I have included Conclusions at the end of the paper.

-- I have included two new references explaining the BGK (Bhatnagar-Gross-Krook) system and its various versions, and I have removed one of my own work [7].

-- I have reordered the reference list.

-- I have corrected minor errors and slightly changed some of the formulations as indicated by the referees.

Reviewer 4 Report

Dear Аuthor,
After reading your manuscript, I have the impression that you have submitted it to an inappropriate journal.
In my opinion, taking into account the specifics of this journal, the formalism you have developed must demonstrate the possibility of solving applied problems and show  its advantages over other approaches, for example, the use of Barnett's equations.
At the same time, I believe that your article is of unquestionable interest for a journal devoted to theoretical problems of kinetic theory.
I also think that sections 2 - 4 can be shortened without losing the meaning and clarity of the presentation.
As for specific comments, I would like to point out  excessive self-citation (almost 43%), unnecessary use of Germanisms in the English text, a typo (line 79), as well as, in my opinion, not quite good formulation “...the results of the above Chapman Enskog procedure above as well as the BGK model suggest a decomposition..." (lines 209 - 210).
Unfortunately, Section 6 (Appendices) contains no information about the results of the comparison of calculations with experimental data. And this does not allow us to judge the adequacy of the proposed approach.

I would advise to avoid unnecessary use of Germanisms in the English text.

Author Response

I thank the referee for the careful reading, the hints and suggestions.

Concerning your comments:

I do not have the impression to have chosen an inappriopriate journal. I did indeed show (in the case of the evaporation condensation problem of section 6.2) that the presented theory yields a solution where none of the others does (at least as far as I know). Furthermore, the new approach avoids some of the deficiencies of others like expansion techniques (see remark at the end of section 6.1) or relaxation schemes.

This technique seems to be superior to others in all cases where portions of flows (like outgoing / reflected flow, different components of gas mixtures etc.) have other bulk velocities or temperatures than others, mainly in the vicinity of interfaces. We have only started with applications, but we feel that these will provide a huge field of research.

The comparison with experimental data is far beyond the scope of the present paper, since this would require a careful adjustment of the collision models to the experimental setting.

For me as a german it is hard to recognize the germanisms in the text. I have changed the part you suggested.

In the new version, I have included the following modifications.

-- In order to meet some of the main concerns of the referees, I have included Conclusions at the end of the paper.

-- I have included two new references explaining the BGK (Bhatnagar-Gross-Krook) system and its various versions, and I have removed one of my own work [7].

-- I have reordered the reference list.

-- I have corrected minor errors and slightly changed some of the formulations as indicated by the referees.

Round 2

Reviewer 4 Report

Dear Author,

I believe that you have provided satisfactory explanations for my comments. The manuscript is now ready for publication in Fluids.

The manuscript is written at an acceptable level for my English