Navier–Stokes Equations and Bulk Viscosity for a Polyatomic Gas with Temperature-Dependent Specific Heats

Round 1

Reviewer 1 Report

The paper is well written and can be published.

Author Response

We wish to thank the reviewer for the very supportive comment.

Reviewer 2 Report

Review Report

Journal: Fluids

Authors: Shingo Kosuge and Kazuo Aoki

Title: Navier–Stokes Equations and Bulk Viscosity for a Polyatomic Gas with Temperature-dependent Specific Heats

Short overview of the paper

The manuscript is devoted to the study of the shock wave structure problem by comparison of two different models for polyatomic gases, ordinary Navier–Stokes equations with the bulk viscosity and Navier–Stokes-type equations with two temperatures, in the case of temperature dependent specific heats. Specific heat at constant volume cv(T) is appropriately extracted from experimental data for specific heat at constant pressure cp(T) and included in the model. Comparison was made and presented for different values of Mach number in front of the shock, with detailed explanations about applicability of compared mathematical models. Different types of possible solution were discussed.

General comment: Generally speaking I have positive opinion about present paper. In my opinion the paper is interesting and well structured. I noticed only few minor issues listed below which I expect to be addressed by the authors.

Minor issues:

Suggestion: I’m not in favor of putting the references of previous work just at the beginning of the abstract!

Keywords are missing!

Equation 4e: Why using two different symbols for degrees of freedom (delta and D)?

Line 130: H-theorem instead H theorem.

Line 248: What do you mean by: “origin X1 = 0 is set appropriately depending on the actual profiles”..?

Line 317: Discussing about type C solutions, are the authors aware of the paper about different types of the shock structure solutions in the case of multi-temperature polyatomic gases?: 

“Madjarević, D.; Pavić-Čolić, M.; Simić, S. Shock Structure and Relaxation in the Multi-Component Mixture of Euler Fluids. Symmetry 2021, 13, 955. https://doi.org/10.3390/sym13060955”

In this paper, type C solution is also spotted for even for weak shocks. Maybe its worth to mention this paper in the context of your research?

Check green color in graphs. Symbols and lines are not exactly in the same green color.

Line 318: Lines of text are condensed probably due to some technical error.

Final comment:

In conclusion, I think that the paper could be accepted for publication after the author has carried out the above issues and suggestions.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The manuscript is well presented and covers a well-considered investigation. The work clearly builds on and extends previous work by the authors and collaborators. The results obtained from the two-temperature NS equations, derived from the ES model with temperature-dependent Cp and Cv are interesting and support conclusions presented.

The manuscript can and needs to be improved in a few areas:

1]  To provide some more context, additional references to work covering multi-temperature NS models are needed. For example, the 'classical' multi-temperature NS-based CFD works in non-equilibrium hypersonic modelling can be mentioned. Even if these models were not formally derived from kinetic models. Furthermore, there are further recent works where multi-temperature models based on polyatomic ES models were derived, that are not included in references.

2] Below Equation (20) on page 6, where the uniqueness for Tint <= Ttr is mentioned - it would be really helpful to add further detail. For example, for Ttr > Tint (e.g. in strong expansions) what is the consequence for a non-unique solution of Equation (19)?

3] For the used two-temperature model, internal models are considered with a single Tint. For context, it would be helpful to add a short discussion about rotational and vibrational models and their role at different temperatures for different gases

Minor comments:

- key words are missing

- abstract is difficult to read due to the multiple references (in a rather long format)

Author Response

Please see the attachment.

Author Response File: Author Response.pdf