Periodic Waves and Ligaments on the Surface of a Viscous Exponentially Stratified Fluid in a Uniform Gravity Field

Round 1

Reviewer 1 Report

Dear Authors,

The manuscript is a very good study on incorporating a density distribution for wave dispersion dynamics. The problem statement is presented in a manner to understand the approach and conveys the message of the improvement required in modeling.

If the below points are addressed in the revised manuscript, it will be ready for acceptance.

1) Please have a figure with the exponential density distribution to make it clear for the readers.

2) How does this compare with linear stratification?

3) Is it possible to have a plot of the velocity profile?

Author Response

Dear Reviewer,

Many thanks for interest in the work and remarks directed to clear the paper text.

Concerning indicated points

1) Please have a figure with the exponential density distribution to make it clear for the readers.

2) How does this compare with linear stratification?

Figure with exponential and linear density stratification is incorporated in the paper text. Some algebraic expressions compared linear and exponential stratification are presented as well.

3) Is it possible to have a plot of the velocity profile?

The paper content is limited by discussion of waves and ligaments dispersion relations. The calculations of physical quantities fields are in progress and will be properly presented in following paper draft.

Reviewer 2 Report

The paper is well-written and well-organized. Results and calculation steps are presented in sufficient details. Graphs are easy to read. Below are weaknesses that have to be addressed

11. On the title, use lower case for “The” and the two “A”s

22. In the Abstract, remove “methods of the” in the first sentence. Also, in the same sentence, replace “of the propagation” with “to study the propagation”

33.    Please define “ligaments” early in the Introduction

44. The introduction needs a lot of work, here are some examples:

a.       The first paragraph is very vague (not clear). Remove the clause “which significantly influence …. human life” – this is unnecessary.

b.       What does it mean by this sentence “when the first fundamental equation … “? If this meant to be historical, please lay out the events one by one.

c.       The second sentence is also not clear. Please remove or rewrite. I would suggest to break it down to two or three sentences.

d.       Remove “However” in the second paragraph

e.       Replace XVIII with 18^th century. Please elaborate the history here. It is not clear.

f.        On the 3^rd paragraph – what does “first publications” mean here ? Please totally revise the paragraph as it is now really confusing. What does “tractate” mean ?

g.       What does it mean by “was rediscovered at least twice” here ? This is very confusing.

h.       And much more …

So please get help from an English writer to assist you to rewrite the whole Introduction. The last half part of the Introduction is not that bad. It’s the first part that is seriously poor.

55.  In line 101, replace XIX with 19^th.

66.      In line 121, use lower case in “Open Ocean”

77.    In lines 136-138, please includes important results laid out in the paper [58]

88.    In the introduction, please add at least five citations that really can tell real-world applications of this project.

99.    At the end of Introduction, please summarize what you want to accomplish in the paper, how you would do that, and brief results.

110.   Your conclusion is too broad. You need to sharpen it.

Author Response

Dear Reviewer,

Many thanks for interest in the work and detail analysis of the paper text.

All your recommendations concerning the grammatical points are introduced in revised test.

11. On the title, use lower case for “The” and the two “A”s
12. In the Abstract, remove “methods of the” in the first sentence. Also, in the same sentence, replace “of the propagation” with “to study the propagation”
13.   Please define “ligaments” early in the Introduction

                  Ligaments are difined.

44. The introduction needs a lot of work, here are some examples:
45. The first paragraph is very vague (not clear). Remove the clause “which significantly influence …. human life” – this is unnecessary.

                  Corrected

1. What does it mean by this sentence “when the first fundamental equation … “? If this meant to be historical, please lay out the events one by one.

               Corrected

1. The second sentence is also not clear. Please remove or rewrite. I would suggest to break it down to two or three sentences.

Corrected

1. Remove “However” in the second paragraph

                  Removed

1. Replace XVIII with 18^thcentury. Please elaborate the history here. It is not clear.

Replaced

1. On the 3^rdparagraph – what does “first publications” mean here ? Please totally revise the paragraph as it is now really confusing. What does “tractate” mean ?

                  Corrected

1. What does it mean by “was rediscovered at least twice” here ? This is very confusing.

Authors did not mentioned the Rayleigh paper in their publications

1. And much more …

So please get help from an English writer to assist you to rewrite the whole Introduction. The last half part of the Introduction is not that bad. It’s the first part that is seriously poor.

55. In line 101, replace XIX with 19^th.

                  Corrected

66.     In line 121, use lower case in “Open Ocean”

                  Corrected

77.   In lines 136-138, please includes important results laid out in the paper [58]

                  Included

88.   In the introduction, please add at least five citations that really can tell real-world applications of this project.

`                 Up to our knowledge there are no applications at the moment

99.   At the end of Introduction, please summarize what you want to accomplish in the paper, how you would do that, and brief results.

                  Summary supplemented

110. Your conclusion is too broad. You need to sharpen it.

Experimental techniques must be developed to investigate properties of a real fluid flow with variable density. Approximation of constant density leads to degeneration of the 3D fundamental equations system (Axioms 2021, 10(4), 286. https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040286.

With best regards.

Yuli D. Chashechkin

Artem A. Otchirov

Reviewer 3 Report

1. This paper used a classical, applied-mathematical way of regular perturbation method to derive the modified dispersion relation between wave length and the frequency for a periodic flow in a viscous exponentially stratified fluid. From the reviewer observation, this kind of mathematical manipulation skill can be hardly found nowadays in the literature and this paper deserves for publication in Axioms.

2. Regarding the detailed derivation process, please mention more specifically what parts of the governing equations and the perturbation gauges (epsilon) were originally proposed by the authors or any related prior papers. Especially for the perturbation gauges, I'm curious about how the authors formulated them? 

3. If possible, please show the experimental evidences of the modified dispersion relations derived in this paper (to justify the data in Figures 1 and 2.) I also found the authors have published some perturbation papers in refs. [66,68,69] related with this paper. Please justify them more.

Author Response

Dear  Reviewer,

We sincerely thank you for your attention to the article and deep comments.

The question of justifying the choice of small parameters of asymptotic expansions  ("perturbation gauges") is very interesting. It can be answered simply: It is so happened, the choice is heuristic, followed by substitution and verification of adequacy.

But in fact, the choice is based on the logic of the transition from the physical model of the process to the mathematical one.

The physical model contains observable physical quantities (i.e., measured in the experiment with an error estimate) that are the frequency, length and amplitude of the wave, the density and magnitude of its gradient, momentum and pressure.

The mathematical model includes only quantities with the dimension of length and time or generally dimensionless. As a result of such a transition, “flow” (which is defined as the transfer of momentum, energy and matter with concomitant changes in the physical parameters of the liquid state) is transformed into “motion” (orthonormal transformation of the problem functional space into itself with the distance conservation) [1, 2].

Each impacting physical quantity in this approach corresponds to an additional dimension in the functional space of the problem and its own characteristic scale. All dimensionless parameters, traditional Reynolds, Froude numbers and others, as well as small parameters of the problem, are defined as ratios of proper scales, or combinations of scales, depending on the type of the governing equations.

Due to the smallness of some coefficients of the equations, the approach allows us to apply the theory of singular perturbations, which is done in this paper for description of surface waves and ligaments. As a result, a reduced system of differential equations was obtained, which is further linearized.

In this formulation, there is no energy and thermodynamics; as a reduced model is used. In the experiment, energy and thermodynamics play an important role, in particular, in droplet impact flows.

The second important element of the theory is explicit the application of the compatibility condition - the requirement to construct a complete solution. Such a procedure was carried out in the problems of generation and propagation of periodic internal wave beams and ligaments in the thickness of a stratified fluid [3].

At the level of our knowledge, a complete description of the constituents of surface periodic flows, including surface waves, is performed for the first time.

After some thought in preparing a response to your comments, we decided to confine ourselves to a statement of the calculations motivation in this message. The text of the manuscript indicates the works in which certain scales appeared and the new parts of the given dispersion relations are noted.

The main result of the work is the conclusion about the need to set up experiments on wave physics in a stratified basin and develop high-resolution tools for recording fields and local values of physical quantities - density, its gradient, momentum, wave parameters - length, frequency, shapes of crests and troughs. In a number of our works, it was previously noted that the fluid velocity is a mathematical concept and an experimentally unobservable quantity, since a «fluid particle» cannot be identified, and any introduced marker uncontrollably changes the flow properties and medium parameters [1]).

References

1. 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. 328 p. P. 61-91. DOI: 10.1007/978-3-319-23564-6-5.
2. Chashechkin Yu.D. Singularly perturbed components of flows – linear precursors of shock waves // Math. Model. Nat. Phenom. 2018. Vol. 13. No. 2. P. 1-29. https://0-doi-org.brum.beds.ac.uk/10.1051/mmnp/2018020
3. Chashechkin Y.D. Foundations of engineering mathematics applied for fluid flows // Axioms. 2021. V. 10. Iss.4 286. https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040286.

With best regards.

Yuli D. Chashechkin

Artem A. Otchirov

Reviewer 4 Report

Review of “Periodic Waves and Ligaments on The Surface of A Viscous Exponentially Stratified Fluid in A Uniform Gravity Field” by Yuli D. Chashechkin  and Artem A. Ochirov.

This paper provides a theoretical analysis of viscous surface waves on a density gradient, for a large range of frequencies. This is something that is important in a wide range of applications.

The background and methodology are both clearly set out and the results are presented and discussed thoroughly and also validated through comparison to known solutions in limiting cases.

This is one of the best papers I have reviewed in a long time and my only suggestion for improvements is that some of the figures (particularly figure 2 which contains a lot of information) are slightly hard to read and would be better presented as full-size figures with part a above part b.

Author Response

Dear Reviewer,

Many thanks for your attention to work and stimulating comments.

With best regards.

Yuli D. Chashechkin

Artem A. Otchirov