Viscous froth model applied to the motion and topological transformations of two-dimensional bubbles in a channel: three-bubble case

The viscous froth model is used to predict rheological behaviour of a two-dimensional (2D) liquid-foam system. The model incorporates three physical phenomena: the viscous drag force, the pressure difference across foam films and the surface tension acting along them with curvature. In the so-called infinite staircase structure, the system does not undergo topological bubble neighbour-exchange transformations for any imposed driving back pressure. Bubbles then flow out of the channel of transport in the same order in which they entered it. By contrast, in a simple single bubble staircase or so-called lens system, topological transformations do occur for high enough imposed back pressures. The three-bubble case interpolates between the infinite staircase and simple staircase/lens. To determine at which driving pressures and at which velocities topological transformations might occur, and how the bubble areas influence their occurrence, steady-state propagating three-bubble solutions are obtained for a range of bubble sizes and imposed back pressures. As an imposed back pressure increases quasi-statically from equilibrium, complex dynamics are exhibited as the systems undergo either topological transformations, reach saddle-node bifurcation points, or asymptote to a geometrically invariant structure which ceases to change as the back pressure is further increased.

The authors delineate the different bubble geometries and possible topological changes as the driving pressure slowly rises. It is undoubtedly careful painstaking work -almost 30 pages of description and a further 37 pages of supplemental material (that I didn't read carefully) to back this up -and so further discussion of the implications of the results would be welcome in the conclusions section to justify such an extensive manuscript.
Further comments on manuscript length/content: Numerical method: the numerics are only referred to obliquely in the main body of the manuscript and delegated to the supplemental material. I think it would help the reader to understand a bit more where and why numerics are required, and how they are implemented, although I agree with the authors that the supplemental material is the right place for the details.
Film coordinates: The other topic that struck me as mysterious without having read the supplemental material is the use of film coordinates (x_{ij}, y_{ij}). I think this need to be defined in the text.
In contrast, I felt that much of section 2(c) was repetition, and by the sixth page of the results (section 3(b)) I was desperate to know what the implications are for real systems, and wonder if the rest of the results (3(b) -3(d)) could go in the supplemental material. Section 3(c) in particular seems not to add very much.
Other than this, I have only minor comments: Abstract: the punctuation in the abstract needs careful revision (and indeed there are many parts of the manuscript which would be easier to read with elimination/redistribution of commas and greater use of hyphens) Section 1(c): could the authors explain what they mean by "quasi(-)statically" changing the driving pressure? Presumably that the changes are slow compared to the relaxation time of the structure to a new steady state? Figure 2: the arrows for labels might give the impression that they indicate movement; the authors might wish to re-draw.
Section 1(d): I think it might help to emphasise that it is not necessarily the bubble size A that is important, but its relation to the channel width A/L^2. At the start of section 2, please note that the symmetry is only the case for p_b = v = 0.
Upper and lower, upwards and downwards: Perhaps it would disentangle some of the sentences (e.g. in section 1(c)) if it was made clear that "lower wall" refers to the one to which the spanning film is attached. Does s_{ij} grow "downwards" (section 2(a)) for films _{12} and _{23} for any p_b, or can the orientation of these films change?
Section 2(d): two methods of parametrizing the system are described, but without explaining their benefits. This only comes much later. Why not help the reader here?
Geometrically-invariant states: Is it possible to show examples of the bubble shape for these? The introduction suggests infinite staircases, as in ref [16], but that is presumably not relevant here. At the end of page 13 it is stated that these "might be reached"; is there any reason why the authors can not be definitive here? Figure 7: the authors state that there is "a reasonable correlation" between figs 7(a) and (b). This is certainly not the case at first glance, although if by correlation the authors mean that the number of regions is the same then I might find this acceptable. I wonder if light shading could be used to indicate the correspondence that is meant? I think it would help to specify in the caption that 7(b) is the result of increasing p_b slowly from zero (if I'm correct).
for publication in this journal. (1) No experimental counterpart to these results is reported, yet the experiments are not very demanding? This is important, as the viscous froth model is clearly an idealisation, with which any given foam may or may not comply. It is always unfair in such a case to say "Go do the experiments.", but at least their prospects and the issues at stake could be addressed?
Without experiments it is difficult to assess what is significant/realistic in the very detailed blow-by-blow accounts of these simulations. It would be much better if they could be presented in a much more condensed form, highlighting that main results, ie, "cutting to the chase". Many details could be relegated to appendices, supplementary material or archive. (3) Personally I would question whether " it is possible to capture the rich properties of liquid foams by using a 2D model known as the viscous froth model" . On the face of it, this is claiming too much. The origin of the viscous effects that lie in at the heart of this model is the drag exerted by the two plates. There is no obvious counterpart in a typical 3d foam: viscous effects must arise otherwise and are likely to be very different in their effects. So I would recommend that less be claimed for the general significance of this work.

06-Oct-2021
Dear Dr Torres Ulloa The Editor of Proceedings A has now received comments from referees on the above paper and would like you to revise it in accordance with their suggestions which can be found below (not including confidential reports to the Editor).
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When revising your paper please ensure that it remains under 28 pages long. In addition, any pages over 20 will be subject to a charge (£150 + VAT (where applicable) per page). Your paper has been ESTIMATED to be 27 pages. Comments to the Author(s) This is an impressive piece of work to catalogue and explain the possible outcomes when a system of three bubbles is driven along a straight channel in a Hele-Shaw cell. It has immediate applications in foam microfluidics (having more bubbles reduces the possibility of changes to the structure during flow) and possible, but perhaps currently more nebulous, applications in oil recovery and soil remediation.
The authors delineate the different bubble geometries and possible topological changes as the driving pressure slowly rises. It is undoubtedly careful painstaking work -almost 30 pages of description and a further 37 pages of supplemental material (that I didn't read carefully) to back this up -and so further discussion of the implications of the results would be welcome in the conclusions section to justify such an extensive manuscript.
Further comments on manuscript length/content: Numerical method: the numerics are only referred to obliquely in the main body of the manuscript and delegated to the supplemental material. I think it would help the reader to understand a bit more where and why numerics are required, and how they are implemented, although I agree with the authors that the supplemental material is the right place for the details.
Film coordinates: The other topic that struck me as mysterious without having read the supplemental material is the use of film coordinates (x_{ij}, y_{ij}). I think this need to be defined in the text.
In contrast, I felt that much of section 2(c) was repetition, and by the sixth page of the results (section 3(b)) I was desperate to know what the implications are for real systems, and wonder if the rest of the results (3(b) -3(d)) could go in the supplemental material. Section 3(c) in particular seems not to add very much.
Other than this, I have only minor comments: Abstract: the punctuation in the abstract needs careful revision (and indeed there are many parts of the manuscript which would be easier to read with elimination/redistribution of commas and greater use of hyphens) Section 1(c): could the authors explain what they mean by "quasi(-)statically" changing the driving pressure? Presumably that the changes are slow compared to the relaxation time of the structure to a new steady state? Figure 2: the arrows for labels might give the impression that they indicate movement; the authors might wish to re-draw.
Section 1(d): I think it might help to emphasise that it is not necessarily the bubble size A that is important, but its relation to the channel width A/L^2. At the start of section 2, please note that the symmetry is only the case for p_b = v = 0.
Upper and lower, upwards and downwards: Perhaps it would disentangle some of the sentences (e.g. in section 1(c)) if it was made clear that "lower wall" refers to the one to which the spanning film is attached. Does s_{ij} grow "downwards" (section 2(a)) for films _{12} and _{23} for any p_b, or can the orientation of these films change?
Section 2(d): two methods of parametrizing the system are described, but without explaining their benefits. This only comes much later. Why not help the reader here?
Geometrically-invariant states: Is it possible to show examples of the bubble shape for these? The introduction suggests infinite staircases, as in ref [16], but that is presumably not relevant here. At the end of page 13 it is stated that these "might be reached"; is there any reason why the authors can not be definitive here? Figure 7: the authors state that there is "a reasonable correlation" between figs 7(a) and (b). This is certainly not the case at first glance, although if by correlation the authors mean that the number of regions is the same then I might find this acceptable. I wonder if light shading could be used to indicate the correspondence that is meant? I think it would help to specify in the caption that 7(b) is the result of increasing p_b slowly from zero (if I'm correct).
Referee: 2 Comments to the Author(s) Report This paper presents simulations of a 2d soap froth in steady motion., for various cases. It uses the so-called 2D viscous froth model. The work has been diligently compiled and presented. I have a number of reservations, which the authors might consider. If adequately implemented, they should make the paper acceptable for publication in this journal.
(1) No experimental counterpart to these results is reported, yet the experiments are not very demanding? This is important, as the viscous froth model is clearly an idealisation, with which any given foam may or may not comply. It is always unfair in such a case to say "Go do the experiments.", but at least their prospects and the issues at stake could be addressed?
(2) Without experiments it is difficult to assess what is significant/realistic in the very detailed blow-by-blow accounts of these simulations. It would be much better if they could be presented in a much more condensed form, highlighting that main results, ie, "cutting to the chase". Many details could be relegated to appendices, supplementary material or archive.
(3) Personally I would question whether " it is possible to capture the rich properties of liquid foams by using a 2D model known as the viscous froth model" . On the face of it, this is claiming too much. The origin of the viscous effects that lie in at the heart of this model is the drag exerted by the two plates. There is no obvious counterpart in a typical 3d foam: viscous effects must arise otherwise and are likely to be very different in their effects. So I would recommend that less be claimed for the general significance of this work.

Is the paper of sufficient general interest? Acceptable
Is the overall quality of the paper suitable? Good Can the paper be shortened without overall detriment to the main message? Yes Do you think some of the material would be more appropriate as an electronic appendix? No

Recommendation? Accept with minor revision (please list in comments)
Comments to the Author(s) See attached file (Appendix B).

Review form: Referee 2
Is the manuscript an original and important contribution to its field? Good

Is the paper of sufficient general interest? Good
Is the overall quality of the paper suitable? Good

Comments to the Author(s)
The authors have addressed the issues raised. I now recommend publication.

22-Dec-2021
Dear Dr Torres Ulloa, On behalf of the Editor, I am pleased to inform you that your Manuscript RSPA-2021-0642.R1 entitled "Viscous froth model applied to the motion and topological transformations of twodimensional bubbles in a channel: Three-bubble case" has been accepted for publication subject to minor revisions in Proceedings A. Please find the referees' comments below.
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Once again, thank you for submitting your manuscript to Proceedings A and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch.  Dear Dr Torres Ulloa I am pleased to inform you that your manuscript entitled "Viscous froth model applied to the motion and topological transformations of two-dimensional bubbles in a channel: Three-bubble case" has been accepted in its final form for publication in Proceedings A.

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06-Oct-2021
Dear Dr Torres Ulloa The Editor of Proceedings A has now received comments from referees on the above paper and would like you to revise it in accordance with their suggestions which can be found below (not including confidential reports to the Editor).
Please submit a copy of your revised paper within four weeks -if we do not hear from you within this time then it will be assumed that the paper has been withdrawn. In exceptional circumstances, extensions may be possible if agreed with the Editorial Office in advance.
Please note that it is the editorial policy of Proceedings A to offer authors one round of revision in which to address changes requested by referees. If the revisions are not considered satisfactory by the Editor, then the paper will be rejected, and not considered further for publication by the journal. In the event that the author chooses not to address a referee's comments, and no scientific justification is included in their cover letter for this omission, it is at the discretion of the Editor whether to continue considering the manuscript.
To revise your manuscript, log into http://mc.manuscriptcentral.com/prsa and enter your Author Centre, where you will find your manuscript title listed under "Manuscripts with Decisions." Under "Actions," click on "Create a Revision." Your manuscript number has been appended to denote a revision.
You will be unable to make your revisions on the originally submitted version of the manuscript. Instead, revise your manuscript and upload a new version through your Author Centre.
When submitting your revised manuscript, you will be able to respond to the comments made by the referee(s) and upload a file "Response to Referees" in Step 1: "View and Respond to Decision Letter". Please provide a point-by-point response to the comments raised by the reviewers and the editor(s). A thorough response to these points will help us to assess your revision quickly. You can also upload a 'tracked changes' version either as part of the 'Response to reviews' or as a 'Main document'.
IMPORTANT: Your original files are available to you when you upload your revised manuscript. Please delete any unnecessary previous files before uploading your revised version.
When revising your paper please ensure that it remains under 28 pages long. In addition, any pages over 20 will be subject to a charge (£150 + VAT (where applicable) per page). Your paper has been ESTIMATED to be 27 pages.
Once again, thank you for submitting your manuscript to Proc. R. Soc. A and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch.

Yours sincerely
Raminder Shergill proceedingsa@royalsociety.org on behalf of Professor Helen Wilson Board Member Proceedings A Dear Editor, Thank you for sending the reviewer reports on the manuscript ID RSPA-2021-0642. We have addressed all the concerns of the reviewers, making the corresponding changes to the documents (main text and supplementary material), and are resubmitting a new version of the manuscript. As with any submission, we confirm that this manuscript has not been previously published, nor is it under consideration for publication by any other journal. Both authors are aware of the manuscript and approve its submission and agree to be accountable for all aspects of the work. For the convenience of the editor and reviewers, changes within the main text and supplementary material are highlighted in colour. We also confirm that we are aware of the article length restrictions applicable to Proc. Roy. Soc. A. In the event that our manuscript is accepted for publication and attracts page charges, we confirm that we agree to meet those charges. To address the concerns of reviewer 1, we have reduced the extension of the results section by moving section 3(c) to the supplementary material. Also, in an effort to avoid repetition in section 2(c), this has been reduced in length. In addition, as reviewer 2 also pointed out, further discussion on the experimental counterpart related to viscous froth model has been included in the introduction section, and more details about the possible experimental implications of the results have also been included in the conclusion section. We have also provided a detailed point-by-point response to follow.

Reviewer(s)' Comments to Author:
Referee: 1 Comments to the Author(s) This is an impressive piece of work to catalogue and explain the possible outcomes when a system of three bubbles is driven along a straight channel in a Hele-Shaw cell. It has immediate applications in foam microfluidics (having more bubbles reduces the possibility of changes to the structure during flow) and possible, but perhaps currently more nebulous, applications in oil recovery and soil remediation.
The authors delineate the different bubble geometries and possible topological changes as the driving pressure slowly rises. It is undoubtedly careful painstaking work -almost 30 pages of description and a further 37 pages of supplemental material (that I didn't read carefully) to back this up -and so further discussion of the implications of the results would be welcome in the conclusions section to justify such an extensive manuscript.
This seems to echo reviewer 2, who mentioned possible issues at stake for experiments as being a point that needed to be mentioned. In our response to reviewer 2 we included a discussion of experimental outlook right at the end of the conclusions, and we feel this may also satisfy the point reviewer 1 is making here.
Further comments on manuscript length/content: Numerical method: the numerics are only referred to obliquely in the main body of the manuscript and delegated to the supplemental material. I think it would help the reader to understand a bit more where and why numerics are required, and how they are implemented, although I agree with the authors that the supplemental material is the right place for the details.
The reason why numerics are needed has been be explained. Specifically a brief explanation was included in section 2(d).
Film coordinates: The other topic that struck me as mysterious without having read the supplemental material is the use of film coordinates (x ij , y ij ). I think this need to be defined in the text. This notation has been defined now in section 2(b).
In contrast, I felt that much of section 2(c) was repetition, and by the sixth page of the results (section 3(b)) I was desperate to know what the implications are for real systems, and wonder if the rest of the results (3(b) -3(d)) could go in the supplemental material. Section 3(c) in particular seems not to add very much.
Section 2(c) has been reduced in length. There were indeed some sentences that were repetitive in section 2(c) and these have been eliminated. Former section 3(c) has been moved to supplementary material, now as section S 4(a). We argue for retaining sections 3(b) and former section 3(d) (now section 3(c)) in main text as they are sections that correspond to measurements an experimentalist would be likely to make (topological break up vs driving pressure; migration velocity vs driving pressure); this is a point we now make in the final paragraph of 3(c)(ii) and also in the final paragraph of the conclusions. Meanwhile the sorts of quantities discussed in former section 3(c) (now supplementary section S 4(a)), namely film turning angles and film lengths, could in principle be measured experimentally e.g. using image processing techniques, but these are arguably not the first measurements an experimentalist would choose to make. Hence we agree this section should be supplementary material.
Other than this, I have only minor comments: Abstract: the punctuation in the abstract needs careful revision (and indeed there are many parts of the manuscript which would be easier to read with elimination/redistribution of commas and greater use of hyphens) The punctuation has been checked throughout. Early on in the abstract, there was a clumsy sentence with far too many commas. We broke this up into two sentences to make it clearer.
Section 1(c): could the authors explain what they mean by "quasi(-)statically" changing the driving pressure? Presumably that the changes are slow compared to the relaxation time of the structure to a new steady state?
This does indeed need to be explained, and the reviewer's explanation is correct. In the quasistatic limit, changes in p b are arbitrarily slow compared to the relaxation time of the structure to steady state, so the system effectively evolves through a sequence of steady states. Section 1(c) is modified accordingly.  Section 1(d): I think it might help to emphasise that it is not necessarily the bubble size A that is important, but its relation to the channel width A/L 2 .
It has now been mentioned in section 1(d) that what matters is bubble size relative to channel size.
At the start of section 2, please note that the symmetry is only the case for p b = v = 0.
This has been stated explicitly at the start of section 2.
Upper and lower, upwards and downwards: Perhaps it would disentangle some of the sentences (e.g. in section 1(c)) if it was made clear that "lower wall" refers to the one to which the spanning film is attached. Does s ij grow "downwards" (section 2(a)) for films  12 and  23 for any p b , or can the orientation of these films change?
The point that the lower wall is the wall to which the spanning film connects has been made clear in section 1(c). In general we have tried to use terminology consistently throughout. We use the term "upper wall" and "lower wall" to refer to the 2D view, whereas we use "top plate" (but never "upper plate") (in section 1) to refer to a 3D view. On the other hand, in section 2(a), "downwards" in this sense for films  12 and  23 means "moving away from V 2 ". The reviewer is correct that there are cases (in Figure S 6 in the supplementary material for instance) in which these films start moving downwards away from vertex V 2 and then swing back upwards to reach vertex V 1 and V 3 , although V 1 and V 3 still end up lower down than V 2 started, so the net motion is still downwards. For avoidance of doubt though we have reworded section 2(a).
Section 2(d): two methods of parametrizing the system are described, but without explaining their benefits. This only comes much later. Why not help the reader here?
This has been mentioned in section 2(d).
Geometrically-invariant states: Is it possible to show examples of the bubble shape for these? The introduction suggests infinite staircases, as in ref [16], but that is presumably not relevant here. At the end of page 13 it is stated that these "might be reached"; is there any reason why the authors can not be definitive here?
In section 1(d) we have reworded to emphasise that in the three-bubble system (unlike the infinite staircase), the structure is not invariant over the full range of back pressures, only in high pressure limit. Also in the first paragraph in section 1(d) the reader is pointed to a figure showing the geometrically invariant state (Figure S 9 in the supplementary material). In addition, "might be reached" has been changed to "can be reached" (in section 2(f)). The original wording "might" was intended to reflect the notion that there is a necessary condition (in terms of the parameters l • 1 and l • 2 ) for existence of the geometrically invariant state, but not all parameter sets meeting that necessary condition attained the state. It is a necessary condition after all, but not sufficient. On balance though we agree with the reviewer that the word "might" is a little distracting in this context, so have changed it. Figure 7: the authors state that there is "a reasonable correlation" between figs 7(a) and (b). This is certainly not the case at first glance, although if by correlation the authors mean that the number of regions is the same then I might find this acceptable.
This has been reworded (in section 3(a)(ii)). It is not just the number of regions that is the same, there is also a correlation in terms of where they are located with respect to each other in the diagram. What differs of course between the diagrams is the size of the regions as section 3(a)(ii) goes on to explain. I wonder if light shading could be used to indicate the correspondence that is meant? I think it would help to specify in the caption that 7(b) is the result of increasing p b slowly from zero (if I'm correct).
The explanation has been given in words (as alluded to above); as mentioned, the correspondence here concerns the spatial arrangement of regions, albeit not their size. We feel shading would be very distracting (particularly when we move from Figure 7(b) to Figure 8 which has a multitude of regions).

Referee: 2
Comments to the Author(s) This paper presents simulations of a 2d soap froth in steady motion., for various cases. It uses the so-called 2D viscous froth model. The work has been diligently compiled and presented. I have a number of reservations, which the authors might consider. If adequately implemented, they should make the paper acceptable for publication in this journal.
(1) No experimental counterpart to these results is reported, yet the experiments are not very demanding? This is important, as the viscous froth model is clearly an idealisation, with which any given foam may or may not comply. It is always unfair in such a case to say "Go do the experiments.", but at least their prospects and the issues at stake could be addressed? This is a valid point and discussion has been added, in the introduction (section 1), in section 2 (see the second paragraph of section 2), in section 3(c)(ii) (see the last paragraph of that section), and in the conclusions (see the last paragraph of the conclusions). Note that a number of the claims we made in section 1 were actually describing experimental systems (not just simulation studies), but it was left implicit: it is now made explicit. One general comment we would make however is that doing an experiment to test (and possibly disprove) a specific set of predictions is better than doing an experiment blindly without any predictions available to test. The intellectual challenge in this manuscript has been to obtain the predictions. An experimental study will no doubt present a different set of intellectual challenges.
(2) Without experiments it is difficult to assess what is significant/realistic in the very detailed blow-byblow accounts of these simulations. It would be much better if they could be presented in a much more condensed form, highlighting that main results, i.e., "cutting to the chase". Many details could be relegated to appendices, supplementary material or archive.
This echoes the view of reviewer 1 that some of the results could be moved to supplementary material, and this has been done. Please see the response to reviewer 1. What we retained in the main text were the results we deemed most likely to appeal to an experimentalist.
(3) Personally I would question whether "it is possible to capture the rich properties of liquid foams by using a 2D model known as the viscous froth model". On the face of it, this is claiming too much. The origin of the viscous effects that lie in at the heart of this model is the drag exerted by the two plates. There is no obvious counterpart in a typical 3d foam: viscous effects must arise otherwise and are likely to be very different in their effects. So I would recommend that less be claimed for the general significance of this work.
We agree with the reviewer: the original claim was overstated. We have changed the wording (in the introduction) from "to capture the rich properties of liquid foams" to "the properties of a foam layer flowing between two plates". The point we were trying to make here is that the viscous froth model captures some of the rich properties of foam (but by no means does it capture all the complexity of 3D foam).
The authors have made considerable changes in response to my earlier comments, for which I thank them. The manuscript is now slightly shorter and more accessible. A highlight is the addition of the last paragraph of the conclusions, which outlines what an experimentalist might measure to verify these predictions, and the difficulties that might have to be overcome to perform the experiments. Nonetheless, this is still a complex piece of theoretical work that I think is worthy of publication in PRSA. I have only minor comments, and don't need see the manuscript again.
Page numbers refer to manuscript pages, rather than pages in the PDF file that I was sent. 2) pg 3, line 20: it might be worth emphasising here that you mean only a straight channel (as the next paragraph makes clear, this is not true in a curved channel). I'm not sure that the statement "bubbles have the exact same shape no matter ... how fast they move" is true -I wondered if you mean that all bubbles have the same shape no matter how fast they move, even if that shape depends on speed, but later it is written that the bubbles move "without deforming". Doesn't this contradict the sort of structures found in ref 16?
Later in this paragraph, the authors have added the phrase "a bamboo foam was obtained experimentally"; I'm not sure what the point is here ... is it that this differs from theoretical predictions?
3) pg 3, line 26: the authors have updated some instances of dimensional quantities being referred to as small/large relative to the channel width, but I think this one, since it occurs early on, could also be changed. Then it might be worth being explicit at the top of page 9 that lengths are scaled by L and areas by L^2. This would help with lines 50-51 on page 14. 4) pg 3, line 47: "beyond", rather than "after", a threshold? 5) pg 4, line 19: I'm not sure that anything was "proven" in [1], only demonstrated. 6) pg 4, fig 2: the authors might consider labelling the vertex, since it helps with the following paragraph. 7) pg 4, line 56: doesn't the use of "side wall" contradict the author's response letter concerning the use of upper and lower only? Shrinks to zero "length"?

Appendix B
> Dear Editor of Proc Roy Soc A, > Thank you for sending the reviewer comments on our manuscript > RSPA-2021-0642.R1 > We have noted that the reviewer has explicitly said > that they do not need to see the manuscript again.
> However for the convenience of the editor, our detailed responses to > the reviewer are given below and also the latest round of changes to > the manuscript have been marked up in colour.
> On the other hand, we also inform that we would like to opt for open access.
> Yours sincerely, > Carlos Torres-Ulloa, Paul Grassia The authors have made considerable changes in response to my earlier comments, for which I thank them. The manuscript is now slightly shorter and more accessible. A highlight is the addition of the last paragraph of the conclusions, which outlines what an experimentalist might measure to verify these predictions, and the difficulties that might have to be overcome to perform the experiments. Nonetheless, this is still a complex piece of theoretical work that I think is worthy of publication in PRSA. I have only minor comments, and don't need see the manuscript again.
Page numbers refer to manuscript pages, rather than pages in the PDF file that I was sent. 2) pg 3, line 20: it might be worth emphasising here that you mean only a straight channel (as the next paragraph makes clear, this is not true in a curved channel). I'm not sure that the statement ``bubbles have the exact same shape no matter ... how fast they move'' is true -I wondered if you mean that all bubbles have the same shape no matter how fast they move, even if that shape depends on speed, but later it is written that the bubbles move ``without deforming''. Doesn't this contradict the sort of structures found in ref 16?
> The comment about the retaining the same shape referred specifically Appendix C > to the structures in Figure 1 in a straight channel, as has been > clarified. > There is no contradiction with ref [16] as that does not concern one > of the shapes in Figure 1.
Later in this paragraph, the authors have added the phrase ``a bamboo foam was obtained experimentally''; I'm not sure what the point is here ... is it that this differs from theoretical predictions?'' > We have expanded the discussion to explain that there is a maximum > theoretical bubble area at which an infinite staircase must convert > to a bamboo (although bamboos are permitted for areas smaller than > this theoretical maximum).
> The reason we included the word ``experimentally'' was to highlight > that experimentally the switch from staircase to bamboo happens at > an area less than the theoretical maximum permitted area.
> We also made some minor modifications to the supplementary material > to mention that the largest bubble areas in a three-bubble > (equilibrium, but not necessarily monodisperse) staircase differ > from largest bubble areas in a (monodispere) infinite staircase 3) pg 3, line 26: the authors have updated some instances of dimensional quantities being referred to as small/large relative to the channel width, but I think this one, since it occurs early on, could also be changed. Then it might be worth being explicit at the top of page 9 that lengths are scaled by L and areas by L^2. This would help with lines 50--51 on page 14.
> We have clarified that bubble size is relative to channel size.
> We have also explicitly mentioned that spatial coordinates are > scaled by L, and areas are scaled by L^2.
> We agree this helps with the text later on, since it is clearer what > small (dimensionless) area and large (dimensionless) area means.