Metachronal Swimming with Rigid Arms near Boundaries

Round 1

Reviewer 1 Report

This manuscript by Hayashi and Takagi investigates the locomotion of a mechanical swimmer situated at an oil-air interface, which actuates two rigid arms in a periodic fashion. The authors explore how changes in the actuation sequence and physical surroundings can affect the overall locomotion of the robot. While the article is interesting and well written, it would benefit from some additional details and discussion, and further analysis in the model.

- The authors use proximity to a nearby wall as a justification for utilising the Stokes equations. However, for many of the organisms mentioned (e.g. copepods, krill, crabs, etc), inertial effects would still be important even when close to a substrate. Moreover, the calculated Reynolds number of 0.19 is not overly small. Can the authors describe which of the effects (if any) are a result of this finite Reynolds number? Would the time-dependence of motor actuation play a role in this experiment?

- Can the authors account for the observation that the propulsive speed in cases 1 and 3 seems to be completely independent of the distance to the wall? At the same time, case 2 exhibits a fairly strong dependence on height h. Can the hydrodynamic model be extended to account for these observations? Given the title of the paper emphasises swimming near boundaries, it would be important to present some physical explanation for why the boundaries affect some gaits but not others.

- It would be useful for the reader to see sinusoidal curves plotted alongside the functions in Figure 4.

- In Figure 3, it appears that the amplitude of the blue and black curves have some modulations on longer timescales. Can the authors determine the origin of these modulations?

- It is shown that the net propulsive speed depends on the difference between \theta_1 and \theta_2, but only one non-zero case is studied. Can this be extended to a more systematic study?

- When there is a boundary nearby, case 2 is the only configuration with net propulsion. Is there any rocking of the body as the arms vary in their distance to the boundary? What does the side view look like?

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Summary

In this paper, the authors experimentally investigate low Reynolds number propulsion by organisms metachronally stroking multiple appendages. This situation is relevant to many crustaceans but, at the low Re investigated here, is most relevant to tiny or larval crustaceans and organisms beating flagella. The authors use a robot on an air bearing with two legs submerged in oil to investigate the effects of leg synchronization (phase difference), the direction in which the legs point, and the effects of confining the robot with walls and a floor. The resulting motion of the robot in response to changing these parameters is quantified and is compared to a slender-body theory they previously developed in 2015. The experimental and model results match well.

The paper is well written and is tightly focused on a couple of key questions. I think the most interesting result is that the leg direction matters, and this point could be developed a bit more – are there animals that adopt this sort of leg orientation? Further, the effect of the boundaries is interesting but is not well explained in the Discussion – why is this increase in speed happening for Case 2? Finally, the paper lays a nice foundation and develops useful tools for further studies looking at more complex problems.

 

Concerns

In the abstract, you use the phrase “orienting the arms in the same direction.” This is confusing until the reader sees what you mean in the Methods. Perhaps find a better way to describe the different arm positioning. Also in the abstract, you say “confining the robot with rigid boundaries has little effect on the results…” I think it would help to be more specific here on which results you mean. The authors miss a key reference in the Introduction. Lim and Demont (2009) also investigated metachronal stroking near a boundary in the following paper: Kinematics, hydrodynamics and force production of pleopods suggest jet-assisted walking in the American lobster (Homarus americanus) Jeanette L. Lim, M. Edwin DeMont, Journal of Experimental Biology 2009 212: 2731-2745; doi: 10.1242/jeb.026922. This reference should be cited, and the authors should interact with its findings in their Results or Discussion. In the Methods, what ratio of leg spacing to leg length was chosen, and why? Why was the leg direction (e.g. median angle) chosen as a parameter? It’s not mentioned in the Introduction, so there is not a clear motivation for varying it. Also, I believe this is this seen in some animals, so it would be worth mentioning. In the Results, it is clear from Figure 3 that the robot is actually drifting back and forth a little bit over the longer term in Case 1, which is not stated/recognized by the authors. Most importantly, I think further discussion/explanation is warranted concerning the effects of the boundaries. Some sort of physical explanation is needed beyond referencing other papers that have found similar results.

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Please see the attached PDF. 

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I thank the authors for revising the manuscript, which now addresses my comments and concerns.

Reviewer 2 Report

The authors have addressed my concerns to my satisfaction.

Reviewer 3 Report

Thanks to the authors for the significant improvement. The concerns and issues are addressed in the new version. The most important issue was addressing the effect of inertia, which the authors have mentioned it in the new version. The results obviously have inertial effects, but the equations are developed for a Low Reynolds Flow without inertia. 

One minor comment: Abstract, L3: "To study the minimal ingredients needed for swimming without inertia, we built an experimental system featuring a robot equipped with a pair of rigid slender arms.": I'd suggest to add 'with negligible inertia' at the end of this sentence.  

Suggestion: I understand this might not be possible at this stage, but if the authors show that the role of inertia (particularly the inertia of the body) is negligible, it could improve the quality of the work. There is enough data in the manuscript to find the acceleration of the body, and the viscous forces on arms can be found using resistive force theory (just for one specific direction). Comparing these two could show how significant is the inertia.