Sébastien Neukirch
Institut Jean le Rond d'Alembert
Centre National de la Recherche Scientifique
Université Pierre et Marie Curie
Paris, France

tel: +33 1 44 27 72 61
fax : +33 1 44 27 52 59
e-mail: sebastien.neukirch (-atat-) upmc.fr

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Elastocapillary coiling of an elastic rod inside a drop

H. Elettro, P. Grandgeorge, and S. Neukirch

Journal of Elasticity, vol. 127 (2017) 235ā€“247

Abstract : Capillary forces acting at the surface of a liquid drop can be strong enough to deform small objects and recent studies have provided several examples of elastic instabilities induced by surface tension. We present such an example where a liquid drop sits on a straight fiber, and we show that the liquid attracts the fiber which thereby coils inside the drop. We derive the equilibrium equations for the system, compute bifurcation curves, and show the packed fiber may adopt several possible configurations inside the drop. We use the energy of the system to discriminate between the different configurations and find a intermittent regime between two-dimensional and three-dimensional solutions as more and more fiber is driven inside the drop.

DOI: 10.1007/s10659-016-9611-4

download the journal version : PDF


Submitted (Jan 25th, 2016)
Reports (received May 16th, 2016)

Comments for the Author:

Reviewer #1: The paper describes an original buckling problem of a rod where the compressive force results from the capillary forces induced by a liquid droplet. This study is motivated by experimental studies from the same group of authors and cited in the bibliography. Experiments also illustrate the present paper. This system exhibits a quite complex buckling behavior that the authors explore numerically. The different steps of the derivations are carefully described. The authors also made the effort of providing physical insights to mathematical quantities. I encourage the publication of this study. However I have a concern with the shape of the droplet. The authors assume that the droplet is spherical, which is in contradiction with experimental observations (fig. 9 bottom). Indeed the wetting properties of the liquid (i.e. the contact angle) are prone to deform the drop. Although I may not change the general features of the analysis, I think that the authors should comment on this issue.
Minor comments:
Figures 2 and 3: the paths are quite intricate and difficult to follow. Using different colors would probably help the reader to track them.
p2, last paragraph of the introduction: coiling of a elastic rod -> an elastic
p9, second paragraph of section 4: emerges and progress toward -> progresses
p12, conclusion: a capillary force that compress -> that compresses

Reviewer #2: This paper treats an interesting mechanical problem, which is worth bringing to the attention of the community. My comments fall into two classes: Substantive remarks on the methodology and minor remarks on the language.
I would welcome a little more discussion of the underlying physics.
In the first paragraph of Section 2, it should be pointed out that that the rod is naturally straight and that the response is linear.
The sentence in the penultimate line of page 2, beginning with "As" is really an assumption, it should be stated thus, and should be justified.
Figure 1 does not suggest that the deformation can be nonplanar. A much more elaborate figure is required.
Figure 1 does not suggest that boundary condition (6c) holds. In general this boundary condition is incompatible with the symmetry asserted in the caption of Figure 1.
When K_1 and K_2 are different there could be lateral instabilities due to alpha not associated with capillarity. This matter could be remarked on.
In the line after (9) the inequality is meaningless, and suggest that the authors do not understand the derivative in (9). For the same reason, the first line on page 5 is unnecessary.
The whole exercise in getting the Euler-Lagrange equations is unnecessarily involved. The one thing it does is get the forces on the rod due to capillarity from its potential energy. It would be far cleaner to find these forces ab initio, and put them into the equilibrium equations for rods, which have been well known for 150 years. Simplifying this matter is important.
Third line of Section 3: Replace U with what?
After (23c), clean up the statement about invariance.
In Abstract, replace 2D with 2-dimensional or two-dimensional, etc.
A very irritating usage, which requires the reader to go back over the sentence to get its intent, appears four lines after (2) and elsewhere. This sentence should be rendered "The rod enters the drop at meniscus point A and exits the drop at meniscus point B." Analogous corrections should be made in the dreadful line 1 of page 4, in two lines after (18b), in the caption of figure 2.
In the paragraph containing (3), the authors use the French notation that ] [ encloses and open set. I suggest that they use the more common ( ) throughout.
After (5), the boundary conditions are not clamped, the end of the rod is. Read "boundary conditions of clamping".
7 lines after (8), replace "As", which is weak and may suggest a limit process to the reader, with "Since" or "Because".
After (20) insert a comma and read: "That is, the"
First line of Section 3. Read: "We restrict our attention".