Fluid ‘rope trick’ investigated
L. Mahadevan, W. S. Ryu, A. D. T. Samuel
Nature 392, 140 ( 1998)
We wish to amend a small mistake in our calculations, although this does not affect the basic idea in our paper, particularly by comparison with experiment. As the longitudinal viscous stress σ in the filament scaled as ∼μ Ur/R2 varies linearly across the cross-section, the integrated stress resultant ∫σdA vanishes. However, the net bending torque due to this viscous stress does not vanish and scales as ∫σrdA∼μ Ur4/R2. The force per unit volume on the fluid due to centripetal and Coriolis accelerations scales as f∼ρΩ2R, so that the bending torque on the whirling filament in the vicinity of the coil scales as fr2R2∼ρΩ2r2 R3. Torque balance, together with the ancillary continuity relations, leads to a scaling law for the coiling frequency
which is slightly different from the result given in our paper, where an erroneous argument confuses the transverse and longitudinal timescales in the filament. Equation (1) can be derived directly using an analogy to the coiling of an elastic rope by simply replacing the elastic bending modulus Er4 in ref. 1 with the ‘viscous bending modulus’ μr4U/ R. A reconsideration of the experimental results leads to data collapse with a power law Ω/Q1.33∼ r−3.45±0.10, in agreement with our argument.
Mahadevan, L. & Keller, J. B. Proc. R. Soc. Lond. A 452, 1679–1694 (1996).
The online version of the original article can be found at 10.1038/32321
About this article
Physical Review Letters (2020)
Physics of Fluids (2017)
Physical Review Fluids (2017)
RSC Advances (2016)
Materials Research Express (2014)