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Membrane-induced hydroelastic migration of a particle surfing its own wave

Abstract

While coupling between fluid flow and soft elastic surfaces is common in biology and engineering, an analytical description is challenging as it often involves non-linear dynamics. Here we show using theory and experiments that a small particle moving along an elastic membrane through a viscous fluid is repelled from the membrane due to hydroelastic forces. The flow field produces an elastic disturbance in the membrane leading to particle–wave coupling. We derive an analytic expression for the particle trajectory and find that the normal migration velocity of the particle is quadratic in its speed and depends on a combination of the tension and bending resistance of the membrane. Experimentally, we measure the normal displacement of spheres sedimenting under gravity along a suspended elastic membrane and find quantitative agreement with the theoretical predictions with no fitting parameters. We experimentally demonstrate that the effect is strong enough for separation and sorting of particles on the basis of both their size and density. We discuss the significance of our results for particles interacting with biological membranes, and propose the use of our model for membrane elasticity measurements.

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Fig. 1: Self-surfing and hydroelastic repulsion of a particle near a thin elastic sheet.
Fig. 2: Size-dependent sorting of spherical particles sedimenting near an elastic membrane.
Fig. 3: Comparison of the experimental results with the theory.
Fig. 4: Typical trajectory of a sphere.
Fig. 5: The normal migration depends on the size and the density of the particles, as well as the properties of the sheet.

Data availability

The experimental data for the plots within this paper are available from the figshare repository27. The same repository contains a movie corresponding to Fig. 1a. Raw image data and other supporting data relevant to this study are available from the authors upon request.

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Acknowledgements

The authors acknowledge support from the National Science Foundation via award DMS-1614907, and partial support from the Carbon Mitigation Initiative of Princeton University. M.Y.B.Z. acknowledges support by the Center for Bio Inspired Energy Sciences, an Energy Frontier Research Center funded by the DOE, Office of Sciences, Basic Energy Sciences, under award DE-SC0000989 (Paul M. Chakin). We thank T. Salez for preliminary discussions, M. Shelley for helpful ideas and J. Nunes, A. Perazzo and Y. E. Yu for their help with the experiments.

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B.R. and N.O. contributed equally to this work. B.R., N.O. and H.A.S conceived the project and developed the theory. B.R., N.O. and M.Y.B.Z. performed the experiments. All authors analysed and interpreted the data and wrote the paper.

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Correspondence to Howard A. Stone.

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Supplementary Information, Supplementary Figures 1–5, Supplementary References 1–7

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Rallabandi, B., Oppenheimer, N., Ben Zion, M.Y. et al. Membrane-induced hydroelastic migration of a particle surfing its own wave. Nature Phys 14, 1211–1215 (2018). https://doi.org/10.1038/s41567-018-0272-z

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