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Viscous Marangoni propulsion

Published online by Cambridge University Press:  19 December 2011

Eric Lauga  and
Anthony M. J. Davis
Eric Lauga*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
Anthony M. J. Davis
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
*
†Email address for correspondence: elauga@ucsd.edu
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Abstract

Marangoni propulsion is a form of locomotion wherein an asymmetric release of surfactant by a body located at the surface of a liquid leads to its directed motion. We present in this paper a mathematical model for Marangoni propulsion in the viscous regime. We consider the case of a thin rigid circular disk placed at the surface of a viscous fluid and whose perimeter has a prescribed concentration of an insoluble surfactant, to which the rest of its surface is impenetrable. Assuming a linearized equation of state between surface tension and surfactant concentration, we derive analytically the surfactant, velocity and pressure fields in the asymptotic limit of low capillary, Péclet and Reynolds numbers. We then exploit these results to calculate the Marangoni propulsion speed of the disk. Neglecting the stress contribution from Marangoni flows is seen to over-predict the propulsion speed by 50 %.

JFM classification

Interfacial Flows (free surface): Capillary flows Low-Reynolds-number flows: Low-Reynolds-number flows Biological Fluid Dynamics: Propulsion
Type
Papers
Information
Journal of Fluid Mechanics , Volume 705: Special issue dedicated to Professor T. J. Pedley on his 70th birthday , 25 August 2012 , pp. 120 - 133
Copyright
Copyright © Cambridge University Press 2011

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References

1. Abramowitz, M. & Stegun, I. A. 1969 Handbook of Mathematical Functions. Dover.Google Scholar
2. Alexander, D. E. 2002a Nature’s Flyers: Birds, Insects, and the Biomechanics of Flight. The Johns Hopkins University Press.Google Scholar
3. Alexander, R. M. 2002b Principles of Animal Locomotion. Princeton University Press.CrossRefGoogle Scholar
4. Bain, C. D., Burnetthall, G. D. & Montgomerie, R. R. 1994 Rapid motion of liquid drops. Nature 372, 414–415.CrossRefGoogle Scholar
5. Barton, K. D. & Subramanian, R. S. 1989 The migration of liquid-drops in a vertical temperature-gradient. J. Colloid Interface Sci. 133, 211–222.CrossRefGoogle Scholar
6. Bassik, N., Abebe, B. T. & Gracias, D. H. 2008 Solvent driven motion of lithographically fabricated gels. Langmuir 24, 12158–12163.Google Scholar
7. Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245–268.CrossRefGoogle Scholar
8. Betz, O. 2002 Performance and adaptive value of tarsal morphology in rove beetles of the genus Stenus (Coleoptera, Staphylinidae). J. Expl Biol. 205, 1097–1113.Google Scholar
9. Billard, G. & Bruylant, C. 1905 Sur un mode particulier de locomotion de certains stenus . C. R. Soc. Biol. 59, 102–103.Google Scholar
10. Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398.Google Scholar
11. Brochard, F. 1989 Motions of droplets on solid surfaces induced by chemical or thermal gradients. Langmuir 5, 432–438.Google Scholar
12. Bush, J. W. M. & Hu, D. L. 2006 Walking on water: biolocomotion at the interface. Annu. Rev. Fluid Mech. 38, 339–369.CrossRefGoogle Scholar
13. Chaudhury, M. K. & Whitesides, G. M. 1992 How to make water run uphill. Science 256, 1539–1541.Google Scholar
14. Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.CrossRefGoogle Scholar
15. Davis, A. M. J. 1991a Shear-flow disturbance due to a hole in the plane. Phys. Fluids 3, 478–480.CrossRefGoogle Scholar
16. Davis, A. M. J. 1991b Slow viscous flow due to motion of an annular disk: pressure-driven extrusion through an annular hole in a wall. J. Fluid Mech. 231, 51–71.CrossRefGoogle Scholar
17. Dossantos, D. F. & Ondarcuhu, T. 1995 Free-running droplets. Phys. Rev. Lett. 75, 2972–2975.CrossRefGoogle Scholar
18. Ellington, C. P 1984 The Aerodynamics of Hovering Insect Flight. The Royal Society.Google Scholar
19. Fauci, L. J. & Dillon, R. 2006 Biofluidmechanics of reproduction. Annu. Rev. Fluid Mech. 38, 371–394.CrossRefGoogle Scholar
20. Fish, F. E. & Lauder, G. V. 2006 Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38, 193–224.CrossRefGoogle Scholar
21. Gibbs, J. W. 1878 On the equilibrium of heterogeneous substances. Trans. Conn. Acad. 3, 343. In Scientific Papers, vol. 1 (1961), p. 272. Dover.Google Scholar
22. Golestanian, R., Liverpool, T. B. & Ajdari, A. 2005 Propulsion of a molecular machine by asymmetric distribution of reaction products. Phys. Rev. Lett. 94, 220801.CrossRefGoogle ScholarPubMed
23. Gray, J. 1968 Animal Locomotion. Norton.Google Scholar
24. Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125–143.CrossRefGoogle Scholar
25. Howse, J. R., Jones, R. A. L., Ryan, A. J., Gough, T., Vafabakhsh, R. & Golestanian, R. 2007 Self-motile colloidal particles: from directed propulsion to random walk. Phys. Rev. Lett. 99, 048102.Google Scholar
26. Iida, K., Suematsu, N. J., Miyahara, Y., Kitahata, H., Nagayama, M. & Nakata, S. 2010 Experimental and theoretical studies on the self-motion of a phenanthroline disk coupled with complex formation. Phys. Chem. Chem. Phys. 12, 1557–1563.Google Scholar
27. Jenkins, M. F. 1960 On the method by which Stenus and Dianous (Coleoptera: Staphylinidae) return to the banks of a pool. Trans. R. Ent. Soc. Lond. 112, 1–14.Google Scholar
28. Joy, N. H. 1910 The behaviour of coleoptera in time of floods. Trans. Ent. Soc. Lond. 58, 379–385.CrossRefGoogle Scholar
29. Kim, S. & Karilla, J. S. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
30. Kitahata, H., Hiromatsu, S.-i., Doi, Y., Nakata, S. & Rafiqul Islam, M. 2004 Self-motion of a camphor disk coupled with convection. Phys. Chem. Chem. Phys. 6, 2409–2414.Google Scholar
31. Kohira, M. I., Hayashima, Y., Nagayama, M. & Nakata, S. 2001 Synchronized self-motion of two camphor boats. Langmuir 17, 7124–7129.CrossRefGoogle Scholar
32. Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
33. Lighthill, J. L. 1975 Mathematical Biofluiddynamics. SIAM.CrossRefGoogle Scholar
34. Linsenmair, K. E. & Jander, R. 1963 Das Entspannungsschwimmen von Velia und Stenus . Naturwissenschaften 50, 231.Google Scholar
35. Maxworthy, T. 1981 The fluid dynamics of insect flight. Annu. Rev. Fluid Mech. 13, 329–350.CrossRefGoogle Scholar
36. Nakata, S. & Arima, Y. 2008 Self-motion of a phenanthroline disk on divalent metal ion aqueous solutions coupled with complex formation. Colloids Surf. A 324, 222–227.Google Scholar
37. Nakata, S., Doi, Y. & Kitahata, H. 2004 Synchronized motion of a mobile boundary driven by a camphor fragment. J. Colloid Interface Sci. 279, 503–508.CrossRefGoogle ScholarPubMed
38. Nakata, S., Iguchi, Y., Ose, S., Kuboyama, M., Ishii, T. & Yoshikawa, K. 1997 Self-rotation of a camphor scraping on water: new insight into the old problem. Langmuir 13, 4454–4458.Google Scholar
39. Nakata, S. & Kirisaka, J. 2006 Characteristic motion of a camphanic acid disk on water depending on the concentration of Triton X-100. J. Phys. Chem. B 110, 1856–1859.Google Scholar
40. Nakata, S., Kirisaka, J., Arima, Y. & Ishii, T. 2006 Self-motion of a camphanic acid disk on water with different types of surfactants. J. Phys. Chem. B 110, 21131–21134.Google Scholar
41. Nakata, S., Komoto, H., Hayashi, K. & Menzinger, M. 2000 Mercury drop ‘attacks’ an oxidant crystal. J. Phys. Chem. B 104, 3589–3593.CrossRefGoogle Scholar
42. Pawar, Y. & Stebe, K. J. 1996 Marangoni effects on drop deformation in an extensional flow: the role of surfactant physical chemistry. Part 1. Insoluble surfactants. Phys. Fluids 8, 1738–1751.Google Scholar
43. Paxton, W. F., Kistler, K. C., Olmeda, C. C., Sen, A., Angelo, S. K. St, Cao, Y. Y., Mallouk, T. E., Lammert, P. E. & Crespi, V. H. 2004 Catalytic nanomotors: autonomous movement of striped nanorods. J. Am. Chem. Soc. 126, 13424–13431.Google Scholar
44. Paxton, W. F., Sundararajan, S., Mallouk, T. E. & Sen, A. 2006 Chemical locomotion. Angew. Chem. Intl Ed. 45, 5420–5429.Google Scholar
45. Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24, 313–358.CrossRefGoogle Scholar
46. Phongikaroon, S., Hoffmaster, R., Judd, K. P., Smith, G. B. & Handler, R. A. 2005 Effect of temperature on the surface tension of soluble and insoluble surfactants of hydrodynamical importance. J. Chem. Engng Data 50, 1602–1607.Google Scholar
47. Rayleigh, L. 1890 Measurements of the amount of oil necessary in order to check the motions of camphor upon water. Proc. R. Soc. Lond. 47, 364–367.Google Scholar
48. Ruckner, G. & Kapral, R. 2007 Chemically powered nanodimers. Phys. Rev. Lett. 98, 150603.Google Scholar
49. Schildknecht, H. 1976 Arthropod defense substances. Chemical ecology: a chapter of modern natural products chemistry. Angew. Chem. 15, 214–222.CrossRefGoogle Scholar
50. Scriven, L. E. & Sternling, C. V. 1960 Marangoni effects. Nature 187, 186–188.Google Scholar
51. Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. North-Holland.Google Scholar
52. Soh, S., Bishop, K. J. M. & Grzybowski, B. A. 2008 Dynamic self-assembly in ensembles of camphor boats. J. Phys. Chem. B 112, 10848–10853.CrossRefGoogle ScholarPubMed
53. Stone, H. A. 1990 A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids 2, 111–112.Google Scholar
54. Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32, 33–53.Google Scholar
55. Tsai, W. & Yue, D. K. P. 1995 Effects of soluble and insoluble surfactant on laminar interactions of vortical flows with a free surface. J. Fluid Mech. 289, 315–349.CrossRefGoogle Scholar
56. Tsemakh, D., Lavrenteva, O. M. & Nir, A. 2004 On the locomotion of a drop, induced by the internal secretion of surfactant. Intl J. Multiphase Flow 30, 1337–1367.CrossRefGoogle Scholar
57. Vogel, S. 1996 Life in Moving Fluids. Princeton University Press.Google Scholar
58. Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210.CrossRefGoogle Scholar
59. Young, N. O., Goldstein, L. S. & Block, M. J. 1959 The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6, 350–356.Google Scholar