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A continuum model for flow induced by metachronal coordination between beating cilia

Published online by Cambridge University Press:  30 August 2011

Jeanette Hussong ,
Wim-Paul Breugem  and
Jerry Westerweel
Jeanette Hussong
Affiliation:
Laboratory for Aero and Hydrodynamics, Delft University of Technology, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands
Wim-Paul Breugem*
Affiliation:
Laboratory for Aero and Hydrodynamics, Delft University of Technology, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands
Jerry Westerweel
Affiliation:
Laboratory for Aero and Hydrodynamics, Delft University of Technology, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands
*
†Email address for correspondence: [email protected]
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Abstract

In this numerical study we investigate the flow induced by metachronal coordination between beating cilia arranged in a densely packed layer by means of a continuum model. The continuum approach allows us to treat the problem as two-dimensional as well as stationary, in a reference frame moving with the speed of the metachronal wave. The model is used as a computationally efficient design tool to investigate cilia-induced transport of a Newtonian fluid in a plane channel. Contrary to prior continuum models, the present approach accounts for spatial variations in the porosity along the metachronal wave and thus ensures conservation of mass within the cilia layer. Using porous-media theory the governing volume-averaged Navier–Stokes (VANS) equations are derived and closure formulations are given explicitly for the model. This makes it possible to investigate cilia-induced flow with a continuum model in both the viscous regime and the inertial regime. The results show that metachronal coordination can act as a transport mechanism in both regimes. Porosity variations appear to be the key mechanism for correct prediction of the fluid transport in the viscous flow regime. The reason is that spatial variations in the porosity break the symmetry of the drag distribution along the metachronal wave. A new insight that has been gained is that the fluid transport reverses, thus switches from plectic to antiplectic metachronism, for the same cilia beat cycle when the wavespeed is increased such that inertial effects occur. Based on a parameter study, the net transport in the channel is described by a power-law relation of the amplitude, length and speed of the metachronal wave. It is found that the wavelength has the strongest effect on the viscosity-dominated fluid transport.

JFM classification

Micro-/Nano-fluid dynamics: Microfluidics Biological Fluid Dynamics: Propulsion
Type
Papers
Information
Journal of Fluid Mechanics , Volume 684 , 10 October 2011 , pp. 137 - 162
Copyright
Copyright © Cambridge University Press 2011

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References

1. Baltussen, M., Anderson, P., Bosab, F. & den Toonder, J. 2009 Inertial flow effects in a micro-mixer based on artificial cilia. Lab on a Chip 9, 2326–2331.Google Scholar
2. Barlow, D. & Sleigh, M. A. 1993 Water propulsion speeds and power output by comb plates of the ctenophore Pleurobrachia pileus under different conditions. J. Expl Biol. 183, 149–163.Google Scholar
3. Barton, C. & Raynor, S. 1967 Analytical investigation of cilia induced mucous flow. Bull. Math. Biophys. 29, 419–428.CrossRefGoogle ScholarPubMed
4. Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2002 Transport Phenomena. Wiley.Google Scholar
5. Blake, J. R. 1972 Model for micro-structure in ciliated organisms. J. Fluid Mech. 55, 1–23.Google Scholar
6. Blake, J. R. & Winet, H. 1980 On the mechanics of muco-ciliary transport. Biorheology 17, 125–134.Google Scholar
7. Brennen, C. 1974 Oscillating-boundary-layer theory for ciliary propulsion. J. Fluid Mech. 65, 799–824.CrossRefGoogle Scholar
8. Brennen, C. & Winet, H. 1977 Fluid-mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398.CrossRefGoogle Scholar
9. Breugem, W.-P. 2007 The effective viscosity of a channel-type porous medium. Phys. Fluids 19, 103104.CrossRefGoogle Scholar
10. Breugem, W.-P. & Boersma, B. J. 2005 Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach. Phys. Fluids 17, 025103.CrossRefGoogle Scholar
11. Breugem, W.-P., Boersma, B. J. & Uittenbogaard, R. E. 2006 The influence of wall permeability on turbulent channel flow. J. Fluid Mech. 562, 35–72.CrossRefGoogle Scholar
12. Dauptain, A., Favier, J. & Bottaro, A. 2008 Hydrodynamics of ciliary propulsion. J. Fluids Struct. 24, 1156–1165.CrossRefGoogle Scholar
13. Fahrni, F., Prins, M. & van Ijzendoorn, L. 2009 Micro-fluidic actuation using magnetic artificial cilia. Lab on a Chip 9, 3413–3421.Google Scholar
14. Gray, W. G. 1975 A derivation of the equations for multi-phase transport. Chem. Engng Sci. 30, 229–233.CrossRefGoogle Scholar
15. Gueron, S. & Levit-Gurevich, K. 1998 Computation of the internal forces in cilia: application to ciliary motion, the effects of viscosity, and cilia interactions. Biophys. J. 74, 1658–1676.CrossRefGoogle ScholarPubMed
16. Gueron, S., Levit-Gurevich, K., Liron, N. & Blum, J. J. 1997 Cilia internal mechanism and metachronal coordination as the result of hydrodynamical coupling. Proc. Natl Acad. Sci. USA 94, 6001–6006.CrossRefGoogle ScholarPubMed
17. Gueron, S. & Liron, N. 1993 Simulations of three-dimensional ciliary beats and cilia interactions. Biophys. J. 65, 499–507.CrossRefGoogle ScholarPubMed
18. Guirao, B. & Joanny, J. F. 2007 Spontaneous creation of macroscopic flow and metachronal waves in an array of cilia. Biophys. J. 92, 1900–1917.CrossRefGoogle Scholar
19. Heys, J. J., Gedeon, T., Knott, B. C. & Kim, Y. 2008 Modeling arthropod filiform hair motion using the penalty immersed boundary method. J. Biomech. 41, 977–984.CrossRefGoogle ScholarPubMed
20. Hussong, J., Schorr, N., Belardi, J., Prucker, O., Rühe, J. & Westerweel, J. 2011 Experimental investigation of the flow induced by artificial cilia. Lab on a Chip 11, 2017–2022.Google Scholar
21. Katz, D. F. 1974 Propulsion of microorganisms near solid boundaries. J. Fluid Mech. 64, 33–49.Google Scholar
22. Khaderi, S. N., Baltussen, M. G. H. M., Anderson, P. D., Ioan, D., den Toonder, J. M. J. & Onck, P. R. 2009 Nature-inspired microfluidic propulsion using magnetic actuation. Phys. Rev. E 79, 046304.Google Scholar
23. Khaderi, S. N., Baltussen, M. G. H. M., Anderson, P. D., den Toonder, J. M. J. & Onck, P. R. 2010 Breaking of symmetry in microfluidic propulsion driven by artificial cilia. Phys. Rev. E 82, 027302.CrossRefGoogle ScholarPubMed
24. Khatavkar, V. V., Anderson, P. D., den Toonder, J. M. J. & Meijer, H. E. H. 2007 Active micromixer based on artificial cilia. Phys. Fluids 19, 0836059.Google Scholar
25. Kim, Y. & Peskin, C. S. 2007 Penalty immersed boundary method for an elastic boundary with mass. Phys. Fluids 19, 053103.CrossRefGoogle Scholar
26. King, M., Agarwal, M. & Shukla, J. B. 1993 A planar model for mucociliary transport: effect of mucus viscoelasticity. Biorheology 30, 49–61.Google ScholarPubMed
27. Lardner, T. J. & Shack, W. J. 1972 Cilia transport. Bull. Math. Biophys. 34, 325–335.Google Scholar
28. Lenz, P. & Ryskin, A. 2006 Collective effects in ciliar arrays. Phys. Biol. 3, 285–294.Google Scholar
29. Liron, N. 1978 Fluid transport by cilia between parallel plates. J. Fluid Mech. 86, 705–726.CrossRefGoogle Scholar
30. Liron, N. & Mochon, S. 1976 Discrete-cilia approach to propulsion of ciliated microorganisms. J. Fluid Mech. 75, 593–607.Google Scholar
31. MacDonald, I. F., Elsayed, M. S., Mow, K. & Dullien, F. A. L. 1979 Flow through porous media: the Ergun equation revisited. Ind. Engng Chem. Fundam. 18, 199–208.CrossRefGoogle Scholar
32. Merz, R. A. & Edwards, D. R. 1998 Jointed setae: their role in locomotion and gait transitions in polychaete worms. J. Exp. Mar. Biol. Ecol. 228, 273–290.CrossRefGoogle Scholar
33. Mitran, S. M. 2007 Metachronal wave formation in a model of pulmonary cilia. Comput. Struct. 85, 763–774.CrossRefGoogle Scholar
34. Nielsen, N. F. & Larsen, P. S. 1993 A note on ciliated plane channel flow with a pressure gradient. J. Fluid Mech. 257, 97–110.Google Scholar
35. Nonaka, S., Yoshiba, S., Watanabe, D., Ikeuchi, S., Goto, T., Marshall, W. F. & Hamada, H. 2005 De novo formation of left–right asymmetry by posterior tilt of nodal cilia. PLOS Biol. 3, e268.CrossRefGoogle ScholarPubMed
36. Oh, K., Chung, J.-H., Devasia, S. & Riley, J. J. 2009 Bio-mimetic silicone cilia for microfluidic manipulation. Lab on a Chip 9, 1561–1566.Google Scholar
37. Peskin, C. S. 1972 Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271.Google Scholar
38. Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 3–11.CrossRefGoogle Scholar
39. Quintard, M. & Whitaker, S. 1994 Transport in ordered and disordered porous media I: the cellular average and the use of weighting functions. Trans. Porous Med. 14, 163–177.Google Scholar
40. Shields, A. R., Fiser, B. L., Evans, B. A., Falvo, M. R., Washburn, S. & Superfine, R. 2010 Biomimetic cilia arrays generate simultaneous pumping and mixing regimes. Proc. Natl Acad. Sci. 107 (36), 15670–15675.Google Scholar
41. Singh, H., Laibinis, P. E. & Hatton, T. A. 2005 Synthesis of flexible magnetic nanowires of permanently linked core-shell magnetic beads tethered to a glass surface patterned by microcontact printing. Nano Lett. 5, 2149–2154.Google Scholar
42. Smith, D. J., Gaffney, E. A. & Blake, J. R. 2007a A viscoelastic traction layer model of muco-ciliary transport. Bull. Math. Biol. 69, 289–327.Google Scholar
43. Smith, D. J., Gaffney, E. A. & Blake, J. R. 2007b Discrete cilia modelling with singularity distributions: application to the embryonic node and the airway surface liquid. Bull. Math. Biol. 69, 1477–1510.CrossRefGoogle Scholar
44. Smith, D. J., Gaffney, E. A. & Blake, J. R. 2008 Modelling mucociliary clearance. Respir. Physiol. Neurobiol. 163, 178–188.Google Scholar
45. Taylor, G. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A Mat. 209, 447–461.Google Scholar
46. den Toonder, J., Bos, F., Broer, D., Filippini, L., Gillies, M., de Goede, J., Mol, T., Reijme, M., Talen, W., Wilderbeek, H., Khatavkar, V. & Anderson, P. 2007 Artificial cilia for active micro-fluidic mixing. Lab on a Chip 8, 533–541.Google Scholar
47. Tuck, E. O. 1968 A note on a swimming problem. J. Fluid Mech. 31, 305–308.Google Scholar
48. Uhlmann, M. 2005 An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209, 448–476.CrossRefGoogle Scholar
49. Vilfan, M., Potocnik, A., Kavcic, B., Osterman, N., Poberjaj, I., Vilfan, A. & Babic, D. 2009 Self-assembled artificial cilia. Proc. Natl Acad. Sci. 5 (5), 1844–1847.Google Scholar
50. Wesseling, P. 2001 Principles of Computational Fluid Dynamics. Springer.Google Scholar
51. Whitaker, S. 1996 The Forchheimer equation: a theoretical development. Transp. Porous Med. 25, 27–61.CrossRefGoogle Scholar
52. Whitaker, S. 1999 The Method of Volume Averaging. Kluwer.Google Scholar
53. Zhou, Z. G. & Liu, Z. W. 2008 Biomimetic cilia based on MEMS technology. J. Bionic Engng 5, 358–365.Google Scholar