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Mixing and transport by ciliary carpets: a numerical study

Published online by Cambridge University Press:  04 March 2014

Yang Ding ,
Janna C. Nawroth ,
Margaret J. McFall-Ngai  and
Eva Kanso
Yang Ding
Affiliation:
Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
Janna C. Nawroth
Affiliation:
Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
Margaret J. McFall-Ngai
Affiliation:
Medical Microbiology and Immunology, University of Wisconsin, Madison, WI 53706, USA
Eva Kanso*
Affiliation:
Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
*
†Email address for correspondence: kanso@usc.edu
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Abstract

We use a three-dimensional computational model to study the fluid transport and mixing due to the beating of an infinite array of cilia. In accord with recent experiments, we observe two distinct regions: a fluid transport region above the cilia and a fluid mixing region below the cilia tip. The metachronal wave due to phase differences between neighbouring cilia is known to enhance the fluid transport above the ciliary tip. In this work, we show that the metachronal wave also enhances the mixing rates in the sub-ciliary region, often simultaneously with the flow rate enhancement. Our results suggest that this simultaneous enhancement in transport and mixing is due to an enhancement in shear flow. As the flow above the cilia increases, the shear rate in the fluid increases and this shear enhances stretching, which is an essential ingredient for mixing. Estimates of the mixing time scale indicate that, compared to diffusion, the mixing due to the cilia beat may be significant and sometimes dominates chemical diffusion.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Mixing: Mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 743 , 25 March 2014 , pp. 124 - 140
Copyright
© 2014 Cambridge University Press 

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References

Ainley, J., Durkin, S., Embid, R., Boindala, P. & Cortez, R. 2008 The method of images for regularized Stokeslets. J. Comput. Phys. 227 (9), 4600–4616.CrossRefGoogle Scholar
Alexeev, A., Yeomans, J. M. & Balazs, A. C. 2008 Designing synthetic, pumping cilia that switch the flow direction in microchannels. Langmuir 24 (21), 12102–12106.Google Scholar
Altura, M. A., Heath-Heckman, E. A. C., Gillette, A., Kremer, N., Krachler, A., Brennan, C., Ruby, E. G., Orth, K. & McFall-Ngai, M. J. 2013 The first engagement of partners in the Euprymna scolopes-Vibrio fischeri symbiosis is a two-step process initiated by a few environmental symbiont cells. Environ. Microbiol. 15 (11), 2937–2950.CrossRefGoogle ScholarPubMed
Aref, H. 1990 Chaotic advection of fluid particles. Phil. Trans. R. Soc. Lond. A 333 (1631), 273–288.Google Scholar
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46 (1), 199–208.Google Scholar
Blake, J. 1972 A model for the micro-structure in ciliated organisms. J. Fluid Mech. 55 (1), 1–23.CrossRefGoogle Scholar
Blake, J. R. & Sleigh, M. A. 1974 Mechanics of ciliary locomotion. Biol. Rev. 49 (1), 85–125.CrossRefGoogle ScholarPubMed
Bloodgood, R. A. 2010 Sensory reception is an attribute of both primary cilia and motile cilia. J. Cell Sci. 123 (4), 505–509.Google Scholar
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9 (1), 339–398.CrossRefGoogle Scholar
Cortez, R. 2001 The method of regularized Stokeslets. SIAM J. Sci. Comput. 23 (4), 1204–1225.Google Scholar
Elgeti, J. & Gompper, G. 2013 Emergence of metachronal waves in cilia arrays. Proc. Natl Acad. Sci. USA 110 (12), 4470–4475.Google Scholar
Eloy, C. & Lauga, E. 2012 Kinematics of the most efficient cilium. Phys. Rev. Lett. 109 (3), 038101.Google Scholar
Fridlyand, L., Kaplan, A. & Reinhold, L. 1996 Quantitative evaluation of the role of a putative $\text {CO}_2$ -scavenging entity in the cyanobacterial $\text {CO}_2$ -concentrating mechanism. Biosystems 37 (3), 229–238.Google Scholar
Fulford, G. R. & Blake, J. R. 1986 Muco-ciliary transport in the lung. J. Theor. Biol. 121 (4), 381–402.CrossRefGoogle ScholarPubMed
Gauger, E. M., Downton, M. T. & Stark, H. 2009 Fluid transport at low Reynolds number with magnetically actuated artificial cilia. Eur. Phys. J. E 28 (2), 231–242.Google Scholar
Gueron, S. & Levit-Gurevich, K. 1999 Energetic considerations of ciliary beating and the advantage of metachronal coordination. Proc. Natl Acad. Sci. USA 96 (22), 12240–12245.Google Scholar
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 (12), 6001–6006.CrossRefGoogle ScholarPubMed
Gueron, S. & Liron, N. 1992 Ciliary motion modeling, and dynamic multicilia interactions. Biophys. J. 63 (4), 1045–1058.Google Scholar
Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (2), 317–328.Google Scholar
Ibañez-Tallon, I., Heintz, N. & Omran, H. 2003 To beat or not to beat: roles of cilia in development and disease. Human Molec. Genet. 12, 1, R27–R35.Google Scholar
Kelley, D. H. & Ouellette, N. T. 2011 Separating stretching from folding in fluid mixing. Nat. Phys. 7 (6), 477–480.CrossRefGoogle Scholar
Khaderi, S. N. & Onck, P. R. 2012 Fluid–structure interaction of three-dimensional magnetic artificial cilia. J. Fluid Mech. 708, 303–328.Google Scholar
Khaderi, S. N., den Toonder, J. M. J. & Onck, P. R. 2011 Microfluidic propulsion by the metachronal beating of magnetic artificial cilia: a numerical analysis. J. Fluid Mech. 688, 44–65.CrossRefGoogle Scholar
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 (8), 083605.Google Scholar
Kremer, N., Philipp, E. E. R., Carpentier, M., Brennan, C. A., Kraemer, L., Altura, M. A., Augustin, R., Häsler, R., Heath-Heckman, E. A. C. & Peyer, S. M. 2013 Initial symbiont contact orchestrates host-organ-wide transcriptional changes that prime tissue colonization. Cell Host. & Microbe 14 (2), 183–194.CrossRefGoogle ScholarPubMed
Leiderman, K., Bouzarth, E. L., Cortez, R. & Layton, A. T. 2013 A regularization method for the numerical solution of periodic Stokes flow. J. Comput. Phys. 236, 187–202.Google Scholar
Lukens, S., Yang, X. & Fauci, L. 2010 Using Lagrangian coherent structures to analyze fluid mixing by cilia. Chaos 20 (1), 017511.Google Scholar
Mathew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211 (1), 23–46.Google Scholar
Michelin, S. & Lauga, E. 2010 Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys. Fluids 22, 111901.CrossRefGoogle Scholar
Michelin, S. & Lauga, E. 2011 Optimal feeding is optimal swimming for all Péclet numbers. Phys. Fluids 23, 101901.CrossRefGoogle Scholar
Osterman, N. & Vilfan, A. 2011 Finding the ciliary beating pattern with optimal efficiency. Proc. Natl Acad. Sci. USA 108 (38), 15727–15732.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Otto, S. R., Yannacopoulos, A. N. & Blake, J. R. 2001 Transport and mixing in Stokes flow: the effect of chaotic dynamics on the blinking stokeslet. J. Fluid Mech. 430, 1–26.Google Scholar
Saltzman, W. M., Radomsky, M. L., Whaley, K. J. & Cone, R. A. 1994 Antibody diffusion in human cervical mucus. Biophys. J. 66 (2), 508–515.CrossRefGoogle ScholarPubMed
Satir, P. & Christensen, S. T. 2007 Overview of structure and function of mammalian cilia. Annu. Rev. Phys. 69, 377–400.Google Scholar
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. USA 107 (36), 15670–15675.CrossRefGoogle ScholarPubMed
Smith, D. J., Blake, J. R. & Gaffney, E. A. 2008 Fluid mechanics of nodal flow due to embryonic primary cilia. J. R. Soc. Interface 5 (22), 567–573.Google Scholar
Smith, D. J., Gaffney, E. A. & Blake, J. R. 2007 Discrete cilia modelling with singularity distributions: application to the embryonic node and the airway surface liquid. Bull. Math. Biol. 69 (5), 1477–1510.Google Scholar
Stone, Z. B. & Stone, H. A. 2005 Imaging and quantifying mixing in a model droplet micromixer. Phys. Fluids 17 (6), 063103.CrossRefGoogle Scholar
Supatto, W., Fraser, S. E. & Vermot, J. 2008 An all-optical approach for probing microscopic flows in living embryos. Biophys. J. 95 (4), L29–L31.CrossRefGoogle ScholarPubMed
Swaminathan, R., Hoang, C. P. & Verkman, A. S. 1997 Photobleaching recovery and anisotropy decay of green fluorescent protein GFP-S65T in solution and cells: cytoplasmic viscosity probed by green fluorescent protein translational and rotational diffusion. Biophys. J. 72 (4), 1900–1907.Google Scholar
Thiffeault, J. L., Gouillart, E. & Dauchot, O. 2011 Moving walls accelerate mixing. Phys. Rev. E 84 (3), 036313.Google Scholar
Wiggins, S. & Ottino, J. M. 2004 Foundations of chaotic mixing. Phil. Trans. R. Soc. Lond. A 362 (1818), 937–970.Google Scholar