Skip to main content Accessibility help

Mixing and transport by ciliary carpets: a numerical study

  • Yang Ding (a1), Janna C. Nawroth (a2), Margaret J. McFall-Ngai (a3) and Eva Kanso (a1)


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.


Corresponding author

Email address for correspondence:


Hide All
Ainley, J., Durkin, S., Embid, R., Boindala, P. & Cortez, R. 2008 The method of images for regularized Stokeslets. J. Comput. Phys. 227 (9), 46004616.
Alexeev, A., Yeomans, J. M. & Balazs, A. C. 2008 Designing synthetic, pumping cilia that switch the flow direction in microchannels. Langmuir 24 (21), 1210212106.
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), 29372950.
Aref, H. 1990 Chaotic advection of fluid particles. Phil. Trans. R. Soc. Lond. A 333 (1631), 273288.
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46 (1), 199208.
Blake, J. 1972 A model for the micro-structure in ciliated organisms. J. Fluid Mech. 55 (1), 123.
Blake, J. R. & Sleigh, M. A. 1974 Mechanics of ciliary locomotion. Biol. Rev. 49 (1), 85125.
Bloodgood, R. A. 2010 Sensory reception is an attribute of both primary cilia and motile cilia. J. Cell Sci. 123 (4), 505509.
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9 (1), 339398.
Cortez, R. 2001 The method of regularized Stokeslets. SIAM J. Sci. Comput. 23 (4), 12041225.
Elgeti, J. & Gompper, G. 2013 Emergence of metachronal waves in cilia arrays. Proc. Natl Acad. Sci. USA 110 (12), 44704475.
Eloy, C. & Lauga, E. 2012 Kinematics of the most efficient cilium. Phys. Rev. Lett. 109 (3), 038101.
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), 229238.
Fulford, G. R. & Blake, J. R. 1986 Muco-ciliary transport in the lung. J. Theor. Biol. 121 (4), 381402.
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), 231242.
Gueron, S. & Levit-Gurevich, K. 1999 Energetic considerations of ciliary beating and the advantage of metachronal coordination. Proc. Natl Acad. Sci. USA 96 (22), 1224012245.
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), 60016006.
Gueron, S. & Liron, N. 1992 Ciliary motion modeling, and dynamic multicilia interactions. Biophys. J. 63 (4), 10451058.
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), 317328.
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, R27R35.
Kelley, D. H. & Ouellette, N. T. 2011 Separating stretching from folding in fluid mixing. Nat. Phys. 7 (6), 477480.
Khaderi, S. N. & Onck, P. R. 2012 Fluid–structure interaction of three-dimensional magnetic artificial cilia. J. Fluid Mech. 708, 303328.
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, 4465.
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.
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), 183194.
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, 187202.
Lukens, S., Yang, X. & Fauci, L. 2010 Using Lagrangian coherent structures to analyze fluid mixing by cilia. Chaos 20 (1), 017511.
Mathew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211 (1), 2346.
Michelin, S. & Lauga, E. 2010 Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys. Fluids 22, 111901.
Michelin, S. & Lauga, E. 2011 Optimal feeding is optimal swimming for all Péclet numbers. Phys. Fluids 23, 101901.
Osterman, N. & Vilfan, A. 2011 Finding the ciliary beating pattern with optimal efficiency. Proc. Natl Acad. Sci. USA 108 (38), 1572715732.
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
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, 126.
Saltzman, W. M., Radomsky, M. L., Whaley, K. J. & Cone, R. A. 1994 Antibody diffusion in human cervical mucus. Biophys. J. 66 (2), 508515.
Satir, P. & Christensen, S. T. 2007 Overview of structure and function of mammalian cilia. Annu. Rev. Phys. 69, 377400.
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), 1567015675.
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), 567573.
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), 14771510.
Stone, Z. B. & Stone, H. A. 2005 Imaging and quantifying mixing in a model droplet micromixer. Phys. Fluids 17 (6), 063103.
Supatto, W., Fraser, S. E. & Vermot, J. 2008 An all-optical approach for probing microscopic flows in living embryos. Biophys. J. 95 (4), L29L31.
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), 19001907.
Thiffeault, J. L., Gouillart, E. & Dauchot, O. 2011 Moving walls accelerate mixing. Phys. Rev. E 84 (3), 036313.
Wiggins, S. & Ottino, J. M. 2004 Foundations of chaotic mixing. Phil. Trans. R. Soc. Lond. A 362 (1818), 937970.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed