Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T20:07:23.481Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Computation of Doubly-Periodic Stokes Flow Bounded by a Plane with Applications to Nodal Cilia

Part of: Biological fluid mechanics Incompressible viscous fluids Biology and other natural sciences

Published online by Cambridge University Press:  06 July 2017

Franz Hoffmann  and
Ricardo Cortez
Franz Hoffmann*
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
Ricardo Cortez*
Affiliation:
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
*
*Corresponding author. Email addresses:fhoffma@tulane.edu (F. Hoffmann), rcortez@tulane.edu (R. Cortez)
*Corresponding author. Email addresses:fhoffma@tulane.edu (F. Hoffmann), rcortez@tulane.edu (R. Cortez)
Get access

Abstract

A numerical method is presented for the computation of externally forced Stokes flows bounded by the plane z=0 and satisfying periodic boundary conditions in the x and y directions. The motivation for this work is the simulation of flows generated by cilia, which are hair-like structures attached to the surface of cells that generate flows through coordinated beating. Large collections of cilia on a surface can be modeled using a doubly-periodic domain. The approach presented here is to derive a regularized version of the fundamental solution of the incompressible Stokes equations in Fourier space for the periodic directions and physical space for the z direction. This analytical expression for û(k,m;z) can then be used to compute the fluid velocity u(x,y,z) via a two-dimensional inverse fast Fourier transform for any fixed value of z. Repeating the computation for multiple values of z leads to the fluid velocity on a uniform grid in physical space. The zero-flow condition at the plane z=0 is enforced through the use of images. The performance of the method is illustrated by numerical examples of particle transport by nodal cilia, which verify optimal particle transport for parameters consistent with previous studies. The results also show that for two cilia in the periodic box, out-of-phase beating produces considerablemore particle transport than in-phase beating.

Keywords

Stokes flowEwald summationFFT

MSC classification

Secondary: 76D07: Stokes and related (Oseen, etc.) flows 76Z99: None of the above, but in this section 92-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 620 - 642
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Boo-Cheong Khoo

References

[1] Ainley, J., Durkin, S., Embid, R., Boindala, P., and Cortez, R.. The method of images for regularized Stokeslets. J. Comp. Phy, 227:4600–4616, 2008.CrossRefGoogle Scholar
[2] Beenakker, C. W. J.. Ewald sum of the Rotne-Prager tensor. J. Chem. Phys, 85(3), 1986.Google Scholar
[3] Blake, J. R.. A note on the image system for a Stokeslet in a no-slip boundary. Proc. Cambridge Phil. Soc, 70:303–310, 1971.Google Scholar
[4] Blake, J. R.. A spherical envelope approach to ciliary propulsion. J. FluidMech, 46(1):199–208, 1971.CrossRefGoogle Scholar
[5] Blake, J. R.. A model for the micro-structure in ciliated organisms. J. Fluid Mech, 55(1):1–23, 1972.CrossRefGoogle Scholar
[6] Breunig, J. J., Arellano, J. I., and Rakic, P.. Cilia in the brain: going with the flow. Nat. Neurosci, 13(6):654–655, 2010.Google Scholar
[7] Chen, C-Y., Chen, C-Y., Lin, C-Y., and Hu, Y-T.. Magnetically actuated artificial cilia for optimum mixing performance in microfluidics. Lab Chip, 13:2834–2839, 2013.Google Scholar
[8] Cortez, R.. The method of regularized Stokeslets. SIAM J. Sci. Comp, 23(4):1204–1225, 2001.Google Scholar
[9] Cortez, R., Fauci, L., and Medovikov, A.. The method of regularized Stokeslets in Three Dimensions: Analysis,Validation and Application to Helical Swimming. Phys. Fluids, 17:1–14, 2005.Google Scholar
[10] Cortez, R. and Hoffmann, F.. A fast numerical method for computing doubly-priodic regularized Stokes flow in 3D. J. Comp. Phy, 258, 2014.Google Scholar
[11] Cortez, R. and Nicholas, M.. Slender body theory for stokes flows with regularized forces. Comm. App.Math. Com. Sc, 7(1):33–62, 2012.Google Scholar
[12] Cortez, R. and Varela, D.. A general system of images for regularized stokeslets and other elements near a plane wall. J. Comp. Phy, 285:41–54, 2015.CrossRefGoogle Scholar
[13] den Toonder, J. M. J. and Onck, P. R.. Microfluidic manipulation with artificial/bioinspired cilia. Trends Biotechnol, 31(2):85–91, 2013.Google Scholar
[14] Ding, Y., Nawroth, J. C., McFall-Ngai, M. J., and Kanso, E.. Mixing and transport by ciliary carpets: a numerical study. J. Fluid Mech, 743:124–140, 2014.CrossRefGoogle Scholar
[15] Downton, M. T. and Stark, H.. Beating kinematics of magneticallly actuated cilia. EPL, 85(4):44002–p1–p6, 2009.CrossRefGoogle Scholar
[16] Elgeti, J. and Gompper, G.. Emergence of metachronal waves in cilia arrays. Proc. Nat. A. Sci, 110(12):4470–4475, 2013.Google Scholar
[17] Fulford, G. R. and Blake, J. R.. Muco-ciliary transport in the lung. Theor. Biol, 121(4):381–402, 1986.Google Scholar
[18] Gauger, E. M., Downton, M. T., and Stark, H.. Fluid transport at low Reynolds number with magnetically actuated artificial cilia. Eur. Phys. J. E, 28:231–242, 2009.CrossRefGoogle ScholarPubMed
[19] Gueron, S. and Levit-Gurevich, K.. Computation of the internal forces in cilia: Application to ciliary motion, the effects of viscosity, and cilia interactions. Biophy. J, 74:1658–1676, 1998.CrossRefGoogle ScholarPubMed
[20] Gueron, S. and Levit-Gurevich, K.. Energetic considerations of ciliary beating and the advantage of metachronal coordination. Proc. Natl. Acad. Sci, 96(22):12240–12245, 1999.Google Scholar
[21] Gueron, S., Levit-Gurevich, K., Liron, N., and Blum, J. J.. Cilia internal mechanism and metachronal coordination as the result of hydrodynamical coupling. Proc. Natl. Acad. Sci, 94.Google Scholar
[22] Hunter, J. D.. Matplotlib: A 2D graphics environment. Comput. Sci. Eng, 9(3):90–95, 2007.Google Scholar
[23] Hussong, J., Schorr, N., Belardi, J., Prucker, O., Rheb, J., and Westerweel, J.. Experimental investigation of the flow induced by artificial cilia. Lab Chip, 11:2017–2022, 2011.CrossRefGoogle ScholarPubMed
[24] Jones, E., Oliphant, T., Peterson, Pearu, et al. SciPy: Open source scientific tools for Python, 2001.Google Scholar
[25] Keißner, A. and Brücker, C.. Directional fluid transport along artificial ciliary surfaces with base-layer actuation of counter-rotating orbital beating patterns. Soft Matter, 8:5342–5349, 2012.Google Scholar
[26] Khaderi, S. N., den Toonder, J. M. J., and Onck, P. R.. Microfluidic propulsion by the metachronal beating of magnetic artificial cilia: a numerical analysis. J. Fluid Mech, 688:44–65, 2011.Google Scholar
[27] Kokot, G., Vilfan, M., Osterman, N., Vilfan, A., Kavčič, B., Poberaj, I., and Babič, D.. Measurement of fluid flow generated by artificial cilia. Biomicrofluidics, 5:034103, 2011.CrossRefGoogle ScholarPubMed
[28] Lenz, P. and Ryskin, A.. Collective effects in ciliar arrays. Phys. Bio, 3:285–294, 2006.CrossRefGoogle ScholarPubMed
[29] Liron, N. and Mochon, S.. The discrete-cilia approach to propulsion of ciliated micro-organisms. J. Fluid Mech, 75(3):593–607, 1975.Google Scholar
[30] Marshall, W. F. and Nonaka, S.. Cilia: Tuning in to the cell's antenna. Curr. Bio, 16(15):604–614, 2006.Google Scholar
[31] Mitran, S. M.. Metachronal wave formation in a model of pulmonary cilia. Comput. Struct, 85:763–774, 2007.CrossRefGoogle Scholar
[32] Nguyen, H-N. and Leiderman, K.. Computation of the singular and regularized image systems for doubly-periodic stokes flow in the presence of a wall. J. of Comp. Phy, 297:442–461, 2015.Google Scholar
[33] Osterman, N. and Vilfan, A.. Finding the ciliary beating pattern with optimal efficency. Proc. Nat. A. Sci, 108(38):15727–15732, 2011.CrossRefGoogle Scholar
[34] Shields, A. R., Fiser, B. L., Evans, B. A., Falco, M. R., Washburn, S., and Superfine, R.. Biometric cilia arrays generate simultaneous pumping and mixing regimes. Proc. Nat. A. Sci, 107(36):15670–15675, 2010.CrossRefGoogle Scholar
[35] Smith, D.J., Blake, J. R., and Gaffney, E. A.. Fluid mechanics of nodal flow due to embryonic primary cilia. J. R. Soc. Interface, 5:567–573, 2008.Google Scholar
[36] Smith, D.J., Gaffney, E. A., and Blake, J. R.. Discrete cilia modelling with singularity distributions: application to the embryonic node and the airway surface liquid. Bull. Math. Bio, 69:1477–1510, 2007.CrossRefGoogle Scholar
[37] Villfan, M., Potočnik, A., Kavačič, B., Osterman, N., Poberaj, I., Vilfan, A., and Babič, D.. Self-assembled artificial cilia. Proc. Nat. A. Sci, 107(5):1844–1847, 2010.Google Scholar
[38] Wang, Y., Gao, Y., Wyss, H., Anderson, P., and den Tonder, J.. Out of the cleanroom, self-assembled magnetic artificial cilia. Lab Chip, 13:3360–3366, 2013.CrossRefGoogle ScholarPubMed
[39] Wang, Y., Gao, Y., Wyss, H. M., Anderson, P. D., and den Toonder, J. M. J.. Artificial cilia fabricated usingmagnetic fiber drawing generate substantial fluid flow. Microfluid Nanofluid, 2014.CrossRefGoogle Scholar