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

  • Franz Hoffmann (a1) and Ricardo Cortez (a1)
Abstract
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.

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Corresponding author
*Corresponding author. Email addresses: fhoffma@tulane.edu (F. Hoffmann), rcortez@tulane.edu (R. Cortez)
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Communicated by Boo-Cheong Khoo

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[1] Ainley J., Durkin S., Embid R., Boindala P., and Cortez R.. The method of images for regularized Stokeslets. J. Comp. Phy, 227:46004616, 2008.
[2] Beenakker C. W. J.. Ewald sum of the Rotne-Prager tensor. J. Chem. Phys, 85(3), 1986.
[3] Blake J. R.. A note on the image system for a Stokeslet in a no-slip boundary. Proc. Cambridge Phil. Soc, 70:303310, 1971.
[4] Blake J. R.. A spherical envelope approach to ciliary propulsion. J. FluidMech, 46(1):199208, 1971.
[5] Blake J. R.. A model for the micro-structure in ciliated organisms. J. Fluid Mech, 55(1):123, 1972.
[6] Breunig J. J., Arellano J. I., and Rakic P.. Cilia in the brain: going with the flow. Nat. Neurosci, 13(6):654655, 2010.
[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, 13:28342839, 2013.
[8] Cortez R.. The method of regularized Stokeslets. SIAM J. Sci. Comp, 23(4):12041225, 2001.
[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, 17:114, 2005.
[10] Cortez R. and Hoffmann F.. A fast numerical method for computing doubly-priodic regularized Stokes flow in 3D. J. Comp. Phy, 258, 2014.
[11] Cortez R. and Nicholas M.. Slender body theory for stokes flows with regularized forces. Comm. App.Math. Com. Sc, 7(1):3362, 2012.
[12] Cortez R. and Varela D.. A general system of images for regularized stokeslets and other elements near a plane wall. J. Comp. Phy, 285:4154, 2015.
[13] den Toonder J. M. J. and Onck P. R.. Microfluidic manipulation with artificial/bioinspired cilia. Trends Biotechnol, 31(2):8591, 2013.
[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, 743:124140, 2014.
[15] Downton M. T. and Stark H.. Beating kinematics of magneticallly actuated cilia. EPL, 85(4):44002–p1–p6, 2009.
[16] Elgeti J. and Gompper G.. Emergence of metachronal waves in cilia arrays. Proc. Nat. A. Sci, 110(12):44704475, 2013.
[17] Fulford G. R. and Blake J. R.. Muco-ciliary transport in the lung. Theor. Biol, 121(4):381402, 1986.
[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, 28:231242, 2009.
[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, 74:16581676, 1998.
[20] Gueron S. and Levit-Gurevich K.. Energetic considerations of ciliary beating and the advantage of metachronal coordination. Proc. Natl. Acad. Sci, 96(22):1224012245, 1999.
[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.
[22] Hunter J. D.. Matplotlib: A 2D graphics environment. Comput. Sci. Eng, 9(3):9095, 2007.
[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, 11:20172022, 2011.
[24] Jones E., Oliphant T., Peterson Pearu, et al. SciPy: Open source scientific tools for Python, 2001.
[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, 8:53425349, 2012.
[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, 688:4465, 2011.
[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.
[28] Lenz P. and Ryskin A.. Collective effects in ciliar arrays. Phys. Bio, 3:285294, 2006.
[29] Liron N. and Mochon S.. The discrete-cilia approach to propulsion of ciliated micro-organisms. J. Fluid Mech, 75(3):593607, 1975.
[30] Marshall W. F. and Nonaka S.. Cilia: Tuning in to the cell's antenna. Curr. Bio, 16(15):604614, 2006.
[31] Mitran S. M.. Metachronal wave formation in a model of pulmonary cilia. Comput. Struct, 85:763774, 2007.
[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, 297:442461, 2015.
[33] Osterman N. and Vilfan A.. Finding the ciliary beating pattern with optimal efficency. Proc. Nat. A. Sci, 108(38):1572715732, 2011.
[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):1567015675, 2010.
[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, 5:567573, 2008.
[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, 69:14771510, 2007.
[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):18441847, 2010.
[38] Wang Y., Gao Y., Wyss H., Anderson P., and den Tonder J.. Out of the cleanroom, self-assembled magnetic artificial cilia. Lab Chip, 13:33603366, 2013.
[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.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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