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Effects of Cilia Movement on Fluid Velocity: II Numerical Solutions Over a Fixed Domain

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Abstract

Cilia are hair-like structures that move in unison with the purpose of propelling fluid. They are found, for example, in the human bronchiole respiratory system and molluscs. Here, we validate a novel model of fluid flow due to the movement of cilia in a fixed computational domain. We consider two domains, a porous medium and a free-fluid domain and numerically solve the Stokes–Brinkman system of equations where the cilia geometry and velocity are input and the velocity of fluid due to the movement of cilia is determined. The cilia velocities and geometry are approximated using human lung cilia experimental data available in the literature. We use a mixed finite element method of Taylor-Hood type to calculate the fluid velocities in a three-dimensional domain. The results are validated in a simple case by comparison with an exact solution with good agreement. This problem can be used as a benchmark for the movement of fluid phases due to the self-propelled movement of a solid phase in a porous medium.

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References

  • Ahmadil, E., Cortezl, R., Fujioka, H.: Boundary integral formulation for flows containing an interface between two porous media. J. Fluid Mech. 816, 71–93 (2017)

    Article  Google Scholar 

  • Barton, C., Raynor, S.: Analytical investigation of cilia induced mucus flow. Bull. Math. Biophys. 29, 419–428 (1967)

    Article  Google Scholar 

  • Bayliss, A., Turkel, E.: Outflow boundary conditions for fluid dynamics. SIAM J. Sci. Stat. Comput. 3(2), 250–259 (1982)

    Article  Google Scholar 

  • Berre, I., Doster, F., Keilegavlen, E.: Flow in fractured porous media: a review of conceptual models and discretization approaches. Transp. Porous Media 130, 215–236 (2019)

    Article  Google Scholar 

  • Bisgrove, B.W., Yost, H.J.: The roles of cilia in developmental disorders and disease. Development 133, 4131–4143 (2006). https://doi.org/10.1242/dev.02595

    Article  Google Scholar 

  • Braess, D.: Finite Elements Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  • Brokaw, C.J.: Bend propagation by a sliding filament model for flagella. J. Exp. Biol. 55(2), 289–304 (1971)

    Google Scholar 

  • Chamsri, K.: Modeling the flow of PCL fluid due to the movement of lung cilia. Ph.D. Thesis, University of Colorado Denver (2012)

  • Chamsri, K.: N-dimensional Stokes–Brinkman equations using a mixed finite element method. Aust. J. Basic Appl. Sci. 8, 30–36 (2014) (Special issue 2014)

    Google Scholar 

  • Chamsri, K.: Formulation of a well-posed Stokes–Brinkman problem with a permeability tensor. J. Math. 1, 1–7 (2015)

    Article  Google Scholar 

  • Chamsri, K., Bennethum, L.S.: Permeability of fluid flow through a periodic array of cylinders. Appl. Math. Model. 39, 244–254 (2015)

    Article  Google Scholar 

  • Chilvers, M., O’Callaghan, C.: Analysis of ciliary beat pattern and beat frequency using digital high speed imaging: comparison with the photomultiplier and photodiode methods. Thorax 55(4), 314–317 (2000). https://doi.org/10.1136/thorax.55.4.314

    Article  Google Scholar 

  • Christoph, G.: Numerical coupling of Navier–Stokes and Darcy flow for soil-water evaporation. Ph.D. Thesis, Universität Stuttgart (2017)

  • Costerton, J.W., Stewart, P.S., Greenberg, E.P.: Bacterial biofilms: a common cause of persistent infections. Science 284, 1318–1322 (1999)

    Article  Google Scholar 

  • Davenport, J.R., Watts, A.J., Roper, V.C., Croyle, M.J., Van Groen, T., Wyss, J.M., Nagy, T.R., Kesterson, R., Yoder, B.K.: Disruption of intraflagellar transport in adult mice leads to obesity and slow-onset cystic kidney disease. Curr. Biol. 17, 1586–1594 (2007)

    Article  Google Scholar 

  • Elgeti, J., Gompper, G.: Emergence of metachronal waves in cilia arrays. Proc. Natl. Acad. Sci. USA 110(12), 4470–4475 (2013)

    Article  Google Scholar 

  • Fulford, G.R., Blake, J.R.: Muco-ciliary transport in the lung. J. Theor. Biol. 121, 381–402 (1986)

    Article  Google Scholar 

  • Goyeau, B., Lhuillier, D., Gobin, D., Velarde, M.G.: Momentum transport at a fluid-porous interface. Int. J. Heat Mass Transf. 46, 4071–4081 (2003)

    Article  Google Scholar 

  • International Commission on Radiological Protection: Measurement: human respiratory tract model for radiological protection. ICRP Publication 66. Annals of the ICRP, vol. 24, p. 1 (1994)

  • Khelloufi, M.K., Loiseau, E., Jaeger, M., Molinari, N., Chanez, P., Gras, D., Viallat, A.: Spatiotemporal organization of cilia drives multiscale mmucus swirls in model human bronchial epithelium. Sci. Rep. 8(2447), 1–10 (2018). https://doi.org/10.1038/s41598-018-20882-4

    Article  Google Scholar 

  • Kiyota, K., Ueno, H., Numayama-Tsuruta, K., Haga, T., Imai, Y., Yamaguchi, T., Ishikawa, T.: Fluctuation of cilia-generated flow on the surface of the tracheal lumen. Am. J. Physiol. Lung Cell. Mol. Physiol. 306(2), L144–L151 (2014). https://doi.org/10.1152/ajplung.00117.2013

    Article  Google Scholar 

  • Koplik, J., Levine, H., Zee, A.: Viscosity renormalization in the Brinkman equation. Phys. Fluids 26(10), 2864–2870 (1983)

    Article  Google Scholar 

  • Lindemann, C.B., Lesich, K.A.: Flagellar and ciliary beating: the proven and the possible. J. Cell Sci. 123, 519–528 (2010)

    Article  Google Scholar 

  • Lowe, R.J., Shavit, U., Falter, J.L., Koseff, J.R., Monismith, S.G.: Modeling flow in coral communities with and without waves: a synthesis of porous media and canopy flow approaches. Limnol. Oceanogr. 53(6), 2668–2680 (2008)

    Article  Google Scholar 

  • Lu, J.G., Hwang, W.R.: On the interfacial flow over porous media composed of packed spheres: part 2-optimal Stokes–Brinkman coupling with effective Navier-Slip approach. Transp. Porous Media 132, 405–421 (2020)

    Article  Google Scholar 

  • Lyons, R.A., Saridogan, E., Djahanbakhch, O.: The reproductive significance of human Fallopian tube cilia. Hum. Reprod. Update 12, 363–372 (2006)

    Article  Google Scholar 

  • Machin, K.E.: Wave propagation along flagella. J. Exp. Biol. 35, 796–806 (1958)

    Google Scholar 

  • Marshall, W.F.: The cell biological basis of ciliary disease. Int. J. Cell Biol. 180, 17–21 (2008)

    Article  Google Scholar 

  • Martys, N.S., Hagedorn, J.G.: Multiscale modeling of fluid transport in heterogeneous materials using discrete Boltzmann methods. Mater. Struct. 35, 650–659 (2002)

    Article  Google Scholar 

  • Mitran, S.: Metachronal wave formation in a model of pulmonary cilia. Comput. Struct. 85(11–14), 763–774 (2007)

    Article  Google Scholar 

  • Motokawa, T., Satir, P.: Laser-induced spreading arrest of MYTILUS gill cilia. J. Cell Biol. 66, 377–391 (1975)

    Article  Google Scholar 

  • Ochoa-Tapia, A.J., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid—I. Theoretical development. Int. J. Heat Mass Transf. 38(14), 2635–2646 (1995a)

    Article  Google Scholar 

  • Ochoa-Tapia, J.A., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid—II. Comparison with experiment. Int. J. Heat Mass Transf. 38(14), 2647–2655 (1995b)

    Article  Google Scholar 

  • Osterman, N., Vilfan, A.: Finding the ciliary beating pattern with optimal efficiency. Proc. Natl. Acad. Sci. USA 108(38), 15727–15732 (2011)

    Article  Google Scholar 

  • Papanastasiou, T., Malamataris, N., Ellwood, K.: A new outflow boundary condition. Int. J. Numer. Methods Fluids 14, 587–608 (1992)

    Article  Google Scholar 

  • Schöberl, J.: Netgen. http://www.hpfem.jku.at/netgen/. Automatic mesh generator (2001)

  • Sears, P.R., Thompson, K., Knowles, M.R., Davis, C.W.: Human airway ciliary dynamics. Am. J. Physiol. Lung Cell. Mol. Physiol. 304(3), L170–L183 (2012). https://doi.org/10.1152/ajplung.00105.2012

    Article  Google Scholar 

  • Smith, D.J., Gaffney, E.A., Blake, J.R.: A viscoelastic traction layer model of muco-ciliary transport. Bull. Math. Biol. 69, 289–327 (2007)

    Article  Google Scholar 

  • Tan, H., Pillai, K.M.: Finite element implementation of stress-jump and stress-continuity conditions at porous-medium, clear-fluid interface. Comput. Fluids 38, 1118–1131 (2009)

    Article  Google Scholar 

  • Wuttanachamsri, K., Schreyer, L.: Derivation of fluid flow due to a moving solid in a porous medium framework. arXiv:submit/3117498 math.NA (2020)

  • Xu, L., Jiang, Y.: Cilium height difference between strokes is more effective in driving fluid transport in mucociliary clearance: a numerical study. Math. Biosci. Eng. 12(5), 1107–1126 (2015)

    Article  Google Scholar 

  • Xu, L., Jiang, Y.: Mathematical modeling of mucociliary clearance: a mini-review. Cells 8(736), 1–15 (2019)

    Google Scholar 

  • Yang, X., Dillon, R.H., Fauci, L.J.: An integrative computational model of multiciliary beating. Bull. Math. Biol. 70, 1192–1215 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by a grant from National Research Council of Thailand (NRCT).

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Authors and Affiliations

  1. King Mongkut’s Institute of Technology Ladkrabang, Bangkok, 10520, Thailand

    Kanognudge Wuttanachamsri

  2. Washington State University, Pullman, WA, 99164-3113, USA

    Lynn Schreyer

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  1. Kanognudge Wuttanachamsri
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  2. Lynn Schreyer
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Correspondence to Kanognudge Wuttanachamsri.

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Wuttanachamsri, K., Schreyer, L. Effects of Cilia Movement on Fluid Velocity: II Numerical Solutions Over a Fixed Domain. Transp Porous Med 134, 471–489 (2020). https://doi.org/10.1007/s11242-020-01455-4

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  • DOI: https://doi.org/10.1007/s11242-020-01455-4

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