Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-30T21:33:30.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Squirming motion in a Brinkman medium

Published online by Cambridge University Press:  19 September 2018

Herve Nganguia  and
On Shun Pak
Herve Nganguia
Affiliation:
Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA 95053, USA
On Shun Pak*
Affiliation:
Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA 95053, USA
*
†Email address for correspondence: opak@scu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Micro-organisms encounter heterogeneous viscous environments consisting of networks of obstacles embedded in a viscous fluid medium. In this paper we analyse the characteristics of swimming in a porous medium modelled by the Brinkman equation via a spherical squirmer model. The idealized geometry allows an analytical and exact solution of the flow surrounding a squirmer. The propulsion speed obtained agrees with previous results using the Lorentz reciprocal theorem. Our analysis extends these results to calculate the power dissipation and hence the swimming efficiency of the squirmer in a Brinkman medium. The analytical solution enables a systematic analysis of the structure of the flow surrounding the squirmer, which can be represented in terms of singularities in Brinkman flows. We also discuss the spatial decay of flows due to squirming motion in a Brinkman medium in comparison with the decay in a purely viscous fluid. The results lay the foundation for subsequent studies on hydrodynamic interactions, nutrient transport and uptake by micro-organisms in heterogeneous viscous environments.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Biological Fluid Dynamics: Micro-organism dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 554 - 573
Check for updates
Copyright
© 2018 Cambridge University Press 

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.)

References

Berg, H. C. & Turner, L. 1979 Movement of microorganisms in viscous environments. Nature 278, 349–351.Google Scholar
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208.Google Scholar
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398.Google Scholar
Brinkman, H. C. 1949 Calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27–34.Google Scholar
Celli, J. P., Turner, B. S., Afdhal, N. H., Keates, S., Ghiran, I., Kelly, C. P., Ewoldt, R. H., McKinley, G. H., So, P., Erramilli, S. & Bansil, R. 2009 Helicobacter pylori moves through mucus by reducing mucin viscoelasticity. Proc. Natl Acad. Sci. USA 106, 14321–14326.Google Scholar
Chattopadhyay, S., Moldovan, R., Yeung, C. & Wu, X. L. 2006 Swimming efficiency of bacterium Escherichia coli . Proc. Natl Acad. Sci. USA 103, 13712–13717.Google Scholar
Childress, S. 1972 Viscous flow past a random array of spheres. J. Chem. Phys. 56, 2527–2539.Google Scholar
Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.Google Scholar
Childress, S. 2012 A thermodynamic efficiency for stokesian swimming. J. Fluid Mech. 705, 77–97.Google Scholar
Chisholm, N. G., Legendre, D., Lauga, E. & Khair, A. S. 2016 A squirmer across Reynolds numbers. J. Fluid Mech. 796, 233–256.Google Scholar
Cortez, R., Cummins, B., Leiderman, K. & Varela, D. 2010 Computation of three-dimensional Brinkman flows using regularized methods. J. Comput. Phys. 229, 7609–7624.Google Scholar
Datt, C., Zhu, L., Elfring, G. J. & Pak, D. S. 2015 Squirming through shear-thinning fluids. J. Fluid Mech. 784, R1.Google Scholar
De Corato, M., Greco, F. & Maffettone, P. L. 2015 Locomotion of a microorganism in weakly viscoelastic liquids. Phys. Rev. E 92, 053008.Google Scholar
Doostmohammadi, A., Stocker, R. & Ardekani, A. M. 2012 Low-Reynolds-number swimming at pycnoclines. Proc. Natl Acad. Sci. USA 109 (10), 3856–3861.Google Scholar
Drescher, K., Goldstein, R. E. & Tuval, I. 2010 Fidelity of adaptive phototaxis. Proc. Natl Acad. Sci. USA 107, 11171–11176.Google Scholar
Drescher, K., Leptos, K. C., Tuval, I., Ishikawa, T., Pedley, T. J. & Goldstein, R. E. 2009 Dancing Volvox: hydrodynamic bound states of swimming algae. Phys. Rev. Lett. 102, 168101.Google Scholar
Durlofsky, L. & Brady, J. F. 1987 Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30, 3329–3341.Google Scholar
Ebbens, S. J. & Howse, J. R. 2010 In pursuit of propulsion at the nanoscale. Soft Matt. 6, 726–738.Google Scholar
Fauci, L. J. & Dillon, R. 2006 Biofluidmechanics of reproduction. Annu. Rev. Fluid Mech. 38, 371–394.Google Scholar
Feng, J., Ganatos, P. & Weinbaum, S. 1998 Motion of a sphere near planar confining boundaries in a Brinkman medium. J. Fluid Mech. 375, 265–296.Google Scholar
Fu, H. C., Shenoy, V. B. & Powers, T. R. 2010 Low-Reynolds-number swimming in gels. Europhys. Lett. 91, 24002.Google Scholar
Gray, J. & Hancock, G. J. 1955 The propulsion of sea-urchin spermatozoa. J. Expl Biol. 32, 802–814.Google Scholar
Hancock, G. J. 1953 The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. Lond. A 217, 96–121.Google Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media. Noordhoff International.Google Scholar
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695–720.Google Scholar
Ho, N. & Olson, S. D. 2016 Swimming speeds of filaments in viscous fluids with resistance. Phys. Rev. E 93, 043108.Google Scholar
Howells, I. D. 1974 Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64, 449–475.Google Scholar
Ishikawa, T. & Hota, M. 2006 Interaction of two swimming paramecia. J. Expl Biol. 209, 4452–4463.Google Scholar
Ishikawa, T. & Pedley, T. J. 2008 Coherent structures in monolayers of swimming particles. Phys. Rev. Lett. 100, 088103.Google Scholar
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160.Google Scholar
Ishimoto, K. & Gaffney, E. A. 2014 Swimming efficiency of spherical squirmers: beyond the Lighthill theory. Phys. Rev. E 90, 012704.Google Scholar
Jabbarzadeh, M., Hyon, Y. & Fu, H. C. 2014 Swimming fluctuations of micro-organisms due to heterogeneous microstructure. Phys. Rev. E 90, 043021.Google Scholar
Jung, S. 2010 Caenorhabditis elegans swimming in a saturated particulate system. Phys. Fluids 22, 031903.Google Scholar
Koch, D. L. & Brady, J. F. 1985 Dispersion in fixed beds. J. Fluid Mech. 154, 399–427.Google Scholar
Koch, D. L., Hill, R. J. & Sangani, A. S. 1998 Brinkman screening and the covariance of the fluid velocity in fixed beds. Phys. Fluids 10, 3035–3037.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Leiderman, K. & Olson, S. D. 2016 Swimming in a two-dimensional Brinkman fluid: computational modeling and regularized solutions. Phys. Fluids 28, 021902.Google Scholar
Leshansky, A. M. 2009 Enhanced low-Reynolds number propulsion in heterogeneous viscous environments. Phys. Rev. E 80, 051911.Google Scholar
Leshansky, A. M., Kenneth, O., Gat, O. & Avron, J. E. 2007 A frictionless microswimmer. New J. Phys. 9, 145.Google Scholar
Li, G.-J. & Ardekani, A. M. 2014 Hydrodynamic interaction of microswimmers near a wall. Phys. Rev. E 90, 013010.Google Scholar
Lighthill, J. 1975 Mathematical Biofluiddynamics. SIAM.Google Scholar
Lighthill, M. J. 1952 On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Maths 5, 109–118.Google Scholar
Long, D. & Ajdari, A. 2001 A note on the screening of hydrodynamic interactions, in electrophoresis, and in porous media. Eur. Phys. J. E 4 (1), 29–32.Google Scholar
Magar, V., Goto, T. & Pedley, T. J. 2003 Nutrient uptake by a self-propelled steady squirmer. Q. J. Mech. Appl. Maths 56, 65–91.Google Scholar
Marchetti, M. C., Joanny, J. F., Ramaswamy, S., Liverpool, T. B., Prost, J., Rao, M. & Simha, R. A. 2013 Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189.Google Scholar
Michelin, S. & Lauga, E. 2010 Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys. Fluids 22, 111901.Google Scholar
Michelin, S. & Lauga, E. 2011 Optimal feeding is optimal swimming for all Péclet numbers. Phys. Fluids 23, 101901.Google Scholar
Mirbagheri, S. A. & Fu, H. C. 2016 Helicobacter pylori couples motility and diffusion to actively create a heterogeneous complex medium in gastric mucus. Phys. Rev. Lett. 116, 198101.Google Scholar
Nelson, B. J., Kaliakatsos, I. K. & Abbott, J. J. 2010 Microrobots for minimally invasive medicine. Annu. Rev. Biomed. Engng 12, 55–85.Google Scholar
Pak, O. S. & Lauga, E. 2014 Generalized squirming motion of a sphere. J. Engng Maths 88, 1–28.Google Scholar
Palaniappan, D. 2014 On some general solutions of transient Stokes and Brinkman equations. J. Theor. Appl. Mech. 52, 405–415.Google Scholar
Pedley, T. J. 2016 Spherical squirmers: models for swimming micro-organisms. IMA J. Appl. Maths. 81, 488–521.Google Scholar
Radolf, J. & Lukehart, S. 2006 Pathogenic Treponema: Molecular and Cellular Biology. Caister Academic Press.Google Scholar
Rutllant, J., Lopez-Bejar, M. & Lopez-Gatius, F. 2005 Ultrastructural and rheological properties of bovine vaginal fluid and its relation to sperm motility fertilization: a review. Rep. Dom. Anim. 40, 79–86.Google Scholar
Saintillan, D. 2018 Rheology of active fluids. Annu. Rev. Fluid Mech. 50, 563–592.Google Scholar
Saintillan, D. & Shelley, M. J. 2015 Theory of Active Suspensions, pp. 319–355. Springer.Google Scholar
Saltzman, W. M., Radomsky, M. L., Whaley, K. J. & Cone, R. A. 1994 Antibody diffusion in human cervical mucus. Biophys. J. 66, 508–515.Google Scholar
Schmitt, M. & Stark, H. 2016 Marangoni flow at droplet interfaces: three-dimensional solution and applications. Phys. Fluids 28, 012106.Google Scholar
Siddiqui, A. M. & Ansari, A. R. 2003 An analysis of the swimming problem of a singly flagellated microorganism in a fluid flowing through a porous medium. J. Porous Media 6, 235–241.Google Scholar
Stone, H. A. & Samuel, A. D. T. 1996 Propulsion of microorganisms by surface distortions. Phys. Rev. Lett. 77, 4102.Google Scholar
Tam, C. K. W. 1969 The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38, 537–546.Google Scholar
Tam, D. & Hosoi, A. E. 2007 Optimal stroke patterns for Purcell’s three-link swimmer. Phys. Rev. Lett. 98, 068105.Google Scholar
Tamm, S. L. 1972 Ciliary motion in paramecium: a scanning electron microscope study. J. Cell Biol. 55, 250–255.Google Scholar
Taylor, G. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461.Google Scholar
Tecon, R. & Or, D. 2016 Bacterial flagellar motility on hydrated rough surfaces controlled by aqueous film thickness and connectedness. Sci. Rep. 6, 19409.Google Scholar
Wang, S. & Ardekani, A. 2012 Inertial squirmer. Phys. Fluids 24, 101902.Google Scholar
Wiezel, O. & Or, Y. 2016 Optimization and small-amplitude analysis of Purcell’s three-link microswimmer model. Proc. R. Soc. Lond. A 472, 20160425.Google Scholar
Wolgemuth, C. W. 2015 Flagellar motility of the pathogenic spirochetes. Semin. Cell Dev. Biol. 46, 104–112.Google Scholar
Yazdi, S., Ardekani, A. M. & Borhan, A. 2015 Swimming dynamics near a wall in a weakly elastic fluid. J. Nonlinear Sci. 25, 1153–1167.Google Scholar
Zhu, L., Lauga, E. & Brandt, L. 2012 Self-propulsion in viscoelastic fluids: pushers versus pullers. Phys. Fluids 24, 051902.Google Scholar
Zlatanovski, T. 1999 Axisymmetric creeping flow past a porous prolate spheroidal particle using the Brinkman model. Q. J. Mech. Appl. Maths 52, 111–126.Google Scholar
Zöttl, A. & Stark, H. 2012 Nonlinear dynamics of a microswimmer in Poiseuille flow. Phys. Rev. Lett. 108, 218104.Google Scholar