Skip to main content Accessibility help

Nutrient uptake in a suspension of squirmers

  • Takuji Ishikawa (a1) (a2), Shunsuke Kajiki (a2), Yohsuke Imai (a1) and Toshihiro Omori (a1)


Nutrient uptake is one of the most important factors in cell growth. Despite the biological importance, little is known about the effect of cell–cell hydrodynamic interactions on nutrient uptake in a suspension of swimming micro-organisms. In this study, we numerically investigate the nutrient uptake in an infinite suspension of squirmers. In the dilute limit, our results are in good agreement with a previous study by Magar et al. (Q. J. Mech. Appl. Maths, vol. 56, 2003, pp. 65–91). When we increased the volume fraction of squirmers, the nutrient uptake of individual cells was enhanced by the hydrodynamic interactions. The average nutrient concentration in the suspension decayed exponentially as a function of time, and the relaxation time could be scaled using the Sherwood number, the Péclet number and the volume fraction of cells. We propose a fitting function for the Sherwood number, which is useful in predicting nutrient uptake in the non-dilute regime. Furthermore, we analyse the swimming energy consumed by individual cells. The results indicate that both the energetic cost and the nutrient uptake increased as the volume fraction of cells was increased, and that the uptake per unit energy was not significantly affected by the volume fraction. These findings are important in understanding the mass transport and metabolism of swimming micro-organisms in nature and for industrial applications.


Corresponding author

Email address for correspondence:


Hide All
Beenakker, C. W. J. 1986 Ewald sum of the Rotne–Prager tensor. J. Chem. Phys. 85, 15811582.
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199208.
Brady, J. F. & Bossis, G. 1985 The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech. 155, 105129.
Brady, J. F., Phillips, R. J., Lester, J. C. & Bossis, G. 1988 Dynamic simulation of hydrodynamically interacting suspensions. J. Fluid Mech. 195, 257280.
Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. E. & Kessler, J. O. 2004 Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103.
Doostmohammadi, A., Stocker, R. & Ardekani, A. M. 2012 Low-Reynolds-number swimming at pycnoclines. Proc. Natl Acad. Sci. USA 109, 38563861.
Dunkel, J. et al. 2013 Fluid dynamics of bacterial turbulence. Phys. Rev. Lett. 110, 228102.
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 2149.
Ermak, D. L. & McCammon, J. A. 1978 Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69, 13521360.
Ferracci, J., Ueno, H., Numayama-Tsuruta, K., Imai, Y., Yamaguchi, T. & Ishikawa, T. 2013 Hydrodynamical entrapment of ciliates at the air–liquid interface. PLoS ONE 8, e75238.
Giacche, D. & Ishikawa, T. 2010 Hydrodynamic interaction of two unsteady model microorganisms. J. Theor. Biol. 267, 252263.
Goldstein, R. E. 2015 Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech. 47, 343375.
Guasto, J. S., Rusconi, R. & Stocker, R. 2012 Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373400.
Hernandez-Ortiz, J. P., Stoltz, C. G. & Graham, M. D. 2005 Transport and collective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett. 95, 204501.
Hill, N. A. & Pedley, T. J. 2005 Bioconvection. Fluid Dyn. Res. 37, 120.
Ishikawa, T. 2009 Suspension biomechanics of swimming microbes. J. R. Soc. Interface 6, 815834.
Ishikawa, T., Yoshida, N., Ueno, H., Wiedeman, M., Imai, Y. & Yamaguchi, T. 2011 Energy transport in a concentrated suspension of bacteria. Phys. Rev. Lett. 107, 028102.
Ishikawa, T. & Hota, M. 2006 Interaction of two swimming paramecia. J. Expl Biol. 209, 44524463.
Ishikawa, T., Locsei, J. T. & Pedley, T. J. 2008 Development of coherent structures in concentrated suspensions of swimming model micro-organisms. J. Fluid Mech. 615, 401431.
Ishikawa, T., Locsei, J. T. & Pedley, T. J. 2010 Fluid particle diffusion in a semidilute suspension of model micro-organisms. Phys. Rev. E 82, 021408.
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119160.
Jepson, A., Martinez, V. A., Schwarz-Linek, J., Morozov, A. & Poon, W. C. K. 2013 Enhanced diffusion of nonswimmers in a three-dimensional bath of motile bacteria. Phys. Rev. E 88, 041002.
Kasyap, T. V., Koch, D. L. & Wu, M. 2014 Hydrodynamic tracer diffusion in suspensions of swimming bacteria. Phys. Fluids 26, 081901.
Kim, S. & Karrila, S. J. 1992 Microhydrodynamics: Principles and Selected Applications. Butterworth Heinemann.
Koch, D. L. & Subramanian, G. 2011 Collective hydrodynamics of swimming microorganisms: living fluids. Annu. Rev. Fluid Mech. 43, 637659.
Kurtuldu, H., Guasto, J. S., Johnson, K. A. & Gollub, J. P. 2011 Enhancement of biomixing by swimming algal cells in two-dimensional films. Proc. Natl Acad. Sci. USA 108, 1039110395.
Lambert, R. A., Picano, F., Breugem, W.-P. & Brandt, L. 2013 Active suspensions in thin films: nutrient uptake and swimmer motion. J. Fluid Mech. 733, 528557.
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, 109118.
Magar, V., Goto, T. & Pedley, T. J. 2003 Nutrient uptake by a self-propelled steady squirmer. Q. J. Mech. Appl. Maths 56, 6591.
Magar, V. & Pedley, T. J. 2005 Average nutrient uptake by a self-propelled unsteady squirmer. J. Fluid Mech. 539, 93112.
Michelin, S. & Lauga, E. 2011 Optimal feeding is optimal swimming for all Peclet numbers. Phys. Fluids 23, 101901.
Michelin, S. & Lauga, E. 2013 Unsteady feeding and optimal strokes of model ciliates. J. Fluid Mech. 715, 131.
Mino, G. L., Dunstan, J., Rousselet, A., Clément, E. & Soto, R. 2013 Induced diffusion of tracers in a bacterial suspension: theory and experiments. J. Fluid Mech. 729, 423444.
Molina, J. J. & Yamamoto, R. 2014 Diffusion of colloidal particles in swimming suspensions. Mol. Phys. 112, 13891397.
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24, 313358.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Chap. 4, Cambridge University Press.
Saintillan, D. & Shelley, M. J. 2007 Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.
Saintillan, D. & Shelley, M. J. 2008a Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 178103.
Saintillan, D. & Shelley, M. J. 2008b Instabilities, pattern formation, and mixing in active suspensions. Phys. Fluids 20, 123304.
Saintillan, D. & Shelley, M. J. 2011 Emergence of coherent structures and large-scale flows in motile suspensions. J. R. Soc. Interface 68, 571585.
Short, M. B., Solari, C. A., Ganguly, S., Powers, T. R., Kessler, J. O. & Goldstein, R. E. 2006 Flows driven by flagella of multicellular organisms enhance long-range molecular transport. Proc. Natl Acad. Sci. USA 103, 83158319.
Sokolov, A., Aranson, I. S., Kessler, J. O. & Goldstein, R. E. 2007 Concentration dependence of the collective dynamics of swimming bacteria. Phys. Rev. Lett. 98, 158102.
Sokolov, A., Goldstein, R. E., Feldchtein, F. I. & Aranson, I. S. 2009 Enhanced mixing and spatial instability in concentrated bacterial suspensions. Phys. Rev. E 80, 031903.
Stocker, R. 2012 Marine microbes see a sea of gradients. Science 338, 628633.
Underhill, P. T., Hernandez-Ortiz, J. P. & Graham, M. D. 2008 Diffusion and spatial correlations in suspensions of swimming particles. Phys. Rev. Lett. 100, 248101.
Wagner, G. L., Young, W. R. & Lauga, E. 2014 Mixing by microorganisms in stratified fluids. J. Mar. Res. 72, 4772.
Wu, X.-L. & Libchaber, A. 2000 Particle diffusion in a quasi-two-dimensional bacterial bath. Phys. Rev. Lett. 84, 30173020.
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