Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T02:20:13.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Bioconvection under uniform shear: linear stability analysis

Published online by Cambridge University Press:  13 December 2013

Yongyun Hwang  and
T. J. Pedley
Yongyun Hwang*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
T. J. Pedley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Present address: Department of Civil and Environmental Engineering, Imperial College London SW7 2AZ, UK. Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The role of uniform shear in bioconvective instability in a shallow suspension of swimming gyrotactic cells is studied using linear stability analysis. The shear is introduced by applying a plane Couette flow, and it significantly disturbs gravitaxis of the cell. The unstably stratified basic state of the cell concentration is gradually relieved as the shear rate is increased, and it even becomes stably stratified at very large shear rates. Stability of the basic state is significantly changed. The instability at high wavenumbers is drastically damped out with the shear rate, while that at low wavenumbers is destabilized. However, at very large shear rates, the latter is also suppressed. The most unstable mode is found as a pair of streamwise uniform rolls aligned with the shear, analogous to Rayleigh–Bénard convection in plane Couette flow. To understand these findings, the physical mechanism of the bioconvective instability is reexamined with several sets of numerical experiments. It is shown that the bioconvective instability in a shallow suspension originates from three different physical processes: gravitational overturning, gyrotaxis of the cell and negative cross-diffusion flux. The first mechanism is found to rule the behaviour of low-wavenumber instability whereas the last two mechanisms are mainly associated with high-wavenumber instability. With the increase of the shear rate, the former is enhanced, thereby leading to destabilization at low wavenumbers, whereas the latter two mechanisms are significantly suppressed. For streamwise varying perturbations, shear with sufficiently large rates is also found to play a stabilizing role as in Rayleigh–Bénard convection. However, at small shear rates, it destabilizes these perturbations through the mechanism of overstability discussed by Hill, Pedley and Kessler (J. Fluid Mech., vol. 208, 1989, pp. 509–543). Finally, the present findings are compared with a recent experiment by Croze, Ashraf and Bees (Phys. Biol., vol. 7, 2010, 046001) and they are in qualitative agreement.

JFM classification

Biological Fluid Dynamics: Bioconvection Biological Fluid Dynamics: Biological Fluid Dynamics Complex Fluids: Complex Fluids Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 738 , 10 January 2014 , pp. 522 - 562
Copyright
©2013 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

Bearon, R. N., Bees, M. A. & Croze, O. A. 2012 Biased swimming cells do not disperse in pipes as tracers: a population model based on microscale behaviour. Phys. Fluids 24, 121902.CrossRefGoogle Scholar
Bees, M. A. & Hill, N. A. 1997 Wavelengths of bioconvection patterns. J. Expl Biol. 10, 1515–1526.CrossRefGoogle Scholar
Bees, M. A. & Hill, N. A. 1998 Linear bioconvection in a suspension of randomly swimming, gyrotactic micro-organisms. Phys. Fluids 10 (8), 1864–1881.CrossRefGoogle Scholar
Bees, M. A., Hill, N. A. & Pedley, T. J. 1998 Analytical approximations for the orientation distribution of small dipolar particles in steady shear flows. J. Math. Biol. 36, 269–298.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Childress, S., Levandowsky, M. & Spiegel, E. A. 1975 Pattern formation in a suspension of swimming micro-organisms. J. Fluid Mech. 69, 591–613.CrossRefGoogle Scholar
Croze, O. A., Ashraf, E. E. & Bees, M. A. 2010 Sheared bioconvection in a horizontal tube. Phys. Biol. 7, 046001.CrossRefGoogle Scholar
Croze, O. A., Sardina, G., Ahmed, M., Bees, M. A. & Brandt, L. 2013 Dispersion of swimming algae in laminar and turbulent channel flows: consequences for photobioreactors. J. Royal Soc. Interface 10, 20121041.CrossRefGoogle ScholarPubMed
Dombrowski, C., Lewellyn, B., Pesci, A. I., Restrepo, J. M., Kessler, J. O. & Goldstein, R. E. 2005 Coiling, entrainment, and hydrodynamic coupling of decelerated fluid plumes. Phys. Rev. Lett. 95, 184501.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Durham, W. M., Kessler, J. O. & Stocker, R. 2009 Disruption of vertical motility by shear triggers formation of thin phytoplankton layers. Science 323, 1067–1070.CrossRefGoogle ScholarPubMed
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.CrossRefGoogle Scholar
Gallagher, A. P. & Mercer, A. McD. 1965 On the behaviour of small disturbance in plane couette flow with a temperature gradient. Proc. R. Soc. Lond. 286, 117–128.Google Scholar
Hill, N. A. & Bees, M. A. 2002 Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys. Fluids 14, 2598–2605.CrossRefGoogle Scholar
Hill, N. A. & Häder, D.-P. 1997 A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186, 503–526.CrossRefGoogle ScholarPubMed
Hill, N. A. & Pedley, T. J. 2005 Bioconvection. Fluid Dyn. Res. 37, 1–20.CrossRefGoogle Scholar
Hill, N. A., Pedley, T. J. & Kessler, J. O. 1989 Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth. J. Fluid Mech. 208, 509–543.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
Ishikawa, T. 2012 Vertical dispersion of model microorganisms in horizontal shear flow. J. Fluid Mech. 705, 98–119.CrossRefGoogle Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Jerome, J. J. S., Chomaz, J.-M. & Huerre, P. 2012 Transient growth in Rayleigh–Benard–Poiseuille/Couette convection. Phys. Fluids 24, 044103.Google Scholar
Kantsler, V., Dunkel, J., Polin, M. & Goldstein, R. E. 2013 Ciliary contact interactions dominate surface scattering of swimming eukaryotes. Proc. Natl Acad. Sci. USA 110, 1187–1192.CrossRefGoogle ScholarPubMed
Kelly, J. E. 1992 The onset and development of thermal convection in fully developed shear flows. Adv. Appl. Mech. 31, 35–112.CrossRefGoogle Scholar
Kessler, J. O. 1984 Gyrotactic buoyant convection and spontaneous pattern formation in algal cell cultures. In Non-Equilibrium Cooperative Phenomena in Physics and Related Fields (ed. Verlarde, M. G.), pp. 241–248. Plenum.CrossRefGoogle Scholar
Kessler, J. O. 1985 Hydrodynamics focusing of motile algal cells. Nature 315, 218–220.CrossRefGoogle Scholar
Kessler, J. O. 1986 Individual and collective dynamics of swimming cells. J. Fluid Mech. 173, 191–205.CrossRefGoogle Scholar
Koch, D. L. & Shaqfeh, E. S. G. 1989 The instability of a dispersion of sedimenting spheroids. J. Fluid Mech. 209, 521–541.CrossRefGoogle Scholar
Malena, A. & Frankel, I. 2003 Generalized Taylor dispersion in suspensions of gyrotactic swimming micro-organisms. J. Fluid Mech. 490, 99–127.Google Scholar
Pedley, T. J. 2010a Collective behaviour of swimming micro-organisms. Exp. Mech. 50, 1293–1301.CrossRefGoogle Scholar
Pedley, T. J. 2010b Instability of uniform microorganism suspensions revisited. J. Fluid Mech. 647, 335–359.CrossRefGoogle Scholar
Pedley, T. J., Hill, N. A. & Kessler, J. O. 1988 The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms. J. Fluid Mech. 195, 223–237.CrossRefGoogle Scholar
Pedley, T. J. & Kessler, J. O. 1987 The orientation of spheroidal microorganisms swimming in a flow field. Proc. R. Soc. Lond. B 231, 47–70.Google Scholar
Pedley, T. J. & Kessler, J. O. 1990 A new continuum model for suspensions of gyrotactic micro-organisms. J. Fluid Mech. 212, 155–182.CrossRefGoogle ScholarPubMed
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming micro-organisms. Annu. Rev. Fluid Mech. 24, 313–358.CrossRefGoogle Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Saintillan, D., Shaqfeh, E. S. G. & Darve, E. 2006 The growth of concentration fluctuations in dilute dispersions of orientable and deformable particles under sedimentation. J. Fluid Mech. 553, 347–388.CrossRefGoogle Scholar
Saintillan, D. & Shelley, M. J. 2007 Orientiational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.CrossRefGoogle ScholarPubMed
Saintillan, D. & Shelley, M. J. 2008 Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 178103.CrossRefGoogle ScholarPubMed
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Simha, R. A. & Ramaswamy, S. 2002 Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101.Google Scholar
Vladimirov, V. A., Denissenko, P. V., Pedley, T. J., We, M. & Mosklaev, I. S. 2000 Algal motility measured by a laser based tracking method. Mar. Freshwat. Res. 51, 589–600.CrossRefGoogle Scholar
Vladimirov, V. A., Denissenko, P. V., Pedley, T. J., We, M. & Zakhidova, I. S. 2004 Measurement of cell velocity distributions in populations of motile algae. J. Expl Biol. 207, 1203–1216.CrossRefGoogle ScholarPubMed
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465–519.CrossRefGoogle Scholar
Williams, C. R. & Bees, M. A. 2011 Photo-gyrotactic bioconvection. J. Fluid Mech. 678, 41–86.CrossRefGoogle Scholar