Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-09-11T20:26:09.835Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel

Published online by Cambridge University Press:  22 May 2014

Yongyun Hwang  and
T. J. Pedley
Yongyun Hwang*
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, South Kensington, London SW7 2AZ, 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
*
†Email address for correspondence: y.hwang@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Hydrodynamic focusing of cells along the region of the most rapid flow is a robust feature in downflowing suspensions of swimming gyrotactic microorganisms. Experiments performed in a downward pipe flow have reported that the focused beam-like structure of the cells is often unstable and results in the formation of regular-spaced axisymmetric blips, but the mechanism by which they are formed is not well understood. To elucidate this mechanism, in this study, we perform a linear stability analysis of a downflowing suspension of randomly swimming gyrotactic cells in a two-dimensional vertical channel. On increasing the flow rate, the basic state exhibits a focused beam-like structure. It is found that this focused beam is unstable with the varicose mode, the spatial structure, wavelength, phase speed and behaviour with the flow rate of which are remarkably similar to those of the blip instability in the pipe flow experiment. To understand the physical mechanism of the varicose mode, we perform an analysis which calculates the term-by-term contribution to the instability. It is shown that the leading physical mechanism in generating the varicose instability originates from the horizontal gradient in the cell-swimming-vector field formed by the non-uniform shear in the base flow. This mechanism is found to be supplemented by cooperation with the gyrotactic instability mechanism observed in uniform suspensions.

JFM classification

Biological Fluid Dynamics: Bioconvection Biological Fluid Dynamics: Biological Fluid Dynamics Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 749 , 25 June 2014 , pp. 750 - 777
Copyright
© 2014 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. 2011 The spatial distribution of gyrotactic swimming micro-organisms in laminar fow fields. J. Fluid Mech. 680, 602–635.Google Scholar
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.Google 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.Google Scholar
Bees, M. A. & Hill, N. A. 1999 Nonlinear bioconvection in a deep suspension of gyrotactic swimming micro-organisms. J. Math. Biol. 38, 135–168.Google Scholar
Childress, S., Levandowsky, M. & Spiegel, E. A. 1975 Pattern formation in a suspension of swimming micro-organisms. J. Fluid Mech. 69, 591–613.Google 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. R. Soc. Interface 10, 20121041.Google Scholar
Denissenko, P. & Lukaschuk, S. 2007 Velocity profiles and discontinuities propagation in a pipe flow of suspension of motile microorganisms. Phys. Lett. A 362, 298–304.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Ezhilan, B., Shelley, M. J. & Saintillan, D. 2013 Instabilities and nonlinear dynamics of concentrated active suspensions. Phys. Fluids 25, 070697.Google Scholar
Ghorai, S. & Hill, N. A. 1999 Development and stability of gyrtatic plummes in bioconvection. J. Fluid Mech. 400, 1–31.Google Scholar
Ghorai, S. & Hill, N. A. 2000 Periodic arrays of gyrotactic plumes in bioconvection. Phys. Fluids 12, 5–22.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.Google Scholar
Hill, N. A. & Pedley, T. J. 2005 Bioconvection. Fluid Dyn. Res. 37, 1–20.Google 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.Google Scholar
Hinch, E. J. & Leal, L. G. 1976 Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations. J. Fluid Mech. 76, 187–208.Google Scholar
Hwang, Y. & Pedley, T. J. 2014 Bioconvection under uniform shear: linear stability analysis. J. Fluid Mech. 738, 522–562.Google 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.Google Scholar
Kessler, J. O. 1985a Co-operative and concentrative phenomena of swimming microorganisms. Contemp. Phys. 26, 147–166.Google Scholar
Kessler, J. O. 1985b Hydrodynamics focusing of motile algal cells. Nature 315, 218–220.Google Scholar
Kessler, J. O. 1986 Individual and collective dynamics of swimming cells. J. Fluid Mech. 173, 191–205.Google Scholar
Lister, J. R. 1987 Long-wavelength instability of a line plume. J. Fluid Mech. 175, 413–428.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.Google Scholar
Pedley, T. J. 2010b Instability of uniform microorganism suspensions revisited. J. Fluid Mech. 647, 335–359.Google 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.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.Google Scholar
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming micro-organisms. Annu. Rev. Fluid Mech. 24, 313–358.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Smith, M. K. 1989 The axisymmetric long-wave instability of a concentric two-phase pipe flow. Phys. Fluids A 1, 494–506.Google 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.Google Scholar