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Bioconvection under uniform shear: linear stability analysis

  • Yongyun Hwang (a1) and T. J. Pedley (a1)
Abstract
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.

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Corresponding author
Present address: Department of Civil and Environmental Engineering, Imperial College London SW7 2AZ, UK. Email address for correspondence: y.hwang@imperial.ac.uk
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R. N. Bearon , M. A. Bees & O. A. Croze 2012 Biased swimming cells do not disperse in pipes as tracers: a population model based on microscale behaviour. Phys. Fluids 24, 121902.

M. A. Bees & N. A. Hill 1998 Linear bioconvection in a suspension of randomly swimming, gyrotactic micro-organisms. Phys. Fluids 10 (8), 18641881.

M. A. Bees , N. A. Hill & T. J. Pedley 1998 Analytical approximations for the orientation distribution of small dipolar particles in steady shear flows. J. Math. Biol. 36, 269298.

K. M. Butler & B. F. Farrell 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 16371650.

S. Childress , M. Levandowsky & E. A. Spiegel 1975 Pattern formation in a suspension of swimming micro-organisms. J. Fluid Mech. 69, 591613.

O. A. Croze , E. E. Ashraf & M. A. Bees 2010 Sheared bioconvection in a horizontal tube. Phys. Biol. 7, 046001.

O. A. Croze , G. Sardina , M. Ahmed , M. A. Bees & L. Brandt 2013 Dispersion of swimming algae in laminar and turbulent channel flows: consequences for photobioreactors. J. Royal Soc. Interface 10, 20121041.

C. Dombrowski , B. Lewellyn , A. I. Pesci , J. M. Restrepo , J. O. Kessler & R. E. Goldstein 2005 Coiling, entrainment, and hydrodynamic coupling of decelerated fluid plumes. Phys. Rev. Lett. 95, 184501.

W. M. Durham , J. O. Kessler & R. Stocker 2009 Disruption of vertical motility by shear triggers formation of thin phytoplankton layers. Science 323, 10671070.

T. Ellingsen & E. Palm 1975 Stability of linear flow. Phys. Fluids 18, 487488.

A. P. Gallagher & A. McD. Mercer 1965 On the behaviour of small disturbance in plane couette flow with a temperature gradient. Proc. R. Soc. Lond. 286, 117128.

N. A. Hill & M. A. Bees 2002 Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys. Fluids 14, 25982605.

N. A. Hill & D.-P. Häder 1997 A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186, 503526.

N. A. Hill & T. J. Pedley 2005 Bioconvection. Fluid Dyn. Res. 37, 120.

N. A. Hill , T. J. Pedley & J. O. Kessler 1989 Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth. J. Fluid Mech. 208, 509543.

T. Ishikawa 2012 Vertical dispersion of model microorganisms in horizontal shear flow. J. Fluid Mech. 705, 98119.

G. B. Jeffery 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161179.

V. Kantsler , J. Dunkel , M. Polin & R. E. Goldstein 2013 Ciliary contact interactions dominate surface scattering of swimming eukaryotes. Proc. Natl Acad. Sci. USA 110, 11871192.

J. E. Kelly 1992 The onset and development of thermal convection in fully developed shear flows. Adv. Appl. Mech. 31, 35112.

J. O. Kessler 1984 Gyrotactic buoyant convection and spontaneous pattern formation in algal cell cultures. In Non-Equilibrium Cooperative Phenomena in Physics and Related Fields (ed. M. G. Verlarde ), pp. 241248. Plenum.

J. O. Kessler 1986 Individual and collective dynamics of swimming cells. J. Fluid Mech. 173, 191205.

D. L. Koch & E. S. G. Shaqfeh 1989 The instability of a dispersion of sedimenting spheroids. J. Fluid Mech. 209, 521541.

T. J. Pedley 2010a Collective behaviour of swimming micro-organisms. Exp. Mech. 50, 12931301.

T. J. Pedley 2010b Instability of uniform microorganism suspensions revisited. J. Fluid Mech. 647, 335359.

T. J. Pedley , N. A. Hill & J. O. Kessler 1988 The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms. J. Fluid Mech. 195, 223237.

T. J. Pedley & J. O. Kessler 1987 The orientation of spheroidal microorganisms swimming in a flow field. Proc. R. Soc. Lond. B 231, 4770.

T. J. Pedley & J. O. Kessler 1990 A new continuum model for suspensions of gyrotactic micro-organisms. J. Fluid Mech. 212, 155182.

T. J. Pedley & J. O. Kessler 1992 Hydrodynamic phenomena in suspensions of swimming micro-organisms. Annu. Rev. Fluid Mech. 24, 313358.

S. C. Reddy & D. S. Henningson 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209238.

D. Saintillan , E. S. G. Shaqfeh & E. Darve 2006 The growth of concentration fluctuations in dilute dispersions of orientable and deformable particles under sedimentation. J. Fluid Mech. 553, 347388.

D. Saintillan & M. J. Shelley 2007 Orientiational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.

D. Saintillan & M. J. Shelley 2008 Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 178103.

P. J. Schmid 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.

P. J. Schmid & D. S. Henningson 2001 Stability and Transition in Shear Flows. Springer.

V. A. Vladimirov , P. V. Denissenko , T. J. Pedley , M. We & I. S. Mosklaev 2000 Algal motility measured by a laser based tracking method. Mar. Freshwat. Res. 51, 589600.

V. A. Vladimirov , P. V. Denissenko , T. J. Pedley , M. We & I. S. Zakhidova 2004 Measurement of cell velocity distributions in populations of motile algae. J. Expl Biol. 207, 12031216.

J. A. C. Weideman & S. C. Reddy 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465519.

C. R. Williams & M. A. Bees 2011 Photo-gyrotactic bioconvection. J. Fluid Mech. 678, 4186.

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Journal of Fluid Mechanics
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