Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T12:24:34.029Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation and stability of downflowing gyrotactic micro-organism suspensions in a vertical pipe

Published online by Cambridge University Press:  16 September 2020

Lloyd Fung Open the ORCID record for Lloyd Fung [Opens in a new window] ,
Rachel N. Bearon  and
Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window]
Lloyd Fung*
Affiliation:
Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK
Rachel N. Bearon
Affiliation:
Department of Mathematical Sciences, University of Liverpool, LiverpoolL69 7ZL, UK
Yongyun Hwang
Affiliation:
Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the experiment that first demonstrated gyrotactic behaviour of bottom-heavy swimming microalgae (e.g. Chlamydomonas), Kessler (Nature, vol. 313, 1985, pp. 218–220) showed that a beam-like structure, often referred to as a gyrotactic plume, would spontaneously appear from a suspension of gyrotactic swimmers in a downflowing pipe. Such a plume is prone to an instability to form blips. This work models the gyrotactic plume as a steady parallel basic state and its subsequent breakdown into blips as an instability, employing both the generalized Taylor dispersion (GTD) theory and the Fokker–Planck model for comparison. Upon solving for the basic state, it is discovered that the steady plume solution undergoes sophisticated bifurcations. When there is no net flow, there exists a non-trivial solution of the plume structure other than the stationary uniform suspension, stemming from a transcritical bifurcation with the average cell concentration. When a net downflow is prescribed, there exists a cusp bifurcation. Furthermore, there is a critical concentration at which the cell concentration at the centre would blow up for the GTD model. The subsequent stability analysis using the steady plume solution shows that the Fokker–Planck model is inconsistent with what was experimentally observed, as it predicts stabilisation of axisymmetric blips at high concentration of the plume and destabilisation of the first non-axisymmetric mode at low flow rates.

JFM classification

Biological Fluid Dynamics: Bioconvection Biological Fluid Dynamics: Micro-organism dynamics Complex Fluids: Suspensions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 902 , 10 November 2020 , A26
Check for updates
Copyright
© The Author(s), 2020. Published by 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

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 (12), 121902.CrossRefGoogle Scholar
Bearon, R. N. & Hazel, A. L. 2015 The trapping in high-shear regions of slender bacteria undergoing chemotaxis in a channel. J. Fluid Mech. 771, R3.CrossRefGoogle Scholar
Bearon, R. N., Hazel, A. L. & Thorn, G. J. 2011 The spatial distribution of gyrotactic swimming micro-organisms in laminar flow fields. J. Fluid Mech. 680, 602–635.CrossRefGoogle Scholar
Bees, M. A. 2020 Advances in Bioconvection. Annu. Rev. Fluid Mech. 52 (1), 449–476.CrossRefGoogle Scholar
Bees, M. A. & Croze, O. A. 2010 Dispersion of biased swimming micro-organisms in a fluid flowing through a tube. Proc. R. Soc. A Maths Phys. Eng. Sci. 466, 2057–2077.Google Scholar
Bees, M. A. & Hill, N. A. 1997 Wavelengths of bioconvection patterns. J. Exp. Biol. 200 (Pt 10), 1515–1526.Google ScholarPubMed
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
Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101 (3), 1–4.CrossRefGoogle ScholarPubMed
Brenner, H. 1980 General theory of Taylor dispersion phenomena. PCH Physicochem. Hydrodyn. 1 (2–3), 91–123.Google Scholar
Croze, O. A., Bearon, R. N. & Bees, M. A. 2017 Gyrotactic swimmer dispersion in pipe flow: testing the theory. J. Fluid Mech. 816, 481–506.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. R. Soc. Interface 10 (81), 20121041.CrossRefGoogle ScholarPubMed
Denissenko, P. & Lukaschuk, S. 2007 Velocity profiles and discontinuities propagation in a pipe flow of suspension of motile microorganisms. Phys. Lett. Sect. A Gen. At. Solid State Phys. 362 (4), 298–304.Google Scholar
Elgeti, J. & Gompper, G. 2013 Wall accumulation of self-propelled spheres. Europhys. Lett. 101 (4), 48003.CrossRefGoogle Scholar
Ezhilan, B. & Saintillan, D. 2015 Transport of a dilute active suspension in pressure-driven channel flow. J. Fluid Mech. 777, 482–522.CrossRefGoogle Scholar
Frankel, I. & Brenner, H. 1991 Generalized Taylor dispersion phenomena in unbounded homogeneous shear flows. J. Fluid Mech. 230, 147–181.CrossRefGoogle Scholar
Frankel, I. & Brenner, H. 1993 Taylor dispersion of orientable brownian particles in unbounded homogeneous shear flows. J. Fluid Mech. 255, 129–156.CrossRefGoogle Scholar
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 (20), 1–4.CrossRefGoogle ScholarPubMed
Hill, N. A. & Bees, M. A. 2002 Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys. Fluids 14 (8), 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 (4), 503–526.CrossRefGoogle ScholarPubMed
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
Hinch, E. J. & Leal, L. G. 1972 a Note on the rheology of a dilute suspension of dipolar spheres with weak Brownian couples. J. Fluid Mech. 56 (4), 803–813.CrossRefGoogle Scholar
Hinch, E. J. & Leal, L. G. 1972 b The effect of Brownian motion on the rheological properties of a suspensions of non-spherical particles. J. Fluid Mech. 52 (4), 683–712.CrossRefGoogle Scholar
Hwang, Y. & Pedley, T. J. 2014 a Bioconvection under uniform shear: linear stability analysis. J. Fluid Mech. 738, 522–562.CrossRefGoogle Scholar
Hwang, Y. & Pedley, T. J. 2014 b Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel. J. Fluid Mech. 749, 750–777.CrossRefGoogle Scholar
Ishikawa, T. & Pedley, T. J. 2007 Diffusion of swimming model micro-organisms in a semi-dilute suspension. J. Fluid Mech. 588, 437–462.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
Jiang, W. & Chen, G. 2019 Dispersion of active particles in confined unidirectional flows. J. Fluid Mech. 877, 1–34.CrossRefGoogle Scholar
Jiang, W. & Chen, G. 2020 Dispersion of gyrotactic micro-organisms in pipe flows. J. Fluid Mech. 889, A18.CrossRefGoogle Scholar
Kessler, J. O. 1984 Gyrotactic buoyant convection and spontaneous pattern formation in algal cell cultures. In Nonequilibrium Cooperative Phenomena in Physics and Related Fields (ed. Velarde, M. G.), pp. 241–248. Springer.CrossRefGoogle Scholar
Kessler, J. O. 1985 a Co-operative and concentrative phenomena of swimming micro-organisms. Contemp. Phys. 26, 147–166.CrossRefGoogle Scholar
Kessler, J. O. 1985 b Hydrodynamic focusing of motile algal cells. Nature 313, 218–220.CrossRefGoogle Scholar
Kessler, J. O. 1986 a Individual and collective fluid dynamics of swimming cells. J. Fluid Mech. 173, 191–205.CrossRefGoogle Scholar
Kessler, J. O. 1986 b The external dynamics of swimming micro-organisms. Prog. Phycol. Res. 4, 257–307.Google Scholar
Kessler, J. O., Hill, N. A. & Hader, D.-P. 1992 Orientation of swimming flagellates by simultaneously acting external factors. J. Phycol. 28 (6), 816–822.CrossRefGoogle Scholar
Koch, D. L. & Shaqfeh, E. S. G. 1989 The instability of a dispersion of sedimenting spheroids. J. Fluid Mech. 209, 521–542.CrossRefGoogle Scholar
Manela, A. & Frankel, I. 2003 Generalized Taylor dispersion in suspensions of gyrotactic swimming micro-organisms. J. Fluid Mech. 490, 99–127.CrossRefGoogle Scholar
Maretvadakethope, S., Keaveny, E. E. & Hwang, Y. 2019 The instability of gyrotactically trapped cell layers. J. Fluid Mech. 868, R5.CrossRefGoogle Scholar
Mehandia, V. & Nott, P. R. 2008 The collective dynamics of self-propelled particles. J. Fluid Mech. 595, 239–264.CrossRefGoogle Scholar
Meseguer, A & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number $10^7$. J. Comput. Phys. 186 (1), 178–197.CrossRefGoogle Scholar
O'Malley, S. & Bees, M. A. 2012 The orientation of swimming biflagellates in shear flows. Bull. Math. Biol. 74 (1), 232–255.CrossRefGoogle ScholarPubMed
Pedley, T. J. 2010 a Collective behaviour of swimming micro-organisms. Exp. Mech. 50 (9), 1293–1301.CrossRefGoogle Scholar
Pedley, T. J. 2010 b Instability of uniform micro-organism 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. 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 microorganisms. Annu. Rev. Fluid Mech. 24 (1), 313–358.CrossRefGoogle Scholar
Saintillan, D. 2018 Rheology of active fluids. Annu. Rev. Fluid Mech. 50 (1), 563–592.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 Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99 (5), 1–4.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 (17), 178103.CrossRefGoogle ScholarPubMed
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences. vol. 142, Springer.CrossRefGoogle Scholar
Spagnolie, S. E. & Lauga, E. 2012 Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105–147.CrossRefGoogle Scholar
Thom, R. 1989 Structural Stability and Morphogenesis. CRC.Google Scholar
Underhill, P. T., Hernandez-Ortiz, J. P. & Graham, M. D. 2008 Diffusion and spatial correlations in suspensions of swimming particles. Phys. Rev. Lett. 100 (24), 1–4.CrossRefGoogle ScholarPubMed
Vennamneni, L., Nambiar, S. & Subramanian, G. 2020 Shear-induced migration of microswimmers in pressure-driven channel flow. J. Fluid Mech. 890, A15.CrossRefGoogle Scholar
Vladimirov, V. A., Denissenko, P. V., Pedley, T. J., Wu, M. & Moskalev, I. S. 2000 Algal motility measured by a laser-based tracking method. Mar. Freshwat. Res. 51 (6), 589–600.CrossRefGoogle Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.CrossRefGoogle Scholar
Zeeman, E. C. 1976 Catastrophe theory. Sci. Am. 234 (4), 65–83.CrossRefGoogle Scholar