Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-29T09:58:56.190Z Has data issue: false hasContentIssue false
  • English
  • Français

Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach

Published online by Cambridge University Press:  02 February 2012

A. Chertock ,
K. Fellner ,
A. Kurganov ,
A. Lorz  and
P. A. Markowich
A. Chertock
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
K. Fellner*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
A. Kurganov
Affiliation:
Mathematics Department, Tulane University, New Orleans, LA 70118, USA
A. Lorz
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
P. A. Markowich
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK Department of Mathematics, College of Sciences, King Saud University, Riyadh, KSA Faculty of Mathematics, University of Vienna, 1090 Wien, Austria
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Aquatic bacteria like Bacillus subtilis are heavier than water yet they are able to swim up an oxygen gradient and concentrate in a layer below the water surface, which will undergo Rayleigh–Taylor-type instabilities for sufficiently high concentrations. In the literature, a simplified chemotaxis–fluid system has been proposed as a model for bio-convection in modestly diluted cell suspensions. It couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier–Stokes equations subject to a gravitational force proportional to the relative surplus of the cell density compared to the water density. In this paper, we derive a high-resolution vorticity-based hybrid finite-volume finite-difference scheme, which allows us to investigate the nonlinear dynamics of a two-dimensional chemotaxis–fluid system with boundary conditions matching an experiment of Hillesdon et al. (Bull. Math. Biol., vol. 57, 1995, pp. 299–344). We present selected numerical examples, which illustrate (i) the formation of sinking plumes, (ii) the possible merging of neighbouring plumes and (iii) the convergence towards numerically stable stationary plumes. The examples with stable stationary plumes show how the surface-directed oxytaxis continuously feeds cells into a high-concentration layer near the surface, from where the fluid flow (recurring upwards in the space between the plumes) transports the cells into the plumes, where then gravity makes the cells sink and constitutes the driving force in maintaining the fluid convection and, thus, in shaping the plumes into (numerically) stable stationary states. Our numerical method is fully capable of solving the coupled chemotaxis–fluid system and enabling a full exploration of its dynamics, which cannot be done in a linearised framework.

JFM classification

Biological Fluid Dynamics: Bioconvection Mathematical Foundations: Computational methods Biological Fluid Dynamics: Micro-organism dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 694 , 10 March 2012 , pp. 155 - 190
Copyright
Copyright © Cambridge University Press 2012

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.)

Footnotes

‡

Current address: Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, 8010 Graz, Austria

§

Current address: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie – Paris 6, 4 place Jussieu, 75252 Paris, CEDEX 05, France

References

1. Busse, F. H. 1985 Transition to turbulence in Rayleigh–Bénard convection. In Hydrodynamic Instabilities and the Transition to Turbulence, Topics in Applied Physics , vol. 45. pp. 97–137. Springer.CrossRefGoogle Scholar
2. Busse, F. H. & Sieber, M. 1991 Regular and chaotic patterns of Rayleigh–Bénard convection. In Bifurcation and Chaos: Analysis, Algorithms, Applications (Würzburg, 1990), International Series of Numerical Mathematics , vol. 97. pp. 79–92. Birkhäuser.CrossRefGoogle Scholar
3. Carrillo, J. A. & Moll, J. S. 2009 Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms. SIAM J. Sci. Comput. 31 (6), 4305–4329.CrossRefGoogle Scholar
4. Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
5. Chapman, C. J. & Proctor, M. R. E. 1980 Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101 (4), 759–782.CrossRefGoogle Scholar
6. Chertock, A. & Kurganov, A. 2008 A positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169–205.CrossRefGoogle Scholar
7. Chertock, A., Kurganov, A., Wang, X. & Wu, Y. 2010 On a chemotaxis model with saturated chemotactic flux. Kinet. Relat. Models 5, 51–95.CrossRefGoogle Scholar
8. Cook, A. W. & Dimotaki, P. E. 2001 Transition stages of Rayleigh–Taylor instability between miscible fluids. J. Fluid Mech. 443, 69–99.CrossRefGoogle Scholar
9. Di Francesco, M., Lorz, A. & Markowich, P. 2010 Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behaviour. Discrete Contin. Dyn. Syst. A 28 (4), 1437–1453.CrossRefGoogle Scholar
10. Dolak, Y. & Schmeiser, C. 2005 The Keller–Segel model with logistic sensitivity function and small diffusivity. SIAM J. Appl. Maths 66 (1), 286–308(electronic).CrossRefGoogle Scholar
11. Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. & Kessler, J. 2004 Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103.CrossRefGoogle ScholarPubMed
12. Duan, R.-J., Lorz, A. & Markowich, P. 2010 Global solutions to the coupled chemotaxis-fluid equations. Commun. Part. Diff. Equ. 35, 1–39.CrossRefGoogle Scholar
13. E, W. & Liu, J.-G. 1996 Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys. 124, 368–382.CrossRefGoogle Scholar
14. Fellner, K. & Raoul, G. 2010 Stable stationary states of non-local interaction equations. Math. Models Meth. Appl. Sci. 20 (12), 2267–2291.CrossRefGoogle Scholar
15. Filbet, F. 2006 A finite volume scheme for the Patlak–Keller–Segel chemotaxis model. Numer. Math. 104 (4), 457–488.CrossRefGoogle Scholar
16. Ghorai, S. & Hill, N. A. 1999 Development and stability of gyrotactic plumes in bioconvection. J. Fluid Mech. 400, 1–31.CrossRefGoogle Scholar
17. Ghorai, S. & Hill, N. A. 2002 Axisymmetric bioconvection in a cylinder. J. Theor. Biol. 219 (2), 137–152.CrossRefGoogle Scholar
18. Gottlieb, S., Shu, C.-W. & Tadmor, E. 2001 Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112.CrossRefGoogle Scholar
19. Haškovec, J. & Schmeiser, C. 2009 Stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system. J. Stat. Phys. 135 (1), 133–151.CrossRefGoogle Scholar
20. Hernandez-Ortiz, J. P., Underhill, P. T. & Graham, M. D. 2009 Dynamics of confined suspensions of swimming particles. J. Phys.: Condens. Matter 21, 204107.Google ScholarPubMed
21. Higueras, I. & Roldán, T. 2006 Positivity-preserving and entropy-decaying IMEX methods. In Ninth International Conference Zaragoza-Pau on Applied Mathematics and Statistics, Monografías del Seminario Matemático “García de Galdeano” , vol. 33. pp. 129–136, Prensas Univ. Zaragoza, Zaragoza.Google Scholar
22. Hill, N. A. & Pedley, T. J. 2005 Bioconvection. Fluid Dyn. Res. 37, 1–20.CrossRefGoogle Scholar
23. Hillesdon, A. J. & Pedley, T. J. 1996 Bioconvection in suspensions of oxytactic bacteria: linear theory. J. Fluid Mech. 324, 223–259.CrossRefGoogle Scholar
24. Hillesdon, A. J., Pedley, T. J. & Kessler, O. 1995 The development of concentration gradients in a suspension of chemotactic bacteria. Bull. Math. Biol. 57, 299–344.CrossRefGoogle Scholar
25. Hopkins, M. & Fauci, L. 2002 A computational model of the collective fluid dynamics of motile micro-organisms. J. Fluid Mech. 455, 149–174.CrossRefGoogle Scholar
26. LeVeque, R. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
27. Lie, K.-A. & Noelle, S. 2003 On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24, 1157–1174.CrossRefGoogle Scholar
28. Lorz, A. 2010 Coupled chemotaxis fluid model. Math. Models Meth. Appl. Sci. 20, 1–17.CrossRefGoogle Scholar
29. Medovikov, A. A. 1998 a DUMKA3 code available at http://dumkaland.org/.Google Scholar
30. Medovikov, A. A. 1998b High order explicit methods for parabolic equations. BIT 38, 372–390.CrossRefGoogle Scholar
31. Metcalfe, A. M. & Pedley, T. J. 1998 Bacterial bioconvection: weakly nonlinear theory for pattern selection. J. Fluid Mech. 370, 249–270.CrossRefGoogle Scholar
32. Metcalfe, A. M. & Pedley, T. J. 2001 Falling plumes in bacterial bioconvection. J. Fluid Mech. 445, 121–149.CrossRefGoogle Scholar
33. Nessyahu, H. & Tadmor, E. 1990 Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463.CrossRefGoogle Scholar
34. Pareschi, L. & Russo, G. 2005 Implicit–explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (1–2), 129–155.Google Scholar
35. Saintillan, D. & Shelley, M. J. 2007 Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.CrossRefGoogle ScholarPubMed
36. Sharp, D. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 3–18.CrossRefGoogle Scholar
37. Sweby, P. 1984 High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011.CrossRefGoogle Scholar
38. Thom, A. 1933 The flow past circular cylinders at low speeds. Proc. R. Soc. Lond. A 141, 651–669.Google Scholar
39. Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C., Kessler, J. & Goldstein, R. 2005 Bacterial swimming and oxygen transport near contact lines. Proc. Natl Acad. Sci. 102, 2277–2282.CrossRefGoogle ScholarPubMed
40. Wolgemuth, C. W. 2008 Collective swimming and the dynamics of bacterial turbulence. Biophys. J. 95 (4), 1564–1574.CrossRefGoogle ScholarPubMed