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. © 2012 Cambridge University Press.
Bibliographical noteKAUST Repository Item: Exported on 2020-04-23
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: The authors acknowledge support from KAUST through P.A.M.'s Investigator Award with Award Number: KUK-I1-007-43. P. A. M. acknowledges support from the Deanship of Scientific Research at King Saud University in Riyadh for funding this work through the research group project NoRGP- VPP-124. P. A. M. also acknowledges support from the Fondation Sciences Mathematiques de Paris, in form of his Excellence Chair 2011/2012 and the support from his Royal Society Wolfson Research Merit Award. A. C. acknowledges support by the NSF Grants DMS-0712898 and DMS-1115682. A. K. acknowledges support by the NSF Grant DMS-1115718. A. L. acknowledges support from the Fondation Sciences Mathematiques de Paris, in form of their Postdoctoral Program. K. F. acknowledges support from NAWI Graz. The authors would like to thank very much Professor T. Pedley and Professor R. Goldstein for many engaging discussions and answered questions. Moreover, the authors are grateful for fruitful discussions with Professor L. Fauci, Professor M. Proctor and Dr I. Tuval.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.