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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: Klemens.Fellner@uni-graz.at
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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

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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

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