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

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 bioconvection 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.

Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system

“…Accordingly, the literature on coupled chemotaxis-fluid systems is yet quite fragmentary, and beyond very interesting numerical findings [5,22], most rigorous analytical results available so far concentrate either on special cases involving somewhat restrictive assumptions, or on variants of (1.2) which contain additional regularizing effects. For instance, a considerable simplification consists in removing the convective term (u · ∇)u from the third equation in (1.2), thus assuming the fluid motion to be governed by the linear Stokes equations.…”

Long-term behaviour in a chemotaxis-fluid system with logistic source

“…This assumption is less natural, at least for the part of the boundary that separates water and air, but so far has been employed in almost all papers dealing with chemotaxis fluid interaction from a mathematical viewpoint (exceptions being early existence results for weak solutions in 2-dimensional bounded domains [25], numerical experiments like in [6] and, most notably, a recent work by Braukhoff [4], where it was shown that in 2-or 3-dimensional convex bounded domains classical or weak solutions, respectively, exist for a chemotaxis-Navier-Stokes model with logistic source if the boundary condition for c is ∂ ν c = 1 − c). Thus, in total the system to be considered here is n t + u · ∇n = ∆n − χ∇ · (n∇c) + κn − µn …”

Global boundedness of solutions for the chemotaxis-Navier–Stokes system in R2

“…The function χ(c) describes the chemotactic sensitivity whereas f (c) is the consumption rate of the bacteria. The chemotaxis-fluid system has been studied in the past several years [2][3][4][5][6]9,20,22,31,32,37]. Lorz [21] showed local existence of weak solutions of (1.1) in a bounded domain in R d , d = 2, 3 with no-flux boundary condition and in R 2 in the case of inhomogeneous Dirichlet conditions for the oxygen.…”

“…6) under homogeneous boundary condition ∂n R ∂ν = ∂c R ∂ν and u R = 0, x ∈ ∂B R , t > 0, (1.7) and the initial data are chosen as…”