Chemotaxis in a fluid

I am working on a coupled chemotaxis-fluid model aimed to describe swimming bacteria, that need oxygen to survive. Moreover, I investigate the delay of blow-up in the Keller-Segel-Fluid system.

Chemotaxis-fluid model

We model swimming bacteria in a water drop that need oxygen to survive.
Experimental set-up: Swimming bacteria are suspended in a drop of water confined within two (vertical and invisible) glass plates 1 mm apart. The bacteria suspension, initially almost homogeneously distributed, evolves as some bacteria swim upwards the oxygen gradient, while other bacteria run out of oxygen and remain immobile. The oxygen itself diffuses into the water through the water surface.

Since the bacteria are a bit denser than water, instabilities develop at the high concentration layer close to the water surface. Bacteria-rich plumes form and start to move sideways along the curved surface. In that way, bio-convective flow patterns on length scales much larger than the bacteria size are created. Due to these large scale fluid motions formerly inactive bacteria are reoxygenated and participate in the established large scale convection pattern.

Goldstein Lab in Cambridge performed the experiments. See their PNAS paper. In this paper, they suggested a PDE model for the experiment just described. It consists of chemotaxis equations coupled with viscous incompressible fluid equations through transport and the gravitational force. I am working on that model.

Keller-Segel-fluid model

When we replace the consumption of the chemical by its production, we obtain a system consisting of Keller-Segel equations coupled to the Stokes equations. We show global-in-time existence of solutions for small initial mass in 2D. In 3D we establish global existence assuming that the initial $L^{3/2}$-norm is small. Moreover, we give numerical evidence that for this extension of the Keller-Segel system in 2D, solutions exist with mass above 8π, which is the critical mass for the system without fluid.

Numerics illustrates the behavior of the Keller-Segel-fluid system. We choose a Gaussian centered in the middle of a unit square as initial datum. We use the mass M = 27 > 8.5π > 8π+1.5 in the computational domain i.e. above the critical mass.

The video shows the time-evolution of the density of the cells in the Keller-Segel-fluid system.

We observe that the solution does not blow up, although the mass is above the critical value of 8π. Moreover, we see that the solution converges to a steady state. The density maximum moves downwards because of gravity but stops since the chemical vanishes at the boundary.One of the reasons for this blow-up delay is that the diffusion of the chemical is enhanced. So more of it is absorbed at the boundary.

More details can be found in the paper.