Induced diffusion of tracers in a bacterial suspension: theory and experiments

The induced diffusion of tracers in a bacterial suspension is studied theoretically and experimentally at low bacterial concentrations. Considering the swimmer-tracer hydrodynamic interactions at low Reynolds number and using a kinetic theory approach, it is shown that the induced diffusion coefficient is proportional to the swimmer concentration, their mean velocity and a coefficient beta, as observed experimentally. This paper shows that beta increases as a result of the interaction with solid surfaces. The coefficient beta scales as the tracer-swimmer cross-section times the mean square displacement produced by single scattering events, which depends on the swimmer propulsion forces. Considering simple swimmer models (acting on the fluid as two monopoles or as a force dipole), it is shown that beta increases for decreasing swimming efficiencies. Close to solid surfaces, the swimming efficiency degrades and, consequently, the induced diffusion increases. Experiments on wild-type Escherichia coli in a Hele-Shaw cell, under buoyant conditions, are performed to measure the induced diffusion on tracers near surfaces. The modification of the suspension pH varies the swimmers' velocity over a wide range, allowing the beta coefficient to be extracted with precision. It is found that solid surfaces modify the induced diffusion: decreasing the confinement height of the cell, beta increases by a factor of 4. The theoretical model reproduces this increase, although there are quantitative differences, probably attributed to the simplicity of the swimmer models and to the estimates for the parameters that model E. coli.