Induced Diffusion of Tracers in a Bacterial Suspension: Theory and Experiments

The induced diffusion of tracers in a bacterial suspension is studied 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 $\beta$. The coefficient scales as the tracer-swimmer cross section times the mean square displacement produced by single scatterings. Considering simple swimmer models 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 W wild-type {\em 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 in a wide range allowing to extract the $\beta$ coefficient with precision. It is found that that the solid surfaces modify the induced diffusion: decreasing the confinement height of the cell, $\beta$ increases by a factor 4. The theoretical model reproduces this increase although there are quantitative differences, due to the simplified model.