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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 $β$, as observed experimentally. The coefficient $β$ scales as the tracer-swimmer cross section times the mean square displacement produced by single scatterings. The displacements depend 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 $β$ increases for decreasing swimming efficiencies. Close to solid surfaces the swimming efficiency degrades and, consequently, the induced diffusion increase. 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 vary the swimmers' velocity in a wide range allowing to extract the $β$ coefficient with precision. It is found that the solid surfaces modify the induced diffusion: decreasing the confinement height of the cell, $β$ increases by a factor 4. The theoretical model reproduces this increase although there are quantitative differences, probably attributed to the simplicity of the swimmer models.