Hydrodynamic tracer diffusion in suspensions of swimming bacteria

We present theoretical predictions, simulations, and experimental measurements of the diffusion of passive, Brownian tracer particles in the bulk of three-dimensional suspensions of swimming bacteria performing run-tumble random walks. In the theory, we derive an explicit expression for the “hydrodynamic” tracer diffusivity that results from the fluid disturbances created by a slender-body model of bacteria by ensemble averaging the mass conservation equation of the tracer over the space of tracer-bacterium interactions which are assumed to be binary. The theory assumes that the orientations of the bacterium before and after a tumble are uncorrelated and the fluid velocity disturbance created by the bacterium is small compared to its swimming speed. The dependence of the non-dimensional hydrodynamic diffusivity \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h}$\end{document}Dh̃ obtained by scaling the dimensional hydrodynamic diffusivity by nL3UsL on the persistence in bacterial swimming and the Brownian diffusivity of the tracer are studied in detail through two nondimensional parameters—a Peclet number Pe = UsL/D which is the ratio of the time scale of bacterial swimming to the tracer diffusion time scale and a non-dimensional persistence time τ* = Usτ/L obtained by scaling the dimensional bacterial persistence time by the time that a bacterium takes to swim over a distance equal to its length. Here, n, Us, τ, and L are the concentration, swimming speed, tumbling time, and the overall length of the bacteria, respectively, and D is the Brownian diffusivity of the tracer. \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h}$\end{document}Dh̃ is found to be a monotonically increasing function of τ* and a non-monotonic function of Pe with a Pe1/2 scaling in the Pe ≪ 1 limit, an intermediate peak and a constant value in the Pe ≫ 1 limit for the typical case of wild-type bacteria with τ* = O(1). In the simulation study we compute the bacterial contribution to the tracer diffusivity from explicit numerical simulations of binary tracer-bacterium interactions to examine the validity of the weak disturbance assumption made in the theory, and to investigate the effects of correlations in the pre- and post-tumble bacterium orientations and the excluded volume (steric) interactions between the bacterium and the tracer. It is found that the weak disturbance assumption does not have a statistically significant effect on \documentclass[12pt]{minimal