Fig. 3.

Hydrodynamic cell rotation hinders transverse transport and leads to giant stream-wise dispersion. (A) Narrowing of experimental probability density functions of bacterial displacement, $p\left(\mathrm{\Delta }x|2\hspace{0.17em}\mathrm{s}\right)$, transverse to the mean flow direction after 2 s indicates hindered transport with increasing shear rate ($\theta =0°$). Inset shows SDs, $\sigma$, of $p\left(\mathrm{\Delta }x|2\mathrm{s}\right)$ for experiments (solid circles) and simulations (open circles). (B) Cell velocity correlations transverse to the flow, ${\mathrm{\Psi }}_{\perp }\left(t\right)=⟨v_{\perp }\left(t\right)\cdot v_{\perp }\left(0\right)⟩$, show increasingly rapid temporal decorrelation with increasing mean shear ($\theta =0°$) for experiments (solid lines) and simulations (dashed lines). (C) Effective cell transport coefficients transverse to the mean flow direction, $D_{\perp }$, from experiments (solid circles) and simulations (open circles), decrease with increasing rotational Péclet number, ${\mathrm{P}\mathrm{e}}_r$, compared with constant diffusion for Brownian tracers (black triangles). $D_{\perp }$ is bounded by $D_{\perp }/D_0=1/\left(1+{\mathrm{P}\mathrm{e}}_r^2\right)$ (dashed black curve), which corresponds to a swimmer in constant vorticity and scales as ${\mathrm{P}\mathrm{e}}_r^{-2}$ for ${\mathrm{P}\mathrm{e}}_r\gg 1$. Stream-wise cell dispersion coefficients, $D_{\parallel }$, from simulations (open squares) rapidly increase with ${\mathrm{P}\mathrm{e}}_r$, which are bounded by ${\mathrm{P}\mathrm{e}}_r^4$ compared with expected ${\mathrm{P}\mathrm{e}}_r^2$ for Brownian tracer dispersion (black diamonds). Transport coefficients are normalized by $D_0$, measured without flow. Experimental error bars are the SD of three replicates. Simulation error bars are the maximum deviation of the Green–Kubo integral in the asymptotic range determined by a 5% convergence tolerance of the velocity correlation function (SI Appendix, Fig. S7).