Run-and-Tumble Particles with Hydrodynamics: Sedimentation, Trapping, and Upstream Swimming

We simulate by lattice Boltzmann the nonequilibrium steady states of run-and-tumble particles (inspired by a minimal model of bacteria), interacting by far-field hydrodynamics, subject to confinement. Under gravity, hydrodynamic interactions barely perturb the steady state found without them, but for particles in a harmonic trap such a state is quite changed if the run length is larger than the confinement length: a self-assembled pump is formed. Particles likewise confined in a narrow channel show a generic upstream flux in Poiseuille flow: chiral swimming is not required.

Figure 1

Steady-state density for particles confined to harmonic traps. Solid lines: LB simulations; dashed lines: numerics of 9. Key: $\zeta =\ell /r*$ values. Note that without HIs particles cannot exceed the trap radius $r^{*}$.

Figure 2

Simulation of ${10}^3$ swimmers in a harmonic trap. The trap radius is indicated by the circle; $\zeta =8$. Swimmers are marked with cones and shaded by local density in units of the mean (key: upper color bar). For the final image, randomly selected particles’ trajectories over the preceding time interval $\Delta t=0.5{\alpha }^{-1}$ are shown. Arrows depict fluid velocity colored by magnitude in units of the swim speed $v$ (key: lower color bar). For a movie see supplementary material 14.

Figure 3

Snapshot showing the early stage instability in a simulation of 1000 swimmers, initially uniformly and randomly distributed on the surface of a trap. Green (light gray) arrows: local flow field. Red lines: swimmer trajectories. Blue (dark gray) arrows: swimming direction. Gray arrows: radial direction. Swimmers in the dense patch ( center) move radially inwards. Those either side are initially advected away from the patch, but rotate their swimming direction towards it.

Figure 4

Main figure: density (left) and orientation (right) profiles. Parameters: $u/v=1/2$, $1/16$, $1/128$ (triangles, circles, squares) and $\ell /h=2$, 16, 128 [green (light gray), blue (dark gray), pink (medium gray)]. Inset: state diagram showing contours of the average swimmer speed, at interval $v/4$. Starting from $+0.5v$ in the top left (solid blue line), the value decreases to zero (thick black line), further reducing to $-0.75v$ on the right (dashed pink line). At the wall, particles feel a truncated LJ potential of range 0.25.