Suppression of bacterial rheotaxis in wavy channels

Controlling the swimming behavior of bacteria is crucial, for example, to prevent contamination of ducts and catheters. We show that bacteria modeled by deformable microswimmers can accumulate in flows through straight microchannels either in their center or on previously unknown attractors near the channel walls. In flows through wavy microchannels we predict a resonance effect for semiflexible microswimmers. As a result, microswimmers can be deflected in a controlled manner so that they swim in modulated channels distributed over the channel cross section rather than localized near the wall or the channel center. Thus, depending on the flow amplitude, both upstream orientation of swimmers and their accumulation at the boundaries, which can promote surface rheotaxis, are suppressed. Our results suggest strategies for controlling the behavior of live and synthetic swimmers in microchannels.

Figure 1

(a) Flexible swimmer (light gray, swimmer size enhanced) swims against the flow at the center of a plane channel or migrates towards the walls (not shown). Driving direction indicated by the red dashed arrow; channel walls are black lines. (b) A wavy channel suppresses this rheotaxis. (c) Deformable swimmer (light gray), composed of $N=5$ beads (dark gray) in the $x\text{−}y$ plane, with its mean orientation angle $\psi$ (blue dashed line). The force dipole at its rear (pair of red arrows) creates a pusher-type flow field (light red dotted arrows) and thrust along the instantaneous driving direction, given by ${\mathbf{F}}_0$.

Figure 2

Cross-stream migration of a semiflexible microswimmer in unbounded planar Poiseuille flow during tumbling. The channel center is at $y=0$ (see gray dashed line) and no interactions with the channel walls are included. Two trajectories are shown for $u_0=0.16$ and $y_{c,0}=0.36d$, with $F_0=0.1$ (top left) and $F_0=0.3$ (bottom left). In the first case, the swimmer migrates away from the channel center. In the second case, it migrates towards the center, transitions to swinging, and finally approaches a state of upstream orientation at the channel center. Right: a repeller separates these two types of trajectories, as shown for $u_0=0.16$ (orange bold line) and $u_0=0.32$ (violet dashed line). Arrows indicate the migration direction.

Figure 3

(a) Comparison of real space trajectories of a semiflexible microswimmer with $F_0=0.6$ and $\kappa =0.5$ in unbounded plane and wavy flows. The fixed point of upstream orientation at the center of a plane channel ($y=0$, gray dashed line), adapted by the violet bold trajectory, is in a wavy flow replaced by a limit cycle (red dashed and blue dotted lines). The maximal ($y_{\text{c}}^{\text{max}}$) and steady-state amplitude ($y_{\text{c}}^{\text{stat}}$) are also shown. (b) Phase space trajectory (black) in wavy flow for initial conditions $(y_{\text{c},0},{\psi }_0)=(0.05d,0)$, approaching the limit cycle (red).

Figure 4

(a) Resonance curve with steady-state ($y_{\text{c}}^{\text{stat}}$, orange bold line, squares) and maximum amplitude ($y_{\text{c}}^{\text{max}}$, red dotted line, circles) vs modulation length $\lambda$ in units of the swimmer length $L_0$. The black shaded rectangle indicates the range of wall modulation ($\varepsilon =0.1$). (b) $y_{\text{c}}^{\text{stat}}$ vs modulation amplitude $\varepsilon$ for $\lambda /L_0=150$ below (red bold lines, circles) and $\lambda /L_0=400$ beyond the resonance wavelength (violet dashed-dotted lines, squares). For $\lambda /L_0=250$ close to ${\lambda }_{\text{res}}$ (orange dotted lines) the swimmer crosses the wall position in the orange range.

Figure 5

Steady-state swinging amplitude $y_{\text{c}}^{\text{stat}}/d$ vs velocity ratio $u_0/v_0$ is larger for stiffer swimmers ($\kappa =3$, blue dotted lines, crosses) than for softer ones ($\kappa =1$, red bold lines, circles), with upstream (white, left panel) and downstream resonance (gray, right panel). The swimmer exceeds the wall position in certain ranges of $u_0/v_0$ (blue dashed boxes for $\kappa =3$ and red double-dashed box for $\kappa =1$).

Figure 6

(a) Individual and (b) mean trajectories for a swimmer starting at $y_{c,0}=-0.6d$ in straight (blue bold line) and wavy streamlines (red dashed line) with $u_0=107v_0$. Black dotted line is the wavy channel boundary in (a) and its mean at $y_{\text{w}}/d=-1$ in (b). The plane boundary is at $y_{\text{w}}/d=-1$ in both cases. (c) Probability distribution $p\left(y_c\right)$ of a swimmer in straight (dashed blue bars) and wavy streamlines (bold red bars) for $u_0/v_0=107$ and in (d) for $u_0/v_0=5.36$.

Figure 7

Segment of the chain around bead $i$ with opening angle ${\alpha }_i$. The torque according to Eq. (A8) penalizes deviations of the orientation vectors ${\mathbf{p}}_i$ (violet) and ${\mathbf{q}}_i$ (orange) from the local unit vectors along and perpendicular to the axis connecting neighboring beads (black dashed line)

Figure 8

Tumbling time $T$ of a passive, stiff chain of beads in a linear shear flow with the shear rate $\dot{\gamma }$ as a function of the aspect ratio $r_{\text{p}}$ for different numbers of beads $N$. Blue dashed lines correspond to $N=5$, red dashed lines to $N=8$, and orange dashed lines to $N=10$. The black bold line shows the theoretical prediction according to Eq. (B1). Larger values of $N$ match the prediction when higher aspect ratios are chosen (intersections with the black line).

Figure 9

(a) Sketch of a swimmer with $N=5$ beads in a linear configuration. The force dipole (red arrows) is located at its left and the resulting swimming velocity $v_0{\widehat{\mathbf{e}}}_x$ is directed along the $x$ axis. (b) Calibration curve for the intrinsic swimming speed $v_0$ (propulsion speed without incident flow) as a function of the active force's absolute value $F_0$ for the stiff swimmer. The results of the analytical calculation (brown dashed line) according to Eq. (C2) and the numerical simulation (blue bold line) are shown.

Figure 10

Absolute value of the flow field $\mathbf{v}\left(\mathbf{r}\right)$ according to Eq. (D3) (red line) for large distances from the swimmer's center ${\mathbf{r}}_c=(x_c,y_c,z_c)$. The flow field decays as the inverse squared distance from the swimmer (see black line) in $x$ direction (a), as well as in $y$ direction (b). Inset in (a) shows numerically obtained bead positions according to Eq. (D1), with beads sketched in gray and the counterforce point in red.

Figure 11

(a) Real space and (b) corresponding phase space trajectory of a flexible (red line) and a stiff swimmer (blue line) in plane Poiseuille flow with initial position $y_{\text{c},0}=0.3d$ and orientation ${\psi }_0=\pm \pi$ (downstream). $L_0$ is the initial swimmer length and $d$ is the channel half width. A stiff swimmer performs a tumbling motion around a fixed off-centered position (periodic phase space orbits). A flexible swimmer tumbles and simultaneously migrates towards the channel center, eventually switching to a swinging motion with decaying amplitude (see also Figs. 2 and 3). (c) Angular velocity $\dot{\psi }$ of a stiff swimmer (red dashed line) vs its instantaneous orientation $\psi$. The result agrees well with the theoretical prediction from the literature [Eq. (E1), orange bold line].

Figure 12

Long-time oscillation amplitude $y_{\text{c}}^{\text{stat}}$ in units of the channel half width $d$ as function of the channel modulation length $\lambda$ in units of the swimmer's initial length $L_0$. The results for three different activities $F_0=0.6$ (red circles), $F_0=1.0$ (blue crosses), and $F_0=1.4$ (orange squares) are shown.

Figure 13

Spectra $P\left(\omega \right)$ of the swimmer's lateral position as function of time, $y_{\text{c}}\left(t\right)$, for $u_0/v_0=0.554$ [(a), off resonant] and $u_0/v_0=0.197$ [(b), close to resonance], and the predictions for ${\omega }_0$ (orange dashed lines) and ${\omega }_{\text{Ch}}$ (pink dotted lines).