Hydrodynamic analysis of flagellated bacteria swimming near one and between two no-slip plane boundaries

The motility of swimming bacteria near solid surfaces has implications in a wide range of scenarios, including water treatment facilities, microfluidics, and biomedical implants. Using the boundary element method to numerically solve the equations of low Reynolds number fluid flow, we investigate the dynamics of a model swimmer propelled by rotating a single helical flagellum. Building on previous simulation results for swimmers near a single plane boundary, we introduce a second, parallel boundary and show that the bacterial trajectories change as the two plates are brought closer together. Analysis of this dynamical system shows that the configuration in the center of the channel and parallel to the walls is an unstable equilibrium state for large plate separations, but it becomes the only stable position for swimmers when the plate separation is reduced to three to four times the cell width. Our model also predicts that transient trajectories, i.e., those not at steady states, can exhibit curvature in the opposite sense to that expected from the well-known explanation for circular bacterial paths near a single wall.

Figure 1

Illustration of model bacterium shape (top) and configuration between parallel plates (bottom). As depicted, $A_1$ and $A_2$ are the lengths of the semimajor and -minor axes of the spheroidal cell body. The body directors ${\mathbf{e}}_j^{\mathrm{B}}$ and tail directors ${\mathbf{e}}_j^{\mathrm{T}}$ are related by a rotation through the angle ${\varphi }^{\mathrm{T}}$ about the ${\mathbf{e}}_1^{\mathrm{B}}$ axis. The flagellum has a cross-sectional radius $a^{\mathrm{T}}$ and grows into a helix of amplitude $a$ and wavelength $\lambda$. The total length of the flagellum is $L$, measured along the curved centreline. The position of the bacterium relative to a fixed reference frame is described by the vector ${\mathbf{x}}^{\mathrm{B}}$. For swimming in the presence of plane boundaries, we track the distance from the wall below, $h$, and the angle of inclination, $\theta$. Between parallel plates, the plate separation is $H$.

Figure 2

Phase plane diagrams for bacteria swimming (a) near a single no-slip plane boundary and (b) between parallel plates with plate separation $H/\overline{a}=15$. Flagellar lengths are (left column) $L/\overline{a}=7.5$, (center column) $L/\overline{a}=6.5$, and (right column) $L/\overline{a}=5$. Stable fixed points are indicated by filled circles, unstable fixed points at $(\theta =0,h/\overline{a}=0)$ are indicated by open circles, and saddle nodes are indicated by crosses $(\times )$. Trajectories entering and leaving saddle nodes are shown with thicker curves.

Figure 3

Phase plane diagrams for bacteria between parallel plates. Flagellar lengths are (left column) $L/\overline{a}=7.5$ and (right column) $L/\overline{a}=5$. Plate separations are (top row) $H/\overline{a}=10$ and (bottom row) $H/\overline{a}=5$. The boundary accumulator $\left(L/\overline{a}=7.5\right)$ is attracted to a stable state at the walls when the plate separation is large but is attracted to the center of the channel when the separation is small. The boundary escaper $\left(L/\overline{a}=5\right)$ oscillates between walls when the plate separation is large but converges to the center when the separation is small.

Figure 4

Instantaneous curvature $\left(\kappa \right)$ maps for $x\text{−}y$ projection of the bacterial path between parallel plates. (a) Boundary accumulator in a channel of height $H/\overline{a}=10$. (b) Boundary escaper in a channel of height $H/\overline{a}=10$. (c) Boundary accumulator in a channel of height $H/\overline{a}=5$. (d) Boundary escaper in a channel of height $H/\overline{a}=5$. Contour curves roughly separate low from high magnitudes of curvature, with threshold $|\kappa \overline{a}|=0.002$. Dashed curves are used for negative curvature (clockwise circles viewed from above) and solid curves are used for positive curvature (counterclockwise). Unshaded regions near $h=0$ and $h=H$ are unreachable due to intersection or close proximity $\left(d_{\min }=0.05\overline{a}\right)$ of the wall and swimmer.

Figure 5

Trajectories of model bacteria using simulations of full dynamics. Oscillations on the time scale of a motor revolution are too small to be visible in these plots. (a) Time series of height from the wall for boundary-accumulating swimmers starting close to respective steady states predicted by phase-averaged dynamics in channels of height $H/\overline{a}=5$ (short-dashed line), $H/\overline{a}=10$ (medium-dashed line), and $H/\overline{a}=15$ (long-dashed line). The time scale, $T^{\mathrm{M}}=2\pi /{\omega }^{\mathrm{M}}$, is the motor period. (b) Time series of height from the wall for boundary escapers in channels of height $H/\overline{a}=5$ (short-dashed line) and $H/\overline{a}=15$ (long-dashed line). (c),(d) Top-down views of trajectories from (a) and (b), respectively, with arrows indicating the direction of motion. Note that although all boundary accumulators were closer to the lower wall than to the upper wall, the path in the channel of height $H/\overline{a}=5$ curved in the opposite direction from those in larger channels.