Nonlinear Dynamics of a Microswimmer in Poiseuille Flow

We study the three-dimensional dynamics of a spherical microswimmer in cylindrical Poiseuille flow which can be mapped onto a Hamiltonian system. Swinging and tumbling trajectories are identified. In 2D they are equivalent to oscillating and circling solutions of a mathematical pendulum. Hydrodynamic interactions between the swimmer and confining channel walls lead to dissipative dynamics and result in stable trajectories, different for pullers and pushers. We demonstrate this behavior in the dipole approximation of the swimmer and with simulations using the method of multiparticle collision dynamics.

Figure 1

Swimmer in Poiseuille flow. (a) Flow profile ${\mathbf{v}}_f(\rho )$, cylindrical coordinate system with coordinate basis ($\widehat{\mathbf{\rho }}$, $\widehat{\mathbf{\phi }}$, $\widehat{\mathit{z}}$) and orientation angle $\Psi$ for the projected orientation into the $\rho \mathrm{\text{−}}z$ plane. When $e_{\phi }=\sin \Theta =0$, the motion is two-dimensional. Note that the sign of vorticity ${\mathit{\Omega }}_f$ changes when crossing the centerline. (b) Cross section of the microchannel. The orientation in $\phi$ direction defines the angle $\Theta$.

Figure 2

Phase spaces $x\mathrm{\text{−}}\Psi$ (left) and typical trajectories $z(x)$ (right) for several flow strengths ${\overline{v}}_f$. All trajectories start at $z=0$. (a) upstream motion, (b) intermediate motion and (c) downstream motion. Note the various scales for the $z$ axis. The arrows indicate the orientation vector $\widehat{\mathbf{e}}$ of the swimmer.

Figure 3

$\rho \mathrm{\text{−}}\Psi \mathrm{\text{−}}\Theta$ phase space. The intersection between $L_z=\mathrm{const}$ (orange) and $H=\mathrm{const}$ (green) gives the phase space trajectory. (a) helical-like swinging motion (blue intersection curve) for $L_z=0.2$, $H=1$. Black curve: fixed-point line corresponds to helical trajectories. (b) helical-like tumbling motion (blue intersection curve) for $L_z=0.2$, $H=3$. (c)–(e) sketch of trajectories in the channel for helical motion (c), helical-like motion (d) and tumbling motion (e).

Figure 4

Phase space trajectories for a microswimmer in a narrow channel for ${\overline{v}}_f=10$ for a puller (left) and a pusher (right). Green and red indicate, respectively, stable and unstable trajectories. (a) and (b) are obtained from Eqs. (8) and (c) and (d) from MPCD simulations where we used the parameters $B_1=0.045$, $\beta =B_2/B_1=\pm 5$, $R_S=6$, $R_{\mathrm{Ch}}=18$, $N_c=30$, $\Delta t=0.02$ setting $a=m=k_{B}T=1$. Each trajectory was obtained by averaging over 10 individual runs.