Oscillatory Flows Induced by Microorganisms Swimming in Two Dimensions

We present the first time-resolved measurements of the oscillatory velocity field induced by swimming unicellular microorganisms. Confinement of the green alga C. reinhardtii in stabilized thin liquid films allows simultaneous tracking of cells and tracer particles. The measured velocity field reveals complex time-dependent flow structures, and scales inversely with distance. The instantaneous mechanical power generated by the cells is measured from the velocity fields and peaks at 15 fW. The dissipation per cycle is more than 4 times what steady swimming would require.

Figure 1

Probability density function (PDF) of flagellar beat frequencies, $f$, measured from unicellular alga (C. reinhardtii) swimming in quasi-2D liquid films ($\overline{f}=53\pm 5\mathrm{Hz}$). Inset: velocity oscillations of a single swimming cell measured at 500 fps (red circle), where the maximum velocity is 4 times the mean value (solid blue line, $134\mu \mathrm{m}/\mathrm{s}$).

Figure 2

(a) The beat-cycle averaged velocity field around a swimming C. reinhardtii (black disc) in the lab frame, where the direction of travel is to the right toward the hyperbolic stagnation point (green diamond). Solid (red) lines are instantaneous streamlines and velocity vectors are shown on a log scale [20]. (b) The fluid velocity magnitude in various directions away from the cell demonstrates the predicted $u\sim r^{-1}$ scaling for a force dipole in 2D. Local minima correspond to stagnation points encountered in some directions.

Figure 3

Time sequence of the velocity field evolution throughout the beat cycle (period $T=18.9\mathrm{ms}$) of C. reinhardtii (oriented to the right) including the hyperbolic stagnation point position (green diamond). Insets show cell speed and beat cycle phase [lower left, see Fig. 4(b) for details], and approximate flagellar shape (lower right, measured separately). (a) Early in the power stroke, the velocity field resembles a (negative) force dipole. (b) At the peak of the power stroke, the vortices lateral to the organism strengthen and sweep toward the posterior. (c)–(e) The vortices then shift to the anterior as the power stroke is completed. (f) At the peak of the recovery stroke, the flow field again takes the shape of a dipole, but with opposite sign. The recovery stroke velocity field (f) is weaker than the forward stroke, but is enhanced by the log scaling [20].

Figure 4

The mechanical power dissipated by the organism is calculated from the velocity fields. (a) The instantaneous dissipation $P_{\mathrm{osc}}(t)$ is maximal when the swimmer speed is highest. The dissipated power time-averaged over the beat cycle $⟨P_{\mathrm{osc}}(t)⟩$ is more than 4 times the dissipation measured from the time-averaged velocity field (i.e., $⟨P_{\mathrm{osc}}(t)⟩/P_{\mathrm{mean}\mathrm{flow}}\approx 4.2$). (b) The measured power scales with the square of the cell body speed, $P_{\mathrm{osc}}\sim U^2$. Inset: mean cell body velocity as a function of beat cycle phase.