Is the Zero Reynolds Number Approximation Valid for Ciliary Flows?

Stokes equations are commonly used to model the hydrodynamic flow around cilia on the micron scale. The validity of the zero Reynolds number approximation is investigated experimentally with a flow velocimetry approach based on optical tweezers, which allows the measurement of periodic flows with high spatial and temporal resolution. We find that beating cilia generate a flow, which fundamentally differs from the stokeslet field predicted by Stokes equations. In particular, the flow velocity spatially decays at a faster rate and is gradually phase delayed at increasing distances from the cilia. This indicates that the quasisteady approximation and use of Stokes equations for unsteady ciliary flow are not always justified and the finite timescale for vorticity diffusion cannot be neglected. Our results have significant implications in studies of synchronization and collective dynamics of microswimmers.

Figure 1

(a) An optically trapped bead measures the local flow around the cilia of a Chlamydomonas reinhardtii cell held by a micropipette. (b) Tracked flagellar shapes from image analysis of a video-recorded beating cycle. (c) Flow velocity field corresponding to the cell in (a), computed by solving Stokes equations.

Figure 2

Comparison between the flow velocity experimentally measured with OTV (gray) and the flow computed with BEM (magenta). (a) Close to the cell, $r=23.0\mu \mathrm{m}$, the measurements appear in phase. (b) At larger distances from the cell center, $r=66.2\mu \mathrm{m}$, the OTV measurements of the velocity is phase delayed compared to what is predicted from solving Stokes equations (inset) OTV raw data (gray), moving-window-averaged data (blue), and the flow computed with BEM (magenta). (c)–(d) Flow velocity $u$ over one period, averaged for $\sim 40$ periods, for the OTV measurements (c) compared to $u_S$ calculated with BEM from the tracked shapes (d). Solid and dashed lines correspond to the velocity at $r=23.0\mu \mathrm{m}$ and $r=66.2\mu \mathrm{m}$, respectively. A beat cycle begins when the cilia reach the most forward position, at the start of the power stroke. The flow velocity predicted by Stokes equations is in phase at different distances from the cell (d), whereas it is phase delayed in our OTV measurements (c).

Figure 3

Flow velocity as a function of the distance to the cell center at separate ciliary phases (a) corresponds to the flow computed with the BEM from the tracked shapes and (b) to the OTV measurements. (a),(b) Velocity profiles are represented at the same stages during the power-recovery strokes. Each color represents different ciliary phases, $\varphi$, as indicated in the legend. OTV measurements (b) reveal the existence of an inversion point, where the direction of the flow reverses, which is not predicted by the BEM computation.

Figure 4

Characteristics of the flow measured with the OTV as a function of $r/\delta$. Different symbols represent datasets for different cells, with beads of $1\mu \mathrm{m}$ (open symbols) or $2.5\mu \mathrm{m}$ (other symbols) radii. Measurements are taken along the $y$ axis (blue) at positions $(x=0,r)$ and along the $x$ axis (red) at positions $(r,y=0)$. We report the average flow velocity $\overline{u}$ (a) and the amplitude of the zero-averaged oscillatory component $\delta u^{\prime }$ (b). Dashed lines show predictions from Stokes equations for a point force located $120\mu \mathrm{m}$ above a solid surface. The decay rate of the oscillatory flow $\delta u^{\prime }$ ($\sim 1/r^3$) is faster compared to the decay rate of the average flow $\overline{u}$ ($\sim 1/r$). (c) Phase shift $\theta$ between the experimental and the computed results. Solid lines show predictions from the unsteady Stokes equations for an oscillating point force.