Enhanced dispersion of active microswimmers in confined flows

Abstract

In the presence of a laminar shear flow, the diffusion of passive colloidal particles is enhanced in the direction parallel to the flow. This classical phenomenon is known as Taylor-Aris dispersion. Besides, microorganisms, such as active microswimmers, exhibit an effective diffusive behavior at long times. Combining the two ingredients above, a natural question then emerges on how the effective diffusion of active microswimmers is altered in shear flows -- a widespread situation in natural environments with practical implications, \textit{e.g.} regarding biofilm formation. In this Letter, we investigate the motility and dispersion of \textit{Chlamydomonas reinhardtii} microalgae, within a rectangular microfluidic channel subjected to a sinusoidal Poiseuille flow. Using high-resolution optical microscopy and a particle-tracking algorithm, we reconstruct individual trajectories in various flow conditions and statistically analyze them through moment theory and sliding windowed demodulation. We find that the velocity fluctuations and the dispersion coefficient increase as the flow amplitude is increased, with only weak dependencies on the flow periodicity. Importantly, our results demonstrate that the generalization of Taylor-Aris law to active particles is valid.

Figures (4)

• Figure 1: Dynamics of microalgae in an oscillatory microfluidic flow.$\mathbf{(a)}$ Schematic of the experimental setup. A suspension of microalgae is confined in a microfluidic channel with rectangular cross-section. A pressure controller at one end generates a sinusoidal Poiseuille flow, while a constant pressure is applied at the other end via a fluid reservoir. The algae are illuminated with red-filtered light and imaged using a high-speed camera (equipped with a $\times 5$ objective). $\mathbf{(b)}$ Representative top-view image of the channel ($\times 5$ magnification; frame rate: $40\,\text{Hz}$; exposure time: $1\,\text{ms}$) with a typical trajectory superimposed. The colormap encodes time along the trajectory. $\mathbf{(c)}$ Slice of a time series of the longitudinal position $x(t)$. The demodulated signal $\hat{x}(t)$ is overlaid. $\mathbf{(d)}$ Probability density function $\text{P}(y)$ of the transverse position $y$, as estimated from 27 trajectories exceeding $20\,\text{s}$ in duration. The magenta dotted line indicates a uniform distribution. The black dashed line shows the best fit to Eq. (\ref{['Brady_Py']}). In panels ($\mathbf{(b)}$,$\mathbf{(c)}$,$\mathbf{(d)}$), the experimental parameters are: Péclet number $\text{Pe} = 9.15$ (i.e.$\underline{P}=1\,\text{mbar}$), and flow-oscillation period $T = 2\,\text{s}$.
• Figure 2: Mean square displacement of microalgae under oscillatory microfluidic flow.$\mathbf{(a)}$ Experimental ensemble-averaged mean square displacements $\text{MSD}[\hat{x}](\tau)$ (circles) and $\text{MSD}[y](\tau)$ (squares), respectively parallel and perpendicular to the flow, as functions of time increment $\tau$, for Péclet number $\text{Pe} = 9.15$, and flow-oscillation period $T = 2\,\text{s}$. A total of 27 trajectories, longer than 20 s each, are included in the average. Green, red, and purple symbol fillings indicate the time intervals used for estimating $\langle V_x^2 \rangle$, $\langle V_y^2 \rangle$, and $D_x$, respectively, and the associated doted and dash-dotted lines indicate the ballistic and diffusive-like asymptotic behaviours, as indicated. The black dashed line corresponds to Eq. (\ref{['msdlong']}), using the best-fit parameters obtained from the fit of the transverse distribution $\text{P}(y)$ shown in Fig. \ref{['figure_1']}($\mathbf{d}$). $\mathbf{(b)}$ Ensemble-averaged $\text{MSD}[\hat{x}](\tau)$ for various Péclet numbers (see legend in next panel). $\mathbf{(c)}$ Ensemble-averaged $\text{MSD}[y](\tau)$ for various Péclet numbers, as indicated.
• Figure 3: Velocity fluctuations of microalgae under oscillatory microfluidic flow.$\mathbf{(a)}$ Experimental ensemble-averaged mean square velocities $\langle V_x^2 \rangle$ (circles) and $\langle V_y^2 \rangle$ (squares), respectively parallel and perpendicular to the flow, as functions of the squared Péclet number $\text{Pe}^2$, for various flow-oscillation periods $T$, as indicated. $\mathbf{(b)}$ Ratio of the mean square velocities $\langle V_x^2 \rangle / \langle V_y^2 \rangle$ as a function of the squared ratio between Péclet number $\text{Pe}$ and active Péclet number romanczuk2012activesolon2015activezottl2023modelinglowen2020inertialsaintillan2018rheology$\text{Pe}_{\mathrm{A}}\equiv (W/D_0)\sqrt{\langle V_y^2 \rangle}\approx 3.2$, for various flow-oscillation periods $T$, as indicated. The circles correspond to experimental data while the diamonds correspond to the results from two-dimensional Langevin-like simulations Brady in oscillatory flows, including one steady-flow reference case. The solid black line corresponds to Eq. (\ref{['prediction_v']}) with $\beta=8/90$ (see SM). Each point represents an ensemble average over trajectories longer than $20\,\mathrm{s}$. Vertical error bars denote one standard deviation, while horizontal error bars account for uncertainties in the applied pressure (as specified by the manufacturer) and in the measurement of the baseline diffusion coefficient $D_0$ (see SM).
• Figure 4: Dispersion of microalgae under oscillatory microfluidic flow.$\mathbf{(a)}$ Experimental ensemble-averaged dispersion coefficient $D_x$ parallel to the flow as a function of the squared Péclet number $\text{Pe}^2$, for various flow-oscillation periods $T$, as indicated. $\mathbf{(b)}$ Ratio $D_x/D_0$ of the parallel dispersion coefficient to the raw effective diffusion coefficient as a function of squared Péclet number $\text{Pe}^2$, for various flow-oscillation periods $T$, as indicated. The circles correspond to experimental data while the diamonds correspond to the results from two-dimensional Langevin-like simulations Brady in oscillatory flows, including one steady-flow reference case. The solid black line corresponds to Eq. (\ref{['TA_ideal']}) with $\alpha=1/210$. Each point represents an ensemble average over trajectories longer than $20\,\mathrm{s}$. Vertical error bars denote one standard deviation, while horizontal error bars account for uncertainties in the applied pressure (as specified by the manufacturer) and in the measurement of the baseline diffusion coefficient $D_0$ (see SM).