Photofocusing: Light and flow of phototactic microswimmer suspension

We explore in this paper the phenomenon of photofocusing: a coupling between flow vorticity and biased swimming of microalgae toward a light source that produces a focusing of the microswimmer suspension. We combine experiments that investigate the stationary state of this phenomenon as well as the transition regime with analytical and numerical modeling. We show that the experimentally observed scalings on the width of the focalized region and the establishment length as a function of the flow velocity are well described by a simple theoretical model.

Figure 1

Probability distribution of angles $\psi (\theta -\pi )$ in the absence of a light source (closed symbols) and when a light is switched on in the direction $\theta =\pi$ (open symbols); the cells are swimming in a fluid at rest. A truncated Lorentzian of width $1.2$ is found to adjust the experimental points.

Figure 2

(a) Experimental setup and (b) coordinate system.

Figure 3

Examples of typical experiments. (a) In the absence of light, microswimmers are dispersed uniformly along the $y$ direction. (b) In the presence of light upstream, the cells are focalized in the central region of the channel. Note that here only zoomed pictures are shown for the sake of clarity, but the experiments have explored a much wider field of view: $1.5\times 20 {\mathrm{mm}}^2$.

Figure 4

Circles show the experimental density distribution of cells deduced from the logarithm of the gray level intensity of the images (with a flow rate of 70 $\mu \mathrm{L}/\min$ and light source upstream, only half of the data points are shown for clarity). The dashed line is the numerical density distribution and the solid line is the analytical expression of the density distribution (9).

Figure 5

Normalized focusing length scale $L_x$ as a function of the normalized flow speed $U_0/v_0$: experimental data (circles) and numerical data (squares). Data are obtained by varying $U_0$. The fit of the experimental data represents a linear scaling as suggested in the text, $L_x\approx U_0/\alpha$.

Figure 6

Half-width of the cell density profile rescaled by the channel half-width as a function of the rescaled flow speed. Circles represent the experimental data points, squares represent the numerical data points, and the solid line is calculated from the variance of the analytical expression of the density distribution (10).