Advective superdiffusion in superhydrophobic microchannels

We consider pressure-driven flows in wide microchannels, and discuss how a transverse shear, generated by misaligned superhydrophobic walls, impacts cross-sectional spreading of Brownian particles. We show that such a transverse shear can induce an advective superdiffusion, which strongly enhances dispersion of particles compared to a normal diffusion, and that maximal cross-sectional spreading corresponds to a crossover between its subballistic and superballistic regimes. This allows us to argue that an advective superdiffusion can be used for boosting dispersion of particles at smaller Péclet numbers compared to known concepts of passive microfluidic mixing. This implies that our superdiffusion scenario allows one efficient mixing of much smaller particles or using much thinner microchannels than methods, which are currently being exploited.

Figure 1

(a) Sketch of the superhydrophobic channel with identical, but misaligned striped textures at the walls. Main flow direction is from left to right. (b) Top view of particle dispersion.

Figure 2

Schematic representation of various diffusion regimes in a microchannel. The colorbar values ascend from bottom to top.

Figure 3

Cross section of the fluid velocity field $(u_y,u_z)$ at $\varphi =0.5$ and $\lambda =0.5$. Colorbar shows the forward velocity $U_x$.

Figure 4

Dispersion ${\sigma }_y/\left(H\mathrm{Pe}\right)$ as a function of time $t/t_d$ simulated for $\varphi =0.5$. The Péclet numbers are (a) $\mathrm{Pe}=200,300,400,500$ (circles, squares, diamonds, triangles) and (b) $\mathrm{Pe}=10,50,75,100$ (circles, squares, diamonds, triangles).

Figure 5

Positions of individual particles at the cross section $\lambda =50$ of a SH channel with $\varphi =0.5$. The Péclet number is 10 (a), 50 (b), and 300 (c) providing different scenarios of advective superdiffusion.

Figure 6

(a) Transverse particle dispersion ${\sigma }_y/H$ as a function of $\mathrm{Pe}$ calculated with $\varphi =0.75$ at $\lambda =25$ (squares), 50 (triangles), and 100 (circles). (b) The same data plotted in scaled by $\lambda$ coordinates.

Figure 7

Rescaled maximal dispersion, ${\sigma }_{\max }/\left(H\lambda \right)$, (a) and the Péclet number, ${\mathrm{Pe}}_{\max }/\lambda$, (b) as a function of the gas area fraction, $\varphi$.

Figure 8

Particle dispersion ${\sigma }_y$ for (a) $\varphi =0.25$, (b) 0.5, (c) 0.75, (d) 0.9 and $\lambda =25,50,100$ (triangles, squares, circles). Solid curves show a fit to $f_{\sigma }$, given by Eq. (A1).