Phoresis in cellular flows: from enhanced dispersion to blockage

Abstract

In this article, we study numerically the dispersion of colloids in a two-dimensional cellular flow in the presence of an imposed mean salt gradient. Owing to the additional scalar, the colloids do not follow exactly the Eulerian flow field, but have a (small) extra-velocity proportional to the salt gradient, $\boldsymbol {v}_{dp}=\alpha \boldsymbol {\nabla } S$ , where $\alpha$ is the phoretic constant and $S$ the salt concentration. We study the demixing of an homogenous distribution of colloids and how their long-term mean velocity $\boldsymbol {V_m}$ and effective diffusivity $D_{eff}$ are influenced by the phoretic drift. We observe two regimes of colloids dynamics depending on a blockage criterion $R=\alpha G L/\sqrt {4 D_cD_s}$ , where $G$ is the mean salt gradient amplitude, $L$ the length scale of the flow and $D_c$ and $D_s$ the molecular diffusivities of colloids and salt. When $R<1$ , the mean velocity is strongly enhanced with $V_m \propto \alpha G \sqrt {Pe_s}$ , ${Pe}_s$ being the salt Péclet number. When $R > 1$ , the compressibility effect due to the phoretic drift is so strong that a depletion of colloids occurs along the separatrices inhibiting cell-to-cell transport.