Phoresis in cellular flows: from enhanced dispersion to blockage

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, $\mathbf{v}_\mathrm{dp}=\alpha\nabla S$, 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 $\mathbf{V_m}$ and effective diffusivity $D_\mathrm{eff}$ 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 P\'eclet 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.