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Shear electroconvective instability in electrodialysis channel under extreme depletion and its scaling laws.

The electroconvective instability (ECI) in an electrodialysis channel under a strong electric field is studied here. The phenomenon of ECI with extreme depletion (ECI-HD) is reported; that is, the overlapping vortices cause the extreme depletion zone to propagate in the horizontal direction. Using scaling theory and direct numerical simulation, we indicate a series of features under the ECI-HD. The decrease in ion transport rate with voltage in ECI-HD is different from the enhancement in the ECI with moderate depletion (ECI-MD), which results in a unique peak in the voltage-current curve. More importantly, we reveal that the ECI is regulated by a scaling factor consisting of the electric field, hydrodynamic coupling coefficient, and Péclet number. For the ECI-HD, the scaling factor has an opposite effect on the vortex size and overlimiting current as that on the ECI-MD. The extreme depletion zone of the ECI-HD also has an uncommon diffusion self-similar dynamics. These unique scaling laws allow one to establish the quantitative bridge between the ion concentration, electric field, and vortex size by the overlimiting current.