In this paper, we explore the electrokinetics in the complex two-dimensional geometries via conformal mapping and experimental comparison. A general theoretical frame of conformal mapping is provided for the application in electrokinetics, and three geometries are taken as an example to derive concentration polarization, potential, and electric field. For an eccentric annulus, the theoretical calculation of limiting current remarkably agrees with the experimental measurement, indicating that conformal mapping as a powerful approach is applicable for the ion transport. In the overlimiting current, the asymmetric electroconvection and deionization shock in experiments are qualitatively consistent with the asymmetric slip velocity associated with the electric field from conformal mapping. Then for the concentric ellipse geometry, when the inner ellipse squashes into a finite stripe, the local electric field at the tips tends to form a singularity, driving the electro-osmotic instability of concentration enrichment. Additionally, for a corner geometry, the intensity of the electric field is analyzed for different shapes. Hence, conformal mapping as a theoretical tool potentially inspires more future work for the electrokinetics in the complicated geometries, while the experimental findings, particularly the stronger concentration depletion induced by eccentricity, hold the promising applications, such as the shock electrodialysis for the deionization and water treatment, and electrophoresis for the particle manipulation in microfluidic devices.

• Received 7 September 2021
• Accepted 22 February 2022

DOI:https://doi.org/10.1103/PhysRevFluids.7.033701