While the electroosmosis of the first kind (equilibrium) is accepted widely, the electroosmosis of the second kind (nonequilibrium) is still controversial. In this work, the theory of electroosmosis slip, of either the first kind or of the second kind at electrolyte membrane system is revisited via our direct numerical simulation. The obtained results show that above a certain voltage threshold, the basic conduction state becomes electroconvectively unstable. This instability provides a mechanism for explaining the over-limiting conductance in concentration polarization at a permselective membrane. The most important work in our study is to examine the famous electroosmosis of the second kind formula suggested by Rubinstein and Zaltzman in 1999. Although their formula has been presented for a long time, there has no work to validate its accuracy experimentally or numerically due to the difficulty in pinpointing exactly the extended space charge layer in their formula. By using direct numerical simulation, we could solve this problem and inspect the application range of their formula. This also helps to strongly confirm the relationship between the electroosmosis of the second kind and the instability in concentration polarization at electrodialysis membranes.
