Instability and Electroconvection at a Electrodialysis Membrane

Electrolyte layer covered electrodialysis membrane under constant drop of potensial is considered. Self-similar solution of one-dimensional problem for second kind elecroosmosis (overlimitiny current) is found. Using special decomposition method analytical asymptotic solution of the problem is obtained; limiting current for the self-similar solution \[ j_\ast =4/\sqrt \pi \approx 2.25. \] Hydrodinamic instability of this solution with respect to linear 2D-perturbations is studied for the full system of equations. In contract to the works of Rubinstein is found that the region of instability is finite with respect to the wavenumber $\alpha $, growth rate $\lambda (\alpha )$ has maximum at some $\alpha =\alpha _m $ and 1D-solution is stable for sufficiently short perturbations. Direct numerical simulation of the full system of equations with a special non-uniform finite-differential grid shows that filtering mechanism of the linear stability singles out from the initial white-noise perturbations the maximum growth rate mode with $\alpha =\alpha _m $. Secondary instability leads to chaostic flow.