Linear and Nonlinear Evolution and Diffusion Layer Selection in   Electrokinetic Instability
  In the present work fournontrivial stages of electrokinetic instability are identified by direct numerical simulation (DNS) of the full Nernst-Planck-Poisson-Stokes (NPPS) system: i) The stage of the influence of the initial conditions (milliseconds); ii) 1D self-similar evolution (milliseconds-seconds); iii) The primary instability of the self-similar solution (seconds); iv) The nonlinear stage with secondary instabilities. The self-similar character of evolution at intermediately large times is confirmed. Rubinstein and Zaltzman instability and noise-driven nonlinear evolution to over-limiting regimes in ion-exchange membranes are numerically simulated and compared with theoretical and experimental predictions. The primary instability which happens during this stage is found to arrest self-similar growth of the diffusion layer and specifies its characteristic length as was first experimentally predicted by Yossifon and Chang (PRL 101, 254501 (2008)). A novel principle for the characteristic wave number selection from the broadbanded initial noise is established. 
E. A. Demekhin, V. S. Shelistov and S. V. Polyanskikh
physics.flu-dyn math-ph math.MP