Self-similar solutions in ion-exchange membranes and their stability

The self-similar character of the one-dimensional time-dependent solution of the problem in the complete statement is demonstrated by the numerical solution in ion-exchange membranes and their stability of the Nernst-Plank-Poisson-Stokes system. The time-dependent processes in the electrolyte solution in the space between the semipermeable membranes are considered. The electroconvective instability prevents a one-dimensional solution to attain the establishment of the one-dimensional solution and this instability manifests itself at the stage of self-similarity and leads to the two-dimensional solution. The results also show that the one-dimensional process takes place self-similarly, a development that is violated by the loss of stability along the longitudinal coordinate leading to the appearance of the electroconvective motion of the liquid.