Self-similar solution to the problem on electro-kinetic instability in semipermeable membranes
Numerical solution of the Nernst-Planck-Poisson-Stokes system near a charge-selective surface shows the creation of coherent structures which have a form of spikes. In these structures liquid is pumped out of a space charge region into an electroneutral region from these spikes. In this work the analytic self-similar solution of the full system is found in a vicinity of the spike tip. In the 2D case, the opening angle of the spikes is equal to 110.23 deg, whereas, in the 3D case, it is equal to 76.78 deg. The values of the opening angles are in good agreement with the numerical results.