Self-similar solution to the problem of electrokinetic 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 $110{,}23^{\circ}$, whereas, in the 3D case, it is equal to $76{,}78^{\circ}$. The values of the opening angles are in good agreement with the numerical results.