DNS of Electrokinetic Flows in Near Ion –Selective Surfaces. Instabilities, bifurcations and transition to chaos.

Study of the space charge in the electric double layer near ion-selective surfaces is
a fundamental problem of modern physics first addressed by Helmholtz.
Experiments show that volt-current (VC)-characteristic for a ion-selective surface
has 3 distinguishable regions: I – for a small voltage it obeys a linear Ohmic
relation (under-limiting currents); II – saturation region with a limiting plateau
(limiting currents) ; III – region with further increasing of current (over-limiting
currents). Deviation from the Ohmic law was explained by Rubinsten and Shtilman
(1979). It was theoretically predicted by Rubinstein and Zaltzman (2000), that a
physical mechanism responsible for the arising of the over-limiting currents is
electro-convective instability.
In our work electro-convective instability and nonlinear evolution to the over-
limiting regimes are considered from the view-point of hydrodynamic stability and
bifurcation theory. Direct numerical simulation of the full Nernst-Planck-Poisson-
Stokes system is fulfilled. Galerkin pseudo-spectral method is applied. Periodic
domain along the membrane surface allows us to utilize Fourier series in this
direction. Chebyshev polynomials are applied in the transverse direction.
Accumulation of zeros of these polynomials near the walls allows us to properly
resolve in thin space charge region. Two types of initial conditions are
superimposed on the initial equilibrium: a) ”forced”, sinusoidal of finite amplitude,
with some wave number n ; b) ”natural”, small amplitude random noise. The
main stages of evolution are clarified. Numerically found VC – characteristics
quantitatively coincide with experimental ones.