(403c) Spatial and Temporal Analysis of 2nd-Kind Electro-Osmotic Instability in Cross-Flow

For a long time, the theory of electrochemical processes was unable to explain experimentally measured electric current above the diffusion limit, the so-called overlimiting current.  According to theory, the diffusion limit specifies the maximum electric current that can pass from an aqueous electrolyte into an adjacent ion-selective surface.  Examples of such ion-selective surfaces include electrodes, ion-selective membranes and nanochannels, which are key components in applications such as electrodialysis, flow batteries and electrodeposition.  Recently, however, 2nd-kind electro-osmotic flow and instability due to a non-equilibrium electric double layer were conclusively tied to overlimiting current.  For example, in quiescent micro-channels, the formation of Rubinstein—Zaltzman vortices enhances ion transport, and creates additional electric current.  Furthermore, temporal linear stability analysis has proved successful in predicting the onset and growth rate of these vortices.  Similar vortices have also been observed in micro-channels with a mean cross-flow (Kwak et al., PRL, 2013), a configuration of some technological importance.  In this case, however, we argue that the spatial branch of the stability problem is more relevant than the temporal branch.  In particular, we apply spatial stability analysis to explore whether the system is absolutely or convectively unstable.  An absolutely unstable system supports self-sustaining oscillations which can be linked to global instability.  Instability in a convectively unstable system, on the other hand, is washed downstream as it grows, implying that unsteadiness must be supported by external perturbations.  We consider the coupled Navier—Stokes and Poisson—Nernst—Planck equations in our model of the developing electro-diffusion layer.  We apply an approximate similarity transform to describe the 2D base state in a single coordinate, which is then discretized by Chebyshev polynomials for spectral accuracy.  Absolute vs. convective instability is determined by location of pinch points of the dispersion relation in the complex wavenumber plane, numerically resolved by solution of a sequence of matrix eigenvalue problems.  The system is studied for different parameters including the dimensionless electrostatic screening length, the applied voltage, and the strength of the cross-flow. We also compare our results to a linear impulse response derived from a full DNS calculation.  Our results are important for the understanding of overlimiting current and provide insight towards system optimization.