Three-dimensional coherent structures of electrokinetic instability

A direct numerical simulation of the three-dimensional elektrokinetic instability near a charge-selective surface (electric membrane, electrode, or system of micro- or nanochannels) has been carried out and analyzed. A special finite-difference method has been used for the space discretization along with a semi-implicit $3\frac{1}{3}$-step Runge-Kutta scheme for the integration in time. The calculations employ parallel computing. Three characteristic patterns, which correspond to the overlimiting currents, are observed: (a) two-dimensional electroconvective rolls, (b) three-dimensional regular hexagonal structures, and (c) three-dimensional structures of spatiotemporal chaos, which are a combination of unsteady hexagons, quadrangles, and triangles. The transition from (b) to (c) is accompanied by the generation of interacting two-dimensional solitary pulses.

Figure 1

Comparison of our numerical predictions, dashed line, with those by Zaltzman and Rubinstein [14], solid line, $\varkappa =0.5$. (a) Space-charge density, $\rho$, vs $y$ for $\Delta V=20$ and $\nu /\sqrt{2}={10}^{-2}$ (lines 1 and 2) and ${10}^{-3}$ (lines 3 and 4). (b) Threshold of the primary instability: triangles stand for our numerics.

Figure 2

Map of regimes and bifurcations, $\Delta V$, vs $\varkappa$ for $\nu ={10}^{-3}$. The solid line is the neutral stability curve [5] which separates the 1D and 2D regimes. Circles correspond to realizations of 2D regimes; triangles correspond to realizations of 3D regimes, so that the 2D–3D transition happens at the dash-dot line.

Figure 3

VC characteristics for $\varkappa =0.1$ and $\nu ={10}^{-3}$: I, II, and III stand for underlimiting, limiting, and overlimiting currents, respectively. The dashed line joining the circles corresponds to our 3D simulations; the dashed line joining the triangles corresponds to 2D simulations. The inset shows three typical electrokinetic patterns: 2D electroconvective rolls, regular 3D hexagonal structures, and chaotic 3D structures.

Figure 4

Electroconvective rolls for $\Delta V=33$, $\varkappa =0.1$, and $\nu ={10}^{-3}$. (a) Distribution of the electric current $j(x,z)$. (b) Profile of $y_m(x,z)$. (c) Cross section of charge density $\rho$ at $z=z_{*}$. Vortex pairs of liquid flow are shown schematically by the arrows.

Figure 5

2D–3D transition, $\Delta V=34$, and $\varkappa =0.1$. Profiles of the ESR length, $y_m(x,z)$, at (a) $t=7.4$. Flat region I near point 1 has a larger charge density than II; hence the Coulomb force of I at this point is larger than at II. This causes a further narrowing of II: (b) $t=7.5$, with eventual coarsening, (c) $t=7.6$. Region III is a nucleating region of a future electroconvective vortex caused by the primary instability.

Figure 6

Hexagonal coherent structures, $\Delta V=50$, $\varkappa =0.1$, and $\nu ={10}^{-3}$. (a) Distribution of the electric current $j(x,z)$ on the membrane surface. (b) Profile of ESC region, $y_m(x,z)$: (c) cross section $z=z_{*}$ of the charge density $\rho$; (d) vicinity of the pyramid top. The arrows schematically show the direction of the liquid flow.

Figure 7

(a) Distribution of the electric current $j(x,z)$ on the membrane surface. (b) Snapshot of $y_m$ for the spatiotemporal chaos: (c) cross section $z=z_{*}$ of the charge density $\rho$; (d) vicinity of the pyramid top. $\Delta V=60$, $\varkappa =0.1$, and $\nu ={10}^{-3}$.