Scaling laws of electroconvective flow with finite vortex height near permselective membranes

In a steady state, the linear scaling laws are confirmed between the intensity characteristics of electroconvective (EC) vortex (including the vortex height and electroosmotic slip velocity) and the applied voltage for the nonshear EC flow with finite vortex height near permselective membranes. This finding in the nonshear EC flow is different from the shear EC flow [Kwak et al., Phys. Rev. Lett. 110, 114501 (2013)] and indicates that the local concentration gradient has a significant improvement in the analysis of slip velocity. Further, our study reveals that the EC vortex is mainly driven by the second peak effect of the Coulomb thrust in the extended space-charge layer, and the linear scaling law exhibited by the Coulomb thrust is an essential reason for the linear scaling laws of vortex intensity. The scaling laws proposed in this paper are supported by our direct numerical simulation data and previous experimental observations [Rubinstein et al., Phys. Rev. Lett. 101, 236101 (2008)].

Figure 1

Linear scaling of the nonequilibrium EC vortex with finite vortex height. (a) Diagram of the physical model. (b) Flow pattern in the steady state [16]. (c) The relation between the vortex height and the applied voltage in the steady state [16].

Figure 2

Scaling law for the nonequilibrium EC flow with finite vortex height. The height of the EC vortex $d_{\mathrm{EC}}$ plotted against the scaling factor $\sqrt{{\kappa }_0}(V-V_{\mathrm{cri}})l_0^{-0.25}$ at various ${\kappa }_0=0.1\sim 1$ and $V=18\sim 30$. The blue line is the theoretical curve of the nonshear EC vortex height, and the data points are extracted from the present DNS data. Here, in DNS simulations, $l_0$ is selected as $20\mu \mathrm{m}$. In the previous experiment, $l_0$ is 500 μm [16].

Figure 3

Scaling law for the slip velocity. The present scaling relation shows that the local concentration gradient has an important effect on the slip velocity. It should be noted that the critical voltages corresponding to ${\kappa }_0=0.25$, 0.5, 0.75, and 1 are, respectively, 23, 19.6, 18.5, and 18.

Figure 4

The physical mechanism behind slip velocity. Distribution of slip velocity and tangential volume force. Although the volume force of the electric double layer is of the same order of magnitude as that of the extended space-charge layer, it does not contribute to the electro-osmotic slip velocity. For the convenience of visualization, different color gamuts are used to distinguish the EDL from the ESCL. These curves clearly show that the velocity peak overlaps with the second peak of the Coulomb thrust. In other words, the peak of the electro-osmotic slip velocity is caused by the peak thrust in the ESCL, that is, the second peak effect. The voltages corresponding to the solid line and the dotted line are $V=21$ and $V=22$, respectively, with ${\kappa }_0=0.75$.

Figure 5

Scaling of thrust. The Coulomb thrust plotted against the scaling factor ${\kappa }_0\left(V-V_{\mathrm{cri}}\right)$ at various ${\kappa }_0=0.1\sim 1$ and $V=18\sim 30$.

Figure 6

Flow pattern in the steady state. Aqueous monovalent salt solution between two horizontal cation-selective membranes subject to the DC field. Surface plots show the anion-concentration fields superimposed with streamlines. For visualization and statistical convenience, we define the area where the ion concentration is less than 0.15 as the vortex depletion layer, and the remaining concentration ranges are not drawn.

Figure 7

Effects of voltage and hydrodynamic coupling coefficients on (a) EC vortex height and (b) slip velocity. These results indicate that the vortex intensity is controlled jointly by the voltage $V$ and hydrodynamic coupling constant ${\kappa }_0$.

Figure 8

Scaling law of overlimiting transport, in which the illustration shows the case where the critical voltage is considered. The red line is the theoretical curve, and the black data points are extracted from the present DNS.

Figure 9

The relations between the nonequilibrium electro-osmotic slip velocity $u_s$ and mean linear velocity $\langle u\rangle$. These DNS data show that $u_s$ and $\langle u\rangle$ have a positive correlation.