Numerical analysis of electroconvection in cross-flow with unipolar charge injection

Electroconvection driven by unipolar charge injection in the presence of cross-flow between two parallel electrodes is investigated in a numerical study. The two-relaxation-time lattice Boltzmann method with a fast Poisson solver is used to resolve the spatiotemporal distribution of flow field, electric field, and charge density. Couette and Poiseuille cross-flows are applied to the solutions with established electroconvective vortices. Increasing cross-flow velocity deforms the vortices and eventually suppresses them when threshold values of velocities are reached. At intermediate flow velocities, partial suppression of the vortices leads to the reduction in electroconvection. This behavior is parameterized by a nondimensional parameter, $Y$—a ratio of the electrical forcing term to the viscous term in the Navier-Stokes equations. For high values of $Y$, the electric force dominates the flow, while for values below the critical threshold, the electric force influence is negligible and the flow is dominated by the shear.

Figure 1

Hydrostatic solution comparison of the TRT LBM and fast Poisson solver [65], unified SRT LBM [19], and the analytical solution [61, 69] for $C=0.1$ and $C=10, Fe=4000$: (a) charge density and (b) electric field.

Figure 2

Charge density and $x$-direction velocity color contours of the EC with cross-flow. Top: Couette flow with $u_{\mathrm{wall}}^{*}=2.0$; one of the two vortices is suppressed. Bottom: Poiseuille flow with $u_{\mathrm{center}}^{*}=2.0$; both vortices are suppressed and displaced towards the walls.

Figure 3

Electric Nusselt number as a function of the electric Rayleigh number $T$ and (a) applied velocity of the upper wall $u_{\mathrm{wall}}^{*}$ for Couette-type cross-flow or (b) applied body force $F_p$ represented by the centerline velocity $u_{\mathrm{center}}^{*}$ for Poiseuille-type cross-flow. Partial suppression of the EC vortices leads to the reduction in electroconvection for the entire range of electric Rayleigh number.

Figure 4

Electric Nusselt vs nondimensional parameters. (a), (c), (e) Couette cross-flow is applied. (b), (d), (f) Poiseuille-type cross-flow is applied. (a), (b) $\mathrm{Ne}/\mathrm{N}{\mathrm{e}}_{\mathrm{Re}0}$ vs Re showing that the flow becomes more stable for increasing Re or cross-flow. (c), (d) $\mathrm{Ne}/\mathrm{N}{\mathrm{e}}_{X\infty }$ vs $X$. (e), (f) $\mathrm{Ne}/\mathrm{N}{\mathrm{e}}_{Y\infty }$ collapses on a single curve for various $T$ and $Y$ indicating that $\mathrm{Ne}/\mathrm{N}{\mathrm{e}}_{Y\infty }$ is a function of $Y$ only for constant C. $\mathrm{N}{\mathrm{e}}_{Re0}=\mathrm{N}{\mathrm{e}}_{X\infty }=\mathrm{N}{\mathrm{e}}_{Y\infty }$ is the electric Nusselt number without cross-flow.

Figure 5

Color contours of vorticity ${\omega }^{*}$, curl of electric force $-\left({\nabla }^{*}{\rho }_{{}_c}^{*}\times {\nabla }^{*}{\phi }^{*}\right)$, diffusion and forcing terms from Eqs. (17) and (18), with and without cross-flow: (a) no cross-flow, both vortices exist; (b) intermediate Couette flow with $u_{\mathrm{wall}}^{*}=2.0$, one of the two vortices is suppressed. (c) strong Couette flow with $u_{\mathrm{wall}}^{*}=4.0$; (d) intermediate Poiseuille flow $u_{\mathrm{center}}^{*}=2.0$, vortices are suppressed and displaced towards the walls; and (e) strong Poiseuille flow with $u_{\mathrm{center}}^{*}=4.0$.

Figure 6

Electrical Nusselt number Ne vs Y. Bifurcation thresholds are (a) Couette cross-flow $Y_c=625.25$ and $Y_f=297.32,$ and (b) Poiseuille cross-flow $Y_f=159.36$ and $Y_c=218.58$.