Stability analysis of electroconvection with a solid-liquid interface via the lattice Boltzmann method

In this paper, the electroconvective flow induced by the unipolar charge injection is extended from single-phase dielectric liquid to the solid-liquid interaction problem. The physical model with fully coupled mathematical equations is built in the liquid, solid, and interface for both the Ohmic and non-Ohmic solid models. An improved lattice Boltzmann model (LBM) is developed with three lattice Boltzmann equations for Poisson's equation, charge conservation equation, and Navier-Stokes equations, respectively. Our codes are first validated by the analytical solutions at the hydrostatic state. It is found that the LBM can well reproduce the discontinuous changes of electrical field and charge density at the interface and agrees well with the analytical results. Then, simulations are conducted under different governing parameters and interface position $f_l$. Results show that the bifurcation of electroconvection in the presence of the solid-liquid interface is still of subcritical type, but both the linear and finite amplitude stability criteria increase due to a voltage drop happening at the solid phase. Besides, the stability criterion expressed by the electrical Rayleigh number $T_c$ increases as the permittivity ratios ${\varepsilon }_r$ and the mobility ratios $K_r$ increase, but $T_c$ decreases with the increasing of dimensionless electric conductivity S and the interface position $f_l$.

Figure 1

Schematic diagram of electroconvective flows with solid-liquid interface. Interface zoom in sketch maps for both the Ohmic and non-Ohmic cases are presented on the right side.

Figure 2

Validation of LBM model for the Ohmic cases at hydrostatic state: (a) electric field at $y$ direction $E_y$ and (b) charge-density distributions for different electric conductivity S. Dashed line is the results of single-phase problem.

Figure 3

(a) Electric-field $E_y$ and (b) charge-density $q$ distributions for different interface position $f_l$ for the Ohmic cases at $S=0.01$.

Figure 4

Validation of LBM model for the non-Ohmic cases at hydrostatic state: (a) electric-field $E_y$ and (b) charge-density distributions for different combinations of permittivity ratio and mobility ratio $\left({\varepsilon }_r{K}_r\right)$.

Figure 5

(a) Electric-field $E_y$ and (b) charge-density $q$ distributions for different interface position $f_l$ for the non-Ohmic cases at ${\varepsilon }_r=2$ and $K_r=10$.

Figure 6

Results of electroconvection for (a) the Ohmic cases at $T=500$, $S=0.1$, and (b) the non-Ohmic cases at $T=1200$, ${\varepsilon }_r=2$, $K_r=10$. In each figure, the left contour map is the charge-density distribution while the right plot is the dimensionless stream function Str with electric field E.

Figure 7

The subcritical bifurcation diagram of electroconvection with solid-liquid interface: (a) Ohmic case at $S=0.1$ and (b) non-Ohmic case at ${\varepsilon }_r=2$ and $K_r=10$.

Figure 8

Regions of stable and unstable in electroconvection under different interface position $f_l$ for the (a) Ohmic model at $S=0.1$ and (b) non-Ohmic model at ${\varepsilon }_r=2$ and $K_r=10$.

Figure 9

The steady-state charge-density distributions and streamlines at four interface position $f_l=0.1$, 0.3, 0.6, and 0.9 for the Ohmic cases. Results are presented only in the liquid region as $q=0$ and u = 0 at the solid phase.

Figure 10

The steady-state charge-density distributions and streamlines at two different interface position $f_l=0.4$, and 0.8 for the non-Ohmic cases.

Figure 11

The solid-liquid interface treatment scheme. (a) The sketch of half-lattice division scheme. (b) The sketch of interface treatments for different fields for the non-Ohmic case. (c) the sketch of interface treatments for the Ohmic solid case.