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Numerical Analysis of Electroconvection Phenomena in Cross-flow

Electroconvective phenomena in the presence of cross-flow between two parallel electrodes is investigated in a numerical study. The two-relaxation-time Lattice Boltzmann Method with fast Poisson solver solves for the spatiotemporal distribution of flow field, electric field, and charge density. Couette and Poiseuille cross-flows are applied to the solutions with electroconvective vortices. Increasing cross-flow velocity deforms the vortices and eventually suppresses them when threshold values of shear stress are reached. This behavior is parameterized by a non-dimensional parameter, Y, formulated as the ratio of the electrical force to the viscous force in the Navier-Stokes Equations. For high values of Y, the electric force dominates the flow, while for Y values below the critical threshold, the electric force influence on the flow is negligible and the flow is dominated by the shear.


I. INTRODUCTION
Eletrohydrodynamics (EHD) is an interdisciplinary field describing the interaction of fluids with an electric field. Insights into complex multiphysics interactions are essential for understanding EHD flows: (1) the electric field from the potential difference between the anode and cathode and its modifications by the space charge effects; (2) the ion motion in the electric field; (3) the interaction between the motion of ions and the neutral molecules; and (4) the inertial and viscous forces in the complex flow. As a subset of EHD, electroconvection (EC) is a phenomenon where convective transport is induced by unipolar discharge into a dielectric fluid [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The EC stability problem was first analyzed by a simplified non-linear hydraulic model [21,22] and linear stability analysis without charge diffusion [23,24]. Atten & Moreau [25] showed that in the weak-injection limit, 1 C , where C is the charge injection level, the flow stability is determined by the criterion 2 c TC , where c T is the linear stability threshold for the electric Rayleigh number Ta ratio between electric force to the viscous force. In the space-charge-limited (SCL) injection, C →, the flow stability is determined by c T only. The experimental observations [26,27]  for the same conditions [25]. Atten suggested that the discrepancy is due to the omission of the charge diffusion term in the analysis [28]. The effect of charge diffusion was investigated by Zhang et al. by employing linear stability analysis with a Poiseuille flow [11] and by non-linear analysis using a multiscale method [16]. The authors found that the charge diffusion has a non-negligible effect on c T and the transient behavior depends on the Reynolds number ( Re ) [11,16].
To gain insight into the complexity of the EC flow, the problem can be investigated by numerical simulations. The earlier finite difference model simulations have shown that the strong numerical diffusivity may contaminate the model [2]. Other numerical approaches include the particle-in-cell method [29], finite volume method with the flux-corrected transport scheme [30], total variation diminishing scheme [4,7,[13][14][15], and the method of characteristics The EC stability problem was shown to be analogous to Rayleigh-Bernard convection (RBC) [20,[31][32][33][34][35][36]. Of particular interest is the suppression of the RBC cells in the crossflow [37]. A non-dimensional group 2 / Gr Re , the ratio of buoyancy to the inertia force, was used to parametrize the effect of the applied shear, where Gr is the Grashof number. For 2 / 10 Gr Re  , the effect of the cross-flow is insignificant, while for 2 / 0.1 Gr Re  , the effect of the buoyancy can be neglected. In EC flow scenario, 2D finite volume simulations of Poiseuille flow show that the critical electric Rayleigh number, c T , depends on the Re and ion mobility parameter, M [12]. In this paper, we parameterize the EC stability in the cross-flow between two parallel electrodes. The segregated solver used in the study combines a two-relaxation-time LBM modeling fluid and charged species transport and a Fast Fourier Transform Poisson solver to solve for the electric field directly [38]. Couette and Poiseuille cross-flow scenarios provide shear stress, the dominant terms are determined from non-dimensional analysis of the governing equations. A subcritical bifurcation is described by the ratio of the electrical force to the viscous force.

II. NON-DIMENSIONAL ANALYSIS
The governing equations for EHD flow include the Navier-Stokes equations (NSE) with the electric forcing term where  is the density,  is the dynamic viscosity, is the velocity vector field, P is the static pressure, b  is the ion mobility, c D is the ion diffusivity, c  is the charge density,  is the electric permittivity, and  is the electric potential. The electric force provides a source term in the momentum equation (Eq. 2) [11,[39][40][41].
( ) where H is the distance between the electrodes (two infinite plates), 0  is the injected charge density at the anode, and 0   is the voltage difference between the electrodes. The physical interpretations of these parameters are as follows: M is the ratio between hydrodynamic mobility and the ionic mobility; T is the ratio between electric force to the viscous force; C is the charge injection level [11,16]; and Fe is the reciprocal of the charge diffusivity coefficient [11,16].
In the presence of cross-flow, the velocity term in the non-dimensional analysis of the momentum equation is modified to account for external flow, as proposed by Guan et al. [40]. Since Re is essentially the ratio of inertia to viscous force and X is the ratio of electric force to inertia, the product of these (denoted as Y) is the ratio of electric force to viscous force: where τ is the shear stress. In Couette flow, τ = constant; in Poiseuille flow, the average value for the channel flow is used.

III. RESULT AND DISCUSSION
To model EC vortices, the hydrostatic base-state is perturbed using wave-form functions with a small amplitude that satisfies the boundary conditions and continuity equation:  FIG. 4). The EC contribution to the flow field is negligible at higher values of shear stress (higher velocity), and the flow field is exactly the same as the applied cross-flow. or / dp dx increases. The applied shear stress stabilizes the EC flow.  [38]) suggesting that the EC stability with cross-flow can be characterized by a single non-dimensional parameter Y , which is inversely proportional to τ .

IV. CONCLUSION
The 2D numerical study extends the EC stability analysis to Couette and Poiseuille flows between two infinitely long parallel electrodes. The numerical approach utilizes the tworelaxation-time LBM to solve the flow and charge transport equations and a Fast Poisson Solver to solve the Poisson equation. Shear stress from applied cross-flow deforms the EC vortices and leads to their suppression. The non-dimensional analysis of the governing equations is used to derive parameter Y, a ratio of electric force to viscous force, in the presence of cross-flow. The non-dimensional parameter Y accounts for the effect of the shear stress, analogous to a non-dimensional group 2 / Gr Re (ratio of buoyancy to the inertial forces) used to parametrize the effect of the applied shear in RBC. The electric Ne , defined as a current ratio, is used as stability criteria. Similar to stability parameter T for the hydrostatic case, a hysteresis loop with subcritical bifurcation Ne=f(Y) is observed. The bifurcation thresholds are 625

VI. REFERENCES
[1] B. Malraison and P. Atten,Chaotic behavior of instability due to unipolar ion injection in a dielectric liquid, Physical Review Letters 49, 723 (1982). [2] A. Castellanos and P. Atten,Numerical modeling of finite amplitude convection of liquids subjected to unipolar injection, IEEE transactions on industry applications, 825 (1987).