Numerical Analysis of Electroconvection Phenomena in Cross-flow

Transcription

1 Numerial Analysis o Eletroonvetion Phenomena in Cross-low Yiei Guan and Igor Novosselov Department o Mehanial Engineering, University o Washington, Seattle, U.S.A Deember 018 Eletroonvetive phenomena in the presene o ross-low between two parallel eletrodes is investigated in a numerial study. The two-relaxation-time Lattie Boltzmann Method with ast Poisson solver solves or the spatiotemporal distribution o low ield, eletri ield, and harge density. Couette and Poiseuille ross-lows are applied to the solutions with eletroonvetive vorties. Inreasing rosslow veloity deorms the vorties and eventually suppresses them when threshold values o shear stress is reahed. This behavior is parameterized by a non-dimensional parameter, Y, ormulated as the ratio o the eletrial ore to the visous ore in the Navier-Stokes Equations. For high values o Y, the eletri ore dominates the low, while or Y values below the ritial threshold, the eletri ore inluene on the low is negligible and the low is dominated by the shear. I. INTRODUCTION Eletrohydrodynamis (EHD) is an interdisiplinary ield desribing the interation o luids with an eletri ield. Insights into omplex multiphysis interations are essential or understanding EHD lows: (1) the eletri ield rom the potential dierene between the anode and athode and its modiiations by the spae harge eets; () the ion motion in the eletri ield; (3) the interation between the motion o ions and the neutral moleules; and (4) the inertial and visous ores in the omplex low. As a subset o EHD, eletroonvetion (EC) is a phenomenon where onvetive transport is indued by unipolar disharge into a dieletri luid [1-0]. The EC stability problem was irst analyzed by a simpliied non-linear hydrauli model [1, ] and linear stability analysis without harge diusion [3, 4]. Atten & Moreau [5] showed that in the weak-injetion limit, C 1, where C is the harge injetion level, the low stability is determined by the riterion TC, where T is the linear stability threshold or the eletri Rayleigh number T a ratio between eletri ore to the visous ore. In the spae-harge-limited (SCL) injetion, C, the low stability is determined byt only. The experimental observations [6, 7] have shown that, or the SCL senario, T = 100, while linear stability analysis suggests T = or the same onditions [5]. Atten suggested that the disrepany is due to the omission o the harge diusion term in the analysis [8]. The eet o harge diusion was investigated by Zhang et al. by employing linear stability analysis with a Poiseuille low [11] and by non-linear analysis using a multisale method [16]. The authors ound that the harge diusion has a non-negligible eet on T and the transient behavior depends on the Reynolds number ( Re ) [11, 16]. To gain insight into the omplexity o the EC low, the problem an be investigated by numerial simulations. The earlier inite dierene model simulations have shown that the strong numerial diusivity may ontaminate the model []. Other numerial approahes inlude the partile-in-ell method [9], inite volume method with the lux-orreted transport sheme [30], total variation diminishing sheme [4, 7, 13-15], and the method o harateristis [3]. Reently, Luo et al. showed that a Lattie Boltzmann model (LBM) ould predit the linear and inite amplitude stability riteria o the subritial biuration in the EC low [17-0] or both D and 3D low senarios. This uniied LBM transorms the ellipti Poisson equation into ivn@uw.edu

2 a paraboli reation-diusion equation and introdues artiiial oeiients to ontrol the evolution o the eletri potential. The EC stability problem was shown to be analogous to Rayleigh-Bernard onvetion (RBC) [0, 31-36]. O partiular interest is the suppression o the RBC ells in the rosslow [37]. A non-dimensional group Gr / Re, the ratio o buoyany to the inertia ore, was used to parametrize the eet o the applied shear, where Gr is the Grasho number. For Gr / Re 10, the eet o the ross-low is insigniiant, while or Gr / Re 0.1, the eet o the buoyany an be negleted. In EC low senario, D inite volume simulations o Poiseuille low show that the ritial eletri Rayleigh number, T, depends on the Re and ion mobility parameter, M [1]. In this paper, we parameterize the EC stability in the ross-low between two parallel eletrodes. The segregated solver used in the study ombines a two-relaxation-time LBM modeling luid and harged speies transport and a Fast Fourier Transorm Poisson solver to solve or the eletri ield diretly [38]. Couette and Poiseuille ross-low senarios provide shear stress, the dominant terms are determined rom non-dimensional analysis o the governing equations. A subritial biuration is desribed by the ratio o the eletrial ore to the visous ore. II. NON-DIMENSIONAL ANALYSIS The governing equations or EHD low inlude the Navier-Stokes equations (NSE) with the eletri oring term F e = in the momentum equation, the harge transport equation, and the Poisson equation or eletri potential. u =0, (1) Du = P + u, Dt () + ( b ) D = 0 t u, (3) =, (4) where is the density, is the dynami visosity, = ( u, x uy ) is the stati pressure, b is the ion mobility, u is the veloity vetor ield, P D is the ion diusivity, is the harge density, is the eletri permittivity, and is the eletri potential. The eletri ore provides a soure term in the momentum equation (Eq ) [11, 39-41]. Non-dimensional analysis o the governing equations (Eq. 1-4) yields: u =0 (5) D u M * = P + u CM, D t T (6) * 1 + ( ) = 0 t Fe (7) * = C, (8) where the asterisk denotes the non-dimensional variables. In the absene o ross-low, nondimensional governing equations yield our dimensionless parameters desribing the system s state [4, 6, 7, 9, 11-0].

3 ( ) 1/ / H 0 0 b0 M =, T =, C =, Fe =, (9) b b 0 De where H is the distane between the eletrodes (two ininite plates), 0 is the injeted harge density at the anode, and 0 is the voltage dierene between the eletrodes. The physial interpretations o these parameters are as ollows: M is the ratio between hydrodynami mobility and the ioni mobility; T is the ratio between eletri ore to the visous ore; C is the harge injetion level [11, 16]; and Fe is the reiproal o the harge diusivity oeiient [11, 16]. In the presene o ross-low, the veloity term in the non-dimensional analysis o the momentum equation is modiied to aount or external low, u ext, while in the previous deinitions (Eq. 9), the veloity term was non-dimensionalized by the drit veloity o harges. Here, we onsider the veloity o the upper wall in Couette low or the enterline veloity or Poiseuille low as u. ext u =0 (10) D u * = P + u, Dt Re uext (11) * 1 + ( ) = 0 t Fe (1) * = C, (13) ext where Re = u H 0 0 and X = as proposed by Guan et al. [40]. Sine Re is essentially u ext the ratio o inertia to visous ore and X is the ratio o eletri ore to inertia, the produt o these (denoted as Y) is the ratio o eletri ore to visous ore: 00H 00 Y = X Re = =, (14) uext τ where τ is the shear stress. In Couette low, τ = onstant; in Poiseuille low, the average value or the hannel low is used. III. RESULT AND DISCUSSION To model EC vorties, the hydrostati base-state is perturbed using wave-orm untions with a small amplitude that satisies the boundary onditions and ontinuity equation: ( ) ( ) u = L sin y / L sin( x / L ) 10 x x y x 3 uy = L y os y / Ly 1 os( x / Lx ) 10 3, (15) The physial domain size Lx = 1.m and Ly = 1m limits the perturbation wavenumber to = / L 5.15(1/ m), yielding the most unstable mode under the onditions x x C = 10, M = 10 and Fe = 4000 [18]. The eletri Nusselt number, Ne = I / I0, serves as a low stability riteria, where I is the athode urrent or a given solution and I 0 is the athode urrent or the hydrostati solution [4, 18]. For ases where EC vorties exist, Ne 1. For a strong ion injetion, the EC stability largely depends on T, so, in this analysis, T is varied, while other non-dimensional parameters are held onstant at C = 10, M = 10, and Fe = 4000.

4 The Couette ross-low is added to the simulation with EC vorties by assigning onstant veloity o the upper wall. To model the Poiseuille low, a body ore in the x-diretion is added. FIG. 1 shows the harge density and x-diretion veloity or Couette ross-low ( uwall = 0.5 m / s ) and Poiseuille ross-low ( uenter = 0.5 m / s ). The Couette ross-low strethes the vorties in the diretion o the bulk low and may eliminate one o the two vorties. In a Poiseuille ross-low, the vortex pair beomes separated; the vorties are pushed toward the opposite walls. For strong ross-low, both vorties in the pair are eliminated, and I=I0, Ne = 1 (see FIG. 4). The EC ontribution to the low ield is negligible at higher values o shear stress (higher veloity), and the low ield is exatly the same as the applied ross-low. FIG. 1. Charge density and x-diretion veloity ontour o the EC with ross-low. Top: Couette low with u = 0.5 m / s ; one o the two vorties is suppressed. Bottom: Poiseuille low with wall u enter = 0.5 m / s ; two vorties are suppressed and pushed towards the walls. FIG. shows the extended stability analysis o EC without ross-low[38] by introduing (a) inite veloity o the upper wall (athode) and (b) a uniorm body ore or pressure driven low dp / dx. For a onstant T, Ne dereases as U wall or dp / dx inreases. The applied shear stress stabilizes the EC low.

5 FIG.. Eletri Nusselt number depends on the eletri Rayleigh number T and applied veloity o the upper wall U or Couette type ross-low or applied body ore dp / dx or Poiseuille type wall ross-low. FIG. 3 shows the dependeny o Ne on non-dimensional parameter Y. For varying values o T, the solutions lie on the same urve o Ne normalized by NeY ( Ne at Y, solutions without ross-low [38]) suggesting that the EC stability with ross-low an be haraterized by a single non-dimensional parameter Y, whih is inversely proportional to τ. FIG. 3. Eletri Nusselt number depends on the non-dimensional parameter Y. (a) Couette type rosslow is applied. (b) Poiseuille type ross-low is applied. FIG. 4 shows Ne = ( Y ) or C = 10, M = 10, T = , and Fe = 4000 or Couette and Poiseuille ross-low. A hysteresis loop with subritial biuration is observed; the biuration thresholds are Y = 65.5, Y = 97.3 or Couette low and Y = 18.58, Y = or Poiseuille low. Similar to stability parameter T or the hydrostati ase (FIG. ), or Y Y, the system does not yield the EC instability, returning to the unperturbed state I = I and Ne = 1). I Y dereases ater the EC vorties are ormed, Ne dereases nonlinearly, ( 0 until Y = Y, then the EC vorties are suppressed; the low is not inluened by the Coulombi ores. FIG. 4. Eletrial Nusselt number Ne versus Y. Biuration thresholds are: (a) Couette ross-low Y = 65.5 and Y = 97.3 ; (b) Poiseuille ross-low Y = and Y =

6 IV. CONCLUSION The D numerial study extends the EC stability analysis to Couette and Poiseuille lows between two ininitely long parallel eletrodes. The numerial approah utilizes the tworelaxation-time LBM to solve the low and harge transport equations and a Fast Poisson Solver to solve the Poisson equation. Shear stress rom applied ross-low deorms the EC vorties and leads to their suppression. The non-dimensional analysis o the governing equations is used to derive parameter Y, a ratio o eletri ore to visous ore, in the presene o ross-low. The non-dimensional parameter Y aounts or the eet o the shear stress, analogous to a non-dimensional group Gr / Re (ratio o buoyany to the inertial ores) used to parametrize the eet o the applied shear in RBC. The eletri Ne, deined as a urrent ratio, is used as stability riteria. Similar to stability parameter T or the hydrostati ase, a hysteresis loop with subritial biuration Ne=(Y) is observed. The biuration thresholds are Y = 65.5, Y = 97.3 or Couette low and Y = 18.58, Y = or Poiseuille low. V. ACKNOWLEDGMENTS This researh was supported by the DHS Siene and Tehnology Diretorate, Homeland Seurity Advaned Researh Projets Ageny, Explosives Division, and UK Home Oie, grant no. HSHQDC C-B0033 and by the National Institutes o Health, grant no. NIBIB U01 EB0193. VI. REFERENCES [1] B. Malraison and P. Atten,Chaoti behavior o instability due to unipolar ion injetion in a dieletri liquid, Physial Review Letters 49, 73 (198). [] A. Castellanos and P. Atten,Numerial modeling o inite amplitude onvetion o liquids subjeted to unipolar injetion, IEEE transations on industry appliations, 85 (1987). [3] K. Adamiak and P. Atten,Simulation o orona disharge in point plane oniguration, Journal o eletrostatis 61, 85 (004). [4] P. Traoréand A. Pérez,Two-dimensional numerial analysis o eletroonvetion in a dieletri liquid subjeted to strong unipolar injetion, Physis o Fluids 4, (01). [5] R. Kwak, V. S. Pham, K. M. Lim, and J. Han,Shear low o an eletrially harged luid by ion onentration polarization: saling laws or eletroonvetive vorties, Physial review letters 110, (013). [6] P. Traoréand J. Wu,On the limitation o imposed veloity ield strategy or Coulombdriven eletroonvetion low simulations, Journal o Fluid Mehanis 77 (013). [7] J. Wu, P. Traoré, P. A. Vázquez, and A. T. Pérez,Onset o onvetion in a inite twodimensional ontainer due to unipolar injetion o ions, Physial Review E 88, (013). [8] S. M. Davidson, M. B. Andersen, and A. Mani,Chaoti indued-harge eletroosmosis, Physial review letters 11, 1830 (014). [9] A. Pérez, P. Vázquez, J. Wu, and P. Traoré,Eletrohydrodynami linear stability analysis o dieletri liquids subjeted to unipolar injetion in a retangular enlosure with rigid sidewalls, Journal o Fluid Mehanis 758, 586 (014). [10] I. Rubinstein and B. Zaltzman,Equilibrium eletroonvetive instability, Physial review letters 114, (015). [11] M. Zhang, F. Martinelli, J. Wu, P. J. Shmid, and M. Quadrio,Modal and non-modal stability analysis o eletrohydrodynami low with and without ross-low, Journal o Fluid Mehanis 770, 319 (015).

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