Coulomb Driven Electro-Convection within Two Stacked Layers of Miscible Dielectric Liquids †

Citation: Traore, P.; Pérez, A.T.; Mondal, S.; Bhattacharya, A.; Vázquez, P.A.; Yan, Z. Coulomb Driven Electro-Convection within Two Stacked Layers of Miscible Dielectric Liquids. Fluids 2024,9, 219. https://doi.org/10.3390/ fluids9090219 Academic Editor: Tim Persoons Received: 31 July 2024 Revised: 10 September 2024 Accepted: 12 September 2024 Published: 19 September 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fluids Article Coulomb Driven Electro-Convection within Two Stacked Layers of Miscible Dielectric Liquids † Philippe Traore 1,*, Alberto T. Pérez 2,* , Subhadeep Mondal 3, Anandaroop Bhattacharya 3, Pedro A. Vázquez 4 and Zelu Yan 5 1Institut PPRIME, Universitéde Poitiers, ISAE-ENSMA, 86073 Futuroscope Chasseneuil, France 2Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, 41012 Sevilla, Spain 3 Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India; [email protected] (S.M.); [email protected] (A.B.) 4Departamento de Física Aplicada III, Universidad de Sevilla, 41004 Sevilla, Spain; [email protected] 5 Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China; [email protected] *Correspondence: [email protected] (P.T.); [email protected] (A.T.P.); Tel.: +33-5-49496930 (P.T.) †This paper is an extended version of paper published in the 22nd International Conference on Dielectric Liquids (ICDL), held in Worcester, MA, USA, 25–28 June 2023. Abstract: This article investigates the behavior of two parallel layers of different miscible dielectric liquids enclosed and sandwiched between two electrodes. By applying an electric potential to one electrode while grounding the other, electro-convection occurs when the electric Rayleigh number exceeds a critical value, setting the fluid into motion and resulting in rapid mixing between the two liquids. A numerical model is developed to account for the varying ionic mobility and permittivity of the two liquids, considering their evolution based on the relative concentration field. The simulations confirm that electro-convection significantly enhances the mixing between the two liquids, as expected. Additionally, intriguing ripples are observed near the initial interface during the early stages of electro-convection instability growth. To explain and describe the flow dynamics in terms of stability analysis, a semi-analytical model is presented. This study provides insights into the mixing behavior and flow dynamics of miscible dielectric liquids under the influence of electro-convection. The findings contribute to a better understanding of the underlying mechanisms and can be valuable for applications such as microfluidics, energy conversion, and mixing processes. Further research is encouraged to explore additional parameters and optimize the control of electro-convection for practical applications. Keywords: miscible dielectric liquids; mass transfer; numerical simulation; Coulomb force driven flow; charge injection; electro-convective instability 1. Introduction Electrohydrodynamics (EHD) is an interdisciplinary field that explores the interaction between fluid flow and electric fields [ 1 ]. EHD techniques offer significant potential in various applications, including flow control, heat and mass transfer enhancement, and micro-electromechanical systems (MEMS) [2]. In this study, two parallel layers of two dielectric liquids are stacked into an enclosed cavity. The two liquids are considered miscible, so that, even without any mechanical action, they tend to mix by molecular diffusion. However, this molecular diffusion may take a long time (depending on the diffusion coefficient between the two fluids) before reaching a homogeneous mixture. That is the reason why, in many industrial processes which deal with fluid mixing, one often uses mechanical devices based on rotative blades to increase the mixing. But this mechanical device has an extra energy cost. Another option could be to use the Coulomb force between two metallic electrodes that sandwiches the two layers to induce electro-convection to mix the two liquids. Fluids 2024,9, 219. https://doi.org/10.3390/fluids9090219 https://www.mdpi.com/journal/fluids Fluids 2024,9, 219 2 of 16 When certain conditions are met, electric forces act on the layers of dielectric liquids, leading to electro-convection. Similar to how electro-convection enhances heat transfer [3–5] , this effect can also be utilized to improve the mixing of two layers of dielectric liquids in contact. In this context, it becomes crucial to determine the optimal electrical conditions, particularly the applied electric potential, to achieve the perfect mixing of the two specific dielectric liquids defined by their diffusion coefficient. A related experiment was conducted by the authors in [ 6 , 7 ], which closely relates to the work presented here. In this study, both liquids are characterized by different electric permittivity and ionic mobility. The objective is to gain a fundamental understanding of how the different properties of the liquids, their density, permittivity and ionic mobility influence the electro-soluto- convection phenomenon. Through this investigation, we aim to enhance our understanding of mass transfer efficiency and mixing enhancement between two miscible liquids with distinct properties. The remainder of this paper is organized as follows: In the next section, the physical and numerical models of two layers of miscible dielectric liquids subjected to an electric field applied between two horizontal parallel electrodes are detailed. Numerical results are presented and discussed in Section 3. Section 4is devoted to the description of a semi-analytical model that confirms the trends observed in the numerical results. Finally, a conclusion is drawn up in Section 5. 2. Theoretical and Numerical Models 2.1. Problem Formuation In this study, we investigate the behavior of two distinct layers of miscible dielectric liquid layers with equal thicknesses. These layers are positioned between two horizontal flat plate electrodes along the entire length of cavity, as depicted in Figure 1. One key aspect of this study is that these two liquids are Newtonian, incompressible, miscible, and perfectly insulating. Fluids 2024, 9, x FOR PEER REVIEW 2 of 19 could be to use the Coulomb force between two metallic electrodes that sandwiches the two layers to induce electro-convection to mix the two liquids. When certain conditions are met, electric forces act on the layers of dielectric liquids, leading to electro-convection. Similar to how electro-convection enhances heat transfer [3– 5], this effect can also be utilized to improve the mixing of two layers of dielectric liquids in contact. In this context, it becomes crucial to determine the optimal electrical conditions, particularly the applied electric potential, to achieve the perfect mixing of the two specific dielectric liquids defined by their diffusion coefficient. A related experiment was conducted by the authors in [6,7], which closely relates to the work presented here. In this study, both liquids are characterized by different electric permittivity and ionic mobility. The objective is to gain a fundamental understanding of how the different properties of the liquids, their density, permittivity and ionic mobility influence the electro-soluto-con- vection phenomenon. Through this investigation, we aim to enhance our understanding of mass transfer efficiency and mixing enhancement between two miscible liquids with distinct properties. The remainder of this paper is organized as follows: In the next section, the physical and numerical models of two layers of miscible dielectric liquids subjected to an electric field applied between two horizontal parallel electrodes are detailed. Numerical results are presented and discussed in Section 3. Section 4 is devoted to the description of a semi-analytical model that confirms the trends observed in the numerical results. Finally, a conclusion is drawn up in Section 5. 2. Theoretical and Numerical Models 2.1. Problem Formuation In this study, we investigate the behavior of two distinct layers of miscible dielectric liquid layers with equal thicknesses. These layers are positioned between two horizontal flat plate electrodes along the entire length of cavity, as depicted in Figure 1. One key aspect of this study is that these two liquids are Newtonian, incompressible, miscible, and perfectly insulating. (a) (b) Figure 1. Schematic diagram of the problem. (a) General electric configuration. (b) Physical properties of each liquid. The two dielectric liquids considered in this study have different physical properties, different density 12 and ρρ , different dynamic viscosity 12 and μμ , different permittivity 12 and εε as well as different ionic mobility 12 and K K. The main point is also that the two liquids are assumed to be miscible. This miscibility allows for the mixing to occur through molecular coefficient diffusion D. Figure 1. Schematic diagram of the problem. (a) General electric configuration. (b) Physical properties of each liquid. The two dielectric liquids considered in this study have different physical properties, different density ρ1and ρ2 , different dynamic viscosity µ1and µ2 , different permittivity ε1and ε2as well as different ionic mobility K1and K2. The main point is also that the two liquids are assumed to be miscible. This miscibility allows for the mixing to occur through molecular coefficient diffusion D. When a sufficiently strong DC electric potential is applied across the two electrodes, charge injection takes place, leading to the generation of Coulomb electric body force. Under some circumstances, this force sets the liquids in motion by electro-convection, initiating the mixing process. This mechanical motion acts like an electric mixer and will Fluids 2024,9, 219 3 of 16 significantly increase the degree of mixing between the two liquids, compared with a situation where molecular diffusion alone was likely to act. The charge injection is assumed to be homogeneous and autonomous, which means the density of injected charges at the emitter electrode remains constant and unaffected by the local electric field and flow motion [8]. As it has been stated in many publications [ 8 – 11 ] the development of a subcritical bifurcation occurs as soon as the electric Rayleigh number Tis above a critical one T c . This subcritical bifurcation results from a strong nonlinear coupling between the electric field and the fluid velocity field. Even with extremely simple geometry configurations, this strong nonlinear interaction between the electric field (Maxwell’s equations) and the velocity field (Navier–Stokes equations) precludes the possibility of using analytical methods to determine the charge density distribution as well as all the dynamics of the flow. In this study, we are going to numerically solve this coupled problem and analyze the enhanced mass transfer phenomena in a binary fluid due to unipolar charge injection. 2.2. Governing Equations The governing equations include the mass conservation equation, the Navier–Stokes equations, the mass transfer equation, a reduced set of Maxwell’s equations, and some equations of state. Some basic assumptions need to be made, namely, (i) there is only one kind of ion in the liquid (unipolar injection); (ii) magnetic effect is negligible; (iii) fluids are perfect insulators; (iv) fluids are isothermal; and (v) the Boussinesq approximation for density is retained. The concept of the Boussinesq approximation, often related to the dependency of density on temperature, is also applied to the dependency of density on concentration. This approximation assumes that all physical properties of the fluid are considered constant, except for the density in the buoyancy term. In the buoyancy term, the fluid density is treated as a linear function of concentration C, and can be expressed as ρ=ρ0(1−βS(C−C0)) , where ρ0 is the reference density of fluid 1 at concentration C0and βSis the volumetric expansion coefficient due to the concentration of fluid 1 [12]. The complete mathematical model leading to this coupled set of equations can be found in references [3,11,13]. ∇·→ u=0 (1) ∂(ρ→ u) ∂t+∇·(ρ→ u→ u) = −∇p+∇· µ ∇→ u+∇→ u T!!+q→ E−E2 2∇ε+ρ→ g(2) ∂C ∂t+→ u·∇C=D∇2C(3) ∂q ∂t+∇·→ u+K→ Eq=0 (4) ∇·(ε∇V)=−q(5) → E=−∇V(6) ρ=ρ0(1−βS(C−C0))(7) µ=Cµ1+ (1−C)µ2(8) ε=Cε1+ (1−C)ε2(9) K=CK1+ (1−C)K2(10) Here, → u and pstand for the velocity and the pressure field, respectively. Cis the concentration field and Dis the molecular diffusion between the two liquids. ρ , µ , εand K are, respectively, the density, the dynamic viscosity, the permittivity and the ionic mobility of the mix. These quantities are related to the density, the dynamic viscosity, the permittivity, and the ionic mobility of each liquid ρi,µi,εi,Kiand concentration C. qstands for charge density, Vstands for electric potential, and → Eis the electric field. Fluids 2024,9, 219 4 of 16 Fluid 1 is taken as the reference fluid to compute the concentration C. For the sake of simplicity, the governing equations are non-dimensionalized based on the following scales: Hfor length, ∆V for electric potential, ∆V/H for electric field, µre f /ρre f H for velocity, ρre f µre f /ρre f H2 for pressure, and q 0 for the charge density. Under these assumptions, the dimensionless governing equations read now as ∇·→ u=0 (11) ∂(→ u) ∂t+∇·(→ u→ u) = −∇p+∇· µ ∇→ u+∇→ u T!!+CinjT2 M2q→ E−T2 M2E2 2∇ε+Ras Sc C→ y(12) ∂C ∂t+→ u·∇C=1 Sc ∇2C(13) ∂q ∂t+∇·→ u+T M2K→ Eq=0 (14) ∇·(ε∇V)=−Cinjq(15) → E=−∇V(16) ρ=1−βSC(17) µ=C+ (1−C)µratio (18) ε=C+ (1−C)εratio (19) K=C+ (1−C)Kratio (20) Under these circumstances, the following set of non-dimensional parameters arise: e RaS=ρ1gβS∆C H3 µre f D; solutal Rayleigh number. e SC=µre f /ρ1D; Schmidt Number. e T=εre f ∆V/(µre f Kre f ) ; electric Rayleigh number, which is the ratio between the Coulomb force and the viscous force. It can be viewed as a representative of the applied voltage. The injection strength number, e Cinj =q0H2/εre f ∆V , is a measure of how the injected charge influences the electric field distribution. The dimensionless mobility number, e M=√εre f /ρ1 Kre f , is defined by fluid properties. Equation (3) is the concentration equation, which accounts for the mass transfer between the two liquids. The Schmidt number is the ratio between kinematic viscosity of fluid 1 and molecular diffusion between the two liquids. It is analogous to the Prandtl number in heat transfer problems. The Schmidt number represents the degree of miscibility between the two liquids. Other dimensionless numbers are introduced: εRatio =ε2/ε1 , KRatio =K2/K1 , and µRatio =µ2/µ1. The reference values are chosen, such as εre f =ε1+ε2 2=ε1(1+εratio) 2;Kre f =K1+K2 2=K1(1+Kratio) 2;µre f =µ1+µ2 2=µ1(1+µratio) 2 In this context, we can define some new effective dimensionless parameters that drive the flow: e T=T2(1+εratio) (1+µratio)(1+Kratio)e CInj =CInj 2 (1+εratio)e M2=M22(1+εratio) (1+Kratio)2 e SC=SC (1+µratio) 2;e RaS=RaS 2 (1+µratio) Fluids 2024,9, 219 5 of 16 where dimensionless numbers Nare defined with the physical properties of liquid 1: T=ε1∆V µ1K1 ;Cinj =q0H2 ε1∆V;M=pε1/ρ1 K1 ;RaS=ρ1gβS∆C H3 µ1D;SC=µ1/ρ1D It is important to observe that even if the input parameters are T , Cinj , M , Ras , Sc , depending on εRatio , KRatio and µRatio , the flow dynamics will be governed by e T , e Cinj , e Me RaS, and e SC. 2.3. Numerical Method To solve the system of governing equations described in previous section, we have developed a custom in-house code. The numerical procedure utilizes a finite volume method with second-order accuracy in both space and time. This approach involves the full and direct integration of the aforementioned equations. For a comprehensive understanding of the entire computational process, including the implementation details, please refer to the detailed description provided in references [11,13]. 2.4. Initial and Boundary Conditions Initially, the fluid is at rest and all variables (velocity, pressure, charge density, electric potential) are set to be zero. The concentration Cfor the reference liquid, the upper layer, is set to one. For the second liquid (lower layer), it set to zero. The dimensionless boundary conditions are classical and are described in Figure 2. Fluids 2024, 9, x FOR PEER REVIEW 6 of 19 It is important to observe that even if the input parameters are ,,,, inj s c TC MRa S, depending on Ratio ε , Ratio K and Ratio μ , the flow dynamics will be governed by T , inj C , M  S R a , and C S . 2.3. Numerical Method To solve the system of governing equations described in previous section, we have developed a custom in-house code. The numerical procedure utilizes a finite volume method with second-order accuracy in both space and time. This approach involves the full and direct integration of the aforementioned equations. For a comprehensive understanding of the entire computational process, including the implementation details, please refer to the detailed description provided in references [11,13]. 2.4. Initial and Boundary Conditions Initially, the fluid is at rest and all variables (velocity, pressure, charge density, electric potential) are set to be zero. The concentration C for the reference liquid, the upper layer, is set to one. For the second liquid (lower layer), it set to zero. The dimensionless boundary conditions are classical and are described in Figure 2. Figure 2. Boundary conditions. 3. Results and Discussions This study aims at defining how the mixing of the two dielectric liquids may be controlled and enhanced by electro-convection. Our investigations focused, too, on examining the development of the electro-convective instability in relation to the evolution of certain parameters such as the Schmidt number, the electric Rayleigh number, and the permittivity and ionic mobility ratios. The strong injection is considered and the non-dimensional parameters Cinj are taken as 10. The mobility parameter M is also taken as 10. The solutal Rayleigh number, RaS, is chosen between 100 and 20000. The electric Rayleigh number T can vary from 0 to 185. Different permittivity and ionic mobility ratios are considered [ ] [ ] 0.25,10 , 0.25,10 ratio ratio K ε ∈∈ . The dynamic viscosity ratio is set to 1 ratio μ =. 3.1. Code Validation The code used for these simulations has been thoroughly validated through many tests and benchmarks [11,13,14]. Comparisons with analytical solution of the hydrostatic case for charge injection in a pure dielectric liquid as well as the determination of the linear and non-linear stability criteria have been undertaken. The new feature that needed to be validated was the code’s ability to simulate cases with varying density, viscosity, permittivity, and ionic mobility. There is an analytical solution for the hydrostatic case of the charge density between two layers of the liquid with different permittivity and different ionic mobility. The hydrostatic case relates to the scenario where charge injection does Dielectric liquid 2 Dielectric liquid 1 0 0/0 /0uv V q y C y== = ∂ ∂= ∂ ∂= 011/0uv V q C y== = = ∂ ∂= 0 /0 /0 /0 /0 u vx Vx qx Cx = ∂∂= ∂∂= ∂∂= ∂∂= 0 /0 /0 /0 /0 u vx Vx qx Cx = ∂∂= ∂∂= ∂∂= ∂∂= Figure 2. Boundary conditions. 3. Results and Discussions This study aims at defining how the mixing of the two dielectric liquids may be controlled and enhanced by electro-convection. Our investigations focused, too, on examining the development of the electro-convective instability in relation to the evolution of certain parameters such as the Schmidt number, the electric Rayleigh number, and the permittivity and ionic mobility ratios. The strong injection is considered and the non-dimensional parameters C inj are taken as 10. The mobility parameter Mis also taken as 10. The solutal Rayleigh number, Ra S, is chosen between 100 and 20000. The electric Rayleigh number Tcan vary from 0 to 185. Different permittivity and ionic mobility ratios are considered εratio ∈[0.25, 10] , Kratio ∈[0.25, 10]. The dynamic viscosity ratio is set to µratio =1. 3.1. Code Validation The code used for these simulations has been thoroughly validated through many tests and benchmarks [ 11 , 13 , 14 ]. Comparisons with analytical solution of the hydrostatic case for charge injection in a pure dielectric liquid as well as the determination of the linear and non-linear stability criteria have been undertaken. The new feature that needed to be validated was the code’s ability to simulate cases with varying density, viscosity, permittivity, and ionic mobility. There is an analytical solution for the hydrostatic case of the charge density between two layers of the liquid with different permittivity and different Fluids 2024,9, 219 6 of 16 ionic mobility. The hydrostatic case relates to the scenario where charge injection does occur but without inducing any fluid motion. In this simplified case, an analytical solution is tractable. The charge density distribution is given by Equations (21) and (22) see Figure 3. Fluids 2024, 9, x FOR PEER REVIEW 7 of 19 occur but without inducing any fluid motion. In this simplified case, an analytical solution is tractable. The charge density distribution is given by Equations (21) and (22) see Figure 3. Figure 3. Hydrostatic solution in the two layers. Coefficient b is the solution of the equation: 3/2 3/2 1/2 3/2 3/ 2 4() () 10 3 inj ratio ratio ratio ratio ratio ratio ratio Cbh b bh K h b b KK εε ε ε      −−−+ + − −=       (23) Equation (23) is solved by Newton–Raphson root-finding methods. Once coefficient b is determined, coefficients a, c and d are given by () 1/2 1/2 2(); ; ratio inj ratio ratio ratio ratio aCbh cK ad b K ε ε ε =−= = Then, the charge density distribution in the two layers is fully determined. Figure 4 shows the comparison between the analytical and numerical solutions for three different and ratio ratio K ε . We can notice a perfect agreement between the numerical solutions and the analytical ones. Figure 3. Hydrostatic solution in the two layers. Coefficient bis the solution of the equation: 4 3 Cinj εratio (b−h)1/2 b3/2 −(b−h)3/2 +pKratioεratio"h+εratio Kratio b3/2 −εratio Kratio b3/2#!−1=0 (23) Equation (23) is solved by Newton–Raphson root-finding methods. Once coefficient b is determined, coefficients a,cand dare given by a=2 εratio Cinj(b−h)1/2 ;c=(Kratioεratio)1/2a;d=εratio Kratio b Then, the charge density distribution in the two layers is fully determined. Figure 4shows the comparison between the analytical and numerical solutions for three different εratio and Kratio . We can notice a perfect agreement between the numerical solutions and the analytical ones. Fluids 2024, 9, x FOR PEER REVIEW 8 of 19 Figure 4. Comparison between analytical vs. numerical solution of the hydrostatic case for different and ratio ratio K ε . 3.2. Mixture Controled by Electro-Convection To quantify the mass transfer rate in the flow domain, we have introduced the Lacey mixing index [15]. This mixing index is defined as follows: 22 1 1() cells N im i cells CC N σ = =−  (24) where i C represents the concentration of liquid 1 in the cell i. m C is the mean expected concentration in the domain. It is equal to 0.5 as the two liquids layers have the same initial volume. σ is the degree of mixing and represents the RMS of the deviation of the concentration from the mean value m C. When the mixture is totally homogenous, 0 σ =. We have computed the evolution of the Lacey index vs. time for different parameters. 3.2.1. Effect of Electro-Convection on the Mixing Please note that from Figures 5–10, the time as well as all quantities displayed are dimensionless. In Figure 5, we have plotted the evolution of Lacey index vs. time. We can observe two parts in this curve. Each part is defined by a certain slope. The first slope (blue) indicates that the two liquids mix solely through molecular diffusion. As the diffusion coefficient D increases, the Schmidt number S c decreases, resulting in an increase in the first slope (blue). When the electric Rayleigh number T increases, the second slope (red) also increases. At T = 0, there is no electro-convection, so the two fluids mix purely via molecular diffusion. When T exceeds the critical value T c = 164, electro-convective cells begin to emerge, significantly enhancing the mixing of the two fluids. As soon as T > T c increases, the second slope (red) becomes steeper. Figure 4. Comparison between analytical vs. numerical solution of the hydrostatic case for different εratio and Kratio. Fluids 2024,9, 219 7 of 16 3.2. Mixture Controled by Electro-Convection To quantify the mass transfer rate in the flow domain, we have introduced the Lacey mixing index [15]. This mixing index is defined as follows: σ2=1 Ncells Ncells ∑ i=1 (Ci−Cm)2(24) where Ci represents the concentration of liquid 1 in the cell i. Cm is the mean expected concentration in the domain. It is equal to 0.5 as the two liquids layers have the same initial volume. σ is the degree of mixing and represents the RMS of the deviation of the concentration from the mean value Cm. When the mixture is totally homogenous, σ=0. We have computed the evolution of the Lacey index vs. time for different parameters. 3.2.1. Effect of Electro-Convection on the Mixing Please note that from Figures 5–10, the time as well as all quantities displayed are dimensionless. In Figure 5, we have plotted the evolution of Lacey index vs. time. We can observe two parts in this curve. Each part is defined by a certain slope. The first slope (blue) indicates that the two liquids mix solely through molecular diffusion. As the diffusion coefficient Dincreases, the Schmidt number S c decreases, resulting in an increase in the first slope (blue). When the electric Rayleigh number Tincreases, the second slope (red) also increases. At T= 0, there is no electro-convection, so the two fluids mix purely via molecular diffusion. When Texceeds the critical value T c = 164, electro-convective cells begin to emerge, significantly enhancing the mixing of the two fluids. As soon as T>T c increases, the second slope (red) becomes steeper. Figure 5. Mixing index vs. time for T= 0 and T= 175: evolution of the two slopes. Fluids 2024,9, 219 8 of 16 Fluids 2024, 9, x FOR PEER REVIEW 9 of 19 Figure 5. Mixing index vs. time for T = 0 and T = 175: evolution of the two slopes. Figure 6. (a) Mixing vs. time for T = 0 and T = 175: identification of electro-convection start with Vmax vs. time (b) and Vmax vs. T (c). Figure 6. (a) Mixing vs. time for T= 0 and T= 175: identification of electro-convection start with Vmax vs. time (b) and Vmax vs. T(c). Figure 7. Influence of Ton the mixing of the liquids. The structure of the concentration field is displayed along the curve to show the structure of the flow at given instants. As Tis increased the slope of the curve increases. Fluids 2024,9, 219 9 of 16 Fluids 2024, 9, x FOR PEER REVIEW 10 of 19 Figure 7. Influence of T on the mixing of the liquids. The structure of the concentration field is displayed along the curve to show the structure of the flow at given instants. As T is increased the slope of the curve increases. Figure 8. Lacey index vs. time for different Ratio K values, T = 190, S c = 1000. Figure 8. Lacey index vs. time for different KRatio values, T= 190, Sc= 1000. Fluids 2024, 9, x FOR PEER REVIEW 11 of 19 Figure 9. Lacey index vs. time for different Ratio ε values, T = 190, S c = 1000. Figure 10. Evolution of Vmax vs. time for different Schmidt numbers. T = 175, 11 ratio ratio K ε == . In Figure 6a, we plotted the evolution of the Lacey index over time, highlighting the onset of electro-convection, which occurs precisely at t = 107.65, as indicated by the red arrow. Figure 6b illustrates the maximum velocity V max over time to demonstrate the development of electro-convective instability for T = 175 and T = 0. For T = 0 (flat red curve), we observe no movement in the liquid, confirming that mixing occurs solely through pure molecular diffusion. Figure 6c presents the stability diagram, where we have plotted the maximum velocity V max at a steady state against different values of the T parameter. We observe that the linear critical value T c for which electro-convection occurs, in this case, is slightly higher than the classical value T c = 164 for homogeneous liquids. Investigations Figure 9. Lacey index vs. time for different εRatio values, T= 190, Sc= 1000. In Figure 6a, we plotted the evolution of the Lacey index over time, highlighting the onset of electro-convection, which occurs precisely at t= 107.65, as indicated by the red arrow. Figure 6b illustrates the maximum velocity V max over time to demonstrate the development of electro-convective instability for T= 175 and T= 0. For T= 0 (flat red curve), we observe no movement in the liquid, confirming that mixing occurs solely through pure molecular diffusion. Figure 6c presents the stability diagram, where we have plotted the maximum velocity V max at a steady state against different values of the T parameter. We observe that the linear critical value T c for which electro-convection occurs, in this case, is slightly higher than the classical value T c = 164 for homogeneous liquids. Investigations are currently underway to explain this discrepancy between homogeneous and non-homogeneous liquids. Fluids 2024,9, 219 16 of 16 to explore the underlying mechanisms and optimize the control of electro-convection for practical applications. Author Contributions: Conceptualization: P.T., A.T.P. and S.M. Methodology: P.T., A.T.P. and P.A.V. Software: P.T., Validation: P.T., A.T.P. and P.A.V. Formal analysis: P.T., A.T.P., S.M., A.B., P.A.V. and Z.Y. Investigation: P.T., A.T.P., S.M., A.B., P.A.V. and Z.Y. Writing—original draft preparation: P.T. and A.T.P. Writing—review and editing: P.T., A.T.P., S.M., A.B., P.A.V. and Z.Y. Funding acquisition: P.T., A.T.P., P.A.V. and A.B. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the France 2030 program (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01). A.T. Pérez. and P. A. Vázquez acknowledge financial support by MCIN/AEI/10.13039/501100011033 by ERDF/EU (Grant number PID2022-138890NB-I00). Data Availability Statement: The availability of the data won’t be public. The policy of the French republic in terms of Protection of scientific and technical potential (https://www.enseignementsuprecherche.gouv.fr/fr/protection-du-potentiel-scientifique-et-technique-et-intelligence-economique- ppst-46370) does not allow us to make this data publicly available. The data could be available only on personal requests. Acknowledgments: AA. Bhattacharya acknowledges the fellowship from Ministry of Education, Govt. of India for graduate students, Mondal. Conflicts of Interest: The authors declare no conflict of interest. References 1. Castellanos, A. Electrohydrodynamics; Springer: New York, NY, USA, 1999. 2. Chang, J.S.; Kelly, A.J.; Crowley, J.M. Handbook of Electrostatic Processes, 2nd ed.; Dekker: New York, NY, USA, 1995. 3. Traoré, P.; Pérez, A.T.; Koulova, D.; Romat, H. Numerical modelling of finite-amplitude electro-thermo convection in a dielectric liquid layer subjected to both unipolar injection and temperature gradient. J. Fluid Mech. 2010,658, 279. 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