Electrohydrodynamic viscous fingering of leaky dielectric fluids in a channel

Viscous fingering is a common interfacial instability phenomenon that takes place when a less viscous fluid is injected into a gap that is initially filled by a more viscous fluid,¹ which was first thoroughly reported and studied by Saffman and Taylor in 1958.² During the displacement, the destabilizing viscous force overcomes the stabilizing surface tension force, forming a fingerlike shape at the fluid interface. In many engineering applications, this interfacial instability is undesirable. For example, the efficiency of enhanced oil recovery by water flooding is decreased significantly due to the emergence of viscous fingering, which leads to nonuniform displacement and residual trapping of oil.^3,4 Similarly, viscous fingering also affects the storage capacity for $CO2$ geological sequestration (a way to mitigate global warming).^5,6 Furthermore, the separation performance of a chromatographic system becomes less efficient and viable as the viscous fingering can lead to peak shape distortion and band broadening.^7–9 However, there are also cases where the viscous fingering can be conducive, such as increasing the mixing efficiency for microfluidic devices,^10,11 and pattern formation for designing different morphologies.^12–15 Therefore, the ability to control the interfacial morphology during fluid displacement processes is essential in order to obtain either a stable or an unstable interface according to different applications.

Although viscous fingering is difficult to control,¹⁶ several strategies have been proposed to realize the manipulation of viscous fingering in recent years. For instance, it has been found that the geometrical heterogeneity, such as a depth gradient in the flow passage¹⁷ and structured porous media,¹⁸ can be employed to control the viscous fingering instability. In Hele–Shaw cells, it was demonstrated that replacing the rigid upper boundary of the flow passage with an elastic membrane provides a stabilizing force at the fluid interface due to the membrane deformation, which consequently suppresses the viscous fingering.^19,20 Furthermore, studies show that increasing the wettability of the displacing fluid enhances cooperative pore filling,^21,22 thus, stabilizing the viscous fingering. Additionally, it has been shown both experimentally and numerically that viscous fingering can be controlled by adjusting the injection rate.^15,23,24

However, in many practical applications, the flow condition and geometry are specified and cannot be arbitrarily modified. Thus, the methods mentioned above are not applicable to control viscous fingering under this condition. To deal with this problem, electrokinetics has been applied as an alternative approach,^16,25–27 where an electric double layer is formed as a result of the dissociation of ionic surface groups. Consequently, the electro-osmotic flow can be induced due to the interaction of the applied external electric field and the electric charges in the electric double layer. Depending on the direction of the electric field, the viscous fingering can either be enhanced or suppressed. Nevertheless, the fluids investigated in these studies were electrolytes, which were perfectly conducting. For the poorly conducting fluids, the behaviors were significantly different from the conducting ones.²⁸ At the same time, it has been shown that although the electrical conductivity of poorly conducting fluids is infinitesimal, it plays a crucial role in the fluid flow under an electric field due to the existence of free charges.²⁸ To account for this effect, the leaky dielectric model was introduced by Taylor in the 1960s,²⁹ which describes the behavior of poorly conducting fluids with small electrical conductivity. For leaky dielectric fluids, the electrical force under an electric field has two sources: one from the polarized charges, i.e., dielectric effect, and the other from the free charges due to the conducting effect.²⁸ An electrical force is induced at the fluid interface, which differs from the aforementioned studies where the viscous fingering of electrolytes (perfect conducting fluids) is controlled by taking advantage of the electro-osmotic flow. Furthermore, it has been demonstrated that various patterns can be designed by applying an electric field.^30–32 

In this paper, the viscous fingering of leaky dielectric fluids under an electric field is investigated. A hybrid numerical model based on the lattice Boltzmann method (LBM) and finite difference method (FDM) is developed and validated. Next, the validated numerical model is applied to study the development of viscous fingering under an electric field for different parameters, including the direction of the electric field, permittivity ratio, conductivity ratio, and the electric field strength. The results show that the effects of a horizontal and vertical electric field on viscous fingering are different. Depending on the permittivity and conductivity ratio, viscous fingering can be promoted or suppressed. The promotion and suppression effect becomes stronger with the increase in the electric field strength. Particularly, different interfacial morphologies (broad fingers and thin jets) are observed when viscous fingering is promoted. A phase diagram for describing the interface stability and interfacial morphology under different permittivity and conductivity ratios is proposed for the horizontal and vertical electric fields, respectively.

The control of viscous fingering of two leaky dielectric fluids involves a coupled system of multicomponent fluid flow and electric field. The governing equations of this system can be expressed as follows.

The governing equations for the multicomponent fluid flow (hydrodynamics) in the presence of an electric field are described by the Navier–Stokes equations (NSE) with a source term representing the electrical force,

where ρ, u, p, and μ represent fluid density, velocity, pressure, and dynamic viscosity, respectively. $γ,κ, and δ$ are the interfacial tension, the local curvature of the interface, and the Dirac-delta function. $Fe$ is the electrical force acting at the fluid–fluid interface.

For the leaky dielectric model, the electric stress induced under an electric field is expressed by the Maxwell stress tensor,^28,33

where E and ε are the electric field strength and fluid permittivity. I is the second-order identity tensor. By taking the divergence of the Maxwell stress tensor, the electrical force can be obtained,

where $qv=∇·(εE)$ represents the volume charge density. The first term on the right-hand side of Eq. (4) is the Coulomb force due to the interaction of the electric charges with the external electric field, which is in the direction of the electric field. The second term is the polarization stress due to the permittivity gradient at the fluid–fluid interface, which acts along the normal direction of the interface.

In order to obtain the electric field strength E, charge density q_v, and then calculate the electrical force $Fe$⁠, the first step is to calculate the electric potential distribution U in the fluids. According to the charge conservation equation, the electric field can be related to the fluid conductivity σ and the electric charge density q_v as follows:

For the leaky dielectric model, it is assumed that the electric relaxation timescale is much smaller than the timescale of fluid flow.²⁸ Consequently, the electric charges accumulate at the interface almost instantaneously. Hence, Eq. (5) can be simplified as

Furthermore, under the assumption that the magnetic induction is ignored and the electric field strength is irrotational, namely $∇×E=0$⁠, the electric field strength can be written in terms of the electric potential as $E=−∇U$⁠. Thus, Eq. (6) can be further expressed as

By solving Eq. (7), the electric potential distribution in the computational domain is obtained. Subsequently, the electric field strength E and electric charges q_v can be calculated. Therefore, the electrical force at the fluid interface due to the interaction between the electric charges and the electric field is obtained.

In this section, the numerical methods used to solve the governing equations for the hydrodynamics and electric field are described in detail.

In this work, the pseudopotential lattice Boltzmann method (LBM) proposed by Shan and Chen,^34,35 which has been applied to study various multicomponent fluid flow problems,^36–38 is employed to simulate the hydrodynamics of the multicomponent fluid flow. The reason why LBM has become popular for simulating fluid flows is that it is numerically efficient and easy to implement. In addition, unlike traditional numerical methods, there is no need for complex interface tracking or capturing techniques for multiphase fluid flow. Fundamentally, LBM is a mesoscopic approach based on the kinetic theory. It describes the fluid flow using the particle distribution function $fi(x,t)$⁠, which represents the density of particles with discrete velocity e_i at position x and time t. Although LBM does not solve NSE directly, it can recover NSE using Chapman–Enskog analysis.³⁹ 

Furthermore, in order to improve the numerical stability and simulate fluid pairs with large kinematic viscosity ratio, multiple-relaxation-time (MRT) collision operator⁴⁰ and explicit forcing scheme for incorporating external force into LBM proposed by Porter et al.⁴¹ are exploited. Thus, the evolution equation of the particle distribution function for SC model with MRT collision operator and explicit forcing scheme can be written as

where $fiα$ is the particle distribution function for $αth$ component. $Δt$ denotes the time step, which is set to be 1 in this study. $fiα,eq$ represents the corresponding equilibrium distribution function, which can be expressed as

where ω_i are the weighting coefficients, and c_s is the speed of sound. For the $D2Q9$ lattice structure considered in this study, ω_i is given by $ω0=4/9, ω1−4=1/9, ω5−8=1/36$⁠.

M in Eq. (8) is the transformation matrix, which for $D2Q9$ structure is defined as

$Sα$ is the diagonal relaxation matrix and is given by

where $Siα$ denotes the inverse of the relaxation time of $fiα$⁠. $Scα$ corresponds to the conserved moments (i.e., density and momentum), and it is set to be 1 here. $Seα, Sϵα, Sqα$⁠, and $Svα$ correspond to the nonconserved moments (i.e., energy, energy squared, energy flux, and stress tensor, respectively). Whereby, the nonconserved moments $Seα, Sϵα$⁠, and $Sqα$ are adjustable parameters in MRT-LBM. A proper choice of these values may enhance the stability and accuracy of the algorithm.⁴² In the current study, we set the nonconserved moments according to previous investigations,⁴³ such that $Seα, Sϵα$⁠, and $Sqα$ equate to 1.64, 1.54, and 1, respectively. Whereas, $Sνα$ is related to the fluid viscosity by

On the right-hand side of Eq. (8), the term $Fiα$ represents the force acting on the $αth$ component and is defined as^41,44

where $Fα=Ffα+Fsα+Feα$ is the total external force with $Ffα, Fsα$⁠, and $Feα$ representing the fluid–fluid force, fluid–solid force, and electrical force, respectively. The fluid–fluid force and fluid–solid force imposed on the α_th component at position x can be expressed as^34,35,45

where G is the interaction strength, and $ψα(x)$ is the interaction potential, which is taken as $ψα(x)=ρα(x)$ in this work.

The macroscopic equilibrium velocity $uαeq$ in Eq. (13) is calculated as

where $ρα$ and $uα$ are the density and velocity of the $αth$ component,

In this study, both fluids are considered as leaky dielectric fluids with constant permittivity ε and conductivity σ in each of the bulk fluids. To account for the variation of the electrical properties at the fluid–fluid interface, the local ε and σ can be calculated by

where ε, σ, and ρ are the permittivity, conductivity, and density with subscripts indicating fluid 1 or fluid 2. The finite difference method (FDM) is used in this study to determine the quasi-steady distribution of electric potential in the fluids and then calculate the electrical force at the fluid-fluid interface. Specifically, second-order central differencing scheme is employed to discretize the electric potential governing equation [Eq. (7)]. The obtained discretized equation is then solved by the Successive-Over-Relaxation (SOR) scheme⁴⁶ at each time step of LBM. The electric potential is considered to have reached the steady state when

where $δU<10−6$ is the tolerance, and n represents the nth iteration. Subsequently, the electric field strength E and electric charge q_v can be obtained. Finally, the external electrical force due to the applied electric field is calculated according to Eq. (4) at each lattice node and incorporated into the collision step of LBM⁴⁷ using the explicit forcing scheme.⁴¹ 

The validation of the hybrid LBM and FDM for simulating multicomponent electrohydrodynamics is conducted using droplet deformation under an electric field. The droplet will be deformed into either an oblate or prolate shape due to different electrical properties of the fluids and the interaction between the electric charges and the electric field. To quantify the droplet deformation in the steady state, a deformation factor D is defined as

where L and H are the lengths of the deformed droplet in the direction parallel and perpendicular to the electric field, respectively.

Based on the first-order model and small deformation theory introduced by Taylor,²⁹ Feng⁴⁸ proposed an improved analytical solution for droplet deformation,

where $CaE=εoE2a/γ$ is the electric capillary number representing the ratio of the electric force to the surface tension, whereby, $εo$ is the permittivity of fluid outside the droplet, and a is the initial radius of the droplet. Moreover, $fd(R,S)=R2+R+1−3S$ is the discriminating function that determines whether the droplet will deform into a prolate shape (⁠$fd>0$⁠) or an oblate shape (⁠$fd<0$⁠). $R=σi/σo$ and $S=εi/εo$ are the conductivity ratio and permittivity ratio, respectively, between the fluid inside and outside the droplet.

In the validation, two typical leaky dielectric systems with $(R,S)=(2.0,0.5)$ and $(R,S)=(0.5,2.0)$ are considered. The same mesh as the fluid flow, i.e., 400 × 400 lu (lattice unit), is used to solve the electric field, which makes the coupling between these two fields more efficient. In this work, the simulation is performed in dimensionless lattice units. Therefore, all the values of the parameters are based on the lattice unit. Half-way bounceback⁴⁹ boundary conditions are applied in both x and y directions for the hydrodynamic field. Whereas, for the electric field, Dirichlet boundary conditions, i.e., constant electric potential, are applied in the y direction and Neuman boundary conditions, i.e., zero electric potential gradient, are applied in the x direction. The results presented in Fig. 1 show a good agreement with the analytical solution [Eq. (22)]⁴⁸ when $|D|<0.1$⁠. For larger deformation, however, it can be observed that the numerical results gradually depart from the analytical solution. The reason is that this analytical solution is only applicable to small deformation, i.e., $|D|<0.1$⁠.⁴⁸ This result is also consistent with the previous study,⁵⁰ where a hybrid immersed boundary and immersed interface method is employed for simulating droplet deformation under an electric field.

The validated numerical model is applied to investigate the effects of an external electric field (including the strength and direction of the electric field), permittivity ratio, and conductivity ratio on the development of viscous fingering. The schematic of the problem under consideration is shown in Fig. 2(a), where a more viscous fluid with viscosity μ₂, permittivity $ε2$⁠, and conductivity σ₂ is displaced by a less viscous fluid with viscosity μ₁, permittivity $ε1$⁠, and conductivity σ₁ (viscosity ratio $M=μ2/μ1=3$⁠) injected from the bottom of the channel with length l = 600 lu and width w = 152 lu at a constant rate u = 0.04 lu. In this work, μ₂, $ε2$⁠, and σ₁ are fixed at 0.5, 0.2, and 0.5, respectively. Other parameters can be obtained based on the corresponding ratio. Constant pressure is applied along the top boundary. Both fluids are immiscible and incompressible with the same density $ρ1=ρ2=1$ lu. The interaction strength between two fluids is set to be $G=−4.4$⁠, which leads to an interfacial tension of 0.0998 lu according to the Young–Laplace test. These conditions correspond to $Re=ρuL/μ=12$ and $Ca=uμ/γ=0.2$⁠, where L and γ represent the characteristic length and surface tension, respectively. Initially, the fluid–fluid interface is assumed to take a sinusoidal shape $y=10+4 sin(2π(x−1/4))$⁠, where $0<x<1$⁠, to introduce the interface perturbation, which is much smaller than the characteristic width of the channel. For the viscous fingering without an electric field, the fluid interface tends to form a fingerlike shape, as shown in Fig. 2(b). In this study, the effect of a horizontal (E_x) and vertical electric field (E_y) on the development of viscous fingering will be investigated separately.

For leaky dielectric fluids, the free and polarized charges accumulate at the fluid interface under an electric field, generating an electrical force that affects the development of viscous fingering. In this section, the effects of a horizontal electric field on viscous fingering are investigated. Two different permittivity ratios $S=ε2/ε1$ are studied to gain a perspective of the effects of permittivity ratio, i.e., S = 4 and S = 1/4. For each case, we study the effect of the conductivity ratio $R=σ2/σ1$ on the viscous fingering. One of the focuses of this work is to investigate how the fluid electrical properties (S and R) impact the stability of the interface, i.e., whether it is suppressing or promoting effects, as reflected by the structure height $Δh$ when compared with that under no electric field. As a result, we have adopted different ranges of conductivity ratio R under different permittivity ratios S such that both the suppressing and promoting regimes are covered in the simulation results (see Figs. 3 and 4). To characterize the viscous fingering instability, the structure height $Δh$ is defined as

where h_m and h_s denote the positions of the middle and side points of the interface, respectively. The positions of these two points are monitored during the development of viscous fingering under an electric field.

For the first case where the permittivity ratio S is 4, with conductivity ratio R varied, the stability of the system is characterized by the structure height $Δh$ at characteristic time $t*=1.7$ [see Fig. 3(a)]. The characteristic time $t*$ is defined as $t*=tc/tf$⁠, where t_c is the current time step, and t_f is the timescale of the flow given as $tf=l/u$⁠. The high electric potential (left wall) is 40 with the low electric potential being 0 (right wall), which corresponds to an electric field strength E = 0.27. Compared with the case without an electric field [the red dashed line in Fig. 3(a)], viscous fingering is promoted when R < 2. However, this instability is suppressed when R > 2. If $Δh$ is less than 0.013, which is the initial structure height at $t*=0$ (the green dashed line), the viscous fingering is considered to be suppressed completely by the electric field, namely, the system is stabilized.

The suppression or promotion effect of the electric field on the viscous fingering is determined by the electrical force along the fluid interface. For leaky dielectric fluids, both free and polarized charges exist. Under a horizontal electric field, the electrical force due to free charges is mainly in the horizontal direction. However, the force originating from the polarized charges is in the vertical direction and responsible for the effect of electric field on the viscous fingering. Therefore, to investigate the reasons on whether the viscous fingering is suppressed or promoted, the electrical force in the y direction along the fluid interface is calculated for different R, as depicted in Fig. 3(b). The force is normalized by $|ε2−ε1|U02/(2w2)$⁠. Despite the fact that the interfacial morphology evolves over time, the electrical force distribution at the initial time can uncover the underlying reason for which the fluid interface develops under the applied electric field. It can be observed from Fig. 3(b) that the viscous fingering is enhanced when R is small (see R = 0.5), as the vertical electrical force along the interface is pushing the interface downward with the magnitude of the force in the center less than that near the wall. However, an increase in R leads to a change of the electrical force distribution and magnitude. Although the direction of the electrical force keeps unchanged (downward), the magnitude of the force in the channel center is greater than that near the wall, imposing a suppression effect on the viscous fingering. At the same time, the difference between the maximum and minimum value of the electrical force becomes greater as R increases, leading to a larger suppression effect. We want to emphasize here that it is the differential electrical force between the maximum and minimum value instead of the absolute value that is actually impacting the development of viscous fingering.

In order to investigate the effects of permittivity ratio on the viscous fingering, simulations for the second case where S is inverted (i.e., S = 1/4) are also carried out with different R, as shown in Fig. 4(a). It is found that compared to the first case, R plays a different role, where the viscous fingering is suppressed at a small R value. However, a large R leads to the promotion of viscous fingering. Similarly, the electrical force in the y direction along the fluid interface [see Fig. 4(b)] is plotted to explain this phenomenon. Different from S = 4, the electrical force points upward when S = 1/4 with a greater magnitude near the wall compared with the channel center for R < 1. Thus, a suppression effect is imposed on the viscous fingering. Whereas, an inverse distribution of the electrical force is observed when R = 1.5, enhancing the development of viscous fingering. Although the electrical force in the y direction along the fluid interface is the underlying reason for affecting the viscous fingering instability, the electrical force in the x direction also plays a role, which will be discussed below. It is the combinational effect of these two forces, viscous force, and capillary force.

Then, the effect of the electric field strength E on the development of viscous fingering is studied for two phenomena, namely, the suppressing and promoting cases. For each phenomenon, two fluid systems with different S and R are investigated, as shown in Table I. For the suppressing case, it is shown in Fig. 5(a) that the horizontal electric field suppresses the viscous fingering when S = 4, R = 5 and S = 1/4, R = 1/2. With an increase in E, the structure height $Δh$ both decreases with the case of S = 1/4, R = 1/2 experiencing a faster drop. Eventually, $Δh$ approaches zero, meaning that the viscous fingering is completely suppressed [see the inset in Fig. 5(a)]. The electrical force in the y direction along the fluid interface for three representative cases shown as black circles in Fig. 5(a) is plotted to reveal the mechanism of this phenomenon. It is clear that the differential electrical force increases with the increase in the electric field strength E when S and R are the same. Furthermore, the differential electrical force is greater for S = 1/4, R = 1/2 compared with S = 4, R = 5 under the same electric field strength E, which also explains the faster decrease in $Δh$ for S = 1/4, R = 1/2 in Fig. 5(a).

For the promoting case, the viscous fingering instability is enhanced more as E is increasing, as can be seen in Fig. 6(a). The interfacial morphology for different fluid systems, however, is different [see the inset in Fig. 6(a)]. When S = 4 and R = 1/2, a broad fingerlike shape is observed. Whereas, a thin jet is formed for S = 1/4 and R = 1.5. The explanation of the development of viscous fingering under a horizontal electric field for different S and R has been discussed by analyzing the electrical force distribution in the y direction along the fluid interface [see Figs. 3(b) and 4(b)]. At the same time, the electrical force in the x direction also plays a role in determining the interfacial morphology when the applied electric field is in the horizontal direction, as depicted in Fig. 6(b). When S = 4 and R = 1/2, the finger is stretched in width according to the electrical force distribution, leading to the formation of a wider finger. On the other hand, this force tends to squeeze the finger when S = 1/4 and R = 1.5. Thus, the fluid interface becomes long and thin.

Based on the simulation results, a phase diagram is proposed to characterize the interfacial morphologies under different conductivity ratios R for four permittivity ratios, namely S = 1/4, 1/3, 3, and 4, as shown in Fig. 7. It is shown that the fluid-–fluid interface becomes flat at small R values when S < 1 and at large R values when S > 1 (solid circle), which means that the viscous fingering is suppressed. However, the viscous fingering is enhanced at large R values when S < 1 and at small R values when S > 1 with different interfacial morphologies for these two cases. For the first case, namely large R values at S < 1, thin jets are formed at the fluid interface. Whereas, broad fingers are observed for another case.

To investigate the effect of the direction of the applied electric field, the viscous fingering under a vertical electric field is analyzed in this section.

Similarly, two cases are considered here. For the first case, S and E are fixed at 4 and 0.27, respectively. The effects of the conductivity ratio R on the viscous fingering are studied by plotting the structure height $Δh$ at $t*=1.7$⁠, and the results are shown in Fig. 8(a). It can be seen that the structure height first decreases with the increase in R and has a minimum value when R = 1, after which $Δh$ increases. Similar results are also obtained in a previous study,⁵¹ where the conductivity ratio R has a non-monotonic aggravation effect of the Rayleigh–Taylor instability under a vertical electric field, although it may actually be considered as suppression effect for the points below the red dashed line that represents the electric field-free case. To explain this phenomenon, the electrical force in the y direction along the fluid interface is plotted in Fig. 8(b), where the force is normalized by $|ε2−ε1|U02/(2l2)$⁠. When R = 1/3, the electrical force is pointing upward (i.e., positive). However, the magnitude of the force is greater in the middle than that on the side. As a result, the interface is dragged upward more in the middle, thereby promoting viscous fingering instability. Whereas, the non-uniformity of the electrical force distribution (or the differential electrical force) decreases with increasing of R (see R = 1 and R = 2), leading to a weaker promotion effect. More obvious non-uniformity is observed when R is further increased with the force pointing downward (see R = 3 and R = 4). Consequently, the promotion effect of the vertical electric field on the viscous fingering becomes stronger. Likewise, it is the differential electrical force that impacts the development of viscous fingering.

For the second case where S and E are fixed at 1/4 and 0.27, the effects of R on the viscous fingering are characterized by using the structure height $Δh$ at $t*=1.7$ for each R value, as shown in Fig. 9(a). Similarly, $Δh$ tends to decrease at first and then increase with the increase in R. However, the $Δh$ for all R values are higher than the case without an electric field, which means that the viscous fingering is promoted regardless of R when S = 1/4. Again, this phenomenon can be explained by the direction and uniformity of the electrical force in the y direction, as depicted in Fig. 9(b).

Finally, the viscous fingering under different electric field strengths E is analyzed. Two cases with different permittivity and conductivity ratios are considered, namely $(S,R)=(4,1/3)$ and $(S,R)=(1/4,1.5)$⁠. By changing E, $Δh$ for each case at $t*=1.7$ is plotted, and the results are shown in Fig. 10(a). Clearly, the promotion effect of a vertical electric field on the viscous fingering becomes stronger as E increases. However, different morphologies can be observed for these two fluid systems. When R = 4 and S = 1/3, the interface develops into a thin jet with the increase in E, whereas a broader fingerlike shape is observed when R = 1/4 and S = 1.5. In this case, the direction and distribution of the electrical force in the vertical direction along the interface [see Fig.10(b)] are responsible for different morphologies, which have been discussed previously for explaining the promotion effect of the vertical electric field on the viscous fingering. The electrical force is positive (pointing upward) at S = 4 and R = 1/3 with the maximum value at the middle point. Therefore, the development of the fluid interface in the middle is accelerated, which renders the interface thin and long. For R = 1/4 and S = 1.5, the electrical force is negative (pointing downward), and the force attains a maximum magnitude at the side points. As a result, this force decelerates the interface development for the side points more than for the middle points, promoting and broadening the viscous fingering at the same time.

To characterize various morphologies of viscous fingering under different permittivity ratios S and conductivity ratios R, a phase diagram is proposed based on the simulation results, as depicted in Fig. 11. Similarly, four permittivity ratios are considered, i.e., S = 1/4, 1/3, 3, and 4. For each S, the interfacial morphology under different conductivity ratios R is obtained. The results show that a thin jet is formed when R is small for all S. Whereas, a broad finger is observed as R becomes larger. No flat interface is observed, which means that the viscous fingering is promoted for all cases under a vertical electric field.

To summarize, a hybrid numerical model based on LBM and FDM is developed to investigate the control of viscous fingering of leaky dielectric fluids in a channel in the presence of an electric field. Systematic simulations are carried out in terms of the effects of the direction of the applied electric field, permittivity ratio, conductivity ratio, and electric field strength on viscous fingering. It is found that viscous fingering can be suppressed or promoted under a horizontal electric field depending on the permittivity and conductivity ratios, which is attributed to the electrical force distribution in the y direction along the fluid interface. The emphasis here is that the differential electrical force is the underlying reason for influencing the development of viscous fingering. Based on the simulation results, a phase diagram for a horizontal electric field is proposed, which describes the interface morphology under different permittivity ratios S and conductivity ratios R during the displacement process. Moreover, the effect of a vertical electric field on the viscous fingering is a non-monotonous function of the conductivity ratio R. The structure height $Δh$ first decreases and then increases with R. At the same time, different morphologies are also observed under a vertical electric field. A phase diagram characterizing the interfacial morphology under different permittivity ratios S and conductivity ratios R for the vertical electric field is also established, which shows various interface shapes when the viscous fingering is enhanced.

We acknowledge the High-Performance Computing facilities at Queensland University of Technology. E. Sauret is the recipient of an Australian Research Council (ARC) Future Fellowship (No. FT200100446), funded by the Australian government. J. Zhao gratefully acknowledges the ARC for support through a Ph.D. scholarship (No. FT200100446).

The authors have no conflicts to disclose.

Jiachen Zhao: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Zhongzheng Wang: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Yuantong Gu: Resources (equal); Supervision (equal); Writing – review & editing (equal). Emilie Sauret: Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.