Electrohydrodynamic instabilities in freely suspended viscous films under normal electric fields

Electrohydrodynamic instabilities of fluid-fluid interfaces can be exploited in various microfluidic applications to enhance mixing, replicate well-controlled patterns, or generate drops of a particular size. In this work, we study the stability and dynamics of a system of three superimposed layers of two immiscible fluids subject to a normal electric field. Following the Taylor-Melcher leaky dielectric model, the bulk remains electroneutral while a net charge accumulates on the interfaces. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow, and finite charge relaxation. Using this model, we perform a linear stability analysis and identify different modes of instability, and we characterize the behavior of the system as a function of the relevant dimensionless groups in each mode. Further, we perform numerical simulations using the boundary element method to study the effect of nonlinearities on long-time interfacial dynamics. We demonstrate how the coupling of flow and surface charge transport in different modes of instability can give rise to nonlinear phenomena such as tip streaming or pinching of the film into droplets.

Figure 1

Problem definition: A liquid film suspended between two semi-infinite immiscible liquid layers is subject to a perpendicular electric field ${\mathit{E}}_{\infty }$.

Figure 2

Streamlines of the flow in the two dominant modes of linear instability: (a) in-phase (sinuous) mode when $R=2$ and (b) antiphase (varicose) mode when $R=0.5$. Other parameters are $(Q,\lambda ,C{a}_E,{\text{Re}}_E)=(1,1,10,1)$ in both systems. The blue region illustrates a typical shape of the film in the dominant mode.

Figure 3

Growth rate versus wave number in the dominant modes of linear instability for $(R,Q,\lambda )=(0.5,1,1)$: (a) effect of electric capillary number when ${\text{Re}}_E=1$. Inset shows the unstable wave numbers. (b) Effect of electric Reynolds number when $C{a}_E=10$.

Figure 4

Wave number $k_{\text{marg}}$ corresponding to marginally stable perturbations (${\sigma }_{\mathrm{marg}}=0$) in different modes of instability as a function of (a) electric capillary number and (b) electric Reynolds number for two different systems with $R=0.5$ and $R=2$ while $(Q,\lambda )=(1,1)$. ${\text{Re}}_E=1$ in (a) and $C{a}_E=10$ in (b).

Figure 5

Maximum growth rate in each mode of instability as a function of (a) electric capillary number and (b) electric Reynolds number for two different systems with $R=0.5$ and $R=2$ while $(Q,\lambda )=(1,1)$. ${\text{Re}}_E=1$ in (a) and $C{a}_E=10$ in (b) (identical systems studied in Fig. 4).

Figure 6

Maximum growth rate in each mode of instability as a function of (a) viscosity ratio $\lambda$ for two different systems with $R=0.5$ and $R=2$, (b) conductivity ratio $R$, and (c) permittivity ratio $Q$. The permittivity ratio is set to $Q=1$ in (a,b), the conductivity ratio is $R=1$ in (c), and $\lambda =1$ in (b,c). In all systems $(C{a}_E,{\text{Re}}_E)=(10,1)$.

Figure 7

Evolution of the shape of both interfaces (left column) and their charge distributions (right column) in different modes of instability: (a,b) antiphase mode for system $\mathit{S}1$, (c,d) in-phase mode for system $\mathit{S}2$. Each snapshot is color-coded based on the time it was taken (from red at $t=0$ to blue). Insets in (b,d) show the average charge densities $\overline{q}$ on each interface as a function of time. In the insets, blue and red curves show the transient average charge on the upper and lower interface, respectively, while the dashed lines show the steady interfacial charges in the base state.

Figure 8

Growth rate $s$ as a function of wave number $k$ in the dominant mode of instability obtained via numerical simulations (NS) and linear stability analysis (LSA) for $\mathit{S}1$ in (a), and $\mathit{S}2$ in (b). In these simulations, the size of the domain is set to $L_p=4\pi k_{\text{max}}^{-1}$, which is two times the wavelength associated with the fastest growing mode.

Figure 9

Snapshots of the tip-streaming jets (left column) formed on the upper interface during antiphase instability with different (a) permittivity ratios in system $\mathit{S}4$, (b) conductivity ratios in system $\mathit{S}3$, and (c) viscosity ratios in system $\mathit{S}5$. Deformation parameter is $\mathcal{D}=5$ for all cases except for $R=5$ in (b), where the minimum thickness of the film reached the local grid size at $\mathcal{D}=2.18$ (disintegration from the base). The evolution of (d) the jet profile, (e) tip curvature, and (f) the vertical tip velocity is shown in the right column for system $\mathit{S}5$ when $\lambda =0.1$. Shaded regions in (e,f) correspond to the time interval shown in (d). Also see videos in the Supplemental Material [37].

Figure 10

Snapshots of the films during in-phase mode instability with different (a) permittivity ratios in system $\mathit{S}4$, (b) conductivity ratios in system $\mathit{S}6$, (c) viscosity ratios in system $\mathit{S}7$. For the cases with inward jets, the minimum thickness of the film is $d=0.2$, or one tenth of the equilibrium thickness. $\mathcal{D}=5$ for outward jets. Also see videos in the Supplemental Material [37].

Figure 11

Fastest-growing wave number in each mode of instability as a function of (a) electric capillary number and (b) electric Reynolds number for two different systems with $R=0.5$ and $R=2$ with $(Q,\lambda )=(1,1)$. ${\text{Re}}_E=1$ in (a) and $C{a}_E=10$ in (b).

Figure 12

Fastest-growing wave number in each mode of instability as a function of (a) viscosity ratio $\lambda$ for two different systems with $R=0.5$ and $R=2$, (b) conductivity ratio $R$, and (c) permittivity ratio $Q$. The permittivity ratio is set to $Q=1$ in (a,b), the conductivity ratio is $R=1$ in (c) and $\lambda =1$ in (b,c). In all systems $(C{a}_E,{\text{Re}}_E)=(10,1)$.

Figure 13

Evolution of the tip streaming jets during antiphase instability as a function of (a,b) permittivity ratio in system $\mathit{S}4$, (c,d) conductivity ratio in system $\mathit{S}3$, and (e,f) viscosity ratio in system $\mathit{S}5$. Left and right columns show the evolution of the tip curvature and vertical tip velocity, respectively.

Figure 14

Evolution of the inward and outward jets during in-phase instability as a function of (a,b) permittivity ratio in system $\mathit{S}4$, (c,d) conductivity ratio in system $\mathit{S}6$, and (e,f) viscosity ratio in system $\mathit{S}7$. Left and right columns show the evolution of the tip curvature and vertical tip velocity, respectively.