5.1.1 Effect of the $Oh^{2}$ number

The monotonic behaviour seen in figures 4 and 5 for the deformation-time curves will be disrupted if $Oh^{2}$ becomes sufficiently small. This parameter controls the response dynamics through the local acceleration term $\unicode[STIX]{x2202}\boldsymbol{u}/\unicode[STIX]{x2202}t$, per (2.1). For the ambient fluid, $1/Oh^{2}$ emerges as the parameter that controls the local acceleration term. However, considering this equation for the drop fluid and balancing the first and last terms suggests that

(5.3)$$\begin{eqnarray}\frac{\widetilde{\unicode[STIX]{x1D70C}}}{\widetilde{\unicode[STIX]{x1D707}}}\left(\frac{1}{Oh^{2}}\right)\end{eqnarray}$$

is the parameter that controls the local acceleration in general. Thus, small $Oh^{2}$, as well as large $\widetilde{\unicode[STIX]{x1D70C}}$ and small $\widetilde{\unicode[STIX]{x1D707}}$, favours an oscillatory behaviour. Generally speaking, smaller $Oh^{2}$ number is tantamount to a larger drop or a less viscous surroundings.

To explore the effect of variation of $Oh^{2}$ on the deformation-time curve, we have performed two other simulations at $Oh^{2}=0.2$ and 0.02 to compare with the reference simulation, $Oh^{2}=2$. Since in these simulations the capillary number $Ca$ is fixed, a smaller $Oh^{2}$ leads to a larger flow Reynolds number $Re_{f}=Ca/Oh^{2}$. Figure 6 compares the effect of $Oh^{2}$ on the deformation-time history for fluid systems $A{-}C$. As is evident, the dynamic response transitions from a monotonic response to an oscillatory one as $Oh^{2}$ decreases from 2 to 0.02. For $Oh^{2}=2$, the dynamic response is overdamped for all the simulations. For $Oh^{2}=0.2$ it is slightly underdamped for fluid system $A$ as the deformation $|{\mathcal{D}}(t)|$ exceeds the steady-state deformation $|{\mathcal{D}}_{ss}|$ at some time, while it is overdamped for fluid systems $B$ and $C$. However, inspection of the deformation-time curves showed the existence of an inflection point during the transient period for these two fluid systems. This suggests that the dynamic response for all the three fluid systems at $Oh^{2}=0.2$ is controlled by more than one time scale. For $Oh^{2}=0.02$, the dynamic response resembles that of an underdamped second-order mechanical system, where the drop settles to its steady state via an oscillatory motion. Here, the deformation has a large initial peak that is followed by a smaller dip, and nearly an imperceptible second peak. The overshoot is the largest for fluid system $B$ and is nearly the same for fluid systems $A$ and $C$. This is plausible since flow Reynolds number $Re_{f}$ for fluid system $B$ is 100, while it is 10 for the other two fluid systems.

Figure 6.

Comparison of deformation-time history as a function of $Oh^{2}$ for fluid systems $A{-}C$, corresponding to (a)–(c), respectively. The $Oh^{2}=2$ curves correspond, respectively, to $R_{1}{-}R_{3}$ reference simulations (table 2). For all the cases $\widetilde{\unicode[STIX]{x1D70C}}=1$ and $\widetilde{\unicode[STIX]{x1D707}}=1$.

Two observations can be made here. First, the steady-state deformation for each set of simulations (fluid system) is essentially independent of $Oh^{2}$, and hence $Re_{f}$, which characterizes the fluid inertia. This is evident from figure 6, where the steady-state deformation remains nearly the same over two orders of magnitude variation in the flow Reynolds number, i.e. $Re_{f}=0.1,1,10$ for fluid systems $A$ and $C$, and $Re_{f}=1,10,100$ for fluid system $B$. Second, the reason for the appearance of the first peak in the $Oh^{2}=0.02$ simulations is that the deformation settles to a steady state corresponding to the time scale of deformation $\unicode[STIX]{x1D70F}_{def}=\unicode[STIX]{x1D707}_{o}a/\unicode[STIX]{x1D6FE}$. However, since in the long term the dynamics is controlled by the momentum diffusion time scale $\unicode[STIX]{x1D70F}_{diff}$ (recall $Oh^{2}=\unicode[STIX]{x1D70F}_{def}/\unicode[STIX]{x1D70F}_{diff}$), the drop settles to its final steady state over a time that is dictated by this time scale. The oscillation period for the $Oh^{2}\ll 1$ simulations is estimated from the inviscid theory to be $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\sqrt{\unicode[STIX]{x1D70C}_{o}a^{3}/\unicode[STIX]{x1D6FE}}$. This time scale can be rewritten as $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\sqrt{\unicode[STIX]{x1D70F}_{def}\unicode[STIX]{x1D70F}_{diff}}$, which asserts that $\unicode[STIX]{x1D70F}_{def}<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}<\unicode[STIX]{x1D70F}_{diff}$ for $Oh^{2}<1$.

Figure 7.

Steady-state drop shape for fluid system $A$ at $Oh^{2}=0.02$. The capillary numbers are 0.02, 0.2, 0.35, 0.6, 0.75, 1, 1.5 and 2, corresponding to (a)–(h).

5.1.2 Effect of the $Ca$ number

The capillary number $Ca$ controls the drop behaviour through the net normal and shear stresses, per (2.2) and (2.3). We study the effect of this parameter on the dynamic response of the drop by performing a few simulations over three orders of magnitude of the capillary number at $Oh^{2}=2$, 0.2 and 0.02. The effect of the capillary number on the equilibrium shape and stability of the drops has been reasonably well studied (Sherwood Reference Sherwood1988; Feng & Scott Reference Feng and Scott1996; Lac & Homsy Reference Lac and Homsy2007). However, not much is known about the impact of this parameter on the transient behaviour of the drops. Here, the literature is limited to only a few studies. For weak electric field strength ($E^{\ast }\ll 1$), Supeene et al. (Reference Supeene, Koch and Bhattacharjee2004) studied the deformation of a drop in a perfectly dielectric and nearly inviscid ($Oh^{2}\ll 1$) fluid system at several imposed electric field strengths and showed that the normalized dynamic response (${\mathcal{D}}/{\mathcal{D}}_{\infty }$ versus $t/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$), which was oscillatory, collapsed to a single curve, ${\mathcal{D}}_{\infty }$ being the steady-state deformation of a perfectly dielectric drop in a weak electric field (O’Konski & Thacher Reference O’Konski and Thacher1953; Allan & Mason Reference Allan and Mason1962). For intermediate ($E^{\ast }<E_{cr}^{\ast }$) and large ($E^{\ast }>E_{cr}^{\ast }$) electric field strengths, Zhang et al. (Reference Zhang, Zahn and Lin2013) performed a semi-numerical study, where they ignored the local acceleration term, but allowed for the surface charge relaxation. For the former, the drop deformed monotonically and settled to a steady-state deformation. For the latter, the deformation proceeded monotonically and slowly in time until it reached a plateau, where the deformation suddenly increased, at which point the drop became unstable.

Figure 7 shows the steady-state drop shapes for fluid system $A$ at $Oh^{2}=0.02$, spread over three orders of magnitude of capillary number ($Ca=0.02{-}2$). Here, the drop elongates in the direction perpendicular to the field. As the capillary number increases, the drop shape evolves from a spheroid to an ellipsoid, and eventually a donut shape. For this fluid system and in an unbounded domain, the drop would remain stable for all the capillary numbers, according to the Ajayi (Reference Ajayi1978) second-order creeping flow solution, i.e. $k_{1}k_{2}<0$. For the finite computational domain used here, the drop will be eventually influenced by the drops in the adjacent periodic cells beyond a capillary number.

Figure 8 shows the deformation-time curves for representative simulations. The unscaled deformation parameter ${\mathcal{D}}$ (a,c,e) highlights the effect of the capillary number on the degree of the deformation and the initial rate of the deformation $\unicode[STIX]{x2202}{\mathcal{D}}/\unicode[STIX]{x2202}t|_{t=0}$, and the normalized deformation ${\mathcal{D}}/{\mathcal{D}}_{ss}$ (b,d,f; ${\mathcal{D}}_{ss}$ being the steady-state deformation) provides the possibility of comparison of the dynamic response with that of a linear system (mass–damper or mass–damper–spring). Considering the left panels, the dynamic response gradually transitions from a monotonic response to an oscillatory one with a decrease in the $Oh^{2}$ number. For the monotonic evolution (a,c), the time constant at which the drop settles to its steady-state deformation decreases with an increase in the capillary number. Similarly, for the oscillatory evolution (e), the increase in the capillary number results in a decrease in the time at which the deformation reaches its first peak. These observations are in accord with intuition and are in line with the non-dimensionalization of the governing equations; as the capillary number increases, the electric pressure increases (per (2.3)) and, therefore, the drop deforms faster.

Figure 8.

Evolution of deformation parameter with time at different capillary numbers for fluid system $A$ for $Oh^{2}=2$, 0.2 and 0.02, corresponding to (a,b), (c,d) and (e,f), respectively. Here, $R_{1}$ in (a,b) denotes the reference simulation. For all the cases, $\widetilde{\unicode[STIX]{x1D70C}}=1$ and $\widetilde{\unicode[STIX]{x1D707}}=1$.

For $Oh^{2}=2$, the result of the reference simulation $R_{1}$, figure 5(a), is identified on the figure. Here, the dynamic response resembles that of a first-order mechanical system, ${\mathcal{D}}={\mathcal{D}}_{ss}[1-\exp (-t/\unicode[STIX]{x1D70F}_{rel})]$, where $\unicode[STIX]{x1D70F}_{rel}$ is the relaxation time (b). The dependency of the relaxation time $\unicode[STIX]{x1D70F}_{rel}$ on the input parameters of the problem is discussed in § 5.1.5. For $Oh^{2}=0.2$, the dynamic response is overdamped for $Ca=0.02$, becomes slightly underdamped for $Ca=0.2$ and 1 and switches back to an overdamped response at $Ca=2$ (d). Here, the dynamic response is controlled by more than one time scale as the inspection of the ${\mathcal{D}}-t$ curves through magnification (not shown here) showed the appearance of an inflection point during the transient period. As such, the dynamic response at $Oh^{2}=0.2$ and low capillary number $Ca\ll 1$ seems to be well represented by that of a second-order mechanical system. For $Oh^{2}=0.02$, the dynamic response is underdamped for all the capillary numbers (f). Here, the normalized peak increases with an increase in the capillary number for $Ca\leqslant 1$ but it decreases for $Ca=2$. We believe the change in the behaviour of the curves for $Ca>1$ in these simulations is due to the deviation of the drop shape from the ellipsoidal shape. This point will be elucidated in § 5.1.5.

Figure 9 shows the steady-state drop shape for fluid system $B$ at $Oh^{2}=0.02$, spread over three orders of magnitude of the capillary number ($Ca=0.02{-}8$). Here, the drop deforms to a prolate and the change in the degree of the deformation with the capillary number is rather slow because the characteristic deformation function $\unicode[STIX]{x1D6F7}$ is low (table 1). For this fluid system and in an unbounded domain, the drop would remain stable for all capillary numbers, according to the Ajayi (Reference Ajayi1978) second-order creeping flow solution, i.e. $k_{1}k_{2}<0$. However, for the finite computational domain used here, the elongation of the drop will eventually be limited by the size of the computational box in the vertical direction beyond a certain capillary number. Figure 10 shows the deformation-time curves for a few representative simulations pertaining to this fluid system. For $Oh^{2}=2$, the result of the reference simulation $R_{2}$, figure 5(b), is identified on the figure. The drop deforms monotonically for both $Oh^{2}=2$ and 0.2, and becomes oscillatory for $Oh^{2}=0.02$. The normalized deformation curves are nearly indistinguishable from one another for $Ca=0.02{-}0.2$ for each set of curves. For $Oh^{2}=0.02$, in particular, the normalized curves are essentially the same over the entire range of capillary number ($Ca=0.02{-}8$). As in fluid system $A$, the time constant for the monotonic (a–d) and the oscillatory (e,f) evolutions correlates positively with the capillary number.

Figure 9.

Steady-state drop shape for fluid system $B$ at $Oh^{2}=0.02$. The capillary numbers are 0.02, 0.2, 1, 2, 3, 4, 5, 6, 7 and 8, corresponding to (a)–(j).

Figure 10.

Evolution of deformation parameter with time at different capillary numbers for fluid system $B$ for $Oh^{2}=2$, 0.2 and 0.02, corresponding to (a,b), (c,d) and (e,f), respectively. Here, $R_{2}$ in (a,b) denotes the reference simulation. For all the cases, $\widetilde{\unicode[STIX]{x1D70C}}=1$ and $\widetilde{\unicode[STIX]{x1D707}}=1$.

Figure 11 shows the steady-state drop shape for fluid system $C$ at $Oh^{2}=0.02$, spread over three orders of magnitude of capillary number ($Ca=0.02{-}0.55$). For this fluid system and in an unbounded domain, the drop could become unstable beyond a critical capillary number, according to the Ajayi (Reference Ajayi1978) second-order creeping flow solution; i.e. $k_{1}k_{2}>0$. As in fluid system $B$, the drop deforms to a prolate. However, there are two main differences between the evolution of the drops in the two fluid systems. First, here, the rate of the deformation with the capillary number $\unicode[STIX]{x2202}{\mathcal{D}}/\unicode[STIX]{x2202}(Ca)$ is higher. Second, until $Ca\leqslant 0.4$ (g) the drop deforms gradually and settles to a steady-state prolate shape that is essentially independent of the domain confinement. However, the deformation incurs a jump at $Ca=0.425$. From this point onward (h–j), the elongation of the drop will be limited by the size of the computational box in the $z$ direction as the drop touches the top and the bottom electrodes. Here, the end-to-end length of the drop in the direction of the electric field $z_{max}$ will remain unchanged. However, since the maximum breadth in the transverse direction $r_{max}$ continues to decrease gradually, the deformation ${\mathcal{D}}$ continues to increase, albeit at a very slow pace. It should be noted that in order to more accurately determine the capillary number at which the jump occurs one needs to run a few simulations at $0.4<Ca<0.425$.

Figure 11.

Steady-state drop shape for fluid system $C$ at $Oh^{2}=0.02$. The capillary numbers are 0.02, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.425, 0.45 and 0.55, corresponding to (a)–(j), respectively. The steady-state deformation increases gradually until $Ca\leqslant 40$, but a jump in the deformation occurs at approximately $0.4<Ca<0.425$.

Figure 12 shows the deformation-time curves for representative simulations. For $Oh^{2}=2$, the result of the reference simulation $R_{3}$ (figure 5c), is identified in the figure. Here, at all the capillary numbers, the evolution of ${\mathcal{D}}$ toward the steady state is monotonic for $Oh^{2}=2$ and 0.2. For $Oh^{2}=2$ and at $Ca=0.02$ and 0.2, the deformation is moderate and, therefore, is not influenced by the box size. However, for $Ca=1$ and 2, the drop grows very fast and touches the upper and lower walls early on. A similar observation is made for $Oh^{2}=0.2$, where for $Ca=0.02$ and 0.2, the deformation is moderate but the drop touches the walls for $Ca\geqslant 0.45$ (at approximately $t^{\ast }=87.5$). For $Oh^{2}=0.02$, the dynamic response is oscillatory for $Ca\leqslant 0.4$, where the deformation is modest and, therefore, the drop settles to an equilibrium shape that is not limited by the domain size in the vertical direction. At $Ca=0.425$, the drop initially deforms gradually but the rate of the deformation suddenly increases until the drop touches the top and bottom walls at approximately $t^{\ast }=237.5$. At this capillary number and beyond, the deformation will be limited by the box size. The evolution of the deformation ${\mathcal{D}}$ with time at $Ca=0.425$ is similar to the one described by Zhang et al. (Reference Zhang, Zahn and Lin2013), regarding the onset of instability in an unbounded domain. There, the deformation proceeded monotonically and slowly in time until it reached a plateau, where the deformation suddenly increased, at which point the drop became unstable. Here, on the other hand, the drop settles to a steady-state deformation shortly after the jump, because of the domain confinement.

Figure 12.

Evolution of deformation parameter with time at different capillary numbers for fluid system $C$ for $Oh^{2}=2$, 0.2 and 0.02, corresponding to (a,b), (c,d) and (e,f), respectively. Here, $R_{3}$ in (a,b) denotes the reference simulation. For all the cases, $\widetilde{\unicode[STIX]{x1D70C}}=1$ and $\widetilde{\unicode[STIX]{x1D707}}=1$.

Figure 13 shows the steady-state deformation versus capillary number for the three fluid systems. Here, we present the results for $Oh^{2}=0.02$ only, since the steady-state deformation is essentially independent of $Oh^{2}$. The results of the first- and second-order asymptotic solutions, equations (3.4) and (3.5), are also added to this figure. Here, the first-order asymptotic results agree with the results of the numerical simulation until ${\mathcal{D}}\approx 0.02$, 0.2 and 0.15, for fluid systems $A{-}C$, respectively. The agreement between the second-order asymptotic results and those of the numerical simulations holds up to 0.2, 2 and 0.25, for fluid systems $A{-}C$, respectively. The reason that the range of agreement of the asymptotic results with the numerical results is the largest for fluid system $B$ is because of the substantially smaller $k_{1}$ for this fluid system. This observation is in line with that of Feng & Scott (Reference Feng and Scott1996); see, their figure (5). For fluid systems $A$ and $B$, the deformation increases monotonically with the capillary number, as it should. On the other hand, for fluid system $C$, it gradually increases until $Ca\approx 0.4$, encounters a jump around $Ca=0.425$ and remains essentially constant due to the confinement effect. The direction of the curvature of the numerical curves (for fluid system $C$ the focus is up to $Ca\lessapprox 0.425$) are in line with the predictions of the second-order asymptotic solution. Here, $k_{2}$ is positive for fluid systems $A$ and $C$ and negative for fluid system $B$. Thus, the numerical curves are concave up for fluid systems $A$ and $C$, while the opposite holds for fluid system $B$. The sign of $k_{1}k_{2}$ provides some insight regarding the stability of the drop in an unbounded domain, where the deformation is not limited by the domain size. If $k_{1}k_{2}<0$, then the drop remains stable as the capillary number is increased; in this case, the drop elongates progressively in the direction of or perpendicular to the field, becoming prolate or oblate, respectively. On the other hand, if $k_{1}k_{2}>0$, the drop remains stable, or becomes unstable at a critical capillary number $Ca_{cr}$. If the drop remains stable, it will move along a hysteresis curve, where ${\mathcal{D}}-Ca$ curve transitions from $k_{1}k_{2}>0$ to $k_{1}k_{2}<0$ via an inflection point. For fluid systems $A$ and $B$, $k_{1}k_{2}<0$, therefore, the drop remains stable. This is hinted at by the signs (direction) of the curvatures of the ${\mathcal{D}}-Ca$ curve in these two cases. For fluid system $C$, the deformation progressively increases up to $Ca\approx 0.4$ and then encounters a jump somewhere between $Ca=0.4$ and 0.425.

Figure 13.

Variation of steady-state deformation ${\mathcal{D}}_{ss}$ with the capillary number $Ca$ for fluid systems $A{-}C$, corresponding to (a)–(c), respectively. For all the simulations, $Oh^{2}=0.02$.

5.1.3 Effect of the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}$

The effect of the viscosity ratio on the transient electrohydrodynamics of leaky dielectric drops has not been studied in detail. However, several authors have explored this issue more thoroughly in the context of perfect dielectric fluid systems. Moriya, Adachi & Kotaka (Reference Moriya, Adachi and Kotaka1986), Nishiwaki, Adachi & Kotaka (Reference Nishiwaki, Adachi and Kotaka1988) and Haywood et al. (Reference Haywood, Renksizbulut and Raithby1991) focused on highly viscous liquid systems at low electric field strength for which deformation was small and the dynamic response was governed by only a time scale, ${\mathcal{D}}={\mathcal{D}}_{ss}[1-\exp (-t/\unicode[STIX]{x1D70F})]$. The authors developed an expression for the time constant $\unicode[STIX]{x1D70F}$ in terms of the capillary time scale $\unicode[STIX]{x1D70F}_{def}=\unicode[STIX]{x1D707}_{o}a/\unicode[STIX]{x1D6FE}$ and the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D707}_{i}/\unicode[STIX]{x1D707}_{o}$, using heuristic arguments and analytical solutions. Supeene et al. (Reference Supeene, Koch and Bhattacharjee2004) studied the evolution of the drops using numerical simulations at low Ohnesorge numbers ($Oh\ll 1$), where the dynamic response was oscillatory. Considering a baseline simulation for which $\unicode[STIX]{x1D707}_{i}=\unicode[STIX]{x1D707}_{o}$, they observed that the dynamic response became more underdamped if they reduced the viscosity of the drop fluid $\unicode[STIX]{x1D707}_{i}$ or that of the ambient fluid $\unicode[STIX]{x1D707}_{o}$, while the latter rendered the dynamic response more oscillatory.

For steady state, the literature is limited to the studies by Feng & Scott (Reference Feng and Scott1996), Ha & Yang (Reference Ha and Yang2000) and Lac & Homsy (Reference Lac and Homsy2007). The focus of the first and the last studies was on the effect of $\widetilde{\unicode[STIX]{x1D707}}$ on the critical capillary number $Ca_{cr}$ at which the drop disintegrates. Feng & Scott (Reference Feng and Scott1996) examined the effect of $\widetilde{\unicode[STIX]{x1D707}}$ on a prolate ($\widetilde{\unicode[STIX]{x1D70E}}=100$, $\widetilde{\unicode[STIX]{x1D716}}=2$, region II) and an oblate ($\widetilde{\unicode[STIX]{x1D70E}}=0.01$, $\widetilde{\unicode[STIX]{x1D716}}=0.5$, region I) drop at zero Reynolds number and concluded that in both cases increasing $\widetilde{\unicode[STIX]{x1D707}}$ shifted the critical field strength to a lower value. Lac & Homsy (Reference Lac and Homsy2007), on the other hand, examined prolate and oblate drops in creeping flow that were susceptible to instability (i.e. drops in regions I$^{-}$, II$^{+}$ and III$^{+}$, for which $k_{1}k_{2}>0$) and concluded that a ‘moderate’ increase in the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}$ shifted the critical field strength to a lower value in regions III$^{+}$ and I$^{-}$, while it had an opposite effect in region II$^{+}$. Ha & Yang (Reference Ha and Yang2000) developed experimentally the deformation-time curves for twenty one fluid systems with $0.001\leqslant \widetilde{\unicode[STIX]{x1D707}}\leqslant 0.874$. However, these authors did not elaborate on the effect of the viscosity ratio on the results. Similarly, no justification was provided by Feng & Scott (Reference Feng and Scott1996) and Lac & Homsy (Reference Lac and Homsy2007) to back up their observations. We note that, at small electric field strength ($Ca\ll 1$), the viscosity ratio has a weak influence on the degree of deformation of the leaky dielectric drops (Taylor Reference Taylor1964; Ajayi Reference Ajayi1978), while for the perfect dielectric fluid systems, the equilibrium shape is independent of the viscosity ratio, since fluid flow ceases to exist at steady state.

Figure 14.

Evolution of the deformation parameter with time at different viscosity ratios $\widetilde{\unicode[STIX]{x1D707}}$ for fluid system $A$. Here, $Oh^{2}=2$, 0.2 and 0.02 for (a,b), (c,d) and (e,f), respectively, and $Ca=0.2$ for all simulations. The deformation parameter has been scaled with the corresponding steady-state deformation ${\mathcal{D}}_{ss}$ for each case.

To explore the effect of the viscosity ratio on the behaviour of the drop, we performed several simulations at viscosity ratios $\widetilde{\unicode[STIX]{x1D707}}$ above and below that of the reference simulations ($\widetilde{\unicode[STIX]{x1D707}}=1$). In these simulations, the viscosity ratio was changed by changing the viscosity of the drop fluid $\unicode[STIX]{x1D707}_{i}$. Figure 14 shows the deformation-time curves for fluid system $A$. For $Oh^{2}=2$, the result of the reference simulation $R_{1}$ (figure 5a) is identified in the figure. Since the ambient viscosity remains intact, the deformation time scale $\unicode[STIX]{x1D70F}_{def}=\unicode[STIX]{x1D707}_{o}a/\unicode[STIX]{x1D6FE}$ is the same for each set of simulations. Thus, the non-dimensionalization of the time does not affect the relative scales of the curves in a set. Furthermore, since the relaxation time to steady-state deformation (in monotonic response) and the oscillation period $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ (in oscillatory response) are inherent functions of viscosity ratio, the deformation curves in each set will not collapse to a single curve, even if their dynamic response corresponds to that of a linear mechanical system. Considering the left panels, it is seen that, for all simulations, the steady-state deformation $|{\mathcal{D}}|$ increases with an increase in $\widetilde{\unicode[STIX]{x1D707}}$. However, the rate of deformation with the viscosity ratio $\unicode[STIX]{x2202}|{\mathcal{D}}|/\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D707}}$ is small, which is hinted at by the asymptotic solution. For $Oh^{2}=2$, the dynamic response remains monotonic over three orders of magnitude of the viscosity ratio. Furthermore, the time constant at which the deformation settles to steady state increases with an increase in $\widetilde{\unicode[STIX]{x1D707}}$, which is in line with (5.2). For $Oh^{2}=0.2$, an order of magnitude decrease in the viscosity ratio from $\widetilde{\unicode[STIX]{x1D707}}=1$ to 0.1 leads to an oscillatory behaviour. However, another order of magnitude reduction does not lead to a major change in the dynamic response. For $Oh^{2}=0.02$, the dynamic response is already underdamped at $\widetilde{\unicode[STIX]{x1D707}}=1$. Here, an order of magnitude decrease in the viscosity ratio from $\widetilde{\unicode[STIX]{x1D707}}=1$ to 0.1 leads to a larger initial normalized peak. On the other hand, an order of magnitude increase in the viscosity ratio from $\widetilde{\unicode[STIX]{x1D707}}=1$ to 10 leads to transition of the dynamic response from an underdamped response to an overdamped one. It is seen that the differences between the dynamic responses for $\widetilde{\unicode[STIX]{x1D707}}=0.1$ and 0.01 in each set of curves are very small. This suggests that the dynamic response will be essentially independent of the viscosity ratio for $\widetilde{\unicode[STIX]{x1D707}}<0.01$. Hence, $\widetilde{\unicode[STIX]{x1D707}}=0.01$ may serve as the threshold for nearly inviscid fluid systems.

Figure 15 shows the deformation-time curves for fluid system $B$. For $Oh^{2}=2$, the result of the reference simulation $R_{2}$, figure 5(b), is identified in the figure. Here, as opposed to fluid system $A$, the deformation decreases with an increase in $\widetilde{\unicode[STIX]{x1D707}}$. However, other features of the evolution are similar to those for fluid system $A$. Figure 16 shows the deformation-time curves for fluid system $C$. For $Oh^{2}=2$, the result of the reference simulation $R_{3}$ (figure 5c) is identified in the figure. Here, similar to fluid system $A$, the deformation increases with $\widetilde{\unicode[STIX]{x1D707}}$ for all fluid systems. Furthermore, the general features of the evolution are similar to those observed for fluid systems $A$ and $B$.

Figure 15.

Evolution of the deformation parameter with time at different viscosity ratios $\widetilde{\unicode[STIX]{x1D707}}$ for fluid system $B$. Here, $Oh^{2}=2$, 0.2 and 0.02 for (a,b), (c,d) and (e,f), respectively, and $Ca=2$ for all the simulations. The deformation parameter has been scaled with the corresponding steady-state deformation ${\mathcal{D}}_{ss}$ for each case.

Figure 16.

Evolution of the deformation parameter with time at different viscosity ratios $\widetilde{\unicode[STIX]{x1D707}}$ for fluid system $C$. Here, $Oh^{2}=2$, 0.2 and 0.02 for (a,b), (c,d) and (e,f), respectively, and $Ca=0.2$ for all the simulations. The deformation parameter has been scaled with the corresponding steady-state deformation ${\mathcal{D}}_{ss}$ for each case.

Figure 17 shows the steady-state deformation ${\mathcal{D}}_{ss}$ versus the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}$ for fluid systems $A{-}C$. Here, we present the results for $Oh^{2}=0.02$ simulations only. This figure puts into perspective some of the features that we discussed earlier. An increase in $\widetilde{\unicode[STIX]{x1D707}}$ leads to an increase in $|{\mathcal{D}}|$ for fluid systems $A$ and $C$, but the opposite holds for fluid system $B$. These observations are in line with those of Lac & Homsy (Reference Lac and Homsy2007) who briefly touched on this issue. The rate of change of the deformation with the viscosity $\unicode[STIX]{x2202}|{\mathcal{D}}|/\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D707}}$ is higher for $0.01<\widetilde{\unicode[STIX]{x1D707}}<1$ compared to $\widetilde{\unicode[STIX]{x1D707}}>1$. We also superimpose the results of the asymptotic solution, equations (3.4) and (3.5), in this figure. Depending on the sign of $k_{1}k_{2}$, the second-order deformation $|{\mathcal{D}}_{\infty }^{\langle 2\rangle }|$ may be smaller or larger than the first-order one $|{\mathcal{D}}_{\infty }^{\langle 1\rangle }|$. Note that the capillary number $Ca$ is $0.2$ for fluid systems $A$ and $C$, while it is $2$ for fluid system $B$. Here, the correction for fluid system $B$ is marginal since $k_{2}<k_{1}\ll 1$. As expected, the second-order results show a closer agreement with the numerical results, compared to the first-order results, and the agreement between the two is better at $\widetilde{\unicode[STIX]{x1D707}}\lessapprox 0.1$. The agreement between the numerical and the analytical results is the best for fluid system $B$. This is primarily because of the reduction of $|{\mathcal{D}}|$ with $\widetilde{\unicode[STIX]{x1D707}}$ for this fluid system, which helps extend the range of applicability of the asymptotic results. In § 5.3 we provide the justification for the observed correlations between ${\mathcal{D}}_{ss}$ and $\widetilde{\unicode[STIX]{x1D707}}$ for the three fluid systems.

Figure 17.

Variation of steady-state deformation ${\mathcal{D}}_{ss}$ with the viscosity ratio $\widetilde{\unicode[STIX]{x1D707}}$ for fluid systems $A{-}C$, corresponding to (a)–(c), respectively. For all the simulations, $Oh^{2}=0.02$.

5.1.4 Ellipsoidal approximation

In the analytical studies of electrohydrodynamics of a liquid drop, the drop shape has been generally considered as an ellipsoid (Allan & Mason Reference Allan and Mason1962; Taylor Reference Taylor1964). This approximation seems to be reasonable for weak electric fields where $|{\mathcal{D}}|\ll 1$, however, the ellipsoidal solution has been also used for strong electric fields, where the deformation is substantial. In particular, one can refer to the analytical solution of Bentenitis & Krause (Reference Bentenitis and Krause2005) who developed the so-called extended leaky dielectric method and solved the creeping flow equations using an ellipsoidal coordinates to study deformation of the drops in strong electric fields, and Zhang et al. (Reference Zhang, Zahn and Lin2013) who extended their approach to study transient deformation of drops, using numerical solution.

To explore whether the ellipsoidal approximation holds here, we found the cross-section of the drop in the $y{-}z$ mid-plane for the simulations shown in figures 7, 9 and 11, and compared them with the contour of the counterpart ellipse,

$$\begin{eqnarray}\frac{y^{2}}{(r_{max})^{2}}+\frac{z^{2}}{(z_{max})^{2}}=1,\end{eqnarray}$$

whose major (minor) and the minor (major) axes were equal to the maximum end-to-end length of the drop in the direction of the electric field $z_{max}$ and its maximum breadth $r_{max}$ in the transverse direction. The relative error in the ellipsoidal approximation was measured by computing $E=1/N\sum _{i=1}^{N}|z_{n_{i}}-z_{a_{i}}|/z_{n_{i}}$, where $z_{n_{i}}$ and $z_{a_{i}}$ are the numerical and the approximated heights at the corresponding $y_{i}$, $N$ being the number of the points. Figure 18 shows the result of this comparison for representative simulations. Since the error increases with an increase in the capillary number, here we only present the results for the higher capillary numbers. Panels (a,b) pertain to fluid system $A$ at $Ca=1$ and 1.5. The agreement between the two curves seems excellent at $Ca=1$ but for $Ca=1.5$, the two differ around the left and the right ends. Here the relative error is approximately 0.37 % and 0.86 %, respectively. For $Ca=2$ (not shown here), the front deviated substantially from the elliptic shape. Panel (c) pertains to fluid system $B$ at $Ca=8$. Here, the two curves are nearly indistinguishable and the relative error is approximately 0.5 %. Panels (d,e) pertain to fluid system $C$ at $Ca=0.4$ and 0.45. For the former, which is before the onset of instability, the agreement between the two curves seems excellent and the relative error is approximately 2 %. For the latter, which is after the jump in the deformation and the squashing of the drop ends by the electrodes, the deviation between the two curves is substantial. Here, the error is 38 %. We speculate that, even for this more extreme elongated shape, the error would have been only a few per cent if it was characterized relative to an effective radius. Furthermore, the observed agreement should not be surprising, since higher modes in spherical harmonics will be suppressed by dissipation; see, for example, Yu et al. (Reference Yu, Lira, Riske, Dimova and Lin2015).

Figure 18.

Comparison of the cross-section of the drop in the $yz$ mid-plane with the corresponding ellipse. Panels (a) and (b) correspond, respectively, to panels (f) ($Ca=1$) and (g) ($Ca=1.5$) of figure 7. Panel (c) corresponds to panel (j) ($Ca=8$) of figure 9. Panels (d) and (e) correspond, respectively, to panels (g) ($Ca=0.4$) and (i) ($Ca=0.45$) of figure 11.

In the analysis of the effect of the capillary number on the dynamic response of the drops (figures 8, 10 and 12) the normalized curves followed the same trend with an increase in the capillary number for fluid system $B$. However, this was not the case for fluid systems $A$ and $C$, where the curves followed two different trends at the low and the high capillary numbers. Considering figure 18, we believe the change in the trend for fluid system $A$ was primarily due to the deviation of the drop shape from the ellipsoidal shape at $Ca>1$. On the other hand, the change in the trend for fluid system $C$ was initially triggered by the transition of the deformation-time curve from a stable regime ($Ca\leqslant 0.4$) to an unstable one ($0.4<Ca\leqslant 0.425$). However, the deviation of the drop shape from the ellipsoidal shape also played a role later on, as soon as the drop began to touch the upper and the lower electrodes.

5.1.5 Deviation from linear response

In the analysis of the dynamic response of the drop at high $Oh^{2}$ numbers ($Oh^{2}=2$) (figure 5) we observed that the numerical and analytical relaxation times had a nearly perfect agreement for the reference simulations $R_{1}$ and $R_{2}$, but the agreement was not so close for the reference simulation $R_{3}$. Recall that these simulations were based on fluid systems $A{-}C$, and their capillary numbers were $Ca=0.2$, 2 and 0.2, respectively.

Since the analytical solution is valid for the creeping flow regime ($Re_{f}\ll 1$) and for small deformation ($Ca\ll 1$), one would normally expect the agreement between the numerical results and the analytical one to improve as the capillary number decreases. This is indeed the case for fluid system $C$ (figure 12b)) where the time constant of the $Ca=0.02$ curve is greater than the time constant of the reference simulation ($Ca=0.2$), which is smaller than the analytical time constant, as per figure 5(c). On the other hand, for fluid systems $A$ and $B$ (figures 8b and 10b) the agreement between the numerical results and the corresponding analytical ones decreases with a decrease in the capillary number, considering the fact that the deviation from the reference simulations $R_{1}$ ($Ca=0.2$) and $R_{2}$ ($Ca=8$) is tantamount to deviation from the analytical solutions, since the reference simulations had near perfect agreement with the corresponding analytical solution.

We believe that the trend seen here is primarily due to the convective effects that are present in the numerical simulations but absent in the analytical solution. While in our computations we do not explicitly account for the surface charge convection, this effect is possibly accounted for (to some degree) in the numerical simulation, where the coordinates of the marker points $\boldsymbol{x}_{f}$ are moved by the kinematic equation $\text{d}\boldsymbol{x}_{f}/\text{d}t=\boldsymbol{u}_{f}$, where $\boldsymbol{u}_{f}$ is the grid velocity at their positions. Specifically, the movement of the marker points could account for the charge convection due to surface dilation (the second term in (2.6)). The presence of convective effects is further evidenced by the fact that in fluid systems $A$ and $B$, which show a similar trend, the direction of the external flow around the drop is the same (poles-to-equator), while in fluid system $C$, which shows the opposite trend, the flow runs in the opposite direction (equator-to-poles). Furthermore, in fluid system $C$, where the agreement between the numerical simulation and the analytical solution improves with the decrease in the capillary number (from 0.2 to 0.02), the results of computer simulations (not presented here) showed that an order of magnitude decrease in the capillary number from 0.02 to 0.002 did not improve the agreement; the results of $Ca=0.002$ simulation was essentially the same as that of the $Ca=0.02$ simulation. This hints at an effect that is present in the numerical simulation but absent in the analytical solution.

For fluid systems $A$ and $B$, the marker points are moved by the fluid flow from the poles toward the equator, while for fluid system $C$, they are moved in the opposite direction. Because of the convective effects, the numerical time constant in general will not agree with the analytical solution, which ignores this effect. As pointed out by Feng (Reference Feng1999), the charge convection tends to decrease the deformation of the drops in regions I and II (fluid systems $A$ and $B$), while it tends to increase it in region III (fluid system $C$). Accordingly, for fluid systems $A$ and $B$, a combination of a capillary number that is not too small, along with the convective effects (which tend to reduce the deformation) leads to a time constant that is close to the analytical one. On the other hand, for fluid system $C$, where the convective effects tend to increase the deformation, a smaller capillary leads to a better agreement.

In summary, because of the sensitivity of the results to the instantaneous evolution of the drop, we believe the relaxation time $\unicode[STIX]{x1D70F}_{rel}$ in the numerical solution ${\mathcal{D}}={\mathcal{D}}_{\infty }(1-\exp (-t/\unicode[STIX]{x1D70F}_{rel}))$ is a function of $\widetilde{\unicode[STIX]{x1D70E}}$, $\widetilde{\unicode[STIX]{x1D716}}$, $Ca$, $\unicode[STIX]{x1D70F}_{def}=\unicode[STIX]{x1D707}_{o}a/\unicode[STIX]{x1D6FE}$ and $\widetilde{\unicode[STIX]{x1D707}}$. On the other hand, the analytical time constant $\unicode[STIX]{x1D70F}_{a}\equiv \unicode[STIX]{x1D70F}$ is a function of $\unicode[STIX]{x1D70F}_{def}=\unicode[STIX]{x1D707}_{o}a/\unicode[STIX]{x1D6FE}$ and $\widetilde{\unicode[STIX]{x1D707}}$ only, as per (5.2). This leads to the observed mismatch. It is worth mentioning that the dependency of the time constant $\unicode[STIX]{x1D70F}$ on the capillary number $Ca$ and $\widetilde{\unicode[STIX]{x1D70E}}$ and $\widetilde{\unicode[STIX]{x1D716}}$ should be potentially unravelled from a second-order transient solution, and while Das & Saintillan (Reference Das and Saintillan2017b) provides such a solution, unfortunately, it is not possible to discern this dependency, since their solution is not in a closed form.