4.1. Critical condition for tip streaming

We begin our study by presenting the stability limit, i.e. the critical condition when tip streaming occurs, of a liquid meniscus in the general case. Figure 3(a) shows the phase diagram as a function of the dimensionless electric capillary number $C{a_E}$ and the opening half-angle ${\theta _0}$ at fixed charge relaxation parameter $\alpha = 10$. It is noted that the critical condition is expected to be independent of the liquid properties, except the surface tension (Rubio et al. Reference Rubio, Rodríguez-Díaz, López-Herrera, Herrada, Gañán-Calvo and Montanero2023). Three distinct regimes are observed: jetting, non-jetting and dripping. Figure 3(b) shows the meniscus shapes in these three regimes at the instants when pinch-off occurs or the meniscus becomes stable. In the non-jetting regime, for example, ${\theta _0} = 90^\circ $ and $C{a_E} = 1.31$, the meniscus will oscillate under the electric and capillary forces. Then the oscillation is damped out due to the viscous effect and the meniscus eventually approaches a stable state. However, when the electric field is slightly improved, $C{a_E} \to 1.32$, the shape of the meniscus tip becomes a Taylor cone. The tip streaming subsequently appears in the meniscus apex, resulting in a continuously growing jet, which is called the jetting regime. It is noted that the jetting regime is defined as the cases with the occurrence of a tip-streaming jet in our study. The radius of the jet is two orders of magnitude smaller than the nozzle radius. Figure 3(a) demonstrates that the increase in ${\theta _0}$ remarkably decreases the critical electric capillary number $C{a_{E,c}}$ required for jetting. This is because the variation of surface energy of the meniscus and electrostatic energy of the electric field are both affected by increasing the opening half-angle ${\theta _0}$, namely, increasing the liquid volume. This reason is discussed in more detail below. When ${\theta _0}$ is increased to be larger than $134^\circ $, the meniscus undergoes a transition from non-jetting to dripping and jetting rather than non-jetting to jetting for ${\theta _0} \le 134^\circ $. In this transitional regime, the system will first enter a dripping regime. Figure 3(b) shows a typical case of the dripping regime, in which the size of the emitted droplet is of the same order of magnitude as the nozzle radius. Only when the electric field is increased further, the jetting can be obtained. Consequently, the critical number $C{a_{E,c}}$ for jetting increases with increasing ${\theta _0}$ for ${\theta _0} > 134^\circ $. In contrast, the critical number $C{a_{E,c}}$ for dripping is decreased with increasing ${\theta _0}$. The transition between these regimes can be understood from the crucial condition for the jetting behaviour: the electric field is high enough to generate a cone jet at the meniscus tip, as well as the requirement that the primary liquid does not detach from the nozzle. To satisfy the latter, the initial volume of the meniscus could not exceed a critical limit, i.e. ${\theta _0} \le 134^\circ $; otherwise, the dripping would occur before jetting.

Figure 3.

(a) $C{a_E}-{\theta _0}$ phase diagram indicating the jetting, non-jetting and dripping regimes. Here, $\alpha = 10$ and the data points represent the transitions between different regimes. The dashed line marks the case of ${\theta _0} = 134^\circ $. (b) Typical cases in these three regimes.

From figure 3(a), it is noted that neither jetting nor dripping is observed for ${\theta _0} < 50^\circ $. This is due to the non-uniform distribution of the electric field in the nozzle-to-plate configuration. For ${\theta _0} < 50^\circ $, when the electric field is continuously increased, the accumulated charge reaches the limit and a conical meniscus will first form near the nozzle rather than at the meniscus tip, which gives rise to the formation of multiple jets (Jaworek & Krupa Reference Jaworek and Krupa1999). This regime is not accounted for in the present study.

In figure 4, we recover the limiting cases of the jetting regime for different ${\theta _0}$ in figure 3(a). Snapshots for each column respectively display the initial and final states of the meniscus at the incipience that the first droplet is emitted. As the liquid volume increases, the shape of each meniscus and its axial length differ drastically from each other, which is determined by the original liquid volume. The shapes of the liquid meniscus for different ${\theta _0}$ have been experimentally reported in the literature (Verdoold et al. Reference Verdoold, Agostinho, Yurteri and Marijnissen2014; Kong et al. Reference Kong, Yang, Wang, Wang, Dong and Wang2022). It is seen that the shapes of the transition region that connects the tip streaming jet and the Taylor cone are similar for all the limiting cases. However, when ${\theta _0} \ge 134^\circ $, the meniscus eventually takes on a spindle-like shape with a conical end at the incipience of pinch-off. A tiny droplet is then emitted from the spindle tip. From figure 4, it is clearly seen that the radius of the neck connecting the conical meniscus and the spindle is very small. The elongated jet resembling a spindle will certainly detach from the liquid meniscus and eventually form one or more satellite droplets. This jetting-before-dripping mode has been reported in the review of Cloupeau & Prunet-Foch (Reference Cloupeau and Prunet-Foch1994). However, in our study, we focus on the limiting cases when tip streaming occurs so that we prefer to treat this case as the jetting mode since the tip streaming has occurred before dripping. As ${\theta _0}$ continues to increase, the size of the spindle is significantly increased, but the neck radius is slightly increased. We thereby conclude that for the single cone jetting, the initial liquid volume cannot exceed a critical value.

Figure 4.

Simulated meniscus shapes for different ${\theta _0}$ and $C{a_{E,c}}$ in the limiting case of the jetting regime. Here, $\alpha = 10$. The upper images show the initial meniscus shapes and the lower images show the emission of the tip-streaming jet and the about-to-form droplet at the incipience of pinch-off.

To qualitatively discuss the limiting cases, we plot in figure 5 the effect of ${\theta _0}$ on the key parameters of the tip streaming processes. Figure 5(a) shows the variation of the pinch-off time ${t_p}$ for different ${\theta _0}$. It might be surprising that the pinch-off of the electrified jet occurs earlier (${t_p} \approx 3$) for ${\theta _0}$ ranging from $60^\circ $ to $90^\circ $. The pinch-off time will slightly increase when ${\theta _0}$ decreases to $50^\circ $. However, ${t_p}$ is significantly increased with increasing ${\theta _0}$ for ${\theta _0} \ge 90^\circ $. The maximum pinch-off time is found to be ${t_p} \approx 20$ for ${\theta _0} = 134^\circ $, which is exactly the critical limit for the transition regime. When ${\theta _0}$ exceeds this critical limit, ${t_p}$ will decrease again. This is primarily because the time scale for the dripping regime is larger than that for the jetting regime. The pinch-off time is therefore observed to reach its maximum near the transitional regime.

Figure 5.

Effect of ${\theta _0}$ on the meniscus shapes at the incipience of pinch-off in the limiting case of jetting. (a) Pinch-off time, (b) size of the first emitted droplet, (c) axial location of the jet's tip and (d) redefined length of the jet. Here, $\alpha = 10$ for all cases and the filled symbol represents the critical case of ${\theta _0} = 134^\circ $.(e) Definition of the jet length ${L_j}$.

As seen in figure 5(b), the size ratio ${R_d}/{R_0}$ is found to be approximately 0.04 for all limiting cases, indicating that ${\theta _0}$ has little effect on the radius ${R_d}$ of the emitted droplet. Here, ${R_d}$ is calculated from ${(3\int_{{z_{neck}}}^{{z_{tip}}} {{\rm \pi} {r^2}\,\textrm{d}z/4{\rm \pi} } )^{1/3}}$, where ${z_{neck}}$ is the axial coordinate of the neck point. Figure 5(c) shows the axial location of the jet's tip as a function of the opening half-angle. As one may expect, an increase in ${\theta _0}$ leads to a longer jet. However, the axial location of the tip cannot reflect the jet's length because the initial values are distinct. The initial meniscus height is determined by the opening half-angle: ${z_0} = {R_0}(1 - \textrm{cos}\,{\theta _0})/\textrm{sin}\,{\theta _0}$. In this case, we redefine the jet length ${L_b}$ as measured from the point where the change of the jet radius starts to be very small (Ismail et al. Reference Ismail, Yao, Xia and Stark2018). The definition of ${L_j}$ is illustrated in figure 5(e), where the chosen point satisfies $\Delta r/\Delta z = 0.2$. In figure 5(d), it is surprising that the value of the redefined jet length is all around 4 and also has little dependence on ${\theta _0}$. These results suggest that ${\theta _0}$ has an effect only on the pinch-off time.

We proceed by theoretically demonstrating the correlation between the critical electric capillary number $C{a_{E,c}}$ and the opening half-angle ${\theta _0}$. For a sessile droplet subjected to a uniform electric field created by two parallel electrodes, Beroz et al. (Reference Beroz, Hart and Bush2019) suggested that the critically stable droplet shapes correspond to the minima of the free energy, i.e. $\textrm{d}F = 0$. The free energy F consists of the surface energy ${F_s} = \gamma R_0^2a(\xi )$ and electrostatic energy ${F_E} = {\varepsilon _0}E_0^2{\mathcal{V}}\nu (\xi )$. Here, $a(\xi )$ and $\nu (\xi )$ are the dimensionless shape functions. Thus, the differential $\textrm{d}F = 0$ becomes $\gamma R_0^2a^{\prime}(\xi ) + {\varepsilon _0}E_0^2{{\mathcal{V}}}\nu ^{\prime}(\xi ) = 0$, or equivalently, $a^{\prime}(\xi )/\nu ^{\prime}(\xi ) + ({\varepsilon _0}E_0^2{R_0}/\gamma )({{\mathcal{V}}}/R_0^3) = 0$. For the critically stable shape, they infer that $a^{\prime}(\xi )/\nu ^{\prime}(\xi )$ is a constant and the formula becomes

(4.1)\begin{equation}\frac{{R_0^3}}{{\mathcal{V}}} = kC{a_E},\end{equation}

where the prefactor k is found to be ${\rm \pi} /2$ from their experimental results. This simple scaling law yields the qualitative relation between the critical electric capillary number $C{a_{E,c}}$ and the opening half-angle ${\theta _0}$, because

(4.2)\begin{equation}\frac{{R_0^3}}{{\mathcal{V}}} = \frac{{3\,\textrm{si}{\textrm{n}^3}\,{\theta _0}}}{{{\rm \pi} {{(1 - \textrm{cos}\,{\theta _0})}^2}(2 + \textrm{cos}\,{\theta _0})}}.\end{equation}

We then replot $C{a_{E,c}}$ as a function of $R_0^3/{{\mathcal{V}}}$, as shown in figure 6(a). Additionally, the plate-to-plate configuration has also been simulated using our numerical model with the same parameters. A horizontal electrode is included to create a uniform electric field. Figure 6(b) shows the distribution of electric field strength $|\boldsymbol{E} |$ inside and outside the droplet (or meniscus) for the plate-to-plate configuration and nozzle-to-plate configuration.

Figure 6.

(a) Critical electric capillary number $C{a_{E,c}}$ as a function of the dimensionless shape function $R_0^3/{{\mathcal{V}}}$. A linear relationship is identified for the plate-to-plate configuration as reported by Beroz et al. (Reference Beroz, Hart and Bush2019). (b) Distribution of electric field strength $|\boldsymbol{E} |$ inside and outside the droplet for the plate-to-plate configuration (left) and the meniscus for the nozzle-to-plate configuration (right). Here, ${\theta _0} = 90^\circ $ and $\alpha = 10$. The white regions represent the nozzle and plate. (c) Electric field strength along the interface and the symmetry axis as a function of the axial coordinate. Here, $z = 0$ denotes the triple contact point.

The red data points in figure 6(a) show a linear relationship between $C{a_{E,c}}$ and $R_0^3/{{\mathcal{V}}}$ with a fitting constant $k = {\rm \pi}/2$ for the plate-to-plate configuration. There is an excellent agreement between the prediction of (4.2) and our numerical results. It is noted that no dripping is observed among these cases, even for ${\theta _0} \gg 134^\circ $. However, for the nozzle-to-plate configuration, which is the one considered in this work, the critical value $C{a_{E,c}}$ appears to be larger. The qualitative relation between $C{a_{E,c}}$ and $R_0^3/{{\mathcal{V}}}$ for our considered model is quite different from that obtained in the theoretical study of Beroz et al. (Reference Beroz, Hart and Bush2019). To this end, we look at the spatial distribution of the electric field in these two cases. Both the contour in figure 6(b) and spatial variation in figure 6(c) indicate that the electric field around the interface for the nozzle-to-plate configuration is smaller than that for the plate-to-plate configuration. This is because the characteristic electric field is chosen to be the value at the nozzle tip for the nozzle-to-plate configuration rather than the uniform far field for the plate-to-plate configuration. This special geometry causes the local electric field to get weaker quickly with the axial distance, as shown in figure 6(c). Since the electric stress $\boldsymbol{T}_i^E = C{a_E}{\varepsilon _i}(\boldsymbol{EE} - {\boldsymbol{E}^2}\boldsymbol{I}/2)$ is determined by the electric fields on the two sides of the interface, a higher value is required for the critical electric capillary number $C{a_{E,c}}$. However, it is noted that a higher $C{a_{E,c}}$ does not indicate a higher electric potential applied on the nozzle because the definitions of characteristic electric field strength are different for the nozzle-to-plate and plate-to-plate configurations. The former is ${E_0} = 2\hat{\phi }/[{R_0}\,\textrm{ln(}4H/{R_0})]$ and the latter is ${E_0} = \hat{\phi }/H$. For example, when ${\theta _0} = 90^\circ $, the critical electric potential difference is calculated as 1374 V for the nozzle-to-plate configuration but 5981 V for the plate-to-plate configuration; this is the reason why the nozzle-to-plate configuration is preferred in most applications.

Next, we turn our attention towards the transition of non-jetting, dripping and jetting regimes. The transitional opening half-angle is found to be $135^\circ $. We therefore plot in figure 7 the temporal dynamics in these three regimes with $C{a_E} = 1.02,\,1.03$ and $1.08$, and ${\theta _0} = 135^\circ $. Figure 7(a) shows the simulated images of the meniscus shapes. Given almost the same $C{a_E}$, the sequence exhibits an entirely different outcome. For $C{a_E} = 1.02$, the electrified meniscus eventually reaches a stable state after an axial oscillation. However, when $C{a_E}$ is slightly increased to $1.03$, dripping occurs. Under the electric field, the meniscus is stretched into a nearly cylindrical liquid that is attached to the capillary at $t = 12$. Then a neck begins to form and continues to thin. The neck quickly narrows down until the drop pinches off. Figure 7(a) shows that tip-streaming can only be observed until $C{a_E}$ is increased to $1.08$. Among this range of $C{a_E}$ that dripping occurs, we find the radius ${R_d}/{R_0}$ of the emitted droplet is approximately $1.17$. It is therefore concluded that this value determines the minimum size of droplets that can be produced by the dripping regime.

Figure 7.

(a) Simulated images of time evolutions of the meniscus shapes in three regimes for ${\theta _0} = 135^\circ $. Effect of $C{a_E}$ on the temporal evolutions of physical quantities, (b) the axial location, (c) the charge density and (d) the curvature at the tip of the meniscus. Here, $\alpha = 10$.

Figure 7(b–d) respectively show the temporal evolutions of the axial location, the charge density and the curvature at the tip for a meniscus deforming via different regimes. During the initial stage, the meniscus height in the jetting regime increases faster than that in the dripping regime. At $t = 13.85$, it rapidly increases from $9.64$ to $13.36$ during a very short period of $\Delta t = 0.3$. This is because the charge density has reached a limit value at this instant, which gives rise to a strong electrostatic force at the tip of the meniscus. The variations of the charge density and curvature further confirm this significant increase in the normal electric stress at the tip of the meniscus. We note that the charge density and the tip curvature simultaneously rise to values almost an order of magnitude higher than that in the dripping or non-jetting regimes. When tip streaming occurs, both the charge and curvature begin to decrease, suggesting that the charged droplet starts to be emitted from the jet's tip. It is noted that the charge density and the curvature are interdependent during the entire process, as shown in figure 7(c,d). From the results above, we can conclude that the key factor that determines the difference between these three regimes is the speed at which the tip charge density rises. To obtain the jetting regime, the surface charge must accumulate quickly enough at the meniscus tip so that the Taylor cone can form prior to dripping.

4.2. Influence of opening half-angle on the tip streaming

After discussing the critical condition for tip streaming by considering the limiting cases in the previous section, we will now elucidate the influences of three key parameters on the first emission of a charged droplet. Figure 8(a,b) shows the influence of the opening half-angle ${\theta _0}$, i.e. the initial volume, on the meniscus shapes during the tip streaming process. Here the electric capillary number $C{a_E}$ is chosen to be $2.0$ for all cases so that jetting can be ensured for all ${\theta _0}$ ranging from $70^\circ $ to $110^\circ $. It is observed that a liquid meniscus with a larger ${\theta _0}$ will form a longer jet when pinch-off occurs. Compared with the limiting cases in figure 4, the cone angle decreases remarkably as the electric capillary number increases, resulting in the cone shape elongated to a larger extent. The subsequently emitted droplet adapts a teardrop shape and the thread connecting the jet and droplet becomes longer as ${\theta _0}$ increases. Figure 8(c) shows that the size of the emitted droplet varies little with the opening half-angle ${\theta _0}$, indicating that it does not depend on the geometry of the liquid meniscus. This result yields the universality in the formation of a cone jet, which becomes valid when the radius of the jet is orders of magnitude smaller than the characteristic length scale. In this case, Gamero-Castaño & Magnani (Reference Gamero-Castaño and Magnani2019) proposed that the characteristic only depends on the physical properties of the liquid meniscus, and is independent of the geometric and electrostatic variables (e.g. the dimensionless parameters ${\theta _0}$ and $C{a_E}$ in this study). The present results therefore verify that the droplet emission from a liquid meniscus is expected to be a universal phenomenon.

Figure 8.

Simulated meniscus shapes for different ${\theta _0}$ at (a) $t = 0$ and (b) the incipience of pinch-off. Here, $C{a_E} = 2.0$ and $\alpha = 10$ for all cases. In panel (b), the upper-right corner shows the magnified views of the tip-streaming jet's tip and the about-to-form droplet. Effect of ${\theta _0}$ on the (c) size of the first emitted droplet and (d) temporal evolutions of the axial location at the tip of the meniscus.

Figure 8(d) shows the effect of ${\theta _0}$ on the temporal evolutions of the axial location at the meniscus tip. It is seen that for ${\theta _0} = 80^\circ $, the value of pinch-off time is the smallest compared with that for the other ${\theta _0}$. This is different from the results of limiting cases in figure 5(a) because the electric capillary number is kept the same in this case. The pinch-off time determines the frequency of the jet formation so that the underlying mechanism has to be answered. The reason for the tendency of pinch-off time can be found in figure 9, where the distributions of the electric field strength for different ${\theta _0}$ are plotted at $t = 0$. Surprisingly, for the nozzle-to-plate configuration considered in our study, the local electric field at the meniscus tip just reaches its maximum when ${\theta _0} = 80^\circ $. The special distribution determines the magnitude of the initial electrical stresses imposed on the liquid interface. According to (2.7), the charge transport by Ohmic conduction depends on the local electric fields on both sides of the interface. Therefore, the higher electric field accelerates the transport of surface charge, which gives rise to earlier emission of the first droplet. This finding reveals the applicability of ${\theta _0}$ as an index for evaluation of the time required for the tip streaming process and hence the frequency of droplet generation.

Figure 9.

(a) Local electric fields at the tip of the meniscus for different ${\theta _0}$. Here the blue and red points respectively represent the electric field strength inside and outside the drop. (b) Electric field strength along the meniscus interface and the symmetry axis as a function of the axial coordinate. Here, $z = 0$ denotes the triple contact point. (c) Distributions of the electric potential $\phi $ (left) and electric field strength $|\boldsymbol{E} |$ (right) at $t = 0$ for ${\theta _0} = 60^\circ $, $90^\circ $ and $120^\circ $.

We then show the distribution of electric potential during the tip streaming process for the special case of ${\theta _0} = 90^\circ $. The contour in figure 10(a) indicates that the liquid seems to be nearly equipotential before the formation of tip streaming ($t = 1.1$). However, the equipotential assumption is no longer valid when tip streaming occurs at $t = 1.15$. The electric potential near the tip of the meniscus gradually reduces as the jet develops. When the first droplet is emitted at $t = 1.54$, the liquid again becomes an equipotential at the two ends of the thread.

Figure 10.

(a) Distribution of electric potential at various times: $t = 1.1$, $t = 1.15$, $t = 1.45$ and $t = 1.54$ in the case of ${\theta _0} = 90^\circ $, $C{a_E} = 2.0$ and $\alpha = 10$. Distributions of (b) charge density and (c) normal and tangential electric stresses on the half-meniscus surface as a function of the normalized arclength ${s^\ast }$ at $t = 1.1$, $t = 1.15$ and $t = 1.45$. Here, ${s^\ast } = 0$ denotes the triple contact point.

Figure 10(b) shows the surface charge density as a function of the normalized arclength of the half-meniscus surface. The charge density at the meniscus tip is expected to increase quickly at $t = 1.15$ and decreases at $t = 1.45$ as a consequence of the jet development. In figure 10(c), we plot the normal and tangential electric stresses along the meniscus surface. The cone formation modifies the overall electric stress distribution. From the figure, the normal stress is always maximum at the tip, which is balanced by the capillary force caused by the sharp tip curvature. The tangential stress, which is nearly zero at $t = 1.1$, increases with time and leads to the tangential fluid flow. A high tangential stress region is created near the tip, which accelerates the flow towards the meniscus tip, thus developing the electrified jet. Both the normal and tangential electric stresses start to decrease with the droplet emission. Moreover, the region with the maximum tangential stress gradually moves away from the tip of the meniscus as shown in the results at $t = 1.45$.

4.3. Influence of electric capillary number on the tip streaming

Further, to demonstrate the effect of electric field strength, the simulations were carried out for various $C{a_E}$ values. Figure 11(a–c) shows the meniscus shapes at the incipience of pinch-off for various values of $C{a_E}$ at ${\theta _0} = 60^\circ $, $90^\circ $ and $120^\circ $, respectively. The jet length is obviously increased with the increase in $C{a_E}$. Gawande et al. (Reference Gawande, Mayya and Thaokar2019) demonstrated that the jet length is governed by a balance of the tangential electric stress and the viscous stress. The formation of a tip-streaming jet requires a strong tangential electric stress. The increase in $C{a_E}$ leads to a growth of the tangential electric stress which accelerates the flow to the tip and results in the reduction of the pinch-off time, as shown in figure 11(d). Moreover, figure 11(e) shows that the size of the emitted droplets slightly increases with $C{a_E}$, but are of the same order of magnitude, which further confirmed the role of universality in determining the formation of emitted droplets as discussed above.

Figure 11.

Simulated meniscus shapes at the incipience of pinch-off for different $C{a_E}$ at (a) ${\theta _0} = 60^\circ $, (b) ${\theta _0} = 90^\circ $ and (c) ${\theta _0} = 120^\circ $. The upper-right corner shows the magnified views of the tip-streaming jet's tip and the about-to-form droplet. Effect of $C{a_E}$ on the (d) pinch-off time and (e) size of the first emitted droplet. Here, $\alpha = 10$ for all cases.

4.4. Influence of charge relaxation on the tip streaming

Likewise, to demonstrate the effect of surface charge relaxation (i.e. the electric conductivity), we plot in figure 12(a–c) the meniscus shapes at the incipience of pinch-off for various values of $\alpha $ at ${\theta _0} = 60^\circ $, $90^\circ $ and $120^\circ $, respectively. It is noted that the choice of different values of $C{a_E}$ for each ${\theta _0}$ is determined to ensure the occurrence of jetting. The value of $\alpha = {t_c}/{t_e}$ represents the ratio of capillary time to the charge relaxation time. In the previous cases, we chose a constant value of $\alpha = 10$ so that the charge relaxation occurs as quickly as the fluid flow. However, as shown in figure 12, the tip-streaming shapes have been significantly altered when the charge relaxation parameter is varied from $\alpha = 0.01$ to $1000$ with other parameters fixed.

Figure 12.

Simulated meniscus shapes at the incipience of pinch-off for different $\alpha $ at (a) ${\theta _0} = 60^\circ $ and $C{a_E} = 2.25$, (b) ${\theta _0} = 90^\circ $ and $C{a_E} = 1.5$, and (c) ${\theta _0} = 120^\circ $ and $C{a_E} = 1.25$. The upper-right corner shows the magnified views of the tip-streaming jet's tip and the about-to-form droplet.

We note that the jet length ${L_j}$ increases with the increase in $\alpha $ and reaches its maximum at $\alpha = 10$, but decreases for higher values of $\alpha $. This non-monotonic dependence is in agreement with the previous studies (Gawande et al. Reference Gawande, Mayya and Thaokar2019). For a higher $\alpha $, a long jet will form at the tip before the emission. Because the radius of the jet ${R_j}$ is determined by the tip curvature of the electrified meniscus when the conical tip approaches the singularity, the liquid meniscus with a higher $\alpha $ will eject a thinner jet and thus a smaller droplet is emitted from the tip. Since the breakup of the jet is primarily induced by the Plateau–Rayleigh instability: ${L_j} > 2{\rm \pi} {R_j}$, the jet length ${L_j}$ therefore decreases with the increasing $\alpha $. However, for lower $\alpha $, the capillary force is dominant (${t_c}/{t_e} \ll 1$) so that the capillary breakup occurs earlier than the formation of the jet. As a result, the thin jet has not formed completely for lower $\alpha $ and the jet length is still relatively short at the incipience of pinch-off.

To quantitatively describe the effect of charge relaxation, figure 13 shows the pinch-off time and the size of the emitted droplet as functions of $\alpha $, respectively. A monotonic decrease in pinch-off time is observed with an increase in $\alpha $. When $\alpha $ is increased by several orders of magnitude, the pinch-off time continuously decreases from ${\sim} 100$ to ${\sim} 1$. This can be attributed to the definition of $\alpha $ mentioned above. Tip streaming occurs only when the charge density at the tip reaches its limit so that the normal electric stress is strong enough to form the Taylor cone. The rate at which charge density rises primarily depends on the value of $\alpha $. Therefore, an increase in $\alpha $ will promote the tip streaming process. Meanwhile, if $\alpha $ is high enough, its effect will become weaker since the transport of charge to the tip has already been as quick as the fluid flow. As shown in figure 13(a), the minimum values of the pinch-off time are approxiamtely $1.1$, $1.4$ and $3.0$ for ${\theta _0} = 60^\circ $, ${\theta _0} = 90^\circ $ and ${\theta _0} = 120^\circ $, respectively.

Figure 13.

Effect of $\alpha $ on the (a) pinch-off time and (b) size ${R_d}$ of the first emitted droplet for ${\theta _0} = 60^\circ $ with $C{a_E} = 2.25$, ${\theta _0} = 90^\circ $ with $C{a_E} = 1.5$ and ${\theta _0} = 120^\circ $ with $C{a_E} = 1.25$.

Similarly, we observe a non-monotonic dependence of ${R_d}$ on the charge relaxation $\alpha $, as shown in figure 13(b). The accurate results show that the size of the first emitted droplet reaches its maximum at $\alpha \sim 5$. When $\alpha > 5$, the droplet size is found to decrease significantly to a small value, for example, the case with $\alpha = 1000$ produces a tiny droplet whose radius is only ${\sim} 0.1\%$ of the meniscus radius, e.g. $0.2\ {\rm \mu}\mathrm{m}$ for a $200\ {\rm \mu}\mathrm{m}$ nozzle. The variation of ${R_d}$ in the limit of a high $\alpha $ is in accordance with the results of Collins et al. (Reference Collins, Jones, Harris and Basaran2008). Moreover, a power-law relationship ${R_d}\, \sim \, {\alpha ^{ - 2/3}}$ is observed, which is consistent with the popular scaling law proposed by de la Mora & Loscertales (Reference de la Mora and Loscertales1994). They suggested that the size of the emitted droplet varies as ${R_d}\, \sim \, {(\gamma /\rho )^{1/3}}t_e^{2/3}$, namely, ${R_d}\, \sim \, {\alpha ^{ - 2/3}}$ in the limit of large conductivities. However, at lower $\alpha $, the size ${R_d}$ is found to decrease slightly with decreasing $\alpha $. The $\alpha $ dependence becomes weaker as the liquid approaches the dielectric limit ($\alpha \sim 0$).

To compare the jet shapes for different $\alpha $, following Collins et al. (Reference Collins, Sambath, Harris and Basaran2013), we plot the simulated profiles of the jets at the incipiences of jet emission and pinch-off for $\alpha $ ranging from $0.01$ to $1000$, as shown in figure 14. Here, ${\theta _0} = 90^\circ $ and $C{a_E} = 1.5$. When tip streaming occurs, the about-to-form jet shrinks as $\alpha $ is decreased to approach the dielectric limit ($\alpha \sim 0$) or increased to approach the perfectly conducting limit ($\alpha \sim \infty $), which is similar to that of the variation of the size of the emitted droplet. This result is beyond the previous studies. The slight decrease in droplet size in the dielectric limit has not been reported before. To identify the underlying mechanism, we plot in figure 15(a) the profiles of the tip-streaming jet's tip for $\alpha = 0.01$, $\alpha = 1$ and $\alpha = 100$, which represent respectively the cases of the dielectric, leaky-dielectric and conductor. It is observed that the jet radius is much smaller for $\alpha = 100$ than for $\alpha = 0.01$ and $\alpha = 1$. Obviously, this is due to the quick accumulation of surface charge since the jet behaves as a perfect conductor. However, when $\alpha $ decreases from $1$ to $0.01$, the jet shape is similar for the two cases, but the jet radius is slightly increased. From the shapes near the jet's tip, it is clear that the tip curvature is varied for all of the cases. According to the stress balance in (2.4), we thus show in figure 15(b–d) the distributions of the surface charge density, the normal electric stress and the tangential electric stress on the surface near the jet's tip. Figure 15(b) shows that the charge density q always reaches its maximum at the tip. However, the variation in ${q_{tip}}$ (charge density at the tip) for different $\alpha $ shows a reversed trend as the jet radius, which results in a difference in the normal electric stresses on the surface, as shown in figure 15(c). The increase in the normal electric stress leads to a higher capillary pressure and thereby a higher tip curvature. This explains the non-monotonic dependence of ${R_d}$ on $\alpha $, as observed in simulations. However, another question must be answered: Why is the charge density at the tip for $\alpha = 1$ smaller than that for $\alpha = 0.01$? Actually, from the charge transport equation (2.7), for low $\alpha $ values, surface charge convection becomes dominant in charge transport compared with conduction (Sengupta, Walker & Khair Reference Sengupta, Walker and Khair2017; Nie et al. Reference Nie, Li, Ma, Fang and Yin2021b). As $\alpha $ is reduced to 0.01, the significance of the convection at the tip exceeds the conduction and leads to a slight increase in charge density compared with the case where $\alpha = 1$. Moreover, figure 15(d) shows the distribution of the tangential electric stress that accelerates the axial flow of the jet, which also indicates the importance of flow-induced charge convection in charge transport. From the discussion above, we propose that to obtain a larger size of the first emitted droplet, the fluid conductivity should be controlled within a specific range.

Figure 14.

Profiles of the tip-streaming jet's tip and the about-to-form droplet at the incipiences of jet emission (upper) and pinch-off (lower) for $\alpha $ values of (a) $ 0.01$, (b) $0.1$, (c) $1$, (d) $10$, (e) $100$ and (f) $1000$. Here, ${\theta _0} = 90^\circ $ and $C{a_E} = 1.5$.

Figure 15.

(a) Profiles of the tip-streaming jet's tip at the incipiences of jet emission for $\alpha $ values of 0.01 (black), 1 (red) and 100 (blue). Here, ${\theta _0} = 90^\circ $ and $C{a_E} = 1.5$. Distributions of (b) surface charge density, (c) normal electric stress and (d) tangential electric stress on the surface near the jet's tip at the incipiences of emission. In panels (b) and (c), the values at the tips have been marked.