2.3.1. Measurements of minimum flow rate of viscous liquids

In this section, we present the experimental data for pure glycerol (Solution I) and salt-doped glycerol solutions (Solutions II $-$ IV) to elucidate the effect of stand-off distance, needle diameter and electrical conductivity of a viscous liquid on the minimum flow rate essential for steady cone-jetting.

To elucidate the effect of stand-off distance on the minimum flow rate, we performed experiments with solution I, varying $H$ from 3 to 18 mm for a given needle. We then repeated the procedure for all the needles and measured the corresponding values of the minimum flow rate. Figure 3 $(a)$ shows the variation of the minimum flow rate with the stand-off distance for all the needles. From figure 3 $(a)$ , we note that $Q_{\textit{min}}$ is independent of the stand-off distance $H$ for a given needle diameter. However, the threshold potential corresponding to the minimum flow rate, that is, the minimum potential difference required for the onset of steady EHD-jetting, increases with increasing stand–off distance for all needle diameters. This trend suggests that, for a given needle diameter, the threshold electric field acting on the working liquid, near the needle tip, remains nearly the same for the range of $H$ considered in the present study. This observation is consistent with our earlier observation of $Q_{\textit{min}}$ being independent of $H$ . The exact dependence of the threshold potential on the geometrical parameters is discussed in detail in § 2.3.3.

Figure 3.

Variation of (a) the minimum flow rate ( $Q_{\textit{min}}$ ) necessary for steady cone-jetting of solution I and (b) the total current carried by the jet at $Q_{\textit{min}}$ with the stand-off distance ( $H$ ). We infer that both minimum flow rate $Q_{\textit{min}}$ and the total current carried by the jet $I$ at $Q_{\textit{min}}$ are independent of stand-off distance $H$ .

However, $Q_{\textit{min}}$ monotonically increases with the needle diameter for all stand-off distances. For each case, we also measured the total current $I$ carried by the jet at the minimum flow rate condition. Figure 3 $(b)$ shows the variation of $I$ with $H$ for solution I for all five needles. From figure 3 $(b)$ , we note that the total current $I$ , similar to $Q_{\textit{min}}$ , is also independent of $H$ for a given needle diameter and increases with increasing needle diameter. Therefore, we can conclude that stand-off distance $H$ does not affect the minimum flow rate necessary for steady cone-jetting of a viscous liquid.

Next, to investigate the effect of the needle diameter and the electrical conductivity of the viscous liquids on the minimum flow rate essential for steady cone-jetting, we kept the stand-off distance fixed at $H=5\,$ mm. Figure 4(a–d) show images of the cone-jets corresponding to solutions I $-$ IV for $d_n=400\,\unicode {x03BC}$ m with minimum flow rate values of 116.7, 41.7, 25.0 and 11.7 nl s⁻¹, respectively. From figure 4(a–d), we infer that the minimum flow rate and the corresponding jet diameter decrease with an increase in the electrical conductivity of the liquid for a fixed needle diameter. Figure 4(e–h) show the images of cone-jets corresponding to solutions I $-$ IV for $d_n=500\,\unicode {x03BC}$ m with minimum flow rate values of 166.7, 58.3, 33.3 and 16.7 nl s⁻¹, respectively. Similar to steady jetting for a needle with $d_n = 400\,\unicode {x03BC}$ m, the minimum flow rate and the corresponding jet diameter decrease with an increase in the liquid conductivity for the needle with $d_n=500\,\unicode {x03BC}$ m.

Figure 4.

Snapshots of steady Taylor cone-jets of solutions I–IV at the minimum flow rate condition. Steady state cone-jet images of solutions I–IV when flown through needles with outer diameters of $d_n=$ (a–d) 400 and (e–h) 500 $\unicode {x03BC}$ m. The minimum flow rates of steady Taylor cone-jets for panels (a–d) and (e–h) are 116.7, 41.7, 25.0 and 11.7 nl s⁻¹, and 166.7, 58.3, 33.3 and 16.7 nl s⁻¹, respectively. From these observations, we infer that the minimum flow rate of Taylor cone-jets of highly viscous leaky dielectric liquid decreases with the increase in the electrical conductivity of the liquids for a particular needle and increases with the increase in the needle diameter for a given solution. Moreover, the jet diameter, similar to the minimum flow rate, decreases with the increases in the electrical conductivity of the liquid for a given needle diameter. and increases with the increase in the needle diameter for a given solution. For $(a-d)$ , the potential differences maintained between the needle tip and the collector necessary for steady cone-jetting are 3.3, 3.5, 3.7 and 3.8 kV, and 3.4, 3.7, 3.7 and 3.8 kV for panels $(e-h)$ , respectively. These potential differences, when changed slightly, resulted in pulsating jets in the respective experiments.

Next, to understand the effect of the needle diameter on the minimum flow rate of a viscous liquid, we refer to figure 3 $(a)$ . Through figure 3 $(a)$ , we have already shown that the minimum flow rate of solution I increases with the increase in the needle diameter for a given stand-off distance. To verify whether this trend holds for the other liquids, we compare the minimum flow rate values of steady cone-jetting across all the needle diameters while keeping the electrical conductivity fixed. We repeat this process for all four liquids. The comparison of figure 4(a–d) with figure 4(e–h) suggests that the minimum flow rate and the corresponding jet diameter increase with increasing needle diameter. Therefore, the increase in the minimum flow rate upon increasing $d_n$ is consistent for all four liquids, which have electrical conductivities spanning over four orders of magnitude.

To further explore the dependence of the minimum flow rate on the needle diameter, in figure 5 $(a)$ , we plot the measured values of the minimum flow rate of all four liquids against the needle diameter on a logarithmic scale. The experimental data suggest that the minimum flow rate increases linearly with the needle diameter on a log–log plot, indicating a power-law variation of $Q_{\textit{min}}$ on $d_n$ with a positive exponent. Moreover, figure 5 $(a)$ also indicates that for a given needle diameter, the minimum flow rate decreases with increasing electrical conductivity of the liquid.

Figure 5.

Variation of the minimum flow rate ( $Q_{\textit{min}}$ ) in steady cone-jetting of highly viscous leaky dielectric liquids with the outer diameter of the needles ( $d_n$ ) and electrical conductivity of the liquids ( $\kappa$ ). (a) Variation of $Q_{\textit{min}}$ with $d_n$ on a log–log plot for solutions I $-$ IV. The experimental data of $Q_{\textit{min}}$ for each of the four liquids closely follow the scaling law $Q_{\textit{min}} \sim d^{1.7}_n$ . (b) Variation of $Q_{\textit{min}}$ with $\kappa$ of the four liquids used in the cone-jetting experiment for all the five needle diameters in a log–log plot. In this case, the same experimental data of $Q_{\textit{min}}$ aligns well with the scaling law $Q_{\textit{min}} \sim \kappa ^{-0.34}$ for each needle diameter. $(c)$ Minimum flow rate value predicted by the empirical correlation $Q^{\textit{corr}}_{\textit{min}}$ (2.1), which is in excellent agreement with the experimental data of the present work and that of Scheideler & Chen (Reference Scheideler and Chen2014).

To obtain the scaling of the minimum flow rate on the electrical conductivity of a solution, we plot the same data of $Q_{\textit{min}}$ in figure 5 $(b)$ against the electrical conductivity of all four liquids on a logarithmic scale. We infer from the plot that the minimum flow rate decreases linearly with the conductivity of the liquid on a log–log plot, suggesting a power-law variation of $Q_{\textit{min}}$ on $\kappa$ but with a negative exponent.

Now that the dependence of the minimum flow rate on the needle diameter and the electrical conductivity of the liquid has been established, we perform a multi-variable regression analysis (Bahga Reference Bahga2021) with the experimental data of the present work to obtain the respective exponents, and subsequently the dimensional correlation for the minimum flow rate in terms of $d_n$ and $\kappa$ , given as

(2.1) \begin{align} Q^{\textit{corr}}_{\textit{min}} = 0.0075\,d_n^{1.7}\,\kappa ^{-0.34}. \end{align}

Note that $Q^{\textit{corr}}_{\textit{min}}$ , $d_n$ and $\kappa$ in (2.1) are in nl s⁻¹, $\unicode {x03BC}$ m and $\unicode {x03BC}$ S m⁻¹, respectively. The correlation, given by (2.1), suggests that the minimum flow rate scales as $Q_{\textit{min}} \sim d^{1.7}_n$ across liquids with varying electrical conductivities. This observation is consistent with the measurements of Scheideler & Chen (Reference Scheideler and Chen2014), who reported a scaling of the form $Q_{\textit{min}} \sim d^{1.45\pm 0.18}_n$ for salt-doped glycerol solutions. Moreover, the empirical correlation also suggests that $Q_{\textit{min}} \sim \kappa ^{-0.34}$ for all the needle diameters. We note that due to the weak dependence of the minimum flow rate on the electrical conductivity of the liquids, specifically $Q_{\textit{min}}\sim \kappa ^{-0.34}$ , Scheideler & Chen (Reference Scheideler and Chen2014), who performed cone-jetting experiments for a narrow range of electrical conductivity, incorrectly concluded that the minimum flow rate is independent of the electrical conductivity of the viscous liquid.

Next, we plot the minimum flow rate values predicted by the empirical correlation in the same plot where the experimental data are already overlayed, as shown in figures 5 $(a)$ and 5 $(b)$ . In these figures, the symbols represent the experimental data, whereas the dashed lines represent the predicted values of the minimum flow rate using the empirical correlation (2.1). To further establish the efficacy of the empirical correlation, in figure 5 $(c)$ , we plot the minimum flow rate values obtained from the empirical correlation against the experimental data of both the present work and Scheideler & Chen (Reference Scheideler and Chen2014) on a logarithmic scale. From figure 5 $(c)$ , we infer that the experimental data collapse on the dashed line, indicating a one-to-one correspondence.

Building on the present work, in § 2.3.2, we present the empirical scaling for the jet diameter at the minimum flow rate condition with $d_n$ and $\kappa$ . Subsequently, in § 2.3.3, we also present the scaling for the onset potential essential for steady cone-jetting at the minimum flow rate condition.

2.3.2. Jet diameter at the minimum flow rate

The jet diameter of a steady cone-jet is an important quantity for applications such as electrospraying and EHD-jet printing because it influences the size of the monodispersed droplets or the resolution of the deposited patterns. In the present study, we observe that cone-jets attain an asymptotic jet diameter far downstream of the cone-tip, within the precision of our measurements. This behaviour is consistent with the ballistic regime described by Gañán-Calvo (Reference Gañán-Calvo2025), which is particularly favourable for steady cone-jets of highly viscous liquids such as those used in the present study. Hereafter, the jet diameter refers to the asymptotic diameter measured downstream of the cone-tip.

Prior work on electrospraying and steady cone-jetting has shown that the jet diameter is positively correlated with the liquid flow rate. Moreover, from figure 4, we note that the jet diameter depends on the needle diameter and the electrical conductivity of the viscous liquid. In particular, the jet diameter decreases with increasing electrical conductivity and increases with the needle diameter for a given liquid. It is important to note that due to the high electrical conductivities of solutions III and IV, the corresponding jet diameters were below the resolution of the imaging system used in the present study. Consequently, the jet diameters for these two liquids could not be measured through imaging.

To explore the dependence of the jet diameter on the needle diameter, in figure 6 $(a)$ , we plot the measured values of the jet diameter for solutions I and II against the outer diameter of the five needles employed in our experiments. The experimental data suggest that the jet diameter at the minimum flow rate increases monotonically with the needle diameter. From figure 6 $(a)$ , we also note that the jet diameter at $Q_{\textit{min}}$ decreases with increasing electrical conductivity of the liquid. Therefore, the jet diameter exhibits a qualitatively similar dependence on the needle diameter and the electrical conductivity, as observed for the minimum flow rate. As a result, the jet diameter, similar to the minimum flow rate, can be represented as $d=c d^a_n \kappa ^b$ , where $a$ , $b$ and $c$ are constants to be determined using multivariable regression analysis (Bahga Reference Bahga2021). The resulting empirical correlation for the jet diameter is given by

(2.2) \begin{align} d_{\textit{corr}} = 0.25 d^{0.7}_n \kappa ^{-0.37}. \end{align}

Here, $d_{\textit{corr}}$ , $d_n$ and $\kappa$ are in $\unicode {x03BC}$ m and $\unicode {x03BC}$ S m⁻¹, respectively. Although the dimensional correlation (2.2) is based on the jet diameter data of two liquids (Solutions I and II) only, we will justify the applicability of the correlation in §§ 4.4 and 4.5 using theoretical scaling analysis.

Figure 6.

(a) Variation of the jet diameter $d$ at the minimum flow rate condition with the outer diameter of the needles ( $d_n$ ) for solutions I and II. $(b)$ Jet diameter predicted by the proposed correlation $d_{\textit{corr}}$ (2.2), which aligns well with the experimental observation.

Moreover, the experimentally measured jet diameter, when plotted against the predicted values in figure 6 $(b)$ , collapses on the $45^\circ$ line indicating a one-to-one correspondence. Therefore, from figures 6 $(a)$ and 6 $(b)$ , we infer that the empirical correlation given by (2.2) provides an excellent prediction of the jet diameter.

2.3.3. Onset potential at the minimum flow rate

To ensure steady cone-jetting at the minimum flow rate condition, the applied potential has to be equal to the critical value known as the onset potential $V_{\textit{onset}}$ . At $V_{\textit{onset}}$ , flow rates at and above $Q_{\textit{min}}$ provide steady cone-jetting, whereas flow rates below $Q_{\textit{min}}$ result in a pulsating cone-jet (Bober & Chen Reference Bober and Chen2011; Chen Reference Chen2011 ). Thus, for a given solution and geometrical configuration, the minimum flow rate alone has significance only if reported along with $V_{\textit{onset}}$ (Bober & Chen Reference Bober and Chen2011; Chen Reference Chen2011). A good initial approximation of $V_{\textit{onset}}$ is therefore essential for locating $Q_{\textit{min}}$ of steady cone-jetting of a solution. In § 2.3.1, we present the experiments with solution I to show that $Q_{\textit{min}}$ is independent of $H$ . For the same set of experiments, we also recorded the onset potential at which steady cone-jetting was obtained. Figure 7 $(a)$ shows the variation of onset potential $V_{\textit{onset}}$ with varying needle-to-collector distance $H$ for all five needles that are employed in the present work for solution I. This figure shows that $V_{\textit{onset}}$ monotonically increases with increasing $H$ for a given value of $d_n$ and also increases with the increasing $d_n$ for a given $H$ . To further elucidate the dependence of $V_{{{onset}}}$ on $H$ and $d_n$ , we scale the data represented in figure 7 $(a)$ with the reference potential $V_{{ref}} =(\gamma d_n/\varepsilon _0)^{1/2}$ (Zeleny Reference Zeleny1935) and plot against the aspect ratio $\chi =(H/d_n)$ in a semi-log plot, as shown in figure 7 $(b)$ . From figure 7 $(b)$ , we observe that all the data points collapse to a single curve showing that the dimensionless onset potential $\tilde {V}_{\textit{onset}}=(V_{\textit{onset}}/V_{{ref}})$ varies logarithmically with the aspect ratio $\chi$ .

Figure 7.

(a) Variation of the onset potential essential for steady cone-jetting at the minimum flow rate ( $Q_{\textit{min}}$ ) condition with the geometrical parameters of a cone-jetting set-up such as $H$ and $d_n$ for solution I. $(b)$ Linear variation of dimensionless onset potential $\tilde {V}_{\textit{onset}}$ with $4\chi$ on a semi-log plot, which suggests a logarithmic dependence.

The logarithmic dependence of $\tilde {V}_{\textit{onset}}$ on $\chi$ can be established by deriving the relationship for the electric field at the needle tip, $E=E(V_{\textit{onset}},d_n, H)$ . From the perspective of electric field distribution, the needle tip and the protruding liquid meniscus jointly can be approximated as a point electrode facing a flat counter electrode. The corresponding electric field at the tip of a needle for a point-plane geometry can be expressed as (Loeb et al. Reference Loeb, Kip, Hudson and Bennett1941; Smith Reference Smith1986),

(2.3) \begin{equation} E\sim \frac {V_{\textit{onset}}}{d_n\ln (4\chi )}. \end{equation}

Next, the potential required to set in the cone-jet mode at the minimum flow rate condition can be approximated by substituting the electric field $E$ in (2.3) with Taylor’s electrostatic field, $E\sim (\gamma /(\varepsilon _0 d_n) )^{1/2}$ (Taylor Reference Taylor1964). Upon substitution, we get

(2.4) \begin{align} \frac {V_{\textit{onset}}}{d_n\ln (4\chi )} \sim \left (\frac {\gamma } {\varepsilon _0 d_n}\right )^{1/2}, \end{align}

which shows that

(2.5) \begin{align} \tilde {V}_{\textit{onset}} \sim \ln (4\chi ). \end{align}

Hence, for a given liquid, $\tilde {V}_{\textit{onset}}$ varies logarithmically with $\chi$ and for a given geometrical configuration, $V_{\textit{onset}}$ depends only on the interfacial tension of the liquid. From figure 7 $(b)$ , we also note that the slope of the line that shows a logarithmic dependence of $\tilde {V}_{\textit{onset}}$ on $\chi$ has a value of 0.5, which is in agreement with the findings of the previous studies (Loeb et al. Reference Loeb, Kip, Hudson and Bennett1941; Smith Reference Smith1986; Gañán-Calvo et al. Reference Gañán-Calvo, Lasheras, Dávila and Barrero1994; Pantano et al. Reference Pantano, Gañán-Calvo and Barrero1994). Therefore, we can rewrite (2.5) as

(2.6) \begin{align} \tilde {V}_{\textit{onset}}=0.5 \ln (4\chi ). \end{align}

Thus, (2.6) provides an estimate of the onset potential required to establish steady cone-jetting of highly viscous liquids for a given geometrical configuration at the minimum flow rate condition, and will be useful in applications such as EHD-jet printing, electrospinning and electrospraying while operating at or near $Q_{\textit{min}}$ .

4.2.1. Insights from the work of Gañán-Calvo et al.

Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) assumed that the stability of the cone-jet mode at the minimum flow rate condition is primarily dependent on the local flow arising in the current transfer region. Furthermore, the characteristic length of the current transfer region is much smaller than those characterising the cone-jetting set-up, i.e. $L\ll d_n$ and $L\ll H$ . Because of the locality of the current transfer region, Gañán-Calvo $et\,al$ . reasoned that the stability of the cone-jet mode at the minimum flow rate condition does not depend on the length scales associated with the cone-jetting set-up.

Next, they suggested that, in the current transfer region, the balance of the primary supporting and resistant forces is essential to achieve and sustain a steady jet. Using $d$ and $L$ as the reference scales for the radial and axial lengths of the current transfer region, respectively, the axial momentum (4.5) suggests the scaling for all the resisting and the supporting forces in the current transfer region as follows.

1. (i) Resisting forces: $\{\text{I}:\gamma d^{-1} L^{-1}, \text{II}:\rho Q^2 d^{-4}L^{-1}, \text{III}:\mu Q d^{-2}L^{-2}\}$ .

2. (ii) Supporting forces: $\{\text{IV}:\varepsilon _0 E_n E_t d^{-1}, \text{V}:\varepsilon _0 E_n^2 L^{-1}, \text{VI}:\varepsilon _0 \beta E_t^2 L^{-1}\}$ .

In particular, Gañán-Calvo et al. argued that in the current transfer region among the supporting forces, the polarisation force is, at best, comparable to the electrostatic suction force, whereas the tangential electric force is significantly smaller than the electrostatic suction force throughout the current transfer region. On the other hand, among the opposing forces, the viscous force is, at best, comparable to the inertia term and the interfacial tension force is negligibly small. Therefore, Ganan-Calvo et al. considered inertia and electrostatic suction forces to be the primary resisting and driving forces at the current transfer region. Subsequently, they derived the scaling relations for jet diameter and the length of the current transfer region as

(4.7a,b) \begin{align} d \sim d_0 \left (\dfrac {Q}{Q_0}\right )^{1/2}\quad \mathrm{and}\quad L\sim d_0\left (\dfrac {Q}{Q_0}\right )\!, \end{align}

by considering the balance of these forces in the current transfer region.

Furthermore, Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) argued that at the minimum flow rate condition, there could be two scenarios for which steady jet emission is not possible.

1. (i) The viscous forces may become comparable with the inertia and resist the jet emission before the polarisation force, referred to as the viscous limit. In the viscous limit, the minimum flow rate and the jet diameter at the minimum flow rate scale as

   (4.8a,b) \begin{align} Q_{\textit{min}}\sim Q_0 {\textit{Re}_0}^{-1} \quad \mathrm{and}\quad d \sim d_0 {\textit{Re}_0}^{-1/2}, \end{align}
   respectively.

2. (ii) The polarisation force may become negative beyond the cone-jet transition region and oppose the driving force first, termed as the polar limit. In the polar limit, the minimum flow rate and the jet diameter at the minimum flow rate scale as

   (4.9a,b) \begin{align} Q_{\textit{min}}\sim Q_0 \beta \quad \mathrm{and}\quad d \sim d_0\beta ^{1/2}, \end{align}
   respectively.

From (4.8a ) and (4.9a ), Ganan-Calvo et al. noted that the ratio of the minimum flow rate for the viscous limit case to the minimum flow rate for the polar limit case equals to $\beta {\textit{Re}_0}$ . Therefore, they concluded that the term $\beta {\textit{Re}_0}$ dictates which limit out of the viscous and polar limits will govern the minimum flow rate of steady cone-jetting. In particular, for $\beta {\textit{Re}_0} \lt 1$ , the viscous limit governs $Q_{\textit{min}}$ and for $\beta {\textit{Re}_0} \gt 1$ , the polar limit governs $Q_{\textit{min}}$ .

To validate the proposed scaling laws, in their study, Ganan-Calvo et al. considered liquids that satisfy both $\beta {\textit{Re}_0}\lesssim 1$ and $\beta {\textit{Re}_0} \gt 1$ conditions. However, our focus here is exclusively on liquids that meet the $\beta {\textit{Re}_0}\lt 1$ criterion, which are tabulated in table 2. Among all liquids considered by Ganan-Calvo et al., 3-ethylene glycol, 4-ethylene glycol and 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl) imide (EMI-Im), which have electrical conductivities of $\mathcal{O}(10^{3}{-}10^{5}\,\unicode {x03BC}\rm {S\,m}^{-1})$ , satisfy the condition $\beta {\textit{Re}_0}\lt 1$ , as shown in table 2. Subsequently, to validate their scaling laws for the minimum flow rate, Ganan-Calvo et al. plotted the experimental data for the flow rate on a two-dimensional (2-D) regime plot, with $Q_{\textit{min}}/(\beta Q_0)$ along the ordinate and $(\beta {\textit{Re}_0})^{-1}$ on the abscissa, as shown in figure 9. In their case, for $\beta {\textit{Re}_0} \lt 1$ , the experimental data for the minimum flow rate lay just above the $(Q_{\textit{min}}/Q_0){\textit{Re}_0} =1$ line, validating the derived scaling law for the minimum flow rate, $Q_{\textit{min}}\sim Q_0 {\textit{Re}_0}^{-1}$ . Moreover, as we are dealing with the minimum flow rate, in figure 9, we only highlighted the lowest value of the flow rate that has been reported in figure 2 of Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013).

Table 2.

Physical properties of the liquids used by Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) that satisfy the $\beta {\textit{Re}_0}\lt 1$ criterion. Also tabulated are the reference scales for the flow rate $Q_0$ and jet diameter $d_0$ , along with the corresponding values of the electrohydrodynamic Reynolds number ${\textit{Re}_0}$ , $\beta {\textit{Re}_0}$ and Ohnesorge number $(\textit{Oh})$ . The Ohnesorge number ( $\textit{Oh}=\mu /\sqrt {\rho \gamma d_n}$ ) reported in the table is evaluated based on the diameter of the needle, $d_n=100\,\unicode {x03BC}$ m, employed in the study of Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013).

Figure 9.

A two-dimensional regime plot for minimum flow rate of different liquids on a $Q_{\textit{min}}/(Q_0\beta )$ versus $1/(\beta {\textit{Re}_0})$ plot. The regime plot is divided into two regions $\beta {\textit{Re}_0}\lt 1$ and $\beta {\textit{Re}_0}\gt 1$ by the vertical boundary line (in blue colour), $\beta {\textit{Re}_0}=1$ . The region $\beta {\textit{Re}_0}\lt 1$ is further divided by the slanted boundary line (in red colour), $(Q_{\textit{min}}/Q_0){\textit{Re}_0} =1$ , into two sub-regions $(Q_{\textit{min}}/Q_0){\textit{Re}_0} \gt 1$ and $(Q_{\textit{min}}/Q_0){\textit{Re}_0} \lt 1$ . The minimum flow rate data from Scheideler & Chen (Reference Scheideler and Chen2014) and present work clearly fall within the unexplored $(Q_{\textit{min}}/Q_0){\textit{Re}_0} \lt 1$ region. The grey hollow circular symbols represent the minimum flow rate data gathered from literature (figure 2 of Gañán-Calvo et al. Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013), which fall in the $(Q_{\textit{min}}/Q_0){\textit{Re}_0} \gt 1$ region close to the boundary 2. The details of the plotted experimental data are provided in table 2, in table 4 and in table 6.

An alternative explanation of the minimum flow rate $Q_{\textit{min}} \sim Q_0 {\textit{Re}_0}^{-1}$ in the viscous limit can be given by considering the ratio of $\tau _{\mu }$ and $\tau _h$ . Here, $\tau _{\mu }$ represents the viscous diffusion time scale across the length of the current transfer region. Using (4.7), we show that

(4.10) \begin{align} \dfrac {\tau _{\mu }}{\tau _{h}}\sim \dfrac {\rho L^2 Q}{\mu L d^2} \sim \dfrac {Q_{\textit{min}} {\textit{Re}_0}}{Q_0} \gtrsim 1. \end{align}

Therefore, $(Q_{\textit{min}}/Q_0){\textit{Re}_0}\gtrsim 1$ is equivalent to stating that at the minimum flow rate condition, the viscous diffusion time scale, $\tau _{\mu } \sim \rho L^2/\mu$ , across the length of the current transfer region is greater than or comparable to the characteristic residence time scale, $\tau _h \sim Ld^2/Q$ .

Alternatively, the limit can be expressed in terms of the ratio of inertia force density ( $F_{\textit{int}}$ , term $\text{II}$ of (4.5)) to the viscous force density ( $F_{v\textit{isc}}$ , term $\text{III}$ of (4.5)) in the current transfer region. Using the respective scales of the force densities and (4.7), we get

(4.11) \begin{align} \left (\dfrac {F_{\textit{int}}}{F_{v\textit{isc}}}\right )\sim \dfrac {\rho Q^2 d^{-4} L^{-1}}{\mu Q d^{-2} L^{-2}} \sim \dfrac {\rho Q L}{\mu d^2}\sim \dfrac {Q_{\textit{min}} {\textit{Re}_0}}{Q_0} \gtrsim 1. \end{align}

Therefore, the assumption of Ganan-Calvo et al. that inertia is the dominant resisting force in the current transfer region is justified for the liquids listed in table 2.

Similarly, if we compare the electrostatic suction force density ( $F_{\textit{es}}$ , term $\text{V}$ of (4.5)) with the polarisation force density ( $F_{\textit{pol}}$ , term $\text{VI}$ of (4.5)) in the current transfer region, we note that

(4.12) \begin{equation} \left (\dfrac {F_{\textit{pol}}}{F_{\textit{es}}}\right )\sim \dfrac {\varepsilon _0 \beta E^2_t L}{L \varepsilon _0 E^2_n} \sim \dfrac {\beta E^2_t}{E^2_n} \sim \dfrac {\beta d^2}{L^2} \sim \dfrac {\beta Q_0}{Q}. \end{equation}

For the $\beta {\textit{Re}_0}\lt 1$ case, the minimum flow rate $Q_{\textit{min}} \sim Q_0 {\textit{Re}_0}^{-1}$ (see (4.8a )) and $(F_{\textit{pol}}/F_{\textit{es}}) \sim \beta {\textit{Re}_0}\lt 1$ at the minimum flow rate condition. Therefore, the assumption of Ganan-Calvo et al. that at the minimum flow rate condition, in the current transfer region, the electrostatic suction force is the dominant driving force is justified for the liquids listed in table 2.

We note that the criteria $(Q_{\textit{min}}/Q_0){\textit{Re}_0} \gtrsim 1$ and $\beta {\textit{Re}_0}\lt 1$ can also be expressed as

(4.13) \begin{equation} \alpha _{\rho } \gtrsim \alpha ^{1/4}_{\mu }\quad \text{and}\quad \dfrac {\alpha _{\rho }}{\left (\beta -1\right )}\gt 1, \end{equation}

respectively, where $\alpha _{\rho }=Q_{\textit{min}}/Q_0$ and $\alpha _{\mu }= \alpha _{\rho }/{\textit{Re}_0}^3$ (Gañán-Calvo Reference Gañán-Calvo2004). These terms are widely used in the literature on Taylor cone-jets. We also note that (4.13) is the well-known criteria for the inertia electrostatic suction regime (IE regime) (Gañán-Calvo Reference Gañán-Calvo2004). Thus, the liquids considered by Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) which satisfy the $\beta {\textit{Re}_0}\lt 1$ criterion follow IE scaling.

4.2.2. Insights from the work of Scheideler and Chen

In their study on the minimum flow rate of steady cone-jetting, Scheideler & Chen (Reference Scheideler and Chen2014) considered highly viscous liquids such as salt-doped glycerol solutions with $\kappa$ of $\mathcal{O}({10^2{-}10^4}\,\unicode {x03BC}\mathrm{S\,m}^{-1})$ . They observed that $Q_{\textit{min}}$ decreases with increasing viscosity of the liquid and increases with the increasing needle diameter. In particular, $Q_{\textit{min}} \sim d^{1.45\pm 0.18}_n$ . Scheideler & Chen (Reference Scheideler and Chen2014) also concluded that $Q_{\textit{min}}$ is independent of the electrical conductivity of the viscous liquid. Based on their experimental observation, Scheideler & Chen (Reference Scheideler and Chen2014) argued that for highly viscous liquids, $Q_{\textit{min}}$ must depend on the dynamic viscosity of the liquid and the diameter of the needle used in the cone-jetting experiments. In particular, Scheideler & Chen (Reference Scheideler and Chen2014) referred to the scaling law, $Q_{\textit{min}} \sim \beta Q_0$ , proposed by Fernández de La Mora (Reference Fernández de la Mora2007) and advocated for a modified scaling law that could better account for their experimental findings. To support their experimental observation and derive the suitable scaling law, Scheideler & Chen (Reference Scheideler and Chen2014) proposed that the viscous diffusion time scale based on the needle diameter, $\tau _{\mu } \sim \rho d^2_n/\mu$ , should become comparable to or smaller than the charge relaxation time scale $\tau _e \sim \beta \varepsilon _0/\kappa$ , which can be written as

(4.14) \begin{align} \tau _{\mu } &\lesssim \tau _e, \notag \\ \dfrac {\rho d^2_n}{\mu } &\lesssim \dfrac {\beta \varepsilon _0}{\kappa }. \end{align}

Subsequently, upon substituting $\beta \varepsilon _0/\kappa$ from (4.14) into the minimum flow rate scaling, $Q_{\textit{min}} \sim \beta Q_0$ , Scheideler & Chen (Reference Scheideler and Chen2014) obtained the new scaling for $Q_{\textit{min}}$ as

(4.15) \begin{align} Q_{\textit{min}} \sim \dfrac {\beta \gamma \varepsilon _0}{\rho \kappa }\sim \dfrac {\gamma \rho d^2_n}{\mu \rho }\sim \dfrac {\gamma d^2_n}{\mu }. \end{align}

We note that the scaling, given by (4.15), includes dynamic viscosity and the needle diameter. In particular, $Q_{\textit{min}} \sim \mu ^{-1}$ and $Q_{\textit{min}} \sim d^2_n$ . However, the assumption (4.14) made by Scheideler & Chen (Reference Scheideler and Chen2014) itself is not satisfied by most of the liquids considered in their study. As a result, the measured value of the minimum flow rate and that predicted by the scaling law (4.15) differ by several orders-of-magnitude (see table 6). In particular, the scaling obtained by Scheideler & Chen (Reference Scheideler and Chen2014), (4.15), is independent of $\kappa$ . However, in § 2.3.1, through experiments, we have shown that $Q_{\textit{min}} \sim \kappa ^{-0.34}$ for the salt-doped glycerol solutions with electrical conductivity spanning over four orders of magnitude, $\kappa \sim \mathcal{O}({10^0-10^3}\,\unicode {x03BC}\mathrm{S\,m}^{-1})$ . This difference in observations is attributed to the relatively narrow range of electrical conductivity of the viscous liquids considered by Scheideler & Chen (Reference Scheideler and Chen2014) in their study.

To find out the regime in which the experimental data of the present work and that of Scheideler & Chen (Reference Scheideler and Chen2014) fall, we plot the experimental data in figure 9. As depicted in figure 9, we observe that all the experimental data of Scheideler & Chen (Reference Scheideler and Chen2014) and the present work satisfy the $\beta {\textit{Re}_0}\lt 1$ criterion, but lie much below the $(Q_{\textit{min}}/Q_0){\textit{Re}_0} =1$ line, suggesting a distinct jetting regime for highly viscous liquids.

5.1.1. Fast charge relaxation

The theoretical scaling analysis presented in § 4.4 assumes that the characteristic time scale for charge relaxation $\tau _e \sim \beta \varepsilon _0/\kappa$ is comparable to or smaller than the characteristic residence time in the current transfer region, $\tau _h \sim Ld^2/Q$ , as represented in (4.1). Therefore, the condition $\tau _e/\tau _h \lesssim 1$ is crucial for the formation of a charge layer on the liquid surface, ensuring steady cone-jetting. By substituting the expressions for $d$ and $L$ from (4.22a ) and (4.22b ) into (4.1), we obtain

(5.1) \begin{align} \dfrac {\tau _e}{\tau _h}\sim \dfrac {\beta \varepsilon _0 Q}{L d^2 \kappa }\sim \dfrac {\beta {\textit{Re}_0}}{\left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )^{1/4}}. \end{align}

In table 3, we present the values of (5.1) corresponding to all four liquids at the minimum flow rate condition. From table 3, we note that the rapid charge relaxation criterion, $\tau _e/\tau _h\lesssim 1$ , is satisfied by all liquids. More importantly, as shown by Ponce-Torres et al. (Reference Ponce-Torres, Rebollo-Muñoz, Herrada, Gañán-Calvo and Montanero2018), the scaling relations for the normal component of the electric field ( $E_n$ ), surface charge density ( $\sigma$ ) and the electrostatic force terms remain valid even when $\tau _e \sim \tau _h$ . Therefore, in the present study, the assumption of fast charge relaxation is justified and the associated scaling analysis remains valid.

Table 3.

Details of the needle diameter and the conductivity of the liquids used in the present work. Additionally, it includes the experimental data for the minimum flow rate and the values of $\tau _e/\tau _h$ , $(F_{\textit{int}}/F_{v\textit{isc}})$ , $(F_{\textit{st}}/F_{v\textit{isc}})$ and $(F_{\textit{pol}}/F_{\textit{es}})_{\textit{ct}}$ . From these values, we infer that viscous force is the dominant resisting force and electrostatic suction is the dominant driving force in the current transfer region. Moreover, viscous and interfacial tension forces are of the same order of magnitude at the cone-tip.

Subsequently, setting $\tau _e/\tau _h \lesssim 1$ in (5.1) at $Q=Q_{\textit{min}}$ , we obtain a limit on $Q_{\textit{min}}$ for which rapid charge relaxation can be assumed,

(5.2) \begin{align} Q_{\textit{min}}\gtrsim \beta ^4 Q_0 {\textit{Re}_0}^3. \end{align}

Therefore, $Q_{\textit{min}}=\beta ^4 Q_0 {\textit{Re}_0}^3$ represents the first constraint for the minimum flow rate and flow rates above this limiting value will satisfy the rapid charge relaxation criterion.

5.1.2. Viscous force is the dominant resisting force

In § 4.3, we assumed that among interfacial tension, inertia and viscous forces, the viscous force is the dominant resisting force in the current transfer region. To justify the assumption, we now compare the viscous force with the inertia and the interfacial tension force in the current transfer region. Using the derived scaling relations for $d$ and $L$ given by (4.22a ) and (4.22b ), we can write in the current transfer region,

(5.3a) \begin{eqnarray} \left (\dfrac {F_{\textit{int}}}{F_{v\textit{isc}}}\right )_{\textit{ct}} \sim \dfrac {\rho L^2 Q}{\mu L d^2}\sim \left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )^{3/4} \end{eqnarray}

and

(5.3b) \begin{equation} \left (\dfrac {F_{\textit{st}}}{F_{v\textit{isc}}}\right )_{\textit{ct}}\sim \dfrac {\gamma d^2 L^2}{d L \mu Q}\sim \dfrac {{\textit{Re}_0}^{1/2}}{\left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )^{1/8}}. \end{equation}

Next, in table 3, we present the values of $(F_{\textit{int}}/F_{v\textit{isc}})_{\textit{ct}}$ and $(F_{\textit{st}}/F_{v\textit{isc}})_{\textit{ct}}$ for all four liquids, based on these relations at the minimum flow rate condition. From table 3, we note that the conditions $(F_{\textit{int}}/F_{v\textit{isc}})_{\textit{ct}}\lt 1$ and $(F_{\textit{st}}/F_{v\textit{isc}})_{\textit{ct}}\lt 1$ are satisfied. Thus, our assumption of viscous force being the dominant resisting force in the current transfer region, while deriving the scaling laws, is justified.

Consequently, setting $(F_{\textit{int}}/F_{v\textit{isc}})_{\textit{ct}} \lesssim 1$ in (5.3a ) at $Q= Q_{\textit{min}}$ , we obtain

(5.4) \begin{align} Q_{\textit{min}} \lesssim Q_0 {\textit{Re}_0}^{-1}. \end{align}

Therefore, we note that $Q_{\textit{min}}= Q_0 {\textit{Re}_0}^{-1}$ represents the second constraint for the minimum flow rate and for flow rates below this limiting value, the viscous force dominates inertia in the current transfer region.

Similarly, setting $(F_{\textit{st}}/F_{v\textit{isc}})_{\textit{ct}} \lesssim 1$ in (5.3b ) at the minimum flow rate condition, we get

(5.5) \begin{align} Q_{\textit{min}} \gtrsim Q_0 {\textit{Re}_0}^{3}. \end{align}

Therefore, $Q_{\textit{min}} =Q_0 {\textit{Re}_0}^3$ represents the third constraint for the minimum flow rate and for flow rates above this limiting value, the viscous force dominates interfacial tension force in the current transfer region.

While deriving the scaling laws, we also considered that at the cone tip, among the resisting forces, the viscous and the interfacial tension forces are of the same order of magnitude and are greater than inertia by at least one order of magnitude. To verify these assumptions, we compare the forces at the cone-tip. We also assumed that the jet diameters at the cone-tip and in the current transfer region are of the same order of magnitude. Therefore, using the derived scaling law for jet diameter $d$ , given by (4.22a ), we can write at the cone-tip,

(5.6a) \begin{align} \left (\dfrac {F_{\textit{int}}}{F_{v\textit{isc}}}\right )_{\kern-1.5pt\textit{cone}}\sim \dfrac {\rho Q d_n}{\mu d^2_{\textit{cone}}} \sim \left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )\left (\dfrac {d_n}{d_0}\right ) \left (\dfrac {d_0}{d_{\textit{cone}}} \right )^2\sim \left (\dfrac {d_n {\textit{Re}_0}}{d_0}\right ) \left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )^{1/4}\!, \end{align}

and

(5.6b) \begin{align} \left (\dfrac {F_{\textit{st}}}{F_{v\textit{isc}}}\right )_{\kern-1.5pt\textit{cone}}\sim \dfrac {\gamma d_n d_{\textit{cone}}}{\mu Q}\sim \left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )^{-1}\left (\dfrac {d_n {\textit{Re}_0}^2}{d_0}\right ) \left (\dfrac {d_{\textit{cone}}}{d_0}\right )\sim \left (\dfrac {d_n {\textit{Re}_0}}{d_0}\right ) \dfrac {{\textit{Re}_0}^{1/2}}{\left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )^{5/8}}. \end{align}

In table 3, we present the values of $(F_{\textit{int}}/F_{v\textit{isc}})_{\textit{cone}}$ and $(F_{\textit{st}}/F_{v\textit{isc}})_{\textit{cone}}$ at the minimum flow rate condition, based on the derived relations for all four liquids. From table 3, we note that $(F_{{int}}/F_{{visc}})_{\textit{cone}}\lt 1$ and $(F_{{st}}/F_{{visc}})_{\textit{cone}} \sim 1$ . Therefore, our assumption of viscous force and interfacial tension forces being of the same order of magnitude at the cone-tip is justified.

5.1.3. Electrostatic suction force is the dominant driving force

In § 4.3, we assume that in the current transfer region, the forces due to tangential electric stress and electrostatic suction are of the same order of magnitude. Additionally, we argued that in the current transfer region, the tangential component of the electric field is much smaller than the normal component of the electric field, that is, $E_t \ll E_n$ . Consequently, the force due to polarisation stress can, at most, match the order of magnitude of the force due to electrostatic suction. Based on these considerations, we identified the electrostatic suction force to be the primary driving force in the current transfer region and derived the corresponding scaling laws.

To verify this assumption, we now compare the various driving forces in the current transfer region. Using the derived scaling laws for $d$ and $L$ , given by (4.22a ) and (4.22b ), we can write

(5.7a) \begin{eqnarray} \left (\dfrac {F_{et}}{F_{\textit{es}}}\right )_{\textit{ct}}\sim \dfrac {\varepsilon _0 E_n E_t L}{{\rm d} \varepsilon _0 E^2_n} \sim \dfrac {E_t L}{{\rm d} E_n} \sim \dfrac {E_t L d}{{\rm d} E_t L}\sim 1 \end{eqnarray}

and

(5.7b) \begin{equation} \left (\dfrac {F_{\textit{pol}}}{F_{\textit{es}}}\right )_{\textit{ct}}\sim \dfrac {\varepsilon _0 \beta E^2_t}{\varepsilon _0 E^2_n}\sim \dfrac {\beta E^2_t}{E^2_t \dfrac {L^2}{d^2}}\sim \dfrac {\beta d^2}{L^2}\sim \dfrac {\beta {\textit{Re}_0}}{\left (\dfrac {Q {\textit{Re}_0}}{Q_0}\right )^{1/4}}. \end{equation}

From (5.7a ), we note that tangential electric stress and electrostatic suction are of the same order of magnitude irrespective of the scaling relation of $d$ and $L$ . Moreover, from (5.1) and (5.7b ), we note that the final scalings for $\tau _e/\tau _h$ and $(F_{\textit{pol}}/F_{\textit{es}})_{\textit{ct}}$ are identical. Therefore, from table 3, we infer that at the minimum flow rate condition, $(F_{\textit{pol}}/F_{\textit{es}})_{\textit{ct}} \lesssim 1$ for all four liquids. This justifies our earlier assumption of electrostatic suction being the dominant driving force in the current transfer region. Furthermore, we check the validity of these assumptions for the experimental data of Scheideler & Chen (Reference Scheideler and Chen2014) and tabulate the findings in table 5. Similar to our experimental data, all the discussed assumptions are also satisfied by the liquids used by Scheideler & Chen (Reference Scheideler and Chen2014).

Therefore, our assumption that for highly viscous liquids, at the minimum flow rate condition, the viscous force acts as the dominant resisting force, while electrostatic suction serves as the dominant driving force in the current transfer region, is verified. Using (5.3a ), (5.3b ) and (5.7b ), and considering $\beta -1\approx \beta$ , we can now collectively represent these criteria as follows:

(5.8) \begin{align} \left (\dfrac {Q_{\textit{min}}{\textit{Re}_0}}{Q_0}\right )^{3/4} \lesssim 1,\quad \dfrac {{\textit{Re}_0}^{1/2}}{\left (\dfrac {Q_{\textit{min}} {\textit{Re}_0}}{Q_0}\right )^{1/8}} \lesssim 1 \quad \text{and} \quad \dfrac {\beta {\textit{Re}_0}}{\left (\dfrac {Q_{\textit{min}}{\textit{Re}_0}}{Q_0}\right )^{1/4}} \lesssim 1, \end{align}

which on further simplification yields

(5.9) \begin{align} \dfrac {Q_{\textit{min}}{\textit{Re}_0}}{Q_0} \lesssim 1,\quad \dfrac {Q_{\textit{min}}}{Q_0 {\textit{Re}_0}^3}\gtrsim 1 \quad \text{and} \quad \dfrac {\left (\dfrac {Q_{\textit{min}}}{Q_0 {\textit{Re}_0}^3}\right )^{1/4}}{\beta -1} \gtrsim 1. \end{align}

Using the notation defined by Gañán-Calvo (Reference Gañán-Calvo2004), we can express (5.9) as

(5.10) \begin{align} \alpha _{\rho }\lesssim \alpha ^{1/4}_{\mu }, \quad \alpha _{\mu } \gtrsim 1 \quad \text{and} \quad \dfrac {\alpha ^{1/4}_{\mu }}{\beta -1}\gtrsim 1. \end{align}

Here, $\alpha _{\rho } =Q_{\textit{min}}/Q_0$ and $\alpha _{\mu } =\alpha _{\rho }/{\textit{Re}^3_0}$ (Gañán-Calvo Reference Gañán-Calvo2004). We note that (5.10) is the criteria for the viscous electrostatic suction regime (VE-regime) for steady cone-jetting of a viscous leaky dielectric liquid. Thus, the liquids considered in the present study and that of Scheideler & Chen (Reference Scheideler and Chen2014) with $\beta {\textit{Re}_0}\lt 1$ , follow the VE scaling at the minimum flow rate condition. However, liquids considered by Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) and Ponce-Torres et al. (Reference Ponce-Torres, Rebollo-Muñoz, Herrada, Gañán-Calvo and Montanero2018) having $\beta {\textit{Re}_0}\lt 1$ , follow the IE scaling.

It is important to note that the term $\beta {\textit{Re}_0} (Q_{\textit{min}} {\textit{Re}_0}/Q_0 )^{-1/4}$ determines whether viscous or polarisation forces limit steady cone-jetting at $Q_{\textit{min}}$ for both highly and less viscous liquids. Specifically, when $\beta {\textit{Re}_0} (Q_{\textit{min}} {\textit{Re}_0}/Q_0 )^{-1/4}\lt 1$ , the viscous force restricts jet emission at $Q_{\textit{min}}$ . Conversely, when $\beta {\textit{Re}_0} (Q_{\textit{min}} {\textit{Re}_0}/Q_0 )^{-1/4}\gt 1$ , the polarisation force opposes the driving force at $Q_{\textit{min}}$ and stalls the jet beyond the current transfer region. Furthermore, by substituting $Q_{\textit{min}} =Q_0 {\textit{Re}_0}^{-1}$ and $Q_{\textit{min}} =\beta Q_0$ , the criteria $\beta {\textit{Re}_0} (Q_{\textit{min}} {\textit{Re}_0}/Q_0 )^{-1/4}\lt 1$ and $\beta {\textit{Re}_0} (Q_{\textit{min}} {\textit{Re}_0}/Q_0 )^{-1/4}\gt 1$ simplify to $\beta {\textit{Re}_0}\lt 1$ and $\beta {\textit{Re}_0}\gt 1$ , respectively, which is the criteria defined by Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) for less viscous liquids.

5.1.4. Discussion on the assumption of Scheideler and Chen

From figure 9, we also note that for the present work and that of Scheideler & Chen (Reference Scheideler and Chen2014), $Q_{\textit{min}}(\beta Q_0)^{-1}\lesssim 1$ , which is in contrast to the observations of Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) for which $Q_{\textit{min}} (\beta Q_0)^{-1}\gt 1$ . To explain this, we refer to the assumption of Scheideler & Chen (Reference Scheideler and Chen2014). They assumed that the viscous diffusion time scale across a length scale of $d_n$ , $\tau _{\mu }\sim \rho d^2_n/\mu$ , should become comparable to or smaller than the charge relaxation time $\tau _e \sim \beta \varepsilon _0/\kappa$ and subsequently derived the scaling law for the minimum flow rate as $Q_{\textit{min}} \sim \gamma d^2_n/\mu$ . However, the assumption is not satisfied by most of the liquids considered by Scheideler & Chen (Reference Scheideler and Chen2014) and the present study. Therefore, we evaluate the viscous diffusion time across the length of the current transfer region $L$ rather than $d_n$ and compare it with the charge relaxation time. Using (4.22b ), we get

(5.11) \begin{align} \dfrac {\tau _{\mu }}{\tau _e}\sim \dfrac {\rho L^2 \kappa }{\mu \beta \varepsilon _0} \sim \dfrac {Q}{\beta Q_0}\lesssim 1. \end{align}

Therefore, at the minimum flow rate condition, for highly viscous liquids, the viscous diffusion time scale across the length of the current transfer region is comparable to or smaller than the charge relaxation time scale. As Gañán-Calvo et al. (Reference Gañán-Calvo, Rebollo-Muñoz and Montanero2013) considered less viscous liquids in their cone-jetting experiments, the viscous diffusion time across the current transfer region was much greater than the charge relaxation time, $\tau _{\mu }/\tau _{e}\gt 1$ , resulting in $Q_{\textit{min}} (\beta Q_0)^{-1}\gt 1$ .