Armstrong liquid bridge: Formation, evolution and breakup

In this paper, we experimentally explore the formation, evolution, and breakup of the Armstrong liquid bridge. The extremely complicated evolution stage is revealed, which involves many coupled processes including the morphology change, current variation, heat transfer, and water evaporation. By focusing on the final fate of this liquid bridge, we observe that the breakup occurs once an effective length ($\tilde{L}$) is reached. This effective length increases linearly with the applied voltage, implying a threshold electric field to sustain the liquid bridge. Moreover, by an introduced external flow, the lifetime of the liquid bridge can be controlled, while the effective length associated with the breakup is independent of the external flow rate. Hence, these findings remarkably demonstrate that the breakup of the liquid bridge is directly correlated with the effective length. In order to understand this correlation, a simplified model of an electrified jet is employed to take the electric field into account. By the linear stability analysis, the attained phase diagram agrees with the experiments well. Although a more comprehensive theory is required to consider other factors such as the surface charges, these results might provide a fresh perspective on the “century-old” Armstrong liquid bridge to further elucidate its underlying physical mechanism.

Figure 1

Formation of the ALB. (a) The sketch of the experimental setup and a magnified view of the ALB suspended between beakers with their small spouts facing closely ($V$ for the applied voltage, $L$ for the distance between the two spouts, $d=2r_0$ for the diameter, $h_1$ and $h_2$ for the climbing height, and the cylindrical beakers with 3.2 cm diameter and 5 cm height). (b) High-speed images for the initial formation of the ALB from a side view ($L=6\mathrm{mm}, V=13$ kV). (c) A top view of the liquid bridge.

Figure 2

Current density. (a) The simultaneous measurement of current and diameter ($V=12$ kV, $L=6$ mm). (b) Current density $j$ fluctuating over the time. (c) The measurement of $j_a$ (the black dots) agreeing with the calculation from the Ohm's law (the red dashed line). Error bars from four measurements.

Figure 3

Temperature evolution. (a) Thermal image of the ALB ($V=12$ kV, $L=6\mathrm{mm}$). (b) Thermal images revealing a waterfall falling from the liquid bridge to one of the beakers and the corresponding temperature distribution. (c) $T_{\max }$ changing over time under different voltages. (d) $T_a$ increasing with the applied voltages (error bars for four measurements).

Figure 4

Mass loss characterized by the reduced water level or the increased climbing height. (a) Photographs showing the change of water level (red dashed lines) inside the beakers with time, and the inadvertent leakage ($L=6\mathrm{mm}, V=16$ kV). (b) The climbing height of water level in two beakers at the anode side ($h_1$) and the cathode side ($h_2$) changing with time. (c) The total climbing height ($h=h_1+h_2$) increasing with time for various voltages. (d) The mass reduction of water due to the evaporation and the leakage (error bars for four measurements).

Figure 5

Breakup of the ALB. (a) High-speed images of the typical breakup ($L=6\mathrm{mm}, V=12$ kV). (b) Diameter of bridge gradually decreasing until the final breakup. (c) At the breakup time ($t_b$), $\tilde{L}=h\left(t_b\right)+L$ almost a constant for various $L$ ($V=12$ kV). (d) Experimentally $\tilde{L}$ increasing linearly with $V$ for various $L$, and its slope indicating a constant electric field $E_{\mathrm{breakup}}=0.52$ kV ${\mathrm{mm}}^{-1}$ associated with the breakup.

Figure 6

Control of the ALB by an external flow. (a) The sketch of manipulating liquid bridge by an external flow through a syringe pump. (b) The evolution of $h+L$ with time under various flow rates. (c) The similar $\tilde{L}=h\left(t_b\right)+L$ upon breakup, nearly independent on the flow rate. (d) The reduction of ${\tau }_{\mathrm{lifetime}}$ by the flow extraction, the experimental data (black squares, error bars for three measurements) fitted well by ${\tau }_{\mathrm{lifetime}}$: Eq. (4) (the blue line). (e) The extension of ${\tau }_{\mathrm{lifetime}}$ over two hours by flow injection ($V=14$ kV, $L=6\mathrm{mm}$).

Figure 7

Linear stability analysis. (a) The dispersion relationship or the growth rate $\omega$ dependent on the wavelength $k$ under different dimensionless electrical field $\mathrm{\Omega }$. $\mathrm{\Omega }=0.01,1,1.4,1.8,2.1$ from the outer line to the inner line, and ${\omega }_{\max }$ for the maximum of $\omega$. (b) The dispersion relationship (${\omega }_{\max }-\mathrm{\Omega }$) showing a critical value of the dimensionless electric field (${\mathrm{\Omega }}^{*}=2.18$). (c) The calculated phase diagram ($E-r_0$) consistent with the experimental observation. The dark square for the initial formation, while the red circle for the final breakup (error bars from experimental measurements).

Figure 8

Initial morphology under various $L$ and the associated $V_c$. (a)–(d) A side view of the ALB morphology captured by the high-speed camera right after formation at the critical $V_c$ for various $L$, while the inset (the yellow rectangular) for a top-down view from the digital camera, $L=1,3,9,12$ mm, and $V_c=6.0,9.3,14.7,16.1$ kV for (a)–(d), respectively. (e) The critical voltage ($V_c$) increasing linearly with $L$, indicating a constant electric field ($E_{\mathrm{formation}}=1$ kV ${\mathrm{mm}}^{-1}$) required for the formation (error bars for fifteen measurements).

Figure 9

Mass transport between two beakers via a bidirectional axial flow. (a) The mass changing at the anode and cathode sides over time under 12 kV. (b) The flow velocity fluctuating around tens of millimeters per second. (c) The average velocity increasing with the applied voltage, on the order of tens of millimeters per second. All the experiments are conducted with the fixed distance $L=6\mathrm{mm}$ and repeated four times ($L=6\mathrm{mm}$).