Jet breakup in superfluid and normal liquid ${}^4\mathrm{He}$

Past studies have shown that liquid jet breakup behavior can be classified into five regimes: Rayleigh, first wind, sinuous, second wind, and atomization. By experimentally examining the breakup of superfluid and normal liquid ${}^4\mathrm{He}$ in an atmosphere of its own vapor, we investigate the evolution of the jet behavior over a large range of the traditional three-dimensional parameter space of the Ohnesorge number $[{\text{Oh}}_l\sim O({10}^{-5}$–${10}^{-2}\left)\right]$, Reynolds number $[{\text{Re}}_l\sim O({10}^2$–${10}^6\left)\right]$, and gas-liquid density ratio $[{\rho }_g/{\rho }_l\sim O({10}^{-4}$–1)]. Using dimensional analysis we find that the transition from Rayleigh to first-wind breakup occurs at a constant liquid Weber number, and that the transitions from first wind to sinuous, and sinuous to second wind occur at constant gas Weber numbers. The proposed transitions, which differ from some previous studies, are well supported by our new experimental data that extend over all three dimensions of the parameter space. We do not observe any obvious effects of superfluidity on the breakup behavior. In addition, we examine the breakup length and comment on the transition of a liquid jet to a gaseous jet as the temperature passes through the critical point.

Figure 1

A cylindrical jet of liquid ${}^4\mathrm{He}$ emanates from a nozzle with inner diameter $52\mu \mathrm{m}$ and breaks up into a stream of droplets or a spray with various characteristics and appearances for differing input parameters. (a) Rayleigh breakup occurs when droplets pinch off near the bottom tip. (b) first-wind breakup occurs at higher velocities and appears similar to Rayleigh breakup, but in this case the jet pinches together at multiple adjacent locations in rapid succession. (c) In sinuous breakup the jet buckles forming a twisting or corkscrewlike structure. (d) Second-wind and (e) atomization breakups occur at the highest velocities, in which the jet fragments into a mist of very small droplets that forms a conical shape. When this breakup occurs downstream of the nozzle (d) it is known as second wind and when breakup begins within about one diameter from the nozzle (e) the name changes to atomization.

Figure 2

A summary of the literature discussing the jet breakup regimes is shown, emphasizing the various regime names, proposed transitions, and regime combinations. The regime names used in this paper are shown at the top of each of the five colored columns in order of increasing velocity from left to right. Below these are the names or descriptions used in each separate study. The white columns between regimes list any proposed transition criteria. Black, horizontal bars indicate that the authors combined regimes. Ohnesorge [13] did not provide quantitative transition criteria with his regime diagram, but others fit equations to his data and their fits are shown. Reitz et al. [6, 18, 27, 28] interpreted the findings of others in their reviews and the original source of their equations are also noted.

Figure 3

The fluid properties along the saturation curve of gaseous and liquid ${}^4\mathrm{He}$ are shown as a function of temperature $T$. Panel (a) shows the dynamic viscosity $\mu$ of the two phases separately, with the viscosity ratio shown on the right-hand $y$ axis. Below the $\lambda$ point ($T=2.17$ K) the liquid viscosity can be defined in two ways, as the viscosity of the normal component ${\mu }_n$, shown by the black dash-dotted line, or the effective total viscosity ${\mu }_l$, shown with the solid black line. Panel (b) shows the gaseous and liquid densities, ${\rho }_g$ and ${\rho }_l$, with the ratio of the two on the right-hand $y$ axis. Panels (c) and (d) show the surface tension coefficient $\sigma$ and saturation pressure $P_{\text{sat}}$. These plots are made from data compiled from previous studies [63, 64, 65].

Figure 4

A schematic of the basic experimental setup is shown. The cryostat consists of an outer chamber that holds a high vacuum. The cooling occurs in stages on the 40, 4, and 1 K plates, which indicate the nominal temperature values. Radiation shields connect to the bottom of the 40 and 4 K plates. The cell connects to the bottom of the 1 K plate to which it thermally sinks. The cell fills from a ${}^4\mathrm{He}$ pressure cylinder through a line that cools on the 40 and 4 K plates through heat exchangers. The same pressure cylinder feeds the nozzle of the jet through a different line that cools on the 40, 4, and 1 K plates. The inset shows a photo of the tip of the glass nozzle. The breakup event is viewed with a high speed camera and long distance microscope lens with a pulsed laser for lighting.

Figure 5

Photos of the progression of jet breakup regimes are shown for three different ${\text{Oh}}_l$ ranges as indicated to the left of each row ($T$ ranges are also shown for each row). The ${\text{Re}}_l$ for each case is shown at the bottom of each frame which generally increases from left to right. The symbols at the top left of each frame indicate the breakup regime classification and correspond to the symbols in the regime diagram in Fig. 7. Blue circles $○$ are Rayleigh breakup, red squares $\square$ are first wind, magenta stars $★$ are sinuous, green triangles $▵$ are second wind, and black pluses $+$ are atomization. Gray circles $○$ surrounding the symbol indicate the approximate quantity of side drops with one circle for some and two for a lot. The dark triangle in the top right of some frames is vignetting from the microscope lens.

Figure 6

Time sequences of four jets are shown. In panel (a) a superfluid jet exits the nozzle at $U_j=0.20$ m/s and $T=1.44$ K which results in Rayleigh breakup at ${\text{Oh}}_l=7.60\times {10}^{-5}$ and ${\text{Re}}_l=1.28\times {10}^4$. In panel (b) a superfluid jet exits the nozzle at $U_j=2.52$ m/s and $T=1.45$ K which results in first wind breakup at ${\text{Oh}}_l=7.90\times {10}^{-5}$ and ${\text{Re}}_l=1.55\times {10}^5$. In panel (c) a superfluid jet exits the nozzle at $U_j=26.35$ m/s and $T=1.47$ K which results in sinuous breakup of the core with a lot of side droplets at ${\text{Oh}}_l=8.51\times {10}^{-5}$ and ${\text{Re}}_l=1.50\times {10}^6$. In panel (d) a normal fluid jet near the critical point exits the nozzle at $U_j=0.61$ m/s and $T=5.17$ K which results in a sinuous breakup near the transition to second wind at ${\text{Oh}}_l=2.54\times {10}^{-2}$ and ${\text{Re}}_l=1.18\times {10}^3$. Some bubble can be seen inside the jet which are more common near the critical point. For scale the nozzle's outer diameter at the tip is $72\mu \mathrm{m}$ in each image. See Supplemental Material videos 1–4 that correspond to panels (a)–(d), respectively [72].

Figure 7

(a) The different modes of jet breakup can be divided on a ${\text{Re}}_l\text{−}{\text{Oh}}_l$ regime diagram similar to the data of Ohnesorge [13] and Saito et al. [17]. As ${\text{Oh}}_l$ and ${\rho }_g/{\rho }_l$ are both functions of $T$ for constant $d_j$ the relationship between the two is shown in panel (b) showing the three-dimensionality of the regime diagram. The right-hand axes of panels (a) and (b) show the corresponding $T$ to the ${\text{Oh}}_l$ and ${\rho }_g/{\rho }_l$ on the left-hand axes. The small circle, square, star, triangle, and plus symbols indicate the mode of breakup of the core of the jet and the medium and large gray circles surrounding these symbols indicate if droplets pinch off from the sides of the jet, with one gray circle indicating some droplets and two indicating a lot as shown in Fig. 5. Gray circles have been omitted from the second wind and atomization regimes to reduce clutter and because side droplets are the cause of the breakup of the core of the jet in these cases. The dashed line “$--$” plots ${\text{We}}_l=8$, the solid line “—” plots ${\text{We}}_g=8$, and the dotted line “...” plots ${\text{We}}_g=120$.

Figure 8

(a) A small bubble inside a jet with ${\text{We}}_l=8$ moves downward relative to the jet interface as indicated by the blue arrows. The velocities of the bubble and neck are shown over time in panel (b). The large red dot in panel (b) indicates the time that the bubble passes through the narrowest portion of the neck. The time $t$ at the bottom of the frames in panel (a) correspond to the time in panel (b). The small stationary spot in panel (a) is a dust particle on the camera lens.

Figure 9

The breakup length of the jet core $L_b$ normalized by the jet diameter $d_j$ is plotted against ${\text{We}}_g$ for three different ranges of ${\text{Oh}}_l$, which correspond to the three rows from Fig. 5. The green line corresponds with row (a), the red line with row (b), and the blue line with row (c). The symbols represent the breakup regime for each measurement as indicated in Fig. 7. The upward arrows associated with some points mark when the jet breakup length is longer than the image frame.

Figure 10

Photos from a single event of a jet in the atomization regime as the cell temperature $T$ increases through the critical point, $T=5.19$ K. The cell temperature is shown at the bottom of each frame.