c 2008 Cambridge University Press J. Fluid Mech. (2009), vol. 619, pp. 261–275. doi:10.1017/S0022112008004436 Printed in the United Kingdom
261
Visual observation of the bubble dynamics in normal 4He, superfluid 4He and superfluid 3 He–4He mixtures H. A B E1 † M. M O R I K A W A1 , T. U E D A1 , R. N O M U R A1 , Y. O K U D A1 A N D S. N. B U R M I S T R O V2 1
Department of Condensed Matter Physics, Tokyo Institute of Technology, 2-12-1, Oh-okayama, Meguro, Tokyo 152-8551, Japan 2
Kurchatov Institute, Moscow 123182, Russia
(Received 1 December 2007 and in revised form 28 September 2008)
In order to compare the bubble dynamics of various quantum liquids, we performed the visual observation of a sound-induced bubble in a normal liquid 4 He, pure superfluid 4 He, and superfluid 3 He–4 He liquid mixtures of saturated and unsaturated 3 He concentrations. When an acoustic wave pulse was applied to these liquids under saturated vapour pressure, a macroscopic bubble was generated on the surface of a piezoelectric transducer. For all liquids, the size of the bubble increased, as a higher voltage was applied to the transducer at a fixed temperature. In the normal 4 He we observed a primary bubble surrounded with many small bubbles which ascended upward together. In contrast to normal phase, only one bubble was generated in the superfluid 4 He, and its shape proved to be highly irregular with an ill-defined surface. In the 3 He saturated superfluid mixture, we also observed a solitary bubble but with a nearly perfect spherical shape. The bubble in this mixture expanded on the transducer surface, grew to a maximum size of the order of 1 mm and then started shrinking. As the bubble detached from the transducer with shrinking, we clearly detected an origination of the upward jet flow which penetrated the bubble. The jet velocity in the liquid mixture was approximately 102 –103 times smaller than in water. At the final stage of the process we could sometimes observe a vortex ring generation. It is interesting that, though the bubble size and time scale of the phenomenon differ from those in water, the behaviour in the collapsing process had much in common with the simulation study of the vortex ring generation in water. In addition, for the mixture with the unsaturated 3 He concentration of about 25 % at 600 mK, the shape of the upward jet was observed distinctly, using more precise measurement with shadowgraph method.
1. Introduction The bubble dynamics in a liquid is a familiar physical phenomenon in fluid physics, and there have been many studies, both experimental and theoretical, on it (Brennen 1995). In particular, the bubble behaviour near a free surface or rigid boundary has interesting aspects, such as microjet, vortex ring formation and shock waves. It is † Email address for correspondence:
[email protected]; Present Address: National Metrology Institute of Japan, AIST, Umezono 1-1-1, Tsukuba, Ibaraki 305-8563, Japan
262 H. Abe, M. Morikawa, T. Ueda, R. Nomura, Y. Okuda, and S. N. Burmistrov 3
SVP (Pa) viscosity (Pa · s) surface tension (N m−1 ) density (g cm−3 ) sound velocity (m s−1 )
He (0.3 K) −1
2.4 × 10 6 × 10−6 1.538 × 10−4 0.082 178
4
He (0.8 K)
Water (293 K)
1.47 20 × 10−6 3.78 × 10−4 0.148 240
2.34 × 103 1.002 × 10−3 (P = 1 atm) 7.3 × 10−2 (P = 1 atm) 0.9982 1500
Table 1. Comparison between liquid He and water
well known that the equations given by Rayleigh (1917) and Plesset (1949) are useful for discussing expansion and contraction of a spherical bubble if one disregards the effect of external boundaries. Studies have not yet been performed on anything but ordinary liquids, e.g. water and liquid nitrogen. The literature contains a great deal of data on water, while those on other liquids are meager. Here we report our visual observation of the bubble cavitation and macroscopic bubble dynamics in helium liquids. Helium possesses two stable isotopes, 3 He and 4 He, and has unique properties at low temperatures; it is the only substance which can exist in the liquid state down to absolute zero. Due to the relatively weak interaction between helium atoms and large zero-point vibrations the quantum effects manifest before the liquid solidifies. Isotopes 3 He and 4 He, having the lowest boiling temperatures, liquefy at Tb = 3.19 K and Tb = 4.22 K at a pressure of 1 atm, respectively. As the temperature lowers, Bose liquid 4 He goes over into the superfluid state below Tλ = 2.17 K at the saturated vapour pressure. Fermi liquid 3 He also converts to the superfluid state but at considerably lower temperatures of about several mK, depending on the pressure. The superfluid state is usually expressed in terms of a two-fluid model with the normal and superfluid components representing the viscous and inviscid motions, respectively. Below 1 K liquid 4 He consists almost entirely of the superfluid component and in various aspects can be regarded an inviscid fluid. This makes it possible for large Reynolds numbers to be realized. The extremely high purity of liquid helium from extraneous impurities is another attractive feature in the study of the intrinsic factors governing the cavitation and dynamics of bubbles in fluids. As a rule, in quantum liquid the boiling temperature, evaporation heat, viscosity, surface tension and density are significantly lower than those in ordinary classical liquids. Each of these physical quantities makes the bubble cavitation easy. Typical values of some physical quantities for water, liquid 3 He and 4 He are compared in table 1. The finite solubility of 3 He in 4 He opens a way for new experiments on the bubble dynamics in quantum liquid mixtures. Of notice is that 3 He impurity atoms in superfluid 4 He are not drawn into the superfluid motion, and all of them contribute to the density of the normal component. Viscosity of liquid 3 He–4 He mixture can easily be changed by the 3 He concentration in the superfluid 4 He. In addition, below the tricritical point of about 0.9 K the liquid mixture separates into the saturated 3 He-dilute phase (d phase) and the 3 He-concentrated phase (c phase). The phase diagram of the mixture is shown in figure 1. At temperatures below ∼100 mK the lighter c phase is almost pure 3 He, while the heavier d phase is a solution of a finite concentration of 3 He in superfluid 4 He. The saturated 3 He concentration in the d
Bubble dynamics in superfluid 3 He–4 He mixture 2.0
263
λ Line
Normal 4He + Normal 3He 1.5 Superfluid 4He + Normal 3He T (K) 1.0
3He
0.5
d phase 3He
c phase
Phase seperation region 0
0.2
0.4
0.6
0.8
1.0
x3
Figure 1. Phase diagram of 3 He–4 He liquid mixture at the saturated vapour pressure. Concentration x3 is a molar fraction of 3 He in liquid 4 He. The λ line shows the superfluid transition in the mixture. The crosses and closed circles are the points at which our measurements were made. We can also show here a point corresponding to superfluid pure 4 He at x3 = 0. Experiment in normal 4 He is carried out above the λ line.
phase is sensitive to the temperature and pressure, changing between 6.7 % and 9.5 % as a function of pressure at zero temperature (0 K). Several papers address the cavitation in pure liquid 3 He and 4 He at negative pressures, e.g. Pettersen, Balibar & Maris (1994), Balibar et al. (1998) and Caupin and Balibar (2001). In these experiments a focused soundwave is used to produce a high-amplitude negative-pressure swing in a small region of the bulk liquid, far from any wall. The cavitation of bubbles is detected with an additional scattering of light from a laser beam. Classen et al. (1998) and Ghosh and Maris (2005) reported a series of experiments on the effect of electrons on the cavitation in liquid helium. It is known that an electron in helium forms a microscopic spherical cavity from which the helium atoms are completely excluded. In the superfluid liquids it is also expected that the cavitation of bubbles can be influenced by the possible presence of quantized vortices (e.g. Pettersen et al. 1994). For superfluids, there are also suggestions that the vortex ring can be generated after the collapse of ultrasound bubbles (e.g. Berloff and Barenghi 2005; Ghosh and Maris 2005). In contrast to the bubble cavitation experiments in liquid 4 He there have been similar experiments on the acoustic nucleation of the solid phase in the superfluid 4 He at pressures close to the melting pressure (Chavanne, Balibar & Caupin 2001; Ishiguro, Caupin & Balibar 2006). Here, unlike cavitation, positive-pressure swings of a focused soundwave are used to trigger the transition from the liquid phase to the solid phase with the formation of a crystal nucleus. A theoretical description of the acoustic crystallization in liquid helium was attempted by Ben Amar, Brenner & Rice (2003). The model describing growth dynamics of a spherical solid nucleus represents
264 H. Abe, M. Morikawa, T. Ueda, R. Nomura, Y. Okuda, and S. N. Burmistrov an extension of the Rayleigh–Plesset approach for an oscillating gaseous bubble in a liquid. Crystal nucleation dynamics was visualized by a high-speed camera in Abe et al. (2005). The main efforts of all previous cavitation studies were aimed at confirming the stochastic nature of the cavitation process in liquid helium and determining the behaviour of the cavitation threshold as a function of the temperature. The clear physical idea for the experiment was to discover the cross-over from the thermally activated cavitation to the quantum nucleation of bubbles, predicted first by Lifshitz & Kagan (1972). Accordingly, only the inception of bubbles or threshold of cavitation was measured in those experiments, and no datum is available on the dynamics of bubbles at the overcritical stage in which the bubbles reach a detectable size. In the present paper we address this lack and report our visual observations in detail on the acoustic nucleation of bubbles and their clear dynamics in normal 4 He, superfluid 4 He and superfluid 3 He–4 He mixtures. The bubbles are generated by an acoustic wave pulse on the surface of a piezoelectric transducer immersed in normal liquid 4 He, superfluid 4 He, the saturated d phase or the superfluid 3 He–4 He mixture with an unsaturated 3 He concentration of 25 %. The acoustic cavitation in a helium isotopic liquid mixture is studied for the first time. To make visualization more distinct, the bubble behaviour during the collapse process in the mixture was observed with the aid of the shadowgraph method. Direct visualization makes it possible to distinguish qualitatively different behaviours of bubbles in these quantum liquids. In figure 1 the circles represent the point at which the measurements were made in mixture. A few preliminary results on the bubble dynamics were reported by Abe et al . (2006, 2007, 2008). In conclusion, we note that similar visualization experiments have recently been performed by Katagiri et al. (2007) in order to investigate the boiling kinetics in normal fluid 3 He. Instead of an acoustic wave pulse a heated copper surface was used for nucleating 3 He gas bubbles. The shape of the bubble on the copper surface was spheroid, and the average size increased as the temperature was lowered. 2. Experiment The experiment was performed in a 3 He–4 He dilution refrigerator with a pair of optical windows (Nomura et al. 2003). The entire set-up is schematically shown in figure 2. The refrigerator, the full length of which was about 2 m, was surrounded by a vacuum can, a helium bath and a nitrogen bath. The inside of the cell could be cooled to about 100 mK with the aid of the heat effect of 3 He dilution. We could observe the inside of the cell through several infrared filters and panes of infrared absorption glass. The effective volume of the cell for the bulk liquid was about 15 cm3 . The cell had two identical ultrasound transducers placed about 10 mm apart from the top and the bottom of the cell; these had a longitudinal mode and were made of LiNbO3 . Their effective diameters with the coaxial electrodes were about 4 mm, and the resonance frequency was 9.3 MHz. The temperature in the cell was measured with a commercial RuO2 thermometer calibrated down to 50 mK, and the pressure was monitored by a capacitive pressure gauge. In the experiments, we used normal 4 He, superfluid 4 He and the superfluid 3 He– 4 He mixture with the 3 He concentration of 25 %. Although the 25 % mixture is one phase above about 500 mK, it separates into the c and d phases below about 500 mK as is shown in figure 1. The bubble was generated by a burst of an acoustic wave pulse for 1 ms into these liquids. As the intensity of the acoustic pulse increased
Bubble dynamics in superfluid 3 He–4 He mixture
5
265
5
1
4
4
3 7
2
8 6
9
Figure 2. Experimental set-up: (1) an area of 3 He–4 He dilution refrigerator; (2) a cell and a pair of sound transducers; (3) high-vacuum can; (4) liquid helium bath (T = 4.2 K); (5) liquid nitrogen bath (T = 77 K); (6) vacuum part; (7) high-speed camera; (8) halogen light; (9) optical windows.
with increase of the voltage applied to the lower transducer, a single bubble was generated on the surface of the transducer at a certain voltage threshold. This threshold depended on temperature and also on which liquids were used as reported in Abe et al. (2008). As the intensity of ultrasound increased above the threshold, the induced bubble was enlarged. The origins of the critical voltage are the energy barriers for the nucleation as studied extensively in the cavitation experiment, using focused sound. However, no prediction of the critical value is possible for the heterogeneous nucleation. In this report we focus on the behaviour of the bubble after it grows to the macroscopic size. The whole behaviour of the bubble evolution was recorded by a CCD camera at a rate of 1000 frames s−1 and 256 × 240 pixels for the resolution. The light source used was a halogen light. Some images were taken in more detail by a high-speed CCD camera at a rate of 3000 frames s−1 and 512 × 512 pixels for the resolution or 6000 frames s−1 and 128 × 512 pixels, using the shadowgraph method. For those, an He–Ne laser was used instead of the halogen lamp, and a flat convex lens was put between the laser and the cryostat. 3. Results 3.1. Bubble behaviour in normal liquid 4 He Figure 3 shows a series of images of the bubble observed when an acoustic wave pulse of 5 ms duration was emitted from the bottom transducer in normal liquid 4 He at about 3.2 K under saturated vapour pressure. Above the voltage of Va = 7.2 V, a relatively large bubble appeared on the centre of the transducer surrounded by many small bubbles. These small bubbles of the order of 70 μm rose from the entire surface
266 H. Abe, M. Morikawa, T. Ueda, R. Nomura, Y. Okuda, and S. N. Burmistrov 0
5
10
15
20
30
40
50
60
70
80
90
100
105
110
120
130
140
150
160
Figure 3. Images of a sound-induced bubble on the surface of the piezoelectric transducer immersed in normal liquid 4 He at 3.2 K. The applied voltage of the acoustic pulse is 7.2 V, and the pulse is emitted from 0 ms to 5 ms. Frame width is 10 mm.
0
1
2
3
4
5
6
7
9
11
13
15
17
19
21
Figure 4. Images of a sound-induced bubble on the surface of the piezoelectric transducer immersed in superfluid 4 He at 300 mK. The applied voltage of the acoustic pulse is 7.2 V, and the pulse is emitted from 0 ms to 5 ms. Frame width is 10 mm.
of the transducer. The velocity of the ascending bubble beside the transducer was about 0.1 m s−1 ; after detaching from the lower transducer, it did not collapse but rose, eventually reaching the upper transducer 10 mm above. Therefore, the behaviour looked not like a cavitation but like boiling by the warming of the transducer during sound pulse. The local overheating, facilitating the nucleation of the boiling state, is likely due to sufficiently low heat conduction in the normal state. 3.2. Bubble behaviour in pure superfluid 4 He By the same set-up as in figure 2, we observed a bubble in a superfluid state below the phase transition temperature 2.17 K. Figures 4 and 5 show a series of images of the bubble observed at different voltages at 300 mK. The duration of the pulse was 5 ms. The bubble was generated not from the entire surface of the transducer like the
Bubble dynamics in superfluid 3 He–4 He mixture 0
5
10
15
20
30
40
50
60
70
80
90
100
105
110
120
130
140
150
160
267
Figure 5. Images of a sound-induced bubble on the surface of the piezoelectric transducer immersed in superfluid 4 He at 300 mK. The applied voltage of the acoustic pulse is 8.2 V, and the pulse is emitted from 0 ms to 5 ms. Frame width is 13 mm.
small bubbles in normal 4 He but from the active part of the transducer. The bubble shape was highly irregular with an ill-defined surface. When the applied voltage was low, the bubble expanded on the transducer, shrank immediately and disappeared as shown in figure 4. When the voltage became larger, the bubble enlarged, re-expanded after the shrinkage and divided into smaller bubbles as shown in figure 5. The bubble motion in the superfluid 4 He is quite different from that in the normal liquid 4 He. One possible mechanism for the bubble generation on the entire surface of the transducer in normal 4 He is the heating of the transducer over a larger area by the voltage applied to it. The poor thermal conductivity in the normal fluid may result in the warming of a larger part of the transducer. Since thermal conductivity is very high in the superfluid 4 He, the heat will be effectively carried away by the superfluid, and the warming of the entire transducer will be avoided. Therefore, the bubble may appear only on the active part of the transducer by the acoustic wave pulse. The very irregular shape of the bubble is probably caused by a turbulent flow around it. In the pure superfluid liquid, the system can enter a highly nonlinear regime due to the small dissipation. 3.3. Bubble behaviour in the d phase Figure 6 shows a series of images for the bubble dynamics observed in the d phase at 400 mK under saturated vapour pressure. The saturated concentration of 3 He in the d phase is about 16 % at 400 mK. The images were taken by a high-speed CCD camera at a rate of 3000 frames s−1 , and the duration of the pulse was 1 ms. Set-up of the experiment was the same as those of the normal liquid and superfluid 4 He, and the mixture was condensed into the same cell in place of pure 4 He. When the voltage Va = 4.3 V was applied to the lower transducer, a single bubble was nucleated. The bubble continued to expand for several milliseconds after switching off the acoustic pulse and reached the maximum size at t = 8 ms. It then began shrinking at around t = 9 ms, detached from the transducer surface and collapsed at t = 14 ms. Shortly thereafter, it re-expanded at t = 15 ms and rose. During the rise the form was
268 H. Abe, M. Morikawa, T. Ueda, R. Nomura, Y. Okuda, and S. N. Burmistrov 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Figure 6. Images of a sound-induced bubble on the surface of the piezoelectric transducer immersed in the d phase with the 16 % 3 He concentration at 400 mK. The time interval is 1 ms. Frame width is 3 mm.
irregular, and the spherical shape was never recovered. As the initiating voltage was increased, the maximum bubble size became larger, and the bubble showed a series of expansions and collapses. These re-expansions decayed due to the dissipation of energy in the bubble environment. The dissipation mechanism may be associated with the viscosity of the d phase. A larger bubble shows more complex behaviour during the expansion and collapsing process. Figure 7 shows the dynamics of a bubble after its first expansion in the saturated d phase with the 3 He concentration of about 10 % for the applied voltage Va = 10.8 V at 200 mK. The images were recorded with a CCD camera at 1 000 frames s−1 and pulse duration of 1 ms. As is seen from the figure, between t = 23 ms and t = 25 ms the shape of the bubble bottom changed while shrinking. This deformation is likely to originate from the flow surrounding the bubble. As the bubble detached from the transducer surface, the upward jet flow penetrated it within t = 26–30 ms and then began to protrude from the bubble top at t = 31 ms. When the upward jet had completely pierced the bubble, the formation of a vortex ring was observed at t = 49 ms before it disappeared. We found that the jet velocity vj,1 , which pierced the inside of the bubble, differed from the velocity vj,2 after penetration; it was roughly estimated at vj,1 1.3 m s−1 and vj,2 0.26 m s−1 from figure 7. As compared with ordinary liquids, e.g. water, the roughly estimated jet velocity in the latter reached about 100 m s−1 (Tomita & Shima 1989). Thus the jet velocities observed in quantum liquids were approximately 102 –103 times lower. Even the re-expansion of a core of the vortex ring was observed in a more violent case. Figure 8 shows the collapsing process for an applied voltage of 6.1 V at 330 mK as observed by the shadowgraph method. The image was recorded with a high-speed CCD camera at a rate of 3000 frames/s and pulse duration of 1 ms. A He–Ne laser was used instead of the halogen lamp as a light source, and a flat-convex lens was put between the laser and the cryostat. As seen in the figure, the second expansion took place after the vortex ring generation at t = 34 ms, and the deformed bubble ascended. The upward jet pierced through the bubble and a hollow vortex was clearly visible at t = 41–46 ms.
Bubble dynamics in superfluid 3 He–4 He mixture 23 ms
24 ms
25 ms
26 ms
27 ms
28 ms
29 ms
30 ms
31 ms
32 ms
33 ms
34 ms
35 ms
36 ms
37 ms
38 ms
39 ms
40 ms
41 ms
42 ms
43 ms
44 ms
45 ms
46 ms
47 ms
48 ms
49 ms
50 ms
269
Transducer
1 cm
Figure 7. Jet formation during the collapsing process of a bubble in the d phase at 200 mK. The time interval shown is 23 ms to 50 ms after the incidence of the 1 ms pulse into the liquid. The applied voltage is 10.8 V. The images were taken with a halogen lamp as a light source at 1 000 frames s−1 .
3.4. Bubble behaviour in the unsaturated 3 He–4 He mixture with the 3 He concentration of 25 % Figure 9 shows a series of images of a bubble in a 25 % 3 He–4 He mixture at 600 mK under saturated vapour pressure. These were taken by a high-speed CCD camera at the rate of 6000 frames s−1 using the shadowgraph method; pulse duration was 1 ms. The liquid jet formation was clearly observed at t = 11–17 ms but did not penetrate
270 H. Abe, M. Morikawa, T. Ueda, R. Nomura, Y. Okuda, and S. N. Burmistrov Phase-seperated interface
25 ms
26 ms
27 ms
28 ms
29 ms
30 ms
31 ms
32 ms
33 ms
33.67 ms
34 ms
34.33 ms
35 ms
36 ms
37 ms
38 ms
39 ms
40 ms
41 ms
42 ms
43 ms
44 ms
45 ms
46 ms
Transducer
1 cm
Figure 8. Jet formation and the re-expansion process of a bubble in the d phase at 330 mK. The time interval shown is 25 ms to 46 ms after the incidence of the 1 ms pulse into the liquid. The applied voltage is 6.1 V. Images were taken by the shadowgraph method using a He–Ne laser as a light source at 3000 frames s−1 .
the bubble’s upper surface as in the d phases of figures 7 and 8. The jet velocity was roughly estimated at vj,1 0.14 m s−1 from figure 9, about 10 times lower than that in figure 7.
Bubble dynamics in superfluid 3 He–4 He mixture 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
271
Figure 9. Images of a sound-induced bubble on the surface of the piezoelectric transducer immersed in superfluid 3 He–4 He mixtures with the 3 He concentration of 25 % at 600 mK. The time interval is 1 ms. Frame width is 0.3 mm.
4. Discussion Benjamin and Ellis (1966) considered how a liquid jet arises from the bubbles when deviations from spherical symmetry are enforced by proximity to a solid boundary. In particular, they predicted that the jet flow creates a hollow vortex ring if it penetrates axisymmetrically through the surface of a spherical bubble. As we observed, the bubble is in a special situation when the expansion and contraction occur in contact with the wall. However, their speculation is useful to consider the generation mechanism of the jet. Let F denote the external force acting on a bubble. Then the Kelvin impulse I , leading to the liquid jet, can be introduced as follows: dI /dt = F.
(4.1)
If the bubble moves along a specific axis and if x0 (t) denotes its axial coordinate, the impulse I can be expressed as I = M x˙0 + J.
(4.2)
272 H. Abe, M. Morikawa, T. Ueda, R. Nomura, Y. Okuda, and S. N. Burmistrov Here the coefficient M can be interpreted as the induced or associated mass of a fluid, which a rigid body entrains during its motion. The impulse J is completely determined by the deformation rate of the bubble shape and is independent of the translational mode. In general, in a superfluid described in terms of two-fluid hydrodynamics the consistent determination of the associated mass is more complicated than in the case of ordinary ideal liquid. The point is that both the superfluid and normal density components contribute to the tensor of the associated mass. In a case in which the motion of the superfluid component has much in common with an ideal liquid, the motion of the normal component corresponds more to a viscous fluid and can be accompanied by dissipative effects. For qualitative consideration, we will neglect the effect of the normal density component, treating liquid 3 He–4 He mixture as a totally ideal liquid. In a recent paper Klaseboer et al. (2005) investigated experimentally and numerically the collapse process of an underwater explosion bubble near a rigid plate. In particular, to simulate the formation of a vortex ring at the final stages of the collapse process and using the boundary-element method (BEM) and the finite-element method (FEM), they put a ring inside the bubble after a liquid jet had pierced it in order to take into account the fluid domain no longer singly connected. As the velocity potential in an incompressible fluid is labelled with φ, the velocity vector is expressed by u = ∇φ. As the jet flow penetrates through the upper surface of a bubble from its bottom and the vortex ring is generated, a discontinuity in the potential should be observed. The discontinuity must be equal to the jump in the potential between the nodes of the jet tip and impact zones, K and L, respectively. The magnitude of the discontinuity is given by Γ = φK − φL ,
(4.3)
where Γ is a parameter determining the circulation of the flow. Eventually, this discontinuity in the potential will govern the generation of the vortex ring. Let us consider this aspect from the viewpoint of the presence of superfluidity in a liquid. The superfluid velocity is proportional to the phase gradient of wave function ψ = |ψ|eiϕ and defined by (Donnelly 1991) us =
~ ∇ϕ. m4
Here m4 and ~ stand for the 4 He mass and Planck constant, respectively. Thus the circulation is given by ~ ~ ϕ. (4.4) ∇ϕ · ds = Γ = us · ds = m4 m4 Since ϕ is the phase change of the wave function along the closed path, it must be an integral multiple of 2π. Thus Γ =
~ ~ ϕ = 2πn = κn , m4 m4
n = 0, 1, 2, . . . ,
(4.5)
Bubble dynamics in superfluid 3 He–4 He mixture
273
where κ = 2π~/m4 ∼ 10−7 m2 s−1 is a quantum of circulation. Therefore, in superfluid liquid (4.3) reduces to κ κ Γ = (ϕK − ϕL ) = ϕK−L = κnK−L , nK−L = 0, 1, 2, . . . . (4.6) 2π 2π Although it is difficult to obtain the accurate value of Γ of the observed vortex rings, let us estimate it from the translational velocity of the vortex ring v. The translation velocity is written as Γ 8R log , (4.7) 4πR a where R and a are radii of the vortex ring and the vortex core (Donnelly 1991). The precise formula of v depends on whether the core is a solid rotating core or a hollow core. We do not know details of the core and adapt (4.7) for the rough estimation of Γ . From figure 8 at around 34 ms, we obtained R 1.4 mm, a 0.1 mm and v 0.22 ± 0.02 m s−1 . Since only the side view is available, we assumed the crosssection of the core to be circular. If the core is deformed due to Bernoulli’s effect and so on, this value is not very accurate but is adequate for the estimation described next, at least in order of magnitude. Using these values we have Γ 2 × 10−3 m2 s−1 and nK−L 2 × 104 . On the other hand, we can also assume Γ to roughly be Γ 2πRvj . From figure 7 we obtained the jet velocity vj,2 0.26 m s−1 , and then a similar value Γ 2×10−3 m2 s−1 was obtained. This supports the picture that a vortex ring was actually formed in this situation. The observed ring seems to have a circulation four orders of magnitude greater than the quantum circulation. Usually quantized vortices are thermodynamically stable only for the lowest value of possible circulation, i.e. for n = 1. Vortices with n > 1 are unstable from an energetic point of view and eventually have to dissociate into several vortices with a single quantum circulation. The observed macroscopic vortex does not necessarily consist of millions of quantized vortices as long as it has a macroscopic hollow core, since it is no longer in the singly connected region. In the final stage of figure 7, however, the hollow core vortex disappears, and many aligned quantized vortex rings are presumably left immediately after the disappearance. They will soon form a tangle of quantized vortices, and eventually the flow will damp out. The decay process of the macroscopic vortex into the quantized vortices is not understood at this moment and is an intriguing problem which will bridge our knowledge between the classical and quantum fluid dynamics. Macroscopic behaviour of superfluid flow is of increasing interest, since the superfluid turbulence is shown to follow Kolmogorov’s energy spectrum on a larger scale than the quantized vortex spacing, which is the most important statistical law in classical fluid dynamics (Araki, Tsubota & Nemirovskii, 2002; Kobayashi & Tsubota 2005). Reynolds number is an important parameter in investigating the character of a flow and is given by LU ρLU = , (4.8) Re = μ ν where ρ, L, U , μ and ν are the density, characteristic length, characteristic velocity, viscosity and kinematic viscosity, respectively. If L is the radius of a bubble, U is growth or upward velocity of the bubble; ρ is density of the liquid; and μ is viscosity of the liquid, then from (4.8), the Reynolds number can be estimated to be of the order of 6 × 104 in the d phase at 330 mK in figure 8. The Reynolds number for the water v
274 H. Abe, M. Morikawa, T. Ueda, R. Nomura, Y. Okuda, and S. N. Burmistrov bubble in Tomita & Shima (1989) is estimated to be of the same order, Re = 7 × 104 , using the parameters L = 0.007 cm, U = 100 × 102 cm s−1 and ν 1.00 × 10−2 cm2 s−1 . 5. Conclusion Dynamics of sound-induced bubbles in a pure normal liquid 4 He, a pure superfluid 4 He and a superfluid 3 He–4 He liquid mixtures have been systematically studied for the first time by direct visualization, and it has been learned that sound-induced bubbles in these quantum fluids behave quite differently. In the normal liquid 4 He, a primary bubble surrounded with small bubbles rises in the liquid and does not disappear; the behaviour resembles boiling as a result of local short-term overheating during sound emission. In the superfluid 4 He, a single bubble is nucleated, but its shape is highly irregular. The system seems to enter a turbulent regime due to the extremely low viscosity and, consequently, large Reynolds number. In the 3 He–4 He mixtures, a single spherical bubble is generated by an acoustic wave pulse, which undergoes an expansion and contraction process and also shows peculiar phenomena such as bubble jetting and vortex ring formation. The vortex ring formation looks similar to that described in water by Tomita & Shima (1989) and Klaseboer et al. (2005), although the time scale of the dynamics is several orders of magnitude longer than that in classical fluids. Reynolds numbers in both cases are of the same order of magnitude, and this is presumably the reason for the similarity in behaviour in the superfluid 3 He–4 He and water. We believe that further experimental studies of the bubble dynamics in a phase-separated 3 He–4 He liquid mixture might discover a structure much richer than that described here. In addition, our experiment is also useful from the viewpoint of varying physical scales in the bubble dynamics. This study was supported by the Fumi Yamamura Memorial Foundation for Female Natural Scientists, in part by the Ground-based Research Announcement for Space Utilization promoted by the Japan Space Forum and by a Grant-in-Aid for Scientific Research on Priority Areas (grant no. 17071004) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. REFERENCES Abe, H., Ogasawara, F., Saitoh, Y., Tatara, T., Kimura, S., Nomura, R. & Okuda, Y. 2005 Nucleation of crystals and superfluid droplets in 4 He induced by acoustic waves Phys. Rev.B 71, 214506. Abe, H., Ogasawara, F., Saitoh, Y., Ueda, T., Nomura, R., & Okuda, Y. 2006 A single bubble nucleated by acoustic waves in 3 He–4 He mixtures. AIP Conf. Proc. 850, 145–151. Abe, H., Saitoh, Y., Ueda, T., nomura, R., Okuda, Y. & Burmistrov, S. N. 2008 Bubble nucleation in a superfluid 3 He–4 He mixture induced by acoustic wave. Fizika Nizkikh Temp. 34, 391–394. Abe, H., Ueda, T., Saitoh, Y., Nomura, R., Okuda, Y., & Burmistrov, S. N. 2007 Visual observation of a sound-induced bubble in liquid 3 He–4 He mixtures. J. Low. Temp. Phys. 148, 133–138. Araki, T., Tsubota, M. & Nemirovskii, K. 2002 Energy spectrum of superfluid turbulence with no normal-fluid component Phys. Rev. Lett. 89, 145301. Balibar, S., Caupin, F., Lambare, H., Roche, P. & Maris, H. J. 1998 Quantum cavitation: a comparison between superfluid helium 4 and normal liquid helium 3. J. Low Temp. Phys. 113, 459–471. Ben Amar, M., Brenner, M. P., & Rice, J. R. 2003 Crystallisation par onde acoustique: le cas de l’helium. Comptes Rendus Mecanique 331, 601–607. Benjamin, T. B. & Ellis, A. T. 1966 The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Phil. Trans. R. Soc. Lond. A 260, 221–240.
Bubble dynamics in superfluid 3 He–4 He mixture
275
Berloff, N. G., Barenghi, C. F. 2005 Evolution of vortex ring after the collapse of ultrasound bubbles in superfluids. J. Low Temp. Phys. 138, 481. Brennen, C. E. 1995 Cavitation and Bubble Dynamics. Oxford University Press. Caupin, F. & Balibar, S., 2001 Cavitation pressure in liquid helium. Phys. Rev. B 64, 064507. Chavanne, X., Balibar, S. & Caupin, F. 2001 Acoustic crystallization and heterogeneous nucleation. Phys. Rev. Lett. 86, 5506–5509. Classen, J., Su, C.-K., Mohazzab, M., & Maris, H. J. 1998 Electrons and cavitation in liquid helium. Phys. Rev. B 57, 3000–3010. Donnelly, R. J. 1991 Quantized vortices in helium II. In Cambridge Studies in Low Temperature Physics (Ed. A. M. Goldman, P. V. E. McClintock & M. Springford). pp. 50–52. Cambridge University Press. Ghosh, A. & Maris, H. J. 2005 Observation of a new type of electron bubble in superfluid helium. Phys. Rev. Lett. 95, 265301. Ishiguro, R., Caupin, F. & Balibar, S. 2006 Homogeneous nucleation of crystals by acoustic waves. Europhys. Lett. 75, 91–97. Katagiri, M., Izumi, S., Hori, J., Fujii, Y. & Hatanaka, K. 2007 Visualization of bubble nucleation in boiling 3 He. J. Low Temp. Phys. 148, 127–132. Klaseboer, E., Huang, K. C., Wang, C., Wang, C. W., Khoo, B. C., Boyce, P., Debono, S., Chalier, H. 2005 Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure. J. Fluid Mech. 537, 387–413. Kobayashi, M. & Tsubota, M. 2005 Kolmogorov spectrum of quantum turbulence J. Phys. Soc. Jpn. 74, 3248. Lifshitz I. M. & Kagan Yu. 1972 Quantum kinetics of phase transitions at the temperatures close to absolute zero. Sov. Phys. JETP 35, 206. Nomura, R., Suzuki, Y., Kimura, S. & Okuda, Y. 2003 Interface motion and nucleation of solid helium-4 induced by acoustic waves. Phys. Rev. Lett. 90, 075301. Pettersen, M. S., Balibar, S. & Maris, H. J. 1994 Experimental investigation of cavitation in superfluid 4 He. Phys. Rev. B 49, 12062–12070. Plesset, M. S. 1949 The dynamics of cavitation bubbles. Phil. Mag. 16, 277–282. Rayleigh, L. 1917 The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Phil. Mag. 34, 94–98. Tomita, Y. & Shima, A. 1989 High-speed photograph observations of laser-induced cavitation bubbles in water. ACUSTICA 71, 161–171.
