Adiabatic flow of He II

Physica B 154 (1988) 116-124 North-Holland, Amsterdam

ADIABATIC FLOW OF HE II II MOTION OF NORMAL FLUID COMPONENT AND VORTICES

Mineo O K U Y A M A , Toshimi SATOH* and Takeo S A T O H Department of Physics, Faculty of Science, Tohoku University, Sendai 980, Japan Received 15 April 1988 Revised manuscript received 24 August 1988

In order to confirm our thermohydrodynamic description of adiabatic flow of He II with the new-type mutual friction force, NMR technique was applied to obtain the average flow velocity of 3He atoms, v3, added to the adiabatic flow system. It was found that the observed net mass flow velocity dependence of v3 shows a close similarity with that of the normal fluid component velocity v, calculated with the new-type mutual friction force. Especially the existence of the characteristic velocity vc2 was confirmed and a good agreement was found in its value. This supports the validity of our thermohydrodynamic description. The results are discussed with the vortex pinning and depinning concepts.

I. I n t r o d u c t i o n

In the preceding paper [1] denoted as paper I hereafter, we have presented a thermohydrodynamic description of adiabatic flow of He II, in which a new-type mutual friction force between the superfluid component and the normal fluid component has been introduced in order to explain the observed temperature distribution along the flow path capillary. In the adiabatic flow, only the superfluid component of He II is forced to flow via a superleak into the system of a chamber and a capillary connected in series. For the details of the experimental arrangement we refer to Paper I. Our new-type mutual friction force is expressed as Fs n

2 BPsPnK 2p L°" (vS - v . ) ,

- 3

(1)

with

Fs n oc (v s -- Vn) 3

Lo = L o ( v ) -- Lo(os) .

(2)

Here Ps, P. and vs, v, are the density and the * Present address: Sumitomo Heavy Tanashi-shi, Tokyo 188, Japan.

transport velocity of the superfluid component and the normal fluid component, respectively. B is the Hall-Vinen coefficient [2] and K the quantum of circulation. L 0 is the steady state vortex line density and our finding was that it is not a function of v s - v. but a function of the net mass flow velocity, o, through the capillary, which is almost equal to v s. As shown in fig. 1, which is the same figure as fig. 18 in Paper I, Lo(v ) was determined up to about v = 100 cm/s which is an order of magnitude higher than usual He II flow experiments, and a very remarkable v-dependence of L 0 can be seen. Furthermore, it is also noted that the fact of the constancy of Lo(v ) along the capillary with a fixed v means that Lo(v ) is temperature insensitive. Since the thermal counterflow experiment by Gorter and Mellink [3], the mutual friction force of Gorter-Mellink type

Industries,

Ltd.,

(3)

has been widely applied to the He II flow phenomena including not only the thermal counterflow but also other type of flows. Although there have been several proposals and considerations concerning a modification of the G o r t e r -

0921-4526/88/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

M. Okuyama et al. / Adiabatic flow of He H, part H

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Mellink type mutual friction force [4], our experiments [1] gave the first clear results, where the new-type mutual friction force of eq. (1) with eq. (2) works fairly well over a wide velocity range up to v---100 cm/s and over a wide temperature range from 1.3 K down to 0.8 K. Therefore it is worthwhile to confirm the validity of our thermohydrodynamic description with eqs. (1) and (2) from other experiments. The present paper gives such a confirmation. Our thermohydrodynamic description of adiabatic flow of H e II is summarized as follows. In a stationary state, the two-fluid hydrodynamic equation for the normal fluid component is written as dT

d---x =

p•

dp

1

ppsS dx + ~ S (F~. - Fn) ,

(4)

where x is taken along the capillary, p is the H e II density, S the specific entropy per unit mass, T the temperature, p the pressure and F, the frictional force for the normal fluid component. From the energy conservation law which takes into account the energy dissipation in the capillary, we approximately have, in the present case,

T

(pA--p)v+pv v• =

f TA

pST

SdT+K

dr dx (5)

where the suffix A denotes the quantities at the chamber and K the thermal kinetic coefficient. The superfluid component velocity v, which appears in F,. is related to o and o n by p v = PsVs Jr P n P n .

(5)

Solving eqs. (4) and (5) simultaneously for d T/ dx and o,, we can obtain the temperature and v, along the capillary at a given mass flow velocity v, stepwise starting from the position of the chamber, x = 0. In our calculation given in Paper I we took a 0.5 cm step for the capillary of 50 cm long. In the procedure, the vortex line density Lo(v ) is a fitting parameter determined to reproduce the experimentally obtained temperature distribution. In Paper I, we mainly discussed about the temperature distribution and the parameter Lo(v ). The present paper concerns with the normal fluid velocity v, thus calculated.

M. Okuyama et al. / Adiabatic flow of He I1, part 11

118

2. Normal fluid component velocity v.

Examples of the calculated normal fluid velocity along the capillary ( C u - N i capillary of i.d. 0.2 mm and length 50 cm) are given in figs. 16am of Paper I at various net mass flow velocities. Referring to these figures it is seen that in the low velocity region v, increases monotonically with x and in the high velocity region v, has a broad or a flat maximum which moves from the region near the entrance of the capillary to the middle of the capillary. In fig. 2, we plotted v, as a function of v at the various positions of the capillary. The boundaries of the regions designated from the behavior of L0(v), as shown in fig. 1, are also shown in the figure. The characteristic feature seen from the figure is that until v reaches around 25 cm/s, which we called vc2 in Paper I, v n remains very small compared with v and when v exceeds vc2, v, increases very rapidly. It is noted that Vc2 locates just on the boundary between the regions (I)" and (I)" and the rapid

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As noted in Paper I, such a v~-dependence of L]/2 0 was obtained in the pure superflow experiment [5] and the coefficient of the proportionality is comparable with that of eq. (7). However, it is very much desirable if we could check the v-dependence of v n directly. Unfortunately, it is almost impossible to measure o, directly in pure 4He. Therefore we took a way to apply a N M R technique to monitor the motion of 3He atoms added to the system [6]. 3He impurities in the mixture play a role of a temperature independent normal fluid component. Further, the interaction between a 3He atom and a vortex ring is known to be of the same order of magnitude as that between a roton and a vortex ring [7]. Therefore it seems reasonable to consider that in adiabatic flow, where the normal fluid component moves due to the action from the superfluid component via vortex lines, the motion of 3He atoms is very similar to that of rotons. In the following sections, we will describe such an experimer~t and give the results and discussions.

o o oo

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3. Motion of 3He atoms in the adiabatic flow of He II

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L 0 (v~) = 1.54 × I0: [ c m - : s]v~.

o

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~ o o o o

increase of v. occurs in the region (I)". Referring to the v, in the middle region of the capillary, it is seen that in region (II) vn is almost parallel to v and in region (III) v n gradually reaches to v. As the calculated temperature distribution along the capillary fits very well the experimental points [1], the normal fluid component velocity thus obtained is expected to represent the real feature of the v (or v~)-dependence of v,. Such an expectation also arises from the following fact. In region (I)', the flow is considered as pure superflow because the calculated v n is almost zero. The v~-dependence of L 0 in this region is expressed as

o

l

I

40

~

I

1

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I

80

I

[

lO0

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t20

V (c=/s)

Fig. 2. Calculated normal fluid component velocity v. as a function of the net mass flow velocity o at various positions along the capillary of i.d. 0.2 mm.

For the apparatus we refer to figs. 1 and 2 of Paper I. In the mixture experiment, we first condensed 3He through the outlet 01. After closing 01 , the outlet O z connected to the super-

M. Okuyama et al. / Adiabatic flow of He 11, part H

leak S 2 was used for the flow of 4He. So the added 3He atoms did not escape from the flow system. In the present experiment, the 3He content was about 6%. For the flow path capillary, we used the Cu-Ni capillaries of i.d. 0.1 mm and 0.2mm, respectively. Both capillaries have a length of 500 ram. It is noted that the experiment described in Paper I was carried out with the Cu-Ni capillary of i.d. 0.2 mm and 500 mm long. The chamber was made from Stycast 1266, on which a N M R coil was wound. The RF frequency used was 10 MHz. The nuclear spin relaxation time of 3He without flow was measured about 40 s in the temperature range investigated. After setting the flow-control needle valve (see fig. 2 of Paper I) to have a desired net mass flow velocity of 4He in the capillary, v4, the flow was started. The chamber temperature T A gradually changed and settled down to a stationary value. During the time of the experiment, a 90°-180 ° pulse sequence was applied every 3 minutes and the spin-echo height was measured. Alternatively, the method to monitor the time dependence of the CW NMR intensity was also applied. The established stationary temperatures of the chamber, T A, are plotted as a function of v 4 in figs. 3a and 4a for the capillaries of i.d. 0.1 mm and 0.2 mm, respectively. A remarkable feature is that T A shows a sharp maximum at /34 = 2 5 c m / s in the 0 . 1 m m capillary and at v4--~ 12 cm/s for the 0.2 mm capillary. We call these characteristic velocities Vc~ in the following. Such behavior is the same as observed by us previously [8]. In the case of pure 4He experiment, we observed a broad peak only when the bath temperature was lower than about 0 . 8 K [8]. The reason for this difference between the mixture and the pure 4He experiments is not as yet completely clear. It is noted that the superfluid critical velocities, vsc, in the capillaries presently used are much smaller than vcl. They are about 3 c m / s and 0 . 8 c m / s for the capillaries of i.d. 0.1 mm and 0.2mm, respectively as determined from the pressure-difference measurements [1, 6]. An example of the time dependence of the spin-echo height, M(t), is shown in fig. 5. In the figure, the correction due to the temperature dependence of the magnetic susceptibility of 3He

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was already made. It is seen that M(t) is approximately expressed with a single exponential function of time. Generally it is expressed as

M(t) - M~ = [M(0) - M~] e-~'

(8)

where a is a function of/.24, =

~(V4).

(9)

After measuring M(t) for hours at each v4, we get a stationary value of M(t) which we call M~. In figs. 3b and 4b, M~ is plotted as a function of v4. Comparing figs. 3a and 4a with figs. 3b and 4b, respectively, we can see that until v4 reaches v~ the 3He content in the chamber does not show any appreciable change. When v 4 exceeds V¢l, the spin-echo height decreases with time as expressed with eq. (8). For higher v4, except for the region near Vcl, M~ is always zero within our experimental accuracy.

M. Okuyama et al. / Adiabatic flow of He H, part H

120

.,. I.t,

where ~ is the chamber volume and A Ca is the cross sectional area of the capillary. In figs. 6 and 7 we plotted v 3 thus obtained as a function of v 4 for the capillaries of i.d. 0 . 1 m m and 0.2mm, respectively. From the figures it is seen that until v 4 reaches vcl, v 3 is almost zero and when v 4 exceeds v¢1, v 3 gradually increases with v4 but still remains very small compared with v4. When v 4 exceeds the value about 50 cm/s in the case of the 0 . 1 m m capillary and about 2 5 c m / s in the case of the 0.2 mm capillary, v3 increases very rapidly with v 4. So these velocities are considered to correspond to the velocity designated as G2 in the pure 4He case shown in fi~. 2. We recognize a close similarity of the net He mass flow velocity dependence between v 3 in figs. 6 and 7 and v, in the middle region of the capillary shown in fig. 2, which may be considered to represent the average behavior of v,. It is remarkable that for the capillary of i.d. 0.2 mm, the values of vc2 in figs. 2 and 7 are in good agreement. The apparent difference between figs. 6 and 7 is the scale of velocity values, that

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vc2(i.d. 0.1 mm) ---2 × v~2 (i.d. 0.2 m m ) .

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Assuming a homogeneous 3He density in the chamber and the capillary, we obtain from eq. (8), an equation for the average flow velocity of 3He in the capillary, v3, as

(11)

This suggests that we may have an approximate relation vc2" d = 0 . 5 [ c m 2 s - l ] .

(12)

tO0 V3

= (~O/Aca)a ,

(10)

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Fig. 6. Velocity of 3He atoms in the capillary of i.d. 0.1 mm as a function of the net mass flow velocity of 4He, 04.

M. Okuyama et al. / Adiabatic flow of He H, part H

measurements is shown in fig. 8. As can be expected, the 3He atoms do not return back to the chamber until v4 is decreased down to the value around Vcl and then they start coming back to the chamber. The very interesting feature seen from fig. 8 is that they return back intermittently. The NMR intensity reaches the stationary value after repeating a sudden recovery followed by a gradual decreasing as a function of time.

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0

10

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Fig. 7. Velocityof 3He atoms in the capillaryof i.d. 0.2 mm as a functionof the net mass flowvelocityof 4He, u4. The scaling factor of about 2 appearing in eq. (11) is also noticed for u¢i as seen from figs. 3 and 4, so we may also approximately have the relation u¢l • d-- 0.25 [cm 2 s - l ] .

(13)

In the experiments mentioned hitherto, we always started the flow of 4He from the state of 04 = 0 with the flow-control needle valve set to have a desired net mass flow velocity 04. This means that the 3He atoms, which had been driven out from the chamber, returned back to the chamber at the beginning of the flow. In order to see the returning behavior of the 3He atoms, we measured the NMR intensity as a function of time with decreasing the net mass flow velocity step by step. An example of such

I0 min. V4-22.3 tc./,i

28.5

121

252

Fig. 8. The recovering behavior of the NMR intensity in the chamber as a function of time during which the net mass flow velocity of 4He, v 4 is decreased step by step, the positions of which are indicated with arrows (capillary of i.d. 0.I ram). Spin-echo height is measured every 3 minutes, which produces the apparent stepwise behavior in the figure.

4. Discussion

The experimental fact that we found the existence of the characteristic velocity Vc2 in the 04 dependence of v3 and the fact that thus obtained values of vc2 for the i.d. 0.2 mm capillary agrees well with the calculated value of vc2 support the reliability of our thermohydrodynamic description with the new-type mutual friction force. Our study described in paper I consists of the measurements of various physical quantities and the numerical fitting procedure with the simultaneous eqs. (4) and (5). So, in the Appendix of Paper I we compared the magnitude of each term which appears on the right-hand side of these equations. It was noted there that for the temperature gradient d T / d x , the term of the mutual friction force makes the most important contribution throughout the capillary in the entire temperature and velocity region investigated. Here, we note, referring to figs. 20a-e in Paper I, that for the normal fluid velocity v. all the three terms on the right-hand side of eq. (5) contribute almost equally. Therefore, the result that the characteristic behavior of v. shown in fig. 2 is verified by the present 3He-impurity experiment may be considered to support the validity of our experimental procedure described in Paper I to measure various physical quantities concerned with. This remark is distinctively important for the thermal kinetic coefficient data, because it does not have any predecessor's data to be compared with and it is very difficult to justify completely the experimental procedure by itslef as discussed in Paper I. As noted in Paper I, our finding eq. (2), means that the interaction between the super-

122

M. Okuyama et al. / Adiabatic flow o f He 11, part H

fluid component and the capillary wall is the most important thing to determine the steady state vortex line density. Such an interaction may come through the wall-pinned vortices. The fact that Lo(vs) is almost constant through the capillary at fixed v (or vs) while the temperature varies along the capillary suggests that the interaction is temperature insensitive at least in the temperature region investigated. In adiabatic flow, the normal fluid component flows due to the action from the superfluid component. This action, the mutual friction force, is transferred through the vortices. That is, the normal fluid component experiences drag force from the flowing vortices. So, the normal fluid component velocities v n a n d / o r v 3 should be related to the average drift velocity v L and the density nL, of the flowing vortices. Therefore, the characteristic behavior of the observed v 3 a n d / o r the calculated u n is considered to reflect that of v L and n L. In region (I)' where the superfluid velocity v s already exceeds the superfluid critical velocity vsc, the vortex line L 0 increases monotonically with increasing v s. In this region the observed average velocity v 3 a n d / o r the calculated v n are almost zero. Therefore it seems reasonable to consider that v L - 0 in the region (I)'. So we should look for the mechanism of increasing the vortex density and that of preventing the vortices from flowing through the capillary. For the superfluid critical velocity in the extrinsic regime, Glaberson and Donnelly [9] proposed a so called vortex-mill model with a pinned vortex. Because the observed values of v~ for the present two capillaries (i.d. 0.1 mm and 0.2 ram, respectively) cannot be simply scaled with their channel size, the vortex-mill model may not be adequate in its simple form. Schwartz [I0] combined the G l a b e r s o n - D o n n e l l y concept with the vortex nucleation mechanisms from a wall imperfection of a small scale compared with the channel size. According to this model the superfluid critical velocity is given by

v~c = 4~-A In

(80

,

(14)

where A is the scale of the size of the imperfection. a o is considered to have the order of the vortex core size and we tentatively take a 0 = 1.3 x 10 8 cm in the following. K is the unit of quantum circulation, h K = - - = 0.997 × 10 -3 [cm 2 s -1] , m4

(15)

where h is the Planck constant and m 4 the mass of the 4He atom. With eq. (14) and the observed values of Vsc the order of the magnitude of A is estimated as A(i.d. 0.1 m m ) - 0 . 3

[cm],

(16)

A(i.d. 0.2 ram) = 1.3 × 10 -3 [cm] .

(17)

x 10 -3

and

If v > Vsc, the nucleated vortex loop will be on the down-hill side of the free energy curve and will continue to grow across the channel completing the phase slip process. Some of them may happen to cross with each other producing vortex rings. In order to increase the steady state vortex line density, which is prevented from flowing through the capillary, one needs some trapping mechanism. Schwartz [11] discussed a pinning site in the form of a local hemispherical protrusion. If the ends of the growing vortex happens to get close enough to the pinning site, it will be captured there resulting in a wall-pinned vortex. Further, he considered various types of crossing counters between a vortex ring and a vortex line. It was shown that almost in all cases the ring is reconnected to the line. So if the line is a wall-pinned vortex, it acts as a trapping center for the flowing ring vortices. As shown in section 3, 3He atoms in the chamber start flowing out when v 4 exceeds vc~. This means that Vcl is the characteristic velocity of the vortices to start running away from the pinning center because we are assuming a homogeneous 3He density in the chamber and the capillary. The intermittent behavior described in section 3 with fig. 8 seems to be reasonable for the beginning of the instability of

M. Okuyama et al. / Adiabatic flow of He H, part H

pinned vortices. Although the existence of the characteristic velocity vc~ was clearly observed only in the 3He-impurity experiments, the value of v~ of the i.d. 0.2 mm capillary locates around the boundary between (I)' and (I)". Therefore the boundary seems to correspond to the point where the pinned vortices start flowing through the capillary. It is noted that in the region (I)", the average normal fluid velocity vn and/or u3 are very small compared with v 4. This may mean that only a very small fraction of the pinned vortices becomes unstable in this region. Schwartz [11] also argued the instability of a pinned vortex interacting with a hemispherical protrusion on the capillary wall. It is important to note that his discussions of the pinning-depinning phenomenon are entirely within the context of the ideal fluid hydrodynamics. He obtained the expression of the depinning critical velocity of a single vortex line as v~.pin = 2~-d In

,

(18)

where d is the channel diameter and b the radius of the hemispherical protrusion. The expression of eq. (18) seems to explain the experimentally observed channel size dependence of v~1, eq. (13). The d-dependence of eq. (18) arises from the assumed configuration in which the pinned vortex spans across the channel cross section. Region (I)" is already a highly supercritical region where the picture of so-called vortex tangle may be appropriate. Even in such a situation the same d-dependence as in eq. (18) may appear as long as the tangle is captured to the wall pinned vortices. Inserting the experimental value of Vc~ into eq. (18), we obtain I n ( b ) = 103 .

(19)

This is a very unreasonable value. Here it is noted that Schwartz's expressions, eq. (14) and eq. (18), are incompatible as they always give V~,pi. < v~¢.

(20)

This is because in both the formulae, Schwartz

123

considered a single quantized vortex line. For v~c, the discussions based on a single quantized vortex line seem reasonable. However, for Us,pi n there is no such a restriction a priori. Although it is not certain what magnitude of the size the pinning protrusion has, it may be the order of A. Assuming, for example,

b~O.ld,

(21)

we obtain I n ( b ) ~ 10.

(22)

The simplest way to obtain such an order of magnitude of ln(b/a) from eq. (18) may be to assume that r in eq. (18) is more than two orders of magnitude larger than that of a single quantized vortex line eq. (15). This means that the relevant pinned and/or depinned vortices are not in the form of a single quantized vortex line as Schwartz considered, but in the form of a bundle composed of hundreds of quantized vortices. It seems very likely that one pinning site of the size of eq. (21) can trap a lot of quantized vortices. Then they may construct a bundle of quantized vortices and act as a whole like a vortex tube. Although the motion of the assembly of many vortex filaments is a complicated unsolved problem in classical fluid dynamics, it is very desirable to extend Schwartz's argument to such a system. In the region (I)", v 3 and/or v, increase rapidly with v4. This may mean the rapid increase of the density of the flowing vortices. In the pinning-depinning vortex-bundle picture mentioned above, each pinning site is considered to trap vortices up to nearly its full capacity, which may depend on the size of the pinning protrusion. When the depinning critical velocity of the pinning center of certain size, v~2 is reached, it is probable that the depinning process proceeds more or less cooperatively. That is, a depinned vortex bundle hits another pinned vortex bundle resulting in its depinning. Such an avalanche of depinning may explain the behavior of v3 and/or vn, and that of L0(Vs) in the region (I)"'.

124

M. Okuyama et al. / Adiabatic flow of He H, part H

The content of the fact that the steady state vortex line density L 0 is a function of G may not be so simple. L 0 depends on G not only directly through vs~ and G,pin, but also indirectly through the density of pinned vortices and that of depinned vortices. The implication of them may explain the observed characteristic behavior of L o ( G ) and o n ( G ) a n d / o r v3(G) as we argued above concerning with the regions (I)', (I)" and (I)'". For the regions (II) and (III) we have the following interpretation which is at present no more than an interesting speculation. When the depinning process proceeds further with increasing G, the chance of crossing of vortices with each other will decrease. This causes, the decrease of the rate of increase of L o a n d / o r the value of L o itself. When almost all pinned vortices are depinned in the higher velocity region, the steady-state vortex line density is mainly determined by the condition just near the capillary entrance. The vortices made their flow through the capillary without changing their density.

(I)". Both vcl and v~2 seem to be scaled with the channel size following eqs. (12) and (13). The fact that L 0 is a function of only v S, and the existence of the scaling of Ucl and G2 mentioned above led us to interpret our results with Schwartz's vortex pinning-depinning model. His formula of the depinning critical velocity was found not to give the right order of magnitude of our observed vcl a n d / o r vc2 as long as we consider a single quantized vortex line. In order to recover the observed order of magnitude of Oct a n d / o r G2, we made a conjecture that the relevant vortex lines are constructing the form of a vortex bundle with hundreds of quantized vortex lines in the presence of pinning centers. Along with this model, we tried to make up physical pictures of the various flow stages designated in fig. 1. In order to get more definite understanding of the flow state of He II in the high velocity region, further study is necessary, This work is supported in part by the Grant-inAid for Scientific Research from the Ministry of Education, Science and Culture under contract no. 63540270.

References 5. Concluding remarks In order to confirm our thermohydrodynamic description of adiabatic flow of He II with the new-type mutual friction force P3roposed in Paper I, the average flow velocity of H e atoms added to the system was measured with N M R technique. We found a close similarity of the net mass flow velocity dependence between the measured 03 and the calculated o n. For the i.d. 0.2 mm capillary, the observed magnitude of oc2 agrees well with that of the calculated G . These facts show that our thermohydrodynamic description is very reliable. In the present 3He impurity experiments, we found not only the existence of the characteristic velocity uc2 but also that of v~l above which 3He atoms start flowing out from the chamber. Furthermore, we observed intermittent behavior of the motion of 3He atoms around vcl. vcl seems to lie on the boundary between regions (I)' and

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