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Superfluid flow through multiply connected geometries
Physica 78 (1974) 245-258 ©North-Holland Publishing Co.
S U P E R F L U I D FLOW T H R O U G H MULTIPLY CONNECTED G E O M E T R I E S * L. J. CAMPBELL
University o f California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544, USA
Received 1 July 1974
Synopsis For conditions opposite those considered by Van Alphen et al., the equations for superfluid flow between two reservoirs joined by multiple parallel paths are presented and solved for the case of two paths. Significant circulation (persistent current) can be created and detected by fluid transfer between reservoirs without external control over the flow paths. Characteristic signatures of circulation that are present in the level oscillations are derived in terms of the system parameters. The types of unexplained frequency shifts and damping maxima observed by Hammel et al. in U-tube oscillations are shown to result from a parallel flow path of relatively small area. 1. I n t r o d u c t i o n . Multiply connect ed flow geometries have proven to be very useful for demonstrating the p h e n o m e n o n o f superflow and studying the conditions for its appearance and decay. Here will be shown some o f the variety of behavior possible when u n d a m p e d gravitational flow o f superfluid is induced over parallel paths of macroscopic length. The circulations considered are very large compared to the 4He q u a n t u m unit o f circulation so that q u a n t u m effects can be ignored. The fact that a superfluid driven to its limiting velocity vc through parallel paths of unequal lengths would possess circulation around the paths was pointed out by Sikora, Hammel and Keller I ). T h e y illustrated their discussion by the h y p o t h e t i c a l use o f pistons to control the fluid flow. Van Alphen, DeBruyn-Ouboter, Taconis and Van Spronsen studied, theoretically and experimentally, persistent flow in an apparatus consisting o f parallel, paths o f packed rouge connecting bulk-liquid reservoirs. Above 1.4 K the thermal conditions o f their e x p e r i m e n t prevented flow oscillation after a level displacement; this allowed them to retain nearly all the circulation created and cons e que nt l y to successfully correlate the
*Work performed under the auspices of the U.S. Atomic Energy Commission. 245
246
L . J . CAMPBELL
resultant persistent flow with the observed critical flow rate during the run-in. At lower temperature, the oscillations were no longer aperiodically damped and the persistent flow was observed to be less than that created during the run-in. It is these conditions, which are so often realized in practice, that are the specific focus of this paper.
2. Theory. Consider reservoirs of helium joined by parallel flow paths which permit only superfluid flow. The superfluid velocity o in each path satisfies the equation, a----O-°q- V (Ia q- ½ Ps oe) = _ Z .~t~v e-VB/Ivl, at p Ivl
(1)
where/~ is the chemical potential that depends on temperature, pressure, height, and, in the case of a film, the Van der Waals potential. The righthand side of eq. (1) is the phenomenological form of the intrinsic dissipation 3 ) where: d = cross section of flow area (cm 2 ); x = unit of circulation ( h i m = 0.001 cm 2 /s); u = effective " a t t e m p t frequency" [(s cm 3)-a] ; vB -- effective "barrier velocity" (cm/s). Integrating eq. (1) from one reservoir of bulk helium to the other gives, for the j t h path, a
IKj - ~ oj q-
2
t~l~ = - s(oj)lDj~.~c ve-0B/l°il.
(2)
Here IKj , the effective kinetic length o f the j th path between reservoirs, and IDj, the effective dissipative length, enter because the crosssectional flow area~Cj(l) will, in general, vary along the path l so that if v/is the velocity corresponding to the minimum area ~¢/ then vy)s~j(l) = v/sO/along the path and IK/=
~j . dl ~/--(r)
(3)
j
The subscript K on the kinetic lengths will hereafter be dropped. Similarly,
IDj --
f
dl s l j ( 1 ) e - v B / l ° j ( l )
e--OB/IOil"
(4)
1
However, the intrinsic dissipation is such a sharply increasing function of v that to very good approximation, l Di is simply equal to that length of the j t h flow path for which ~¢j (l) is smallest. The factor s(v/) in eq. (2) is equal to the sign of v/since the dissipation force must oppose the flow.
SUPERFLUID FLOW T H R O U G H MULTIPLY CONNECTED GEOMETRIES
247
Assuming equal temperatures and pressures at the reservoirs, the chemical-potential difference reduces to the difference in height z of the bulk liquid levels,/a[~ = gz, where g is the gravitational acceleration. Eq. (2) can be rewritten as
+ (g/l/)z = -Dj(vp,
(5)
where D/(v/) is the dissipative term. Ignoring any mass transport by the vapor, the rate of change in z is simply related to the v i by
Ps ~. sglvi = p A i ,
(6)
1
where A is the reduced area of the reservoirs which have areas A~, A2,
A =A1 A : / ( A 1
+A2).
Given the initial conditions, the dynamics o f flow is completely specified by eqs. (5) and (6) for any number o f paths when the latter are sufficiently long to produce circulations large compared to him. The circulation Cii around paths i and j is defined as Ci/= oil i - off/. From eq. (5)
Cij = -Dill + Dill,
(7)
which re-expresses the fact, obvious from eq. (1), that dissipation is necessary (but not sufficient) for circulation to change. The total energy + ~I pAgz 2 , is always diminished by the of the system, E = 71 Ps Y_,.~/ljv~ . dissipation: !
---ps
!
/jo/Dj.
(8)
Since the level difference z(t) is the quantity easiest to measure it is useful to consider the equation for 2, obtained from eqs. (5) and (6),
+ ¢O2oz = - ( p s / p A ) ~ .~gID/, 1
where
¢o2o = (psglpA ) Z
(dS/tS)-
l Note that the undamped frequency 6Oo of the level oscillation is independent o f the division of a path into paths o f smaller area (and
(9)
248
L. J. C A M P B E L L
equal length) b u t the dissipation is not. The dissipation associated with N paths of equal area is 1/N that for one path o f the same total area. In this sense the dissipation function of eq. (1) is not intrinsic since its dependence on the channel geometry can be significant. Should the flow paths merge before entering the reservoirs the inertia of the film in the c o m m o n path will act to decrease the undamped frequency. This will be the only effect unless dissipation occurs in the c o m m o n path in which case the driving force seen b y the individual paths is reduced. This is easily avoided experimentally and will be ignored here.
3. Solutions. The presence of the dissipation term in the previous equations seems to preclude a general, analytic solution. Fortunately it is straightforward to obtain numerical solutions for particular cases. Examples of such calculations will be found in figs. 1 - 3 , 6 and 7. However, it is also possible to use a simple approximation which allows the equations to be solved in closed form without sacrificing the salient features o f the process. This will be used to illuminate the numerical solutions. The approximation consists of solving the equation piecewise b y taking 10/I = 0 whenever Iojl -- oc a n d / 9 / = 0 whenever Io/I < vc. This is suggested by the empirical observation that moderate variations in driving force cause only small variations in the limiting flow velocity. It is assumed that once the velocity along a particular path reaches o c it does not change until the driving force reverses. It is convenient, though not necessary, to further assume that oc is the same for each path. The equation for z becomes a simple oscillator equation with a frequency determined by the flow paths for which Iojl
(10.1)
where
co2 = (Ps/P) (g/A) ~/. (~¢//~.),
Iojl <
oc.
(10.2)
Van Alphen et aL achieved the same frequency change by increasing the dissipation in one path until the limiting velocity uc and its time derivative tic were zero. The solution z(t) is thus linear (if w 2 = 0) or sinusoidal in segments with different frequencies. The segments are joined b y requiring continuity in z and ~ at the time boundary. Since energy is lost at the rate = --psg]OcZl~.sgi, 1
Ioil = Oc,
(1 1)
SUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES
249
the behavior of the system is irreversible until enough energy is lost that all velocities remain below o c at which time the oscillation frequency remains at its maximum. Experimentally, the oscillations will continue to decay due to heat conduction between the reservoirs: a process b o t h well understood and irrelevant to the present considerations. The remaining discussion will be limited to the case of two paths for which 12 1> Ii. In fig. 1, curve I shows the numerical solution for z(t) for the case T ~ 1.2 K (v = 1024 cm -3 s-1 , o13 = 1670 c m / s ) : 6 =~¢2 = 1.31 × 10 "~ cm 2, ll = 1 cm, 12 = 80 cm, and A = 0.0615 cm 2. Initially, z(0) = 0.15 cm and ol (0) = 02 (0) = 0. After an acceleration period (off scale in fig. 1 ) during which Ol and 02 reach their limiting values, z decreases approximately linearly until it crosses the equilibrium level at point a. After point a the superfluid velocities in b o t h paths begin to reverse due to the reversal of driving force and at point b the velocity o f fluid in the shorter leg, which has the smaller inertia, has reached its limiting value in the opposite direction. The inertia o f the fluid in the longer leg is o
-i\o o.oo
'I°:
rt
-
oo __ -0.04 o
_ Ioo
200
300
t(s) Fig. 1. Liquid-reservoir level vs. time. dl/,-~¢2 = 1; ll/12 = 0.0125. Curve I: run-in and oscillations starting f r o m a level displacement of 0.15 cm; zero initial velocities. Curve II: initial displacement of 0.05 cm in same direction as in I; initial circulation equal to final circulation of I. Curve III: o p p o s i t e displacement of 0.05 cm; initial circulation equal to final circulation o f I.
250
L. J.
CAMPBELL
sufficiently large, in this instance, to prevent its velocity 02 from being reversed. The behavior of the circulation, ol lx -0212, is shown by curve I o f fig. 2 and the velocities vl and v2 are plotted in fig. 3. Note that the limiting velocity in the short path is larger (~ 12%) than that o f the long path. After point c neither velocity again reaches its limiting value and the circulation present at point c will persist even though, in reality, the oscillations eventually disappear due to the thermal conduction between reservoirs mentioned above. The presence of circulation in the system is easily detected by probing with small changes in the level difference z in opposite directions. Curves II o f figs. 1 and 2 show the result o f a second change in level by 0.05 cm in the same direction as for curve I. Since this merely reinforces the circulation already present the behavior is identical to I after (and somewhat before) point a. By contrast, a change in level o f 0.05 cm in the opposite direction as shown in curves III of figs. 1 and 2 acts to cancel the circulation remaining in I or II and results in quite different behavior: the acceleration period is longer, the first half period is not distorted and the amplitude o f the oscillations is appreciably greater. The latter effect is due to the transfer of energy from the circulating current to the oscillating current. (For convenient comparison, curve III of fig. 1 is 4OOO
I 3000
E
20O0
(.~ v (J
I000
0
0
I00
200
300
t(s) Fig. 2. Circulation
vs.
time. Curve labels correspond to those of fig. 1.
SUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES 60
1
1
1
1
1
1
1
1
1
I
I
I
1
1
I
I
I
I
Vl
I
- -
1
251
4O
2O
E u
0
-20
-40 m m
-6o
I 0
I
I
I
I
I
I00
I 2O0
I
I
300
t(s)
Fig. 3. Velocity figs. 1 and 2.
vs.
time. Short path ol and long path 02 velocities correspond to I of
shown inverted from its physical appearance corresponding to an opposite level displacement.) The behavior of the system will n o w be analyzed using the approximation described above. After point a, z obeys 5 + ~ 2 z = 0, which has the solution
z(t) = - Z a s i n wt,
(12)
where Za is determined by matching the slope at a with the linear solution corresponding to both paths having the limiting velocity vc. This gives Za = Uc(Ps/PA co) (.~1 +~¢2 ). After a the velocities satisfy t)j = -gz/lj which has the solution vj(t) = - o c + (gZa/ljco) ( 1 - c o s cot) until loft = o c which occurs first for the short path. This defines point b and time tb (measured from a) ol (tb) = o c = - o c + (gZa/llw) (1 - c o s Wtb),
(13)
252
L.J. CAMPBELL
which gives cos cot b = ( 1 - a - 2 1 ) / ( a + 1),
(14)
where a -= J 1 / s¢ 2 and l -- ll/12 <~ 1. A f t e r p o i n t b, 01 = v c, so ~ + co~z = 0 where co~ = (psg/pA) (J2/12) which has the solution (15)
z = - Z b sin (w2 t + 40. Measuring time f r o m b and m a t c h i n g z and 2 with eq. (12) at t b the m a g n i t u d e Z b and phase ~ are f o u n d to be ~2 2[l(1-l)] tan ¢ = - -co tan cot b 1-a-2l
Zb = Z a
1+
O92
- 1
7
,
cos 2 cot b.
(16)
(17)
If cos w t b < 0 the m a x i m u m d e p t h o f the first h a l f cycle is Z a; otherwise it is Z b. Curves of c o n s t a n t q~ are p l o t t e d in fig. 4 as a f u n c t i o n o f a and l according to eq. (16).. Note t h a t q~, the phase with which the ~ 2 cycle begins, is a very sensitive f u n c t i o n of a ~ 1 w h e n l "~ 1. It is obvious t h a t v2 c a n n o t reach v c after vl does since the total energy would t h e n be greater t h a n or equal to its value at point a which is impossible. Therefore the solution given by eq. (15) is valid to point c. The value o f v2 at c is given b y v2(tb) = vc (--a + [ 4 a l + (a--1)21~ }.
(18)
The ratio R of the circulation at c to the circulation at a is
R = (l + a - [ 4 a l + ( a - l ) 2 ] ~ ) / ( 1 - / ) .
(19)
Curves of c o n s t a n t R are plotted in fig. 4. It can be seen t h a t r e t e n t i o n o f the circulation established during the run-in is favored by small l and by a n o t less t h a n unity. I f R > 0 t h e n circulation present at c will remain constant, otherwise it will decrease f u r t h e r in absolute m a g n i t u d e over the following cycles as shown later. The c o n d i t i o n s on l and a for R > 0, which is equivalent to I011 < Vc over the second h a l f cycle, are / < 2a-l,
a>½.
The persistent c o m p o n e n t o f the velocity in the long leg V2p as a
(20)
SUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES
253
1.0
-'r
O'8I--~"" J, 0.6
",
0,4~1'\%' 0.2 o.o
0
1" I
a
2
3
Fig. 4. Lines of constant circulation ratio (sofid) and phase (dashed) as a function of length l = ll/12 and area a = ~1/~'2 ratios. Reading counterclockwise, the ratio of circulation at end to that at beginning of first half-cycle is 0.95, 0.75, 0.50, 0.25, 0.0, -0.25, -0.50, -0.75, -0.95; phase q~(in deg.) is 170.0, 157.5, 135.0, 112.5, 90.0, 67.5, 45.0, 22.5. fraction Jr of its limiting value during the run-in has been used by Van Alphen et al. to characterize the performance of the system. At the end of the first half cycle, eq. (18) shows
Jr ~ °2p/Vc = a{ 1 - [ 4 a l + ( a - 1) 2 ] ~}/(a + l).
(21)
This is, of course, smaller than the value obtained by Van Alpen et al. who assumed that when oscillations are suppressed there would be no change in circulation after the run-in:
Jr = a ( 1 - 1 ) / ( l + a ) , Jr = a,
a < 1--2l,
a > 1-2l,
(22.1) (22.2)
For a >> 1, there is relatively little time for dissipation in the short path
254
L.J. CAMPBELL
[cos cotb ~ - 1 from eq. (14)] so eqs. (21) and (22.1) become equal. In fig. 5, Jr from eq. (21 ) is plotted for various l with the same format as fig. 4 of ref. 2. For smaller a the final circulation is reached on successively higher cycles; extensions o f the curves in fig. 5 would oscillate around zero with diminishing amplitude and period as a approached zero. As mentioned previously, a level displacement opposite in direction from the initial run-in will cancel the remnant circulation. (Curve III shows a partial cancellation.) The magnitude of the opposite displacement needed to cancel, at the first zero of z, the long path velocity given b y eq. (18) is given by: ocwl 2 Z0 = - g
[la(2a-3al-1)]~ a+l
(23)
The ratio of the undamped amplitude of z after this displacement to that following a long run-in is (24)
Z I I I / Z I = a/[4al + ( a - 1) 2 ] ~.
In the absence of any initial circulation or flow the minimum displacement required to drive the velocity to its limiting value in b o t h paths is ,
Lo
I
I
--
I I II 0.001
If-
I~I /
I_J._--~--~
: : '.,,i
-1 --I
0.8--
/
0.4--
0.2--
o.o
0.1
I
I
I
I.
I0.
Q
Fig. 5. v2p/v c vs. ~l/,Rl2 for various Ii/12. V2p is the persistent circulating component of v2 at the end of the first half:cycle.
sUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES
Z c = (Oc~12/g)l ~,
255 (25)
As a final illustration, the behavior o f the system when ~¢1 "~ ~¢2 and ll "~ 12 will be considered. This is interesting because the unfamiliar effects of this geometry have been previously observed b y Hammel, Keller and Sherman 4) but n o t understood. Other experimenters s ) failed to see the effect due to the absence o f a second flow path (i. e. leak) in their apparatus. The amount of energy that can be dissipated in a flow path is proportional to its volume, so for l ,~ 1, a "~ 1 it is possible that the short path will be dissipative for many cycles before the energy falls to a value consistent with potential flow. During the dissipative cycles, the frequency will be characteristic of the long path since the velocity in the short path will have its limiting value most of the time. After sufficient energy is lost so that ol remains below o c the frequency will increase to that appropriate to potential flow in parallel paths. The result of numerically integrating the equations of the previous section are shown in fig. 6 for d l = 0.142 × 10 -6 cm 2 , ~12 = 0.283 × 10 -s cm 2 , ll = 0.05 cm, 12 = 1 cm and other quantities unchanged. If the damping of the level oscillations shown in fig. 6 is expressed as e - a t then a changes each half-cycle as shown in fig. 7 where the observed frequency 0.06
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
L
I
I
I
I
I
0.04
E 0.02 n4
0.00
-0.020
I
I
I
~)0
I
I 200
I
300
t(s) Fig. 6. Level difference vs. time.~¢l/d2 = 0.05; ll/12 = 0.05.
256
L.J. CAMPBELL
~
1I I I I I I I I I I I
0.03
Ill
Ill
I Ill
I1 , ).050
,,
oo2 / A
).045
--
0.040
0.01
0.00
0
I00
t(s)
200
Fig. 7. Decay constant (A) and frequency (o) vs. time. Points correspond to each half-cycle of fig. 6. of each half-cycle is also plotted. The damping factor rises to a maximum and then falls to a low value while the frequency abruptly increases to a constant value. To obtain quantitative insight into the process it is necessary to integrate the equations for z and 02 over a half-cycle in the same manner as done previously. If 02/is the velocity in the long path at the end of the ]th half-cycle then after another half-cycle, rlj+l = [4al + (a-r//.) 2 ] ~- - a ;
,/j -102/I/Oc,
(26)
which is a more general form o f eq. (18). The fractional velocity r// decreases each half-cycle until r/k ~ l which is the minimum velocity 02 needed to drive o l to its limiting value o c. The differential solution to eq. (26) is 1-~j + l ln[(1-l)/(rti-l)]
= 2aj.
(27)
The apparent decay constant per half-cycle c~/can be found b y relating
SUPERFLUID FLOW THROUGH MULTIPLY cONNECTED GEOMETRIES
257
r/j to Z! by equating the energy at maxima and zeros o f z and assuming that Ol has reached v c when Izl is m a x i m u m (which is true except for,possibly the last one or two dissipative half-cycles). This gives:
Z] = Vcr//(ps~¢212/pgA )~ ,
(28)
or
a z j = -oc2a(1-l/r/)
(ps ¢212 I p g A
,
(29)
where r/---- (r/j + r/i_ 1)/2. For the first few oscillations r / ~ 1 and since, by assumption l ,~ 1, eq. (29) predicts that the early oscillation amplitudes decrease by almost a constant absolute amount every half-cycle rather than by the constant percentage amount characteristic o f a damped harmonic oscillator. This can be seen in the data of ref. 4. Defining t~ by
A Z / Z = e-,~*/2 --1, where r is the full period o f oscillation, it follows that ½t~r = - I n {1 + (2a/r/)[(l/r/)-l]),
(30)
which has a differential m a x i m u m of (~Otr)ma x = - l n (1-a/21),
(31)
at ,7 = 2l. The observed damping m a x i m u m will be progressively less than this as the time interval o f analysis is made longer, a feature also noticed by Hammel et al.
4. Conclusions. The present analysis shows that the generation, detection, and quantitative study o f persistent currents are possible under quite general conditions. It is not necessary to prevent oscillatory flow over the circuit, to externally block the flow in one o f the paths, nor to rotate the apparatus. Other than permitting only superfluid flow, the nature of path is unrestricted. Even knowledge o f the exact form o f the intrinsic dissipation is not important for a quantitative understanding of most of the features. The techniques described here seem particularly promising in studying saturated-film flow for which the inertia is too small for the use o f gyroscopic techniques. In principle, unsaturated films could also be used if pressure reservoirs were substituted for the bulk-liquid reservoirs.
258
L.J. CAMPBELL
Another possible application is the identification o f induced circulation effects in flow geometrics that are geometrically simply connected but in which hydrodynamics favors effective multiple-path flow. A c k n o w l e d g e m e n t s . It is a pleasure to thank Dr. W. E. Keller for his support and interest in this work.
Note added in proof. It is gratifying that evidence for the waveform distortions predicted here due to trapped circulation can be seen in a recent paper by H. J. Verbeek, E. van Spronsen, H. Mars, H. van Beelen, R. de Bruyn Ouboter and K. W. Taconis, Physica 73 (1974) 621. Compare their fig. 2 between t i and tf with curves I or II of fig. 1. REFERENCES 1) Sikora, P., Hammel, E. F. and Keller, W. E., Physica 32 (1966) 1693. 2) Van Alphen, W. M., De Bruyn Ouboter, R., Taconis, K. W. and Van Spronsen, E., Physica 39 (1968) 109. 3) Langer, J. S. and Reppy, J. D., Progr. low Temp. Phys., C. J. Gorter, ed., NorthHolland Publ. Comp. (Amsterdam, 1970). Vol. 6, Ch. 1. 4) Hammel, E. F., Keller, W. E. and Sherman, R. H., Phys. Rev. Letters 24 (1970) 712. 5) Hallock, R. B. and Flint, E. B., Phys. Letters 45A (1973) 245.
S U P E R F L U I D FLOW T H R O U G H MULTIPLY CONNECTED G E O M E T R I E S * L. J. CAMPBELL
University o f California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544, USA
Received 1 July 1974
Synopsis For conditions opposite those considered by Van Alphen et al., the equations for superfluid flow between two reservoirs joined by multiple parallel paths are presented and solved for the case of two paths. Significant circulation (persistent current) can be created and detected by fluid transfer between reservoirs without external control over the flow paths. Characteristic signatures of circulation that are present in the level oscillations are derived in terms of the system parameters. The types of unexplained frequency shifts and damping maxima observed by Hammel et al. in U-tube oscillations are shown to result from a parallel flow path of relatively small area. 1. I n t r o d u c t i o n . Multiply connect ed flow geometries have proven to be very useful for demonstrating the p h e n o m e n o n o f superflow and studying the conditions for its appearance and decay. Here will be shown some o f the variety of behavior possible when u n d a m p e d gravitational flow o f superfluid is induced over parallel paths of macroscopic length. The circulations considered are very large compared to the 4He q u a n t u m unit o f circulation so that q u a n t u m effects can be ignored. The fact that a superfluid driven to its limiting velocity vc through parallel paths of unequal lengths would possess circulation around the paths was pointed out by Sikora, Hammel and Keller I ). T h e y illustrated their discussion by the h y p o t h e t i c a l use o f pistons to control the fluid flow. Van Alphen, DeBruyn-Ouboter, Taconis and Van Spronsen studied, theoretically and experimentally, persistent flow in an apparatus consisting o f parallel, paths o f packed rouge connecting bulk-liquid reservoirs. Above 1.4 K the thermal conditions o f their e x p e r i m e n t prevented flow oscillation after a level displacement; this allowed them to retain nearly all the circulation created and cons e que nt l y to successfully correlate the
*Work performed under the auspices of the U.S. Atomic Energy Commission. 245
246
L . J . CAMPBELL
resultant persistent flow with the observed critical flow rate during the run-in. At lower temperature, the oscillations were no longer aperiodically damped and the persistent flow was observed to be less than that created during the run-in. It is these conditions, which are so often realized in practice, that are the specific focus of this paper.
2. Theory. Consider reservoirs of helium joined by parallel flow paths which permit only superfluid flow. The superfluid velocity o in each path satisfies the equation, a----O-°q- V (Ia q- ½ Ps oe) = _ Z .~t~v e-VB/Ivl, at p Ivl
(1)
where/~ is the chemical potential that depends on temperature, pressure, height, and, in the case of a film, the Van der Waals potential. The righthand side of eq. (1) is the phenomenological form of the intrinsic dissipation 3 ) where: d = cross section of flow area (cm 2 ); x = unit of circulation ( h i m = 0.001 cm 2 /s); u = effective " a t t e m p t frequency" [(s cm 3)-a] ; vB -- effective "barrier velocity" (cm/s). Integrating eq. (1) from one reservoir of bulk helium to the other gives, for the j t h path, a
IKj - ~ oj q-
2
t~l~ = - s(oj)lDj~.~c ve-0B/l°il.
(2)
Here IKj , the effective kinetic length o f the j th path between reservoirs, and IDj, the effective dissipative length, enter because the crosssectional flow area~Cj(l) will, in general, vary along the path l so that if v/is the velocity corresponding to the minimum area ~¢/ then vy)s~j(l) = v/sO/along the path and IK/=
~j . dl ~/--(r)
(3)
j
The subscript K on the kinetic lengths will hereafter be dropped. Similarly,
IDj --
f
dl s l j ( 1 ) e - v B / l ° j ( l )
e--OB/IOil"
(4)
1
However, the intrinsic dissipation is such a sharply increasing function of v that to very good approximation, l Di is simply equal to that length of the j t h flow path for which ~¢j (l) is smallest. The factor s(v/) in eq. (2) is equal to the sign of v/since the dissipation force must oppose the flow.
SUPERFLUID FLOW T H R O U G H MULTIPLY CONNECTED GEOMETRIES
247
Assuming equal temperatures and pressures at the reservoirs, the chemical-potential difference reduces to the difference in height z of the bulk liquid levels,/a[~ = gz, where g is the gravitational acceleration. Eq. (2) can be rewritten as
+ (g/l/)z = -Dj(vp,
(5)
where D/(v/) is the dissipative term. Ignoring any mass transport by the vapor, the rate of change in z is simply related to the v i by
Ps ~. sglvi = p A i ,
(6)
1
where A is the reduced area of the reservoirs which have areas A~, A2,
A =A1 A : / ( A 1
+A2).
Given the initial conditions, the dynamics o f flow is completely specified by eqs. (5) and (6) for any number o f paths when the latter are sufficiently long to produce circulations large compared to him. The circulation Cii around paths i and j is defined as Ci/= oil i - off/. From eq. (5)
Cij = -Dill + Dill,
(7)
which re-expresses the fact, obvious from eq. (1), that dissipation is necessary (but not sufficient) for circulation to change. The total energy + ~I pAgz 2 , is always diminished by the of the system, E = 71 Ps Y_,.~/ljv~ . dissipation: !
---ps
!
/jo/Dj.
(8)
Since the level difference z(t) is the quantity easiest to measure it is useful to consider the equation for 2, obtained from eqs. (5) and (6),
+ ¢O2oz = - ( p s / p A ) ~ .~gID/, 1
where
¢o2o = (psglpA ) Z
(dS/tS)-
l Note that the undamped frequency 6Oo of the level oscillation is independent o f the division of a path into paths o f smaller area (and
(9)
248
L. J. C A M P B E L L
equal length) b u t the dissipation is not. The dissipation associated with N paths of equal area is 1/N that for one path o f the same total area. In this sense the dissipation function of eq. (1) is not intrinsic since its dependence on the channel geometry can be significant. Should the flow paths merge before entering the reservoirs the inertia of the film in the c o m m o n path will act to decrease the undamped frequency. This will be the only effect unless dissipation occurs in the c o m m o n path in which case the driving force seen b y the individual paths is reduced. This is easily avoided experimentally and will be ignored here.
3. Solutions. The presence of the dissipation term in the previous equations seems to preclude a general, analytic solution. Fortunately it is straightforward to obtain numerical solutions for particular cases. Examples of such calculations will be found in figs. 1 - 3 , 6 and 7. However, it is also possible to use a simple approximation which allows the equations to be solved in closed form without sacrificing the salient features o f the process. This will be used to illuminate the numerical solutions. The approximation consists of solving the equation piecewise b y taking 10/I = 0 whenever Iojl -- oc a n d / 9 / = 0 whenever Io/I < vc. This is suggested by the empirical observation that moderate variations in driving force cause only small variations in the limiting flow velocity. It is assumed that once the velocity along a particular path reaches o c it does not change until the driving force reverses. It is convenient, though not necessary, to further assume that oc is the same for each path. The equation for z becomes a simple oscillator equation with a frequency determined by the flow paths for which Iojl
(10.1)
where
co2 = (Ps/P) (g/A) ~/. (~¢//~.),
Iojl <
oc.
(10.2)
Van Alphen et aL achieved the same frequency change by increasing the dissipation in one path until the limiting velocity uc and its time derivative tic were zero. The solution z(t) is thus linear (if w 2 = 0) or sinusoidal in segments with different frequencies. The segments are joined b y requiring continuity in z and ~ at the time boundary. Since energy is lost at the rate = --psg]OcZl~.sgi, 1
Ioil = Oc,
(1 1)
SUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES
249
the behavior of the system is irreversible until enough energy is lost that all velocities remain below o c at which time the oscillation frequency remains at its maximum. Experimentally, the oscillations will continue to decay due to heat conduction between the reservoirs: a process b o t h well understood and irrelevant to the present considerations. The remaining discussion will be limited to the case of two paths for which 12 1> Ii. In fig. 1, curve I shows the numerical solution for z(t) for the case T ~ 1.2 K (v = 1024 cm -3 s-1 , o13 = 1670 c m / s ) : 6 =~¢2 = 1.31 × 10 "~ cm 2, ll = 1 cm, 12 = 80 cm, and A = 0.0615 cm 2. Initially, z(0) = 0.15 cm and ol (0) = 02 (0) = 0. After an acceleration period (off scale in fig. 1 ) during which Ol and 02 reach their limiting values, z decreases approximately linearly until it crosses the equilibrium level at point a. After point a the superfluid velocities in b o t h paths begin to reverse due to the reversal of driving force and at point b the velocity o f fluid in the shorter leg, which has the smaller inertia, has reached its limiting value in the opposite direction. The inertia o f the fluid in the longer leg is o
-i\o o.oo
'I°:
rt
-
oo __ -0.04 o
_ Ioo
200
300
t(s) Fig. 1. Liquid-reservoir level vs. time. dl/,-~¢2 = 1; ll/12 = 0.0125. Curve I: run-in and oscillations starting f r o m a level displacement of 0.15 cm; zero initial velocities. Curve II: initial displacement of 0.05 cm in same direction as in I; initial circulation equal to final circulation of I. Curve III: o p p o s i t e displacement of 0.05 cm; initial circulation equal to final circulation o f I.
250
L. J.
CAMPBELL
sufficiently large, in this instance, to prevent its velocity 02 from being reversed. The behavior of the circulation, ol lx -0212, is shown by curve I o f fig. 2 and the velocities vl and v2 are plotted in fig. 3. Note that the limiting velocity in the short path is larger (~ 12%) than that o f the long path. After point c neither velocity again reaches its limiting value and the circulation present at point c will persist even though, in reality, the oscillations eventually disappear due to the thermal conduction between reservoirs mentioned above. The presence of circulation in the system is easily detected by probing with small changes in the level difference z in opposite directions. Curves II o f figs. 1 and 2 show the result o f a second change in level by 0.05 cm in the same direction as for curve I. Since this merely reinforces the circulation already present the behavior is identical to I after (and somewhat before) point a. By contrast, a change in level o f 0.05 cm in the opposite direction as shown in curves III of figs. 1 and 2 acts to cancel the circulation remaining in I or II and results in quite different behavior: the acceleration period is longer, the first half period is not distorted and the amplitude o f the oscillations is appreciably greater. The latter effect is due to the transfer of energy from the circulating current to the oscillating current. (For convenient comparison, curve III of fig. 1 is 4OOO
I 3000
E
20O0
(.~ v (J
I000
0
0
I00
200
300
t(s) Fig. 2. Circulation
vs.
time. Curve labels correspond to those of fig. 1.
SUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES 60
1
1
1
1
1
1
1
1
1
I
I
I
1
1
I
I
I
I
Vl
I
- -
1
251
4O
2O
E u
0
-20
-40 m m
-6o
I 0
I
I
I
I
I
I00
I 2O0
I
I
300
t(s)
Fig. 3. Velocity figs. 1 and 2.
vs.
time. Short path ol and long path 02 velocities correspond to I of
shown inverted from its physical appearance corresponding to an opposite level displacement.) The behavior of the system will n o w be analyzed using the approximation described above. After point a, z obeys 5 + ~ 2 z = 0, which has the solution
z(t) = - Z a s i n wt,
(12)
where Za is determined by matching the slope at a with the linear solution corresponding to both paths having the limiting velocity vc. This gives Za = Uc(Ps/PA co) (.~1 +~¢2 ). After a the velocities satisfy t)j = -gz/lj which has the solution vj(t) = - o c + (gZa/ljco) ( 1 - c o s cot) until loft = o c which occurs first for the short path. This defines point b and time tb (measured from a) ol (tb) = o c = - o c + (gZa/llw) (1 - c o s Wtb),
(13)
252
L.J. CAMPBELL
which gives cos cot b = ( 1 - a - 2 1 ) / ( a + 1),
(14)
where a -= J 1 / s¢ 2 and l -- ll/12 <~ 1. A f t e r p o i n t b, 01 = v c, so ~ + co~z = 0 where co~ = (psg/pA) (J2/12) which has the solution (15)
z = - Z b sin (w2 t + 40. Measuring time f r o m b and m a t c h i n g z and 2 with eq. (12) at t b the m a g n i t u d e Z b and phase ~ are f o u n d to be ~2 2[l(1-l)] tan ¢ = - -co tan cot b 1-a-2l
Zb = Z a
1+
O92
- 1
7
,
cos 2 cot b.
(16)
(17)
If cos w t b < 0 the m a x i m u m d e p t h o f the first h a l f cycle is Z a; otherwise it is Z b. Curves of c o n s t a n t q~ are p l o t t e d in fig. 4 as a f u n c t i o n o f a and l according to eq. (16).. Note t h a t q~, the phase with which the ~ 2 cycle begins, is a very sensitive f u n c t i o n of a ~ 1 w h e n l "~ 1. It is obvious t h a t v2 c a n n o t reach v c after vl does since the total energy would t h e n be greater t h a n or equal to its value at point a which is impossible. Therefore the solution given by eq. (15) is valid to point c. The value o f v2 at c is given b y v2(tb) = vc (--a + [ 4 a l + (a--1)21~ }.
(18)
The ratio R of the circulation at c to the circulation at a is
R = (l + a - [ 4 a l + ( a - l ) 2 ] ~ ) / ( 1 - / ) .
(19)
Curves of c o n s t a n t R are plotted in fig. 4. It can be seen t h a t r e t e n t i o n o f the circulation established during the run-in is favored by small l and by a n o t less t h a n unity. I f R > 0 t h e n circulation present at c will remain constant, otherwise it will decrease f u r t h e r in absolute m a g n i t u d e over the following cycles as shown later. The c o n d i t i o n s on l and a for R > 0, which is equivalent to I011 < Vc over the second h a l f cycle, are / < 2a-l,
a>½.
The persistent c o m p o n e n t o f the velocity in the long leg V2p as a
(20)
SUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES
253
1.0
-'r
O'8I--~"" J, 0.6
",
0,4~1'\%' 0.2 o.o
0
1" I
a
2
3
Fig. 4. Lines of constant circulation ratio (sofid) and phase (dashed) as a function of length l = ll/12 and area a = ~1/~'2 ratios. Reading counterclockwise, the ratio of circulation at end to that at beginning of first half-cycle is 0.95, 0.75, 0.50, 0.25, 0.0, -0.25, -0.50, -0.75, -0.95; phase q~(in deg.) is 170.0, 157.5, 135.0, 112.5, 90.0, 67.5, 45.0, 22.5. fraction Jr of its limiting value during the run-in has been used by Van Alphen et al. to characterize the performance of the system. At the end of the first half cycle, eq. (18) shows
Jr ~ °2p/Vc = a{ 1 - [ 4 a l + ( a - 1) 2 ] ~}/(a + l).
(21)
This is, of course, smaller than the value obtained by Van Alpen et al. who assumed that when oscillations are suppressed there would be no change in circulation after the run-in:
Jr = a ( 1 - 1 ) / ( l + a ) , Jr = a,
a < 1--2l,
a > 1-2l,
(22.1) (22.2)
For a >> 1, there is relatively little time for dissipation in the short path
254
L.J. CAMPBELL
[cos cotb ~ - 1 from eq. (14)] so eqs. (21) and (22.1) become equal. In fig. 5, Jr from eq. (21 ) is plotted for various l with the same format as fig. 4 of ref. 2. For smaller a the final circulation is reached on successively higher cycles; extensions o f the curves in fig. 5 would oscillate around zero with diminishing amplitude and period as a approached zero. As mentioned previously, a level displacement opposite in direction from the initial run-in will cancel the remnant circulation. (Curve III shows a partial cancellation.) The magnitude of the opposite displacement needed to cancel, at the first zero of z, the long path velocity given b y eq. (18) is given by: ocwl 2 Z0 = - g
[la(2a-3al-1)]~ a+l
(23)
The ratio of the undamped amplitude of z after this displacement to that following a long run-in is (24)
Z I I I / Z I = a/[4al + ( a - 1) 2 ] ~.
In the absence of any initial circulation or flow the minimum displacement required to drive the velocity to its limiting value in b o t h paths is ,
Lo
I
I
--
I I II 0.001
If-
I~I /
I_J._--~--~
: : '.,,i
-1 --I
0.8--
/
0.4--
0.2--
o.o
0.1
I
I
I
I.
I0.
Q
Fig. 5. v2p/v c vs. ~l/,Rl2 for various Ii/12. V2p is the persistent circulating component of v2 at the end of the first half:cycle.
sUPERFLUID FLOW THROUGH MULTIPLY CONNECTED GEOMETRIES
Z c = (Oc~12/g)l ~,
255 (25)
As a final illustration, the behavior o f the system when ~¢1 "~ ~¢2 and ll "~ 12 will be considered. This is interesting because the unfamiliar effects of this geometry have been previously observed b y Hammel, Keller and Sherman 4) but n o t understood. Other experimenters s ) failed to see the effect due to the absence o f a second flow path (i. e. leak) in their apparatus. The amount of energy that can be dissipated in a flow path is proportional to its volume, so for l ,~ 1, a "~ 1 it is possible that the short path will be dissipative for many cycles before the energy falls to a value consistent with potential flow. During the dissipative cycles, the frequency will be characteristic of the long path since the velocity in the short path will have its limiting value most of the time. After sufficient energy is lost so that ol remains below o c the frequency will increase to that appropriate to potential flow in parallel paths. The result of numerically integrating the equations of the previous section are shown in fig. 6 for d l = 0.142 × 10 -6 cm 2 , ~12 = 0.283 × 10 -s cm 2 , ll = 0.05 cm, 12 = 1 cm and other quantities unchanged. If the damping of the level oscillations shown in fig. 6 is expressed as e - a t then a changes each half-cycle as shown in fig. 7 where the observed frequency 0.06
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
L
I
I
I
I
I
0.04
E 0.02 n4
0.00
-0.020
I
I
I
~)0
I
I 200
I
300
t(s) Fig. 6. Level difference vs. time.~¢l/d2 = 0.05; ll/12 = 0.05.
256
L.J. CAMPBELL
~
1I I I I I I I I I I I
0.03
Ill
Ill
I Ill
I1 , ).050
,,
oo2 / A
).045
--
0.040
0.01
0.00
0
I00
t(s)
200
Fig. 7. Decay constant (A) and frequency (o) vs. time. Points correspond to each half-cycle of fig. 6. of each half-cycle is also plotted. The damping factor rises to a maximum and then falls to a low value while the frequency abruptly increases to a constant value. To obtain quantitative insight into the process it is necessary to integrate the equations for z and 02 over a half-cycle in the same manner as done previously. If 02/is the velocity in the long path at the end of the ]th half-cycle then after another half-cycle, rlj+l = [4al + (a-r//.) 2 ] ~- - a ;
,/j -102/I/Oc,
(26)
which is a more general form o f eq. (18). The fractional velocity r// decreases each half-cycle until r/k ~ l which is the minimum velocity 02 needed to drive o l to its limiting value o c. The differential solution to eq. (26) is 1-~j + l ln[(1-l)/(rti-l)]
= 2aj.
(27)
The apparent decay constant per half-cycle c~/can be found b y relating
SUPERFLUID FLOW THROUGH MULTIPLY cONNECTED GEOMETRIES
257
r/j to Z! by equating the energy at maxima and zeros o f z and assuming that Ol has reached v c when Izl is m a x i m u m (which is true except for,possibly the last one or two dissipative half-cycles). This gives:
Z] = Vcr//(ps~¢212/pgA )~ ,
(28)
or
a z j = -oc2a(1-l/r/)
(ps ¢212 I p g A
,
(29)
where r/---- (r/j + r/i_ 1)/2. For the first few oscillations r / ~ 1 and since, by assumption l ,~ 1, eq. (29) predicts that the early oscillation amplitudes decrease by almost a constant absolute amount every half-cycle rather than by the constant percentage amount characteristic o f a damped harmonic oscillator. This can be seen in the data of ref. 4. Defining t~ by
A Z / Z = e-,~*/2 --1, where r is the full period o f oscillation, it follows that ½t~r = - I n {1 + (2a/r/)[(l/r/)-l]),
(30)
which has a differential m a x i m u m of (~Otr)ma x = - l n (1-a/21),
(31)
at ,7 = 2l. The observed damping m a x i m u m will be progressively less than this as the time interval o f analysis is made longer, a feature also noticed by Hammel et al.
4. Conclusions. The present analysis shows that the generation, detection, and quantitative study o f persistent currents are possible under quite general conditions. It is not necessary to prevent oscillatory flow over the circuit, to externally block the flow in one o f the paths, nor to rotate the apparatus. Other than permitting only superfluid flow, the nature of path is unrestricted. Even knowledge o f the exact form o f the intrinsic dissipation is not important for a quantitative understanding of most of the features. The techniques described here seem particularly promising in studying saturated-film flow for which the inertia is too small for the use o f gyroscopic techniques. In principle, unsaturated films could also be used if pressure reservoirs were substituted for the bulk-liquid reservoirs.
258
L.J. CAMPBELL
Another possible application is the identification o f induced circulation effects in flow geometrics that are geometrically simply connected but in which hydrodynamics favors effective multiple-path flow. A c k n o w l e d g e m e n t s . It is a pleasure to thank Dr. W. E. Keller for his support and interest in this work.
Note added in proof. It is gratifying that evidence for the waveform distortions predicted here due to trapped circulation can be seen in a recent paper by H. J. Verbeek, E. van Spronsen, H. Mars, H. van Beelen, R. de Bruyn Ouboter and K. W. Taconis, Physica 73 (1974) 621. Compare their fig. 2 between t i and tf with curves I or II of fig. 1. REFERENCES 1) Sikora, P., Hammel, E. F. and Keller, W. E., Physica 32 (1966) 1693. 2) Van Alphen, W. M., De Bruyn Ouboter, R., Taconis, K. W. and Van Spronsen, E., Physica 39 (1968) 109. 3) Langer, J. S. and Reppy, J. D., Progr. low Temp. Phys., C. J. Gorter, ed., NorthHolland Publ. Comp. (Amsterdam, 1970). Vol. 6, Ch. 1. 4) Hammel, E. F., Keller, W. E. and Sherman, R. H., Phys. Rev. Letters 24 (1970) 712. 5) Hallock, R. B. and Flint, E. B., Phys. Letters 45A (1973) 245.