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Combined measurements on various types of stationary flow of superfluid 4He
Physrca 96B (1979) 312-324 0 North-Holland Pubhshmg Company
COMBINED MEASUREMENTS
R. R. IJSSELSTEIN, Kamerhngh
FLOW OF SUPERFLUID 4He
M. P. de GOEJE and H. C. KIUMERS
Onnes Laboratorium
(Communrcatton
ON VARIOUS TYPES OF STATIONARY
der Rllksunrversrtelt Lerden, Leaden, The Netherlands
No. 437~)
Received 25 January 1979
Experunents are carried out on the stationary flow of superfluid 4He through caprllarres of 620 pm diameter. In many flow srtuations (the transport velocities of the superfluid and the normal fluid were mdependently adJustable) at 1 50 K, four quantrtres have been measured the attenuation and dispersion of second sound m resonance, parallel to the stationary flow, together with the temperature and chemrcal-potentral difference between the two ends of the flow crrcurt. By combmmg the results of the varrous quantttres unexpected results are obtamed. In the second part, specral attentron 1s given (at a number of temperatures) to two types of flow. pure superfluid flow and pure normal-fluid flow. A short descrrptron of the experimental detatls ISalso grven 1. Introduction
All three methods provrde, 111prmcrple, different mformatron. The choice of one of the-three methods implies, m general, also a hmrtatron of the range of values for the dlmensron of the flow pipe. Method one grves most accurate results m narrow caprllaries (d 6 500 pm) whrle for methods 2 and 3 relatively wide tubes (d Z 5 mm) are often necessary As far as we know, experiments where two of these methods are used m the same geometry were performed almost exclusrvely by Vmen [5], who measured the temperature difference together with the second-sound attenuation. Wrth above consrderatrons, an experiment was set up 111whrch four quantities could be measured m flow situations obtamed by adjustmg mdependently the velocity of the normal component and of the superfluid component (of course, as u-r every flow experiment, only the average value of the velocity with respect to the cross-section could be controlled) Every combmatron of us and u,, could be established m this experiment, for whrch the total mass flow had the same drrectron as the normal-fluid flow. The quantities to be measured were the secondsound attenuatron Q’ together with the dispersion cr” and the difference m temperature and chemical potential (AT and &, respectively) between the two ends of the flow crrcurt. The dispersion was basically
One of the thin s that experimental research on flow of superfluid $ He needs most 1s a systematrc mvestrgatron of the relation between the measured quantrtres, and of then dependence on the externally controllable parameters such as the geometry, both transport velocrtres, and the temperature. A severe restrictron m almost all experiments 1s the fmed relatron between the two velocities m the flow (e.g. pure counterflow, pure superfluid flow). This laboratory 1s one of the few places where measurements are done on many different combmatrons of these (refs. 1,2,
394). A second restrictron 1s the choice of one quantity (m some cases two quantrtres) that is going to be measured. In almost all expenments, one of three methods 1s used to mvestrgate the flow. 1. determmatron of the temperafure difference between the two ends of the flow pipe, sometunes together with the chemical-potential difference or the pressure difference, 2. measurement of the attenuation of second sound due to the flow (sometimes the change of the second-sound velocity 1s mvestrgated); 3. the use of ions as probes m the flow, this method 1s rather new and gives very interesting results. 312
R R IJsselstem et al /Flow of superfluid 4He
measured as a change of the resonance frequency of the resonator The best compromrse in looking for a geometry in whrch all four quantities could be measured with acceptable accuracy was found 111a caprllary of drameter 600 pm and length about 40 cm. In order to be able to brmg the second sound mto resonance m such a geometry, a Helmholtz resonator was used with a resonance frequency m the order of 100 Hz. In such a resonator, the direction of the wave vector 1s always parallel to the flow dnectron. In thus article the results of the measurements are presented. It 1s preceded by a short description of the experimental set-up and the measuring methods. An attempt at interpretation of the results 1s m progress and will be published later.
-w
2. Experimental
313
set-up
a) A schematic drawmg of the apparatus used for the measurements 1s shown m fig. 1. Basrcally it consists of a closed flow crrcurt, partially filled with hqurd helium The cncurt 1s thermally msulated from the helium rn the cryostat by a vacuum chamber (VC), except at one point where a heat exchanger (HE) grves a thermal contact. This was made of two blocks of smtered copper soldered into an oxygen free copper block The crrcurt further consists of a superleak (S), the series of stamless-steel caprllarres (CAP) m whrch the flow is investigated, an evaporator, and a condensor at the heat exchanger. The mner diameter of the caprllarres 1s 620 pm In this system, a mass flow can be generated by producmg heat m one or both of two heaters, 1 and 2, placed before and after the caprllarres. The heat energy cannot be transported to the heat exchanger vra the superleak, so rt must cause evaporation in the left side of the apparatus, whrch rs followed by a condensation at the right side Then there wrll be a mass flow of vapour from left to right vra the top of the cucurt, which must be accompamed by a fluid flow from right to left vra the capillaries. For a stationary situation, one can deduce the energy balance from thermo-dynamrcal arguments:
HE
ci1+ & = APu(L + ST),
_L 52
f
--CAP -vc
Fig. 1. SchematIc drawmg of the apparatus CAP ISthe flowcapillary system, S ISthe superleak m the flow &cult, R the Helmholtz resonator, HE ISa heat exchanger, D ISa film constrIctIon, 1 and 2 mdrcate the pomts at the ends of the captiry system where heaters can generate stationary flows, Sl and S2 are glass standplpes m which the hehum levels can be observed. For further explanation see text
(1)
with bl and b2 the energy produced per second 111 heaters 1 and 2, A the cross-sectional area of the capillary, pu the mass current through the capillary, L the heat of evaporation, S the entropy per unit mass, T the temperature. Equation (1) thus grves the mass flow in terms of the total energy added to the system. The normalfluid flow rn the caprllary is determined by the heat transport through the capillary whrch 1s equal to the heat productron by heater 2 Q2 un=-_ ApST From (1) and (2) one can calculate the superflurd flow velocrty wrth the aid of PU = pnun + P,U;
01+d2 us=Aps(L+ST)-p,A~*
Pn
02
(3)
R R IJsselstern et al /Flow of superfluId 4He
314
In this way, any combination of vn and us can be estabhshed with the only restriction that pv > 0 and vn > 0 always hold (see fig. 2). More details on the prmciples and the use of a snnilar apparatus can be found m ref. 6. b) The Helmholtz resonator. In order to investigate the nature of the flows through the capillaries with the aid of second sound, a Helmholtz resonator (R m fig. 1) was made part of the capillary system. This resonator consists of a capdlary closed by two equal, relatively wide volumes. The second-sound wave in the capillary ISalways parallel to the stationary flow. When there is no net flow through the resonator (upper index 0), the second sound wdl be m resonance at a frequency 08, where I-(0$%42 II z,v 2 ---
[
_ps P 6r
2 _ _ 1 ( w& 12 ~11 1
)I
1 WgL 2 12 ( un
(4)
U$ = (p,/p,)(S2T/c,) is the second-sound velocity in bulk helium, 1,~ the length of the resonator capillary, and V the content of one volume. The three terms between the parentheses are first-order corrections. The first term arises from contributions of the viscous mode and of fust sound, which must both be present in order to satisfy all boundary conditions in the resonator. 6 = 4(217/w&) is the viscous penetration depth and r ISthe radius of the capillary. The last two terms are corrections determined by the ratio of the length ZR(L) of the capillary (volume) and 1/2n times the wavelength.
cm
SI 2
\
In a second-sound wave, m which all quantities are assumed to be proportional to exp (-iot + I/CC)when travelling in the pontlve x-chrectlon, the wave vector, still wthout a net flow, can be written as 2
6
1 f- Ps -+1-
~OC!_
4
(
pr
6 Ps - . pr 1
The Imaginary part gives the attenuation. This results in a quality factor Q;=$f.
s
The expementally found quahty factor Q” is smaller than the value calculated from (6). This IS due to addltlonal thermal energy losses into the walls of the resonator, which are proportional to d(2X/wp&. Here X ISthe thermal conductivity and pwc the specific heat per unit volume of the material of the wall. The same correction should be added to the real part of the wave vector. These corrections are assumed not to be altered by a stationary flow through the resonator, and wfl be left out. When there ISa stationary flow, then m general k will change and can be written m resonance as i-z+i(i+:)],
VS
t
(7)
CY’ and CX” representing the extra attenuation and dispersion, respectively. These quantities will be determmed expenmentally as a function of the flow velocities by looking at the real part (z,) and the imaginary part (zo) of the received second sound signal z at frequency ~8. Without a flow zi = 0 and z& gves the absolute value of the signal. The values of a’ and CY” can be expressed in terms of these measured quantities:
\
(8)
\ ‘A2 b
(9
4
cm s
---_ ---3-v=o
Fig. 2. Velocity values obtamable with the apparatus at 1.50 K.
a”=_
z;
cd;-zo
-.
Q" q,,
z,
A description of the principles of Helmholtz resonators can be found 111ref. 7.
(9)
R R Ihselstem et al /Flow of superfluzd 4He
c) In the same flow srtuatron where the secondsound quantities are measured also the difference m chemical potentral and temperature between both ends of the caprllary system are determined. The temperature difference AT IS directly measured with two Allen & Bradley carbon thermometers. The chemical potential difference & is determined by measurmg the height of the hehum levels m two glass standprpes (Sl and S2 m fig 1) connected to the apparatus via two superleaks. The temperature m the standprpes was always equal to the bath temperature, so the level difference, written as a pressure difference, lmmedrately gives the chemical potential difference between the standprpes. Because AP over the connectmg superleaks to the ends of the caprllarres always equals zero, rt also gives the chemical-potential drop over the capillary & and AT are defined as the value at end 1 mmus the value at end 2 of the caprllary system (see fig. 1). The diameter of the caprllary m the Helmholtz resonator was always the same as the one of the other flow caprllaries. During the measurmg runs, the temperature in the resonator was kept constant. Two versions of the apparatus have been used; m the first one, the diameter of the caprllaries was 620 m, the total length 44 cm, and the length of the caprllary of the resonator was equal to 1 cm. In the second version, the capillaries had the same diameter, a total length of 41 cm, and the resonator caprllary was 2 cm long. The two volumes of the resonator m both versions were equal to 0.034 cm3, yreldmg a resonance frequency of about 130 Hz and 85 Hz, respectively. The second-sound heater was always placed m the volume on the downstream side, the thermometer in the other. Thrs 1s done m order to avoid an extra energy flow through the resonator.
3. Results of the measurements at 1 SO K At 1 SO K, measurements are performed to mvestrgate systematically the whole velocity region that could be covered with the apparatus (see fig. 2). The total length 1 of the caprllaries (diameter 620 run) was 44 cm. The capillary of the Hehnholtz resonator was 1 cm long. At thrs temperature, the resonance frequency without a statronary flow (a@ was found at 132.0 Hz.
315
In every stationary flow situation, the extra attenuatron cr’ and the extra drspersron o” of the second sound m the resonator are measured, as well as the chemrcal-potential drop & and temperature drop AT over all caprllarres. The accuracy of the data for CY’/W~ rs 1% and for CY”/W~about 5% (but not better than 10e3 absolute) To calculate these quantities, the quahty factor without a stationary flow (Q”) was used; thrs could be determined wrthm about 2%. The p& and pSAT data could be measured with an accuracy of 0.5 dyne/cm2 and 1.5 dyne/cm2, respectnely. The results for runs with a constant normal-fluid flow velocity u,, and varymg superflmd flow velocity u, are given m figs. 3a, b, c, d. When u,, = 0, Q’ mcreases approxrmately with the square of the superfluid velocity. Q” remams zero wrthm the measurmg accuracy until us has reached a value of about 1.8 cm/s and then rt nses m the positive drrection. p& and PSAT have approximately a third power dependence on us. At flows where u,, 1s kept constant at non-zero value, the run could be started at negative values of us, the mmmum value bemg determined by the condition that the total mass flow be greater than zero. Going to u, = 0, wrth a constant un of 3 cm/s, 0~’drops to a mrmmum value, ~1”changes sign and remams at a constant negative value, p& drops to zero, and the absolute value of pSAT decreases slightly. Then going to posrtrve us, the flow enters a region where oscrllatrons occur 111all four quantities. For (IL’and CY”,the amplitudes are grven by bars through the symbols (figs. 3a, b). The period 1s about 15 seconds. The oscrllatrons are of the same type as reported by Van der Heilden et al. [2] and equal to one of the two types measured by Slegtenhorst et al. [4]. In ref 3 the oscillations were explained by noting that mterpolating & measurements on both sides of the oscrllatory region, an mcreasing negative flow resistance for the superfluid would be obtamed, grvmg an unstable situation. Above values for uQof 1 cm/s, the attenuation increases agam, whrle the extra dispersion passes through a negative mrmmum value at a relative velocity un - us of about 1.5 cm/s. At the same value of vi, - us, pc\lr 1s found to go through zero, whrle -PSAT passes through a maxrmum. Then at hrgher u,, all quantities follow the same quahtatrve behaviour as for the pure superflurd flow. When u, is larger (M cm/s), no oscillatory region
R R IJsselstein et al /Flow of superflutd 4He
316
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Fig. 3. The extra attenuation (a), extra dlsperslon (b), chemical-potential difference (c) and temperature difference (d) as a func tion of us for various runs with v,, kept constant. T= 1.50 K, d = 620 pm, 1 = 44 cm, ~8 = 132.0 Hz
R R IJsselstem et
al
1s found. Instead, a certam positrve value for us 1s found where all four measured quantrtres decrease sharply wrthm a small velocity regron. Thus drop takes place at higher values of us when u,, 1s larger A way to present all data m one figure 1s to draw lines m the urn u, plane, along which the value of the quantrty does not change. Wrarda [ 1,7] has grven such a plot for the extra attenuation m a 1.05 mm diameter caprllary, whrle m ref. 2, a survey 1s given m this way of the measurements of pAp, #AT, and hp over a capillary of diameter 294 pm. The results of the attenuation and the dispersion in the present expenment are given in figs. 4a and 4b, respectively. The plots for pAp and PSAT are not presented here, but are very slmrlar to those of ref. 2. The actual dependences on the velocrtres are not very clear from those plots, but one can make some general remarks. The extra attenuation and drspersron at velocity combmatrons far away from the hne u,., = us (whrch m these figures 1s mamly on the right side) are almost completely determined by the relative velocrty In the area around u,, = us, (Y’seems to be determined by the mass flow velocrty u (the curves tend to be parallel to u = 0), whrle a” again 1s determined mamly by u, - us. Between the lines u,, - us = 0 and u, - u, = 3.4 cm/s, (Y”has negative values, outside thrs area it 1s positive. Fmally the area 111wluch oscrllatrons have been observed can be shown (dotted bulbs m figs. 4a and b). In order to investigate more precisely the role of the relative velocity m the various quantities, another way to assemble all data is used, in whrch cz’,(Y”,PAM, and pSAT are plotted agamst u, - u,. These plots are shown for the attenuation and dispersion cases (figs. 5a and 5b, respectively). In these graphs, all measurmg runs that are made at 1SO K are collected, 1.e. four runs with constant u, (including pure superfluid flow), pure normal-flmd flow, and two runs with constant mass flow. As mentioned in the mtroductron, only flows with normalfluid flow and mass flow 111the same drrectron could be produced. All points of hrghest relative velocrty m runs of constant un (these pomts represent flow with no net mass transport, or pure counterflow) coincide with points of the pure normal-fluid flow. The plots show two totally different velocity regions. When u, - u, is hrgher than 3 cm/s, all data appear to fit the same curve. When un - us has lower
/FZowof superjluld
317
4He
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Fig. 4. Lmes of constant extra attenuation (a) and extra dispersion (b) m the (u,, us) plane The dotted area gives the region 111wluch oscillations were found. T = 1 50 K, d =620jSn,z=44cnl,v8=132OHz.
values than 3 cm/s, thrs tendency to depend only on un - us is completely lost. In fig. 4a, we saw that m this region cr’ was mainly determmed by the mass flow velocity.
4. Combinations of the measured quantities One of the reasons to set up these experiments was to mvestrgate the relation between the gradients of thermodynamrc quantities on the one hand (i.e. pAl_c and PSAT, AP, equal to pAp + PSAT, was m most cases too small to determine wrth acceptable accuracy) and the influence of the flows on the second sound on the other hand. Although the four measured quantities themselves appear to be comphcated functions of the velocitres
R R IJsselstetn et al /Flow of superflutd 4He
318
I
(s-‘)
v,= 3 v,= 4
0
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Fig 5 The extra attenuation (a) and dlsperslon (b) for all measuring runs as a function of the relative velocity The bars through the symbols give the amphtudes of the oscillations. T = 1.50 K,d = 620 pm, I = 44 cm, 4 = 132 0 Hz.
in the flmd, combmations of some quantities give much more simple results. In fig. 6, for instance, the ratio between the gradient of the chemical potential and the extra attenuation is plotted against the relative velocity. Note that this quantity has the dlmension of a velocity. The graph sphts mto three branches which fit all measurmg pomts! The first branch, at relative velocities higher than 3.4 cm/s, coincides wrth the turbulent parts of pure normal-fluid flow and of flow without net mass transport (pure counterflow). The second branch unifies almost all measuring points with vn - v, less than 3.4 cm/s. In this branch, A/J changes sign at v, - us = 1.5 cm/s. Other authors (refs. 2 and 3) also reported that all situations with ,$J = 0 were found at one relative velocity. A pecuharity is that by extrapolating the first branch, one finds the same intersection with the horizontal axis. At negative velocity, the second branch joins the third one. The third branch consrsts mostly of data from the pure superfluid flow. The separate character
of this part follows more clearly from the following consrderatrons. When v, and us are replaced by -vn and -us respectively, the ratio of A$ and (~I/271also changes sign. Consequently all curves are symmetric with respect to the origin. Pure superfluid flow could only be produced m the direction defined as positive, but through the above argument we are allowed to plot also data for negative us (see dashed hne m fig. 6), which now clearly show up as a separate branch. This 1s the only branch grvmg a simple lmear relation between (A&J/((u’/2n) and u, - vs. Because the second and third branch seem to jom at negatrve vn - us, one can speculate whether the first and the thrrd one should also jam at hrgh positwe relative velocity. If one assumes that the third branch can be extrapolated linearly, they show mdeed a tendency m this direction. Unfortunately hrgher velocrtres could not be reached to check on this. In the next section, this matter will be given more attentron, also at other temperatures. Two remarks should be made: firstly
R R IJsselstem et al /Flow of superjluld 4He I
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(cm ) 5
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Fig. 6. The ratlo of the chemlcalqotentlal gradlent and the extra attenuation for all runs as a function of the relative velocity. The bars through the symbols mdlcate when OS& latlons were found. For the dashed hne see text. T = 1.50 K, d = 620 pm, I= 44 cm, ~8 = 132.0 Hz.
when there 1s no turbulence, which means laminar flow, both & and (Y’are identically zero, so these srtuatrons cannot be incorporated m this graph; secondly the runs that go through the oscillatory regron do not follow the high velocity part of the second branch. Coming from the first branch, (b/Z)/ (01’/211)falls directly to zero at u, - us = 3.4 cm/s, A/J and CX’start to oscillate at values lower than 3 cm/s and only below 2 cm/s the second branch 1s followed. The ratio of & and (Y’in the oscillatory regron could not be determined. The existence of three separate branches includes the necessity of at least two criteria on which one of the branches is chosen. The first one is obvrous from
319
fig. 6. un - us = 3.4 cm/s separates branches one and two. The second critermm is harder to find because of the lack of data for whrch the total mass current and normal fluid velocrty have opposrte sign. This region could not be reached with the present apparatus. On one side, the region 1s bounded by un = 0 and on the other side by u = 0. For these two types of flow, the data are located in branch 3 and 1, respectively. If the same branches exrst also m the unexplored region, one should find a lump somewhere when travelling through the area e.g wrth constant relative velocity from un = 0 to u = 0 The most probable place for such alump seems to be where the mass current changes sign, especially when u,, - us < 3.4 cm/s With the available data, however, it was not possible to find definitely a second crrtermm that should mdicate whether the flow srtuatron belongs 111 branch one or two on one hand, or m the third branch on the other hand. Another combinatron of measured quantities that appears to have a direct relation with the relative velocity 1s the ratio of the extra drspersron and the extra attenuation, (Y”/oL’.Its dependence on u, - us 1s shown 111fig. 7. Followmg the descrrptron of (A@/ (a’/2n), one can recognize the same three parts of fig. 6, but the separation 1s somewhat less pronounced. The second branch 1s characterized by the negative values of the ratio CIL”/OL’. This part shows on the average a linear dependence on un - us, with a negative slope, but is rounded off where the transition to the first branch at 3.4 cm/s 1s approached Instead of the gap at the transrtron point m the prevrous figure, here a very steep change from negative to positive values 1s found before the first branch 1s reached. not only has the sign of the values for CY”/QI’ itself changed, but also the srgn of its denvative, which gives a completely different behavrour of the two branches The thud branch could not be measured with sufficrent accuracy, so not much can be sard about thrs (there seems to be a fimte velocity regron for the pure superfluid flow where a”/&’ remams zero; this 1s more clearly found at other temperatures). As sard m the introductron, efforts ‘to interpret these results are bemg made, and wrll be published m a following article, but one should expect that (A&)/ (ol’/27r) and cr”/o’, or perhaps other combmatrons of the four measured quantities, can grve basic mformatron on the processes governing the flow.
R R IJsselstem et al /Flow of superflwd 4He
320 I
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2
3
4
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6
Fig. 7. The ratlo of the extra dlspewon and the extra attenuation for all runs as a function of V, - us. The bars through
the symbols gve a rough indlcatlon of the amphtude of osclllatlon. T = 1.50 K, d = 620 pm, I = 44 cm, ~8 = 132.0 Hz.
5. Flow with either us = 0 or un = 0 a) In the previous sectlon we saw that data for flows with u, = 0 and with us = 0 were located in the third and first branches respectively in the plots of (&/Z)/(ol’/27r) and CY’/CY’ against u, - us (figs. 6 and 7). The question as to whether or not the two branches would melt together at lugh relative velocities could not be settled there. This was a reason to study the flows with either u, = 0 or us = 0 more thoroughly. The best measurements with these types of flow were obtamed at 1.70 K at a zero resonant frequency ~00equal to 87.67 Hz and a zero attenuation v~/Q: of 2.55 s-l. The total length of the captiarles was in tlus case 41 cm, the capillary of the Helmholtz resonator was 2 cm long. In fig. 8, the gradlent of the
-J
Fig. 8. The gradient of the chemical potential, the extra attenuation, and extra chsperslon versus v,, for a pure normalfluid flow at 1.70 K. d = 620 pm, 1= 41 cm, ~8 = 87.67 Hz.
chemical potential, the extra attenuation, and extra dispersion are gven for pure normal-fluid flow. At low velocltles (un G 1.8 cm/s) this flow 1s lammar and &A, Q’ and CY”remam zero. At velocltles above 3 cm/s pAp and CX’ can approxnnately be described by the often used thud and second power dependence on u,. In between, there 1s a flow region m which CX’rises rather smoothly to the values of the lughest regon and pAp first rises slowly and then jumps to the highest branch. With pure counterflow m tubes with circular cross-section, Ladner and Tough [ 81 found in then temperature and pressure difference data also two different regions (apart from the laminar part) which are probably analogous. In the mtermedlate region no extra dlsperslon 1s found, but above 3 cm/s (Y”rises contmuously from zero m the posltlve direction. The total dispersion, mcluding thermal effects, can be wrltten as &Q” - cu”/2n, with Q” the experimentally found quality factor without a stationary flow (m tti case
R R IJsselsteln et al /Flow of superfluid 4He
equal to 25). In fig. 8, one finds that r&Q0 - a”/2n changes sign at u,, = 3.8 cm/s, grvmg a negative drsperson of the wave vector at larger flow velocities. Takmg the sound velocrty u2 equal to the real part of w/k, one finds m that region, m a lmear approxrmatron, a value for u2 hrgher than the sound velocity uII m quiescent bulk helium, whrch seems rather peculiar. It appears, however, that rf one includes quadratic terms m the attenuation a value smaller than that of un is always found. b) One of the guide-lmes at the mvestrgatron of u, = 0 and us = 0 flows was contamed m a short paper of our group [9]. There measurements on the attenuatron m the two types of flow through a caprllary of 1 05 mm diameter at seven temperatures between 1.4 K and 2.0 K could be brought together m an unexpected way. It appeared that, plotted on a logarrthmrc scale, the data for the total attenuation m the fluid (equal to cr’/2n + $/Q$ in normal flow, and data for the extra attenuation a’/2n m superflow, could be fitted with the same straight lme. II DO-
I
I
321
In the prevrous section of the present article, the extra attenuation was always considered, whrch m fact lmphes an assumption that the vrscous damping of the second sound at the wall, present without stationary flow, 1s not affected by a superimposed net flow. If this 1s not a necessary assumption, one can mterpret the former treatment of ref. 9 by the following reasonmg’ when only the superfluid component has an average flow velocrty, the mitral attenuation is not affected, and so only the extra attenuation is related to the turbulence; m pure normal-fluid flow, however, there 1s a crrtrcal velocrty region below whrch the attenuation remams equal to the zero effect, the flow bemg lammar, and above whrch that effect is replaced by the attenuation connected with the turbulence. These arguments also appear to lead to mterestmg results for the present measurements, as is shown m fig. 9a for the attenuation at 1.70 K, and m fig. 9b for the data at 1 SO K presented m the prevrous section. At velocrtres above 3 cm/s, the data for the normal flow indeed follow the same path as those for the superfluid flow. One of the remarkable thmgs
/
T=170K b-‘1
1001
“0 +=
255s-’
/
I
(s-‘)
50-
20-
IO-
.--1 _ I”a
5-
OO 0
(a:)
OO
2I l1
VsCVn) cm/s
I 2
I
5
10
Fig. 9. Extra attenuation for pure superfluld flow versus us and the total attenuation for pure normal-flmd flow versus vn. The dashed lme mdlcates the value of the zero attenuation I&Q!. a) T = 1 70 K, d = 620 Nrn, I = 41 cm, ~8 = 87.67 Hz, the slope of the full line = 2.1. b) T = 1 50 K, d = 620 rm, I = 44 cm, u8 = 132.0 Hz, the slope IS 1.8.
322
R R IJsselstem et al /Flow of superjluld 4He
about these plots 1s that the value of us for whrch a’/27r becomes equal to the zero attenuation rmmedlately gives the value of ur, above whrch the total attenuation devrates from the zero value. The slopes of the lines are found to be 1.8 at 1.50 K and 2.1 at 1.70 K, and there is a tendency to a slowly rrsmg power dependence on the velocrty with rrsmg temperature. In classical hydrodynamrcs, an analogous behavrour 1s found for the pressure drop over a caprllary. There a lammar and a turbulent branch also exrst, aP bemg determmed by the one giving the highest value, the cntical velocity 1s grven by the mtersectron. In helium, then, there 1s an extra freedom: below the critical velocity, both branches can be followed, depending on the type of flow produced. It would be very interesting to mvestigate whether the same behavrour 1s found for AP with the two flows m helium.
lo-
5-
2-
l-
T .5-
=170K
0
'.'O -255s-' F71
(0V,=O)
ov,=o 2-
Vs(Vn) cm/s
c) For the drspersron, the same procedure can be used, but only with an extra assumption instead of taking the total drspersron equal to V&Q: - cy”/2n, the sum of the two contnbutrons must be taken. Then again, the two flows can be fitted by one line as 1s shown m fig. 10. As with the attenuation, the velocrty at which the superfluid flow data reach the value of the zero dispersion grves the velocrty at whrch the drspersron m the normal fluid flow starts to devrate from the mitral value. Note that this 1s a different velocity than m the case of the attenuation (compare figs. 9a and 10). There rt was the velocrty where the transitron from lammar flow to the intermediate region took place, while with the drspersron data it 1s the transrtron pomt between the mtermedrate and the highest region. The two “critrcal” velocrtres at 1.70 K are located at 1 4 cm/s and 3.0 cm/s. The slope of the line m fig. 10, which gives the power dependence on u,, and us, 1s 3.9. The agreement 1s also rather good at 1 50 K, but much less at 1.40 K. At 1 90 K the drspersron in the normal fluid flow first decreases after the “crrtrcal” velocity, and only later increases agam, wrth a tendency to join the data of the superfluid flow. Ajustrficatron for the above assumption concerning the change of the sign has not yet been found d) At 1.70 K, the values of p& for both types of flow can also be described by one relation with the
,11
-
1c
Fg. 10. Extra dlsperslon for pure superfhud flow versus us and the sum of zero and extra dlsperslon for pure normalfluid flow versus v,. The dashed lme mdlcates the value of the zero dispersion V&Q$ T = 1 70 K, d = 620 pm, I = 41 cm, v8 = 87 67 Hz The slope of the full lme IS 3 9.
velocrty, as 1s shown m fig. 11 (the slope 1s 3.1, exactly 1 lugher than for the attenuatron) One may then expect that when the same procedure 1s apphed m the graphs for (A/.1/C)/(ol’/2n) and CX”/CY’, the thud and the first branch should jom above 3 cm/s. Thrs is indeed confirmed in figs. 12 and 13. (In fig. 12, the full line does not go through the origm, this 1s due to the fact that & 1s found to be posrtrve at very low velocitres in pure superflurd flow. This 1s probably due to an error in the measuring mechanism, and was only found 111the results for the caprllanes of 41 cm total length.) e) Up to now, no real ins&t has been obtamed mto the way m whrch the mitral viscous attenuatron at the wall is influenced by the flow. The zero drspersron 1s equal to the zero value of the attenuatron, so one can hope that a thorough study of the drspersron data can contribute to an answer to thrs questron.
R R IJsselstem et al /Flow of superjluld 4He 5-
323
I
(cm/s 1
I
I
T
=
I
170K
4LJ
1 200-
1
[PAPI
3-
t
T =
170K 2-
l0 0
-I 1
z
Fig 11. The absolute value of ence versus the absolute value pure superflmd flow and pure I = 41 cm, ~8 = 87 67 Hz. The
3
1 1U
the chemical-potential dlfferof the relative velocity for both normal-fluld flow. d = 620 Mm, slope of the full hne IS 3.1.
6. Concluding remarks Many drfferent flow srtuatrons m a capillary with an mner drameter of 620 fl have been investigated by measurmg the four mdependant quantities &, AT, cr’and CX”.Each quantity appears to be a complicated function of the two average velocities u, and us, that could be adjusted mdependently. Choosing any two hnear combinations of u,, and us to descnbe the results cannot essentrally simplify the results. In spite of this it is found that the ratios of Ap and (Y’,and of on and o’ can be described wrth only one velocity, namely the relative velocity u,, - us. Thrs srgruficant result, which does not allow for a trrvral explanation, might rndicate m what way an analysis of the flow phenomena should be pursued. It also should be remembered that an analogous result, obtained with measurements of & and AT on a 294 pm capillary, 1s reported m ref. 2. The data for the two ratios &la’ and (Y”/(Y’are
00
0
(cm/, 1
0
0’0
’
Fig. 12. gradient (un) for d = 620
I
I
I
I
1
2
3
4
Ratio of the absolute value of the chemical-potentlal and the extra (total) attenuation as a function of us pure superfluid (normal fluid) flow at 1.70 K. pm, 2 = 41 cm, ~8 = 87.67 Hz.
located on three separate branches m such a way that for most values of un - us, there are two possible values for A/.&Y’and &“/a’. There seems to be a tendency toward a unique relation at higher relative velocities. Investrgatrons at other temperatures seem to indicate that this can happen at lower un - us when the temperature 1s hrgher. The way rn whrch the zero attenuation 1s rnfluenced by the flow seems to be different m the various branches. Possibly a deeper analysis of the dispersion data can bring more light in thrs matter. At some velocity combmatrons, devratrons from the three branches were seen. Thrs was in the region rn which oscillatrons occurred. In a capillary of 216 /.un (ref. 4), two types of oscrllations were found m measurements of A/J and AT, together wrth an extra region in whrch stable situations were found. Thrs may indicate that in narrow capillaries, more branches are found rn plots of A&Y’ and (~“/a’ m the nerghbourhood of the ongm. To check on these speculations experiments like the present one should be done in other geometries of the flow caprllary, and
324
R R IJsselstem et al /Flow of superfluid 4He
of the data for the drspersron, and of the combination of the varrous measured quantities, might give more interesting results. Thrs mvestrgatron ISpart of the research program of the “Strchtmg voor Fundamenteel Onderzoek der Materie (F.O.M.)“, whrch is financrally supported by the “Nederlandse Orgamsatie voor Zumer-Wetenschappehjk Onderzoek (Z.W.O.)“.
References ill T. M. Wrarda, Thesis, Leaden (1967). VI G. van der Heyden, A. G. M van der Boog and H. C. 131 I41
Vs( Vn) cm/S
o.o,1
2
2
5
10
Frg. 13. Ratio of extra (extra plus zero) drspersron and extra (total) attenuatron as a function of us (un) for pure superflmd (normal fluid) flow. d = 620 pm, 1= 41 cm, ~8 = 87.67 Hz The slope of the full hne IS 1.8.
wrth larger velocities, A further study of the large amount of information obtained from the present experiment IS strll m progress. Especially an analysis
ISI 161
[71 181 PI
Kramers, Physica 77 (1974) 487. (Commun. Kamerhngh Onnes Lab., Leaden No. 411~). W. de Haas and H. van Beelen, Physica 83B (1976) 129 (Commun. Kamerhngh Onnes Lab., Leaden No. 422b). R. P Slegtenhorst and H. van Beelen, Physrca 90B (1977) 245 (Commun. Kamerhngh Onnes Lab., Leaden No 4293). W. F Vmen, Proc. Roy. Sot A 240 (1957) 114 G van der Heqden, W J. P. de Voogt and H. C. Kramers, Physrca 59 (1972) 473 (Commun. Kamerlmgh Omres Lab., Leaden No 392a). H. C Kramers, J. Phys. Radium 23 (1962) 326. D. R. Ladner and J. T. Tough, Phys. Rev. B17 (1978) 1455. H. C. Kramers, T. M. Wrarda and G. van der Hegden, Physma 69 (1973) 245 (Commun,Kamerhngh Onnes Lab., Leaden, suppl. No. 129).
COMBINED MEASUREMENTS
R. R. IJSSELSTEIN, Kamerhngh
FLOW OF SUPERFLUID 4He
M. P. de GOEJE and H. C. KIUMERS
Onnes Laboratorium
(Communrcatton
ON VARIOUS TYPES OF STATIONARY
der Rllksunrversrtelt Lerden, Leaden, The Netherlands
No. 437~)
Received 25 January 1979
Experunents are carried out on the stationary flow of superfluid 4He through caprllarres of 620 pm diameter. In many flow srtuations (the transport velocities of the superfluid and the normal fluid were mdependently adJustable) at 1 50 K, four quantrtres have been measured the attenuation and dispersion of second sound m resonance, parallel to the stationary flow, together with the temperature and chemrcal-potentral difference between the two ends of the flow crrcurt. By combmmg the results of the varrous quantttres unexpected results are obtamed. In the second part, specral attentron 1s given (at a number of temperatures) to two types of flow. pure superfluid flow and pure normal-fluid flow. A short descrrptron of the experimental detatls ISalso grven 1. Introduction
All three methods provrde, 111prmcrple, different mformatron. The choice of one of the-three methods implies, m general, also a hmrtatron of the range of values for the dlmensron of the flow pipe. Method one grves most accurate results m narrow caprllaries (d 6 500 pm) whrle for methods 2 and 3 relatively wide tubes (d Z 5 mm) are often necessary As far as we know, experiments where two of these methods are used m the same geometry were performed almost exclusrvely by Vmen [5], who measured the temperature difference together with the second-sound attenuation. Wrth above consrderatrons, an experiment was set up 111whrch four quantities could be measured m flow situations obtamed by adjustmg mdependently the velocity of the normal component and of the superfluid component (of course, as u-r every flow experiment, only the average value of the velocity with respect to the cross-section could be controlled) Every combmatron of us and u,, could be established m this experiment, for whrch the total mass flow had the same drrectron as the normal-fluid flow. The quantities to be measured were the secondsound attenuatron Q’ together with the dispersion cr” and the difference m temperature and chemical potential (AT and &, respectively) between the two ends of the flow crrcurt. The dispersion was basically
One of the thin s that experimental research on flow of superfluid $ He needs most 1s a systematrc mvestrgatron of the relation between the measured quantrtres, and of then dependence on the externally controllable parameters such as the geometry, both transport velocrtres, and the temperature. A severe restrictron m almost all experiments 1s the fmed relatron between the two velocities m the flow (e.g. pure counterflow, pure superfluid flow). This laboratory 1s one of the few places where measurements are done on many different combmatrons of these (refs. 1,2,
394). A second restrictron 1s the choice of one quantity (m some cases two quantrtres) that is going to be measured. In almost all expenments, one of three methods 1s used to mvestrgate the flow. 1. determmatron of the temperafure difference between the two ends of the flow pipe, sometunes together with the chemical-potential difference or the pressure difference, 2. measurement of the attenuation of second sound due to the flow (sometimes the change of the second-sound velocity 1s mvestrgated); 3. the use of ions as probes m the flow, this method 1s rather new and gives very interesting results. 312
R R IJsselstem et al /Flow of superfluid 4He
measured as a change of the resonance frequency of the resonator The best compromrse in looking for a geometry in whrch all four quantities could be measured with acceptable accuracy was found 111a caprllary of drameter 600 pm and length about 40 cm. In order to be able to brmg the second sound mto resonance m such a geometry, a Helmholtz resonator was used with a resonance frequency m the order of 100 Hz. In such a resonator, the direction of the wave vector 1s always parallel to the flow dnectron. In thus article the results of the measurements are presented. It 1s preceded by a short description of the experimental set-up and the measuring methods. An attempt at interpretation of the results 1s m progress and will be published later.
-w
2. Experimental
313
set-up
a) A schematic drawmg of the apparatus used for the measurements 1s shown m fig. 1. Basrcally it consists of a closed flow crrcurt, partially filled with hqurd helium The cncurt 1s thermally msulated from the helium rn the cryostat by a vacuum chamber (VC), except at one point where a heat exchanger (HE) grves a thermal contact. This was made of two blocks of smtered copper soldered into an oxygen free copper block The crrcurt further consists of a superleak (S), the series of stamless-steel caprllarres (CAP) m whrch the flow is investigated, an evaporator, and a condensor at the heat exchanger. The mner diameter of the caprllarres 1s 620 pm In this system, a mass flow can be generated by producmg heat m one or both of two heaters, 1 and 2, placed before and after the caprllarres. The heat energy cannot be transported to the heat exchanger vra the superleak, so rt must cause evaporation in the left side of the apparatus, whrch rs followed by a condensation at the right side Then there wrll be a mass flow of vapour from left to right vra the top of the cucurt, which must be accompamed by a fluid flow from right to left vra the capillaries. For a stationary situation, one can deduce the energy balance from thermo-dynamrcal arguments:
HE
ci1+ & = APu(L + ST),
_L 52
f
--CAP -vc
Fig. 1. SchematIc drawmg of the apparatus CAP ISthe flowcapillary system, S ISthe superleak m the flow &cult, R the Helmholtz resonator, HE ISa heat exchanger, D ISa film constrIctIon, 1 and 2 mdrcate the pomts at the ends of the captiry system where heaters can generate stationary flows, Sl and S2 are glass standplpes m which the hehum levels can be observed. For further explanation see text
(1)
with bl and b2 the energy produced per second 111 heaters 1 and 2, A the cross-sectional area of the capillary, pu the mass current through the capillary, L the heat of evaporation, S the entropy per unit mass, T the temperature. Equation (1) thus grves the mass flow in terms of the total energy added to the system. The normalfluid flow rn the caprllary is determined by the heat transport through the capillary whrch 1s equal to the heat productron by heater 2 Q2 un=-_ ApST From (1) and (2) one can calculate the superflurd flow velocrty wrth the aid of PU = pnun + P,U;
01+d2 us=Aps(L+ST)-p,A~*
Pn
02
(3)
R R IJsselstern et al /Flow of superfluId 4He
314
In this way, any combination of vn and us can be estabhshed with the only restriction that pv > 0 and vn > 0 always hold (see fig. 2). More details on the prmciples and the use of a snnilar apparatus can be found m ref. 6. b) The Helmholtz resonator. In order to investigate the nature of the flows through the capillaries with the aid of second sound, a Helmholtz resonator (R m fig. 1) was made part of the capillary system. This resonator consists of a capdlary closed by two equal, relatively wide volumes. The second-sound wave in the capillary ISalways parallel to the stationary flow. When there is no net flow through the resonator (upper index 0), the second sound wdl be m resonance at a frequency 08, where I-(0$%42 II z,v 2 ---
[
_ps P 6r
2 _ _ 1 ( w& 12 ~11 1
)I
1 WgL 2 12 ( un
(4)
U$ = (p,/p,)(S2T/c,) is the second-sound velocity in bulk helium, 1,~ the length of the resonator capillary, and V the content of one volume. The three terms between the parentheses are first-order corrections. The first term arises from contributions of the viscous mode and of fust sound, which must both be present in order to satisfy all boundary conditions in the resonator. 6 = 4(217/w&) is the viscous penetration depth and r ISthe radius of the capillary. The last two terms are corrections determined by the ratio of the length ZR(L) of the capillary (volume) and 1/2n times the wavelength.
cm
SI 2
\
In a second-sound wave, m which all quantities are assumed to be proportional to exp (-iot + I/CC)when travelling in the pontlve x-chrectlon, the wave vector, still wthout a net flow, can be written as 2
6
1 f- Ps -+1-
~OC!_
4
(
pr
6 Ps - . pr 1
The Imaginary part gives the attenuation. This results in a quality factor Q;=$f.
s
The expementally found quahty factor Q” is smaller than the value calculated from (6). This IS due to addltlonal thermal energy losses into the walls of the resonator, which are proportional to d(2X/wp&. Here X ISthe thermal conductivity and pwc the specific heat per unit volume of the material of the wall. The same correction should be added to the real part of the wave vector. These corrections are assumed not to be altered by a stationary flow through the resonator, and wfl be left out. When there ISa stationary flow, then m general k will change and can be written m resonance as i-z+i(i+:)],
VS
t
(7)
CY’ and CX” representing the extra attenuation and dispersion, respectively. These quantities will be determmed expenmentally as a function of the flow velocities by looking at the real part (z,) and the imaginary part (zo) of the received second sound signal z at frequency ~8. Without a flow zi = 0 and z& gves the absolute value of the signal. The values of a’ and CY” can be expressed in terms of these measured quantities:
\
(8)
\ ‘A2 b
(9
4
cm s
---_ ---3-v=o
Fig. 2. Velocity values obtamable with the apparatus at 1.50 K.
a”=_
z;
cd;-zo
-.
Q" q,,
z,
A description of the principles of Helmholtz resonators can be found 111ref. 7.
(9)
R R Ihselstem et al /Flow of superfluzd 4He
c) In the same flow srtuatron where the secondsound quantities are measured also the difference m chemical potentral and temperature between both ends of the caprllary system are determined. The temperature difference AT IS directly measured with two Allen & Bradley carbon thermometers. The chemical potential difference & is determined by measurmg the height of the hehum levels m two glass standprpes (Sl and S2 m fig 1) connected to the apparatus via two superleaks. The temperature m the standprpes was always equal to the bath temperature, so the level difference, written as a pressure difference, lmmedrately gives the chemical potential difference between the standprpes. Because AP over the connectmg superleaks to the ends of the caprllarres always equals zero, rt also gives the chemical-potential drop over the capillary & and AT are defined as the value at end 1 mmus the value at end 2 of the caprllary system (see fig. 1). The diameter of the caprllary m the Helmholtz resonator was always the same as the one of the other flow caprllaries. During the measurmg runs, the temperature in the resonator was kept constant. Two versions of the apparatus have been used; m the first one, the diameter of the caprllaries was 620 m, the total length 44 cm, and the length of the caprllary of the resonator was equal to 1 cm. In the second version, the capillaries had the same diameter, a total length of 41 cm, and the resonator caprllary was 2 cm long. The two volumes of the resonator m both versions were equal to 0.034 cm3, yreldmg a resonance frequency of about 130 Hz and 85 Hz, respectively. The second-sound heater was always placed m the volume on the downstream side, the thermometer in the other. Thrs 1s done m order to avoid an extra energy flow through the resonator.
3. Results of the measurements at 1 SO K At 1 SO K, measurements are performed to mvestrgate systematically the whole velocity region that could be covered with the apparatus (see fig. 2). The total length 1 of the caprllaries (diameter 620 run) was 44 cm. The capillary of the Hehnholtz resonator was 1 cm long. At thrs temperature, the resonance frequency without a statronary flow (a@ was found at 132.0 Hz.
315
In every stationary flow situation, the extra attenuatron cr’ and the extra drspersron o” of the second sound m the resonator are measured, as well as the chemrcal-potential drop & and temperature drop AT over all caprllarres. The accuracy of the data for CY’/W~ rs 1% and for CY”/W~about 5% (but not better than 10e3 absolute) To calculate these quantities, the quahty factor without a stationary flow (Q”) was used; thrs could be determined wrthm about 2%. The p& and pSAT data could be measured with an accuracy of 0.5 dyne/cm2 and 1.5 dyne/cm2, respectnely. The results for runs with a constant normal-fluid flow velocity u,, and varymg superflmd flow velocity u, are given m figs. 3a, b, c, d. When u,, = 0, Q’ mcreases approxrmately with the square of the superfluid velocity. Q” remams zero wrthm the measurmg accuracy until us has reached a value of about 1.8 cm/s and then rt nses m the positive drrection. p& and PSAT have approximately a third power dependence on us. At flows where u,, 1s kept constant at non-zero value, the run could be started at negative values of us, the mmmum value bemg determined by the condition that the total mass flow be greater than zero. Going to u, = 0, wrth a constant un of 3 cm/s, 0~’drops to a mrmmum value, ~1”changes sign and remams at a constant negative value, p& drops to zero, and the absolute value of pSAT decreases slightly. Then going to posrtrve us, the flow enters a region where oscrllatrons occur 111all four quantities. For (IL’and CY”,the amplitudes are grven by bars through the symbols (figs. 3a, b). The period 1s about 15 seconds. The oscrllatrons are of the same type as reported by Van der Heilden et al. [2] and equal to one of the two types measured by Slegtenhorst et al. [4]. In ref 3 the oscillations were explained by noting that mterpolating & measurements on both sides of the oscrllatory region, an mcreasing negative flow resistance for the superfluid would be obtamed, grvmg an unstable situation. Above values for uQof 1 cm/s, the attenuation increases agam, whrle the extra dispersion passes through a negative mrmmum value at a relative velocity un - us of about 1.5 cm/s. At the same value of vi, - us, pc\lr 1s found to go through zero, whrle -PSAT passes through a maxrmum. Then at hrgher u,, all quantities follow the same quahtatrve behaviour as for the pure superflurd flow. When u, is larger (M cm/s), no oscillatory region
R R IJsselstein et al /Flow of superflutd 4He
316
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1
2
Fig. 3. The extra attenuation (a), extra dlsperslon (b), chemical-potential difference (c) and temperature difference (d) as a func tion of us for various runs with v,, kept constant. T= 1.50 K, d = 620 pm, 1 = 44 cm, ~8 = 132.0 Hz
R R IJsselstem et
al
1s found. Instead, a certam positrve value for us 1s found where all four measured quantrtres decrease sharply wrthm a small velocity regron. Thus drop takes place at higher values of us when u,, 1s larger A way to present all data m one figure 1s to draw lines m the urn u, plane, along which the value of the quantrty does not change. Wrarda [ 1,7] has grven such a plot for the extra attenuation m a 1.05 mm diameter caprllary, whrle m ref. 2, a survey 1s given m this way of the measurements of pAp, #AT, and hp over a capillary of diameter 294 pm. The results of the attenuation and the dispersion in the present expenment are given in figs. 4a and 4b, respectively. The plots for pAp and PSAT are not presented here, but are very slmrlar to those of ref. 2. The actual dependences on the velocrtres are not very clear from those plots, but one can make some general remarks. The extra attenuation and drspersron at velocity combmatrons far away from the hne u,., = us (whrch m these figures 1s mamly on the right side) are almost completely determined by the relative velocrty In the area around u,, = us, (Y’seems to be determined by the mass flow velocrty u (the curves tend to be parallel to u = 0), whrle a” again 1s determined mamly by u, - us. Between the lines u,, - us = 0 and u, - u, = 3.4 cm/s, (Y”has negative values, outside thrs area it 1s positive. Fmally the area 111wluch oscrllatrons have been observed can be shown (dotted bulbs m figs. 4a and b). In order to investigate more precisely the role of the relative velocity m the various quantities, another way to assemble all data is used, in whrch cz’,(Y”,PAM, and pSAT are plotted agamst u, - u,. These plots are shown for the attenuation and dispersion cases (figs. 5a and 5b, respectively). In these graphs, all measurmg runs that are made at 1SO K are collected, 1.e. four runs with constant u, (including pure superfluid flow), pure normal-flmd flow, and two runs with constant mass flow. As mentioned in the mtroductron, only flows with normalfluid flow and mass flow 111the same drrectron could be produced. All points of hrghest relative velocrty m runs of constant un (these pomts represent flow with no net mass transport, or pure counterflow) coincide with points of the pure normal-fluid flow. The plots show two totally different velocity regions. When u, - u, is hrgher than 3 cm/s, all data appear to fit the same curve. When un - us has lower
/FZowof superjluld
317
4He
2
1 Y t 0
0
cm S
/
1
2
I
I
3
4
cm/,
I $-vn-vs= 0 104 a” --()-=
/
5
I
const
WO
/
25-m
/
/
8
4 1
/
0
-4 -8
c-
/
-
-60 gp
-40
-20
L_
(b)
80
/
/
/
Vs
,.
/
I
/
0
,;_-gjk H
---_ L"-o>---_o
0
V”
I
I
1
2
/
320
8 ---_
3
I
4
cm/s
5
Fig. 4. Lmes of constant extra attenuation (a) and extra dispersion (b) m the (u,, us) plane The dotted area gives the region 111wluch oscillations were found. T = 1 50 K, d =620jSn,z=44cnl,v8=132OHz.
values than 3 cm/s, thrs tendency to depend only on un - us is completely lost. In fig. 4a, we saw that m this region cr’ was mainly determmed by the mass flow velocity.
4. Combinations of the measured quantities One of the reasons to set up these experiments was to mvestrgate the relation between the gradients of thermodynamrc quantities on the one hand (i.e. pAl_c and PSAT, AP, equal to pAp + PSAT, was m most cases too small to determine wrth acceptable accuracy) and the influence of the flows on the second sound on the other hand. Although the four measured quantities themselves appear to be comphcated functions of the velocitres
R R IJsselstetn et al /Flow of superflutd 4He
318
I
(s-‘)
v,= 3 v,= 4
0
0
0 20
v,=
*
l
v,=
+
v
‘“071
V
=
4
c@ +
475
(s-'1
0
160
I
1
I
I
0
v,=o
A
V”
0
v,=3
P
V”
0 l
V, vs
IVC
I
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=
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=
4 l
=475 =
0
+
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=
1 60
71 i
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I
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-a’ 2rc
2
64
(b)
_/ 10 a
0 0
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5-
“8,
x
xxx
A
0
0
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OfJ+ 0 I -3
-2
I -1
I
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I
I
,
,
,
123456
-1
Fig 5 The extra attenuation (a) and dlsperslon (b) for all measuring runs as a function of the relative velocity The bars through the symbols give the amphtudes of the oscillations. T = 1.50 K,d = 620 pm, I = 44 cm, 4 = 132 0 Hz.
in the flmd, combmations of some quantities give much more simple results. In fig. 6, for instance, the ratio between the gradient of the chemical potential and the extra attenuation is plotted against the relative velocity. Note that this quantity has the dlmension of a velocity. The graph sphts mto three branches which fit all measurmg pomts! The first branch, at relative velocities higher than 3.4 cm/s, coincides wrth the turbulent parts of pure normal-fluid flow and of flow without net mass transport (pure counterflow). The second branch unifies almost all measuring points with vn - v, less than 3.4 cm/s. In this branch, A/J changes sign at v, - us = 1.5 cm/s. Other authors (refs. 2 and 3) also reported that all situations with ,$J = 0 were found at one relative velocity. A pecuharity is that by extrapolating the first branch, one finds the same intersection with the horizontal axis. At negative velocity, the second branch joins the third one. The third branch consrsts mostly of data from the pure superfluid flow. The separate character
of this part follows more clearly from the following consrderatrons. When v, and us are replaced by -vn and -us respectively, the ratio of A$ and (~I/271also changes sign. Consequently all curves are symmetric with respect to the origin. Pure superfluid flow could only be produced m the direction defined as positive, but through the above argument we are allowed to plot also data for negative us (see dashed hne m fig. 6), which now clearly show up as a separate branch. This 1s the only branch grvmg a simple lmear relation between (A&J/((u’/2n) and u, - vs. Because the second and third branch seem to jom at negatrve vn - us, one can speculate whether the first and the thrrd one should also jam at hrgh positwe relative velocity. If one assumes that the third branch can be extrapolated linearly, they show mdeed a tendency m this direction. Unfortunately hrgher velocrtres could not be reached to check on this. In the next section, this matter will be given more attentron, also at other temperatures. Two remarks should be made: firstly
R R IJsselstem et al /Flow of superjluld 4He I
I
I
4
(cm ) 5
3
I
I
O
v,= 0
A
v,=
2
0
v,=
3
v
v,=4
0 0 +
v,= v,=
X
Vs160
v
I-
l l
J 475 0
=
‘0
I
s
071 i # d+
2
1
0
;4 Ok+ -1
P
X0
0 8 -2
Vn-Vs I
I
-3
km/,)
0
-2
-1
0
I
I
I
I
I
c
1
2
3
4
5
6
Fig. 6. The ratlo of the chemlcalqotentlal gradlent and the extra attenuation for all runs as a function of the relative velocity. The bars through the symbols mdlcate when OS& latlons were found. For the dashed hne see text. T = 1.50 K, d = 620 pm, I= 44 cm, ~8 = 132.0 Hz.
when there 1s no turbulence, which means laminar flow, both & and (Y’are identically zero, so these srtuatrons cannot be incorporated m this graph; secondly the runs that go through the oscillatory regron do not follow the high velocity part of the second branch. Coming from the first branch, (b/Z)/ (01’/211)falls directly to zero at u, - us = 3.4 cm/s, A/J and CX’start to oscillate at values lower than 3 cm/s and only below 2 cm/s the second branch 1s followed. The ratio of & and (Y’in the oscillatory regron could not be determined. The existence of three separate branches includes the necessity of at least two criteria on which one of the branches is chosen. The first one is obvrous from
319
fig. 6. un - us = 3.4 cm/s separates branches one and two. The second critermm is harder to find because of the lack of data for whrch the total mass current and normal fluid velocrty have opposrte sign. This region could not be reached with the present apparatus. On one side, the region 1s bounded by un = 0 and on the other side by u = 0. For these two types of flow, the data are located in branch 3 and 1, respectively. If the same branches exrst also m the unexplored region, one should find a lump somewhere when travelling through the area e.g wrth constant relative velocity from un = 0 to u = 0 The most probable place for such alump seems to be where the mass current changes sign, especially when u,, - us < 3.4 cm/s With the available data, however, it was not possible to find definitely a second crrtermm that should mdicate whether the flow srtuatron belongs 111 branch one or two on one hand, or m the third branch on the other hand. Another combinatron of measured quantities that appears to have a direct relation with the relative velocity 1s the ratio of the extra drspersron and the extra attenuation, (Y”/oL’.Its dependence on u, - us 1s shown 111fig. 7. Followmg the descrrptron of (A@/ (a’/2n), one can recognize the same three parts of fig. 6, but the separation 1s somewhat less pronounced. The second branch 1s characterized by the negative values of the ratio CIL”/OL’. This part shows on the average a linear dependence on un - us, with a negative slope, but is rounded off where the transition to the first branch at 3.4 cm/s 1s approached Instead of the gap at the transrtron point m the prevrous figure, here a very steep change from negative to positive values 1s found before the first branch 1s reached. not only has the sign of the values for CY”/QI’ itself changed, but also the srgn of its denvative, which gives a completely different behavrour of the two branches The thud branch could not be measured with sufficrent accuracy, so not much can be sard about thrs (there seems to be a fimte velocity regron for the pure superfluid flow where a”/&’ remams zero; this 1s more clearly found at other temperatures). As sard m the introductron, efforts ‘to interpret these results are bemg made, and wrll be published m a following article, but one should expect that (A&)/ (ol’/27r) and cr”/o’, or perhaps other combmatrons of the four measured quantities, can grve basic mformatron on the processes governing the flow.
R R IJsselstem et al /Flow of superflwd 4He
320 I
I
I
I
I
’
I
0 18 V,=2
0
v,=
3
v 0
v,=4 v,=
475
012-
0 +
v,=o v90.71
OIO-
X
v
-
a” 7 1
008-
N
I”
170K
0 0
0 0
#G 9 vt
Vo 10 I-
160 +d
0-
f
A -&-
3
5 0
(s-‘1
‘I t
2(3
004
Y
X
b
Ox
t
I
v, = 0
15
006-
002
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A
014
I
v,=o
0
0 16 -
6.
Oo
0 L
l(1
(1
-0
06
-0
08
-0.10
-3
-2
-1
0
1
2
3
4
5
6
Fig. 7. The ratlo of the extra dlspewon and the extra attenuation for all runs as a function of V, - us. The bars through
the symbols gve a rough indlcatlon of the amphtude of osclllatlon. T = 1.50 K, d = 620 pm, I = 44 cm, ~8 = 132.0 Hz.
5. Flow with either us = 0 or un = 0 a) In the previous sectlon we saw that data for flows with u, = 0 and with us = 0 were located in the third and first branches respectively in the plots of (&/Z)/(ol’/27r) and CY’/CY’ against u, - us (figs. 6 and 7). The question as to whether or not the two branches would melt together at lugh relative velocities could not be settled there. This was a reason to study the flows with either u, = 0 or us = 0 more thoroughly. The best measurements with these types of flow were obtamed at 1.70 K at a zero resonant frequency ~00equal to 87.67 Hz and a zero attenuation v~/Q: of 2.55 s-l. The total length of the captiarles was in tlus case 41 cm, the capillary of the Helmholtz resonator was 2 cm long. In fig. 8, the gradlent of the
-J
Fig. 8. The gradient of the chemical potential, the extra attenuation, and extra chsperslon versus v,, for a pure normalfluid flow at 1.70 K. d = 620 pm, 1= 41 cm, ~8 = 87.67 Hz.
chemical potential, the extra attenuation, and extra dispersion are gven for pure normal-fluid flow. At low velocltles (un G 1.8 cm/s) this flow 1s lammar and &A, Q’ and CY”remam zero. At velocltles above 3 cm/s pAp and CX’ can approxnnately be described by the often used thud and second power dependence on u,. In between, there 1s a flow region m which CX’rises rather smoothly to the values of the lughest regon and pAp first rises slowly and then jumps to the highest branch. With pure counterflow m tubes with circular cross-section, Ladner and Tough [ 81 found in then temperature and pressure difference data also two different regions (apart from the laminar part) which are probably analogous. In the mtermedlate region no extra dlsperslon 1s found, but above 3 cm/s (Y”rises contmuously from zero m the posltlve direction. The total dispersion, mcluding thermal effects, can be wrltten as &Q” - cu”/2n, with Q” the experimentally found quality factor without a stationary flow (m tti case
R R IJsselsteln et al /Flow of superfluid 4He
equal to 25). In fig. 8, one finds that r&Q0 - a”/2n changes sign at u,, = 3.8 cm/s, grvmg a negative drsperson of the wave vector at larger flow velocities. Takmg the sound velocrty u2 equal to the real part of w/k, one finds m that region, m a lmear approxrmatron, a value for u2 hrgher than the sound velocity uII m quiescent bulk helium, whrch seems rather peculiar. It appears, however, that rf one includes quadratic terms m the attenuation a value smaller than that of un is always found. b) One of the guide-lmes at the mvestrgatron of u, = 0 and us = 0 flows was contamed m a short paper of our group [9]. There measurements on the attenuatron m the two types of flow through a caprllary of 1 05 mm diameter at seven temperatures between 1.4 K and 2.0 K could be brought together m an unexpected way. It appeared that, plotted on a logarrthmrc scale, the data for the total attenuation m the fluid (equal to cr’/2n + $/Q$ in normal flow, and data for the extra attenuation a’/2n m superflow, could be fitted with the same straight lme. II DO-
I
I
321
In the prevrous section of the present article, the extra attenuation was always considered, whrch m fact lmphes an assumption that the vrscous damping of the second sound at the wall, present without stationary flow, 1s not affected by a superimposed net flow. If this 1s not a necessary assumption, one can mterpret the former treatment of ref. 9 by the following reasonmg’ when only the superfluid component has an average flow velocrty, the mitral attenuation is not affected, and so only the extra attenuation is related to the turbulence; m pure normal-fluid flow, however, there 1s a crrtrcal velocrty region below whrch the attenuation remams equal to the zero effect, the flow bemg lammar, and above whrch that effect is replaced by the attenuation connected with the turbulence. These arguments also appear to lead to mterestmg results for the present measurements, as is shown m fig. 9a for the attenuation at 1.70 K, and m fig. 9b for the data at 1 SO K presented m the prevrous section. At velocrtres above 3 cm/s, the data for the normal flow indeed follow the same path as those for the superfluid flow. One of the remarkable thmgs
/
T=170K b-‘1
1001
“0 +=
255s-’
/
I
(s-‘)
50-
20-
IO-
.--1 _ I”a
5-
OO 0
(a:)
OO
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VsCVn) cm/s
I 2
I
5
10
Fig. 9. Extra attenuation for pure superfluld flow versus us and the total attenuation for pure normal-flmd flow versus vn. The dashed lme mdlcates the value of the zero attenuation I&Q!. a) T = 1 70 K, d = 620 Nrn, I = 41 cm, ~8 = 87.67 Hz, the slope of the full line = 2.1. b) T = 1 50 K, d = 620 rm, I = 44 cm, u8 = 132.0 Hz, the slope IS 1.8.
322
R R IJsselstem et al /Flow of superjluld 4He
about these plots 1s that the value of us for whrch a’/27r becomes equal to the zero attenuation rmmedlately gives the value of ur, above whrch the total attenuation devrates from the zero value. The slopes of the lines are found to be 1.8 at 1.50 K and 2.1 at 1.70 K, and there is a tendency to a slowly rrsmg power dependence on the velocrty with rrsmg temperature. In classical hydrodynamrcs, an analogous behavrour 1s found for the pressure drop over a caprllary. There a lammar and a turbulent branch also exrst, aP bemg determmed by the one giving the highest value, the cntical velocity 1s grven by the mtersectron. In helium, then, there 1s an extra freedom: below the critical velocity, both branches can be followed, depending on the type of flow produced. It would be very interesting to mvestigate whether the same behavrour 1s found for AP with the two flows m helium.
lo-
5-
2-
l-
T .5-
=170K
0
'.'O -255s-' F71
(0V,=O)
ov,=o 2-
Vs(Vn) cm/s
c) For the drspersron, the same procedure can be used, but only with an extra assumption instead of taking the total drspersron equal to V&Q: - cy”/2n, the sum of the two contnbutrons must be taken. Then again, the two flows can be fitted by one line as 1s shown m fig. 10. As with the attenuation, the velocrty at which the superfluid flow data reach the value of the zero dispersion grves the velocrty at whrch the drspersron m the normal fluid flow starts to devrate from the mitral value. Note that this 1s a different velocity than m the case of the attenuation (compare figs. 9a and 10). There rt was the velocrty where the transitron from lammar flow to the intermediate region took place, while with the drspersron data it 1s the transrtron pomt between the mtermedrate and the highest region. The two “critrcal” velocrtres at 1.70 K are located at 1 4 cm/s and 3.0 cm/s. The slope of the line m fig. 10, which gives the power dependence on u,, and us, 1s 3.9. The agreement 1s also rather good at 1 50 K, but much less at 1.40 K. At 1 90 K the drspersron in the normal fluid flow first decreases after the “crrtrcal” velocity, and only later increases agam, wrth a tendency to join the data of the superfluid flow. Ajustrficatron for the above assumption concerning the change of the sign has not yet been found d) At 1.70 K, the values of p& for both types of flow can also be described by one relation with the
,11
-
1c
Fg. 10. Extra dlsperslon for pure superfhud flow versus us and the sum of zero and extra dlsperslon for pure normalfluid flow versus v,. The dashed lme mdlcates the value of the zero dispersion V&Q$ T = 1 70 K, d = 620 pm, I = 41 cm, v8 = 87 67 Hz The slope of the full lme IS 3 9.
velocrty, as 1s shown m fig. 11 (the slope 1s 3.1, exactly 1 lugher than for the attenuatron) One may then expect that when the same procedure 1s apphed m the graphs for (A/.1/C)/(ol’/2n) and CX”/CY’, the thud and the first branch should jom above 3 cm/s. Thrs is indeed confirmed in figs. 12 and 13. (In fig. 12, the full line does not go through the origm, this 1s due to the fact that & 1s found to be posrtrve at very low velocitres in pure superflurd flow. This 1s probably due to an error in the measuring mechanism, and was only found 111the results for the caprllanes of 41 cm total length.) e) Up to now, no real ins&t has been obtamed mto the way m whrch the mitral viscous attenuatron at the wall is influenced by the flow. The zero drspersron 1s equal to the zero value of the attenuatron, so one can hope that a thorough study of the drspersron data can contribute to an answer to thrs questron.
R R IJsselstem et al /Flow of superjluld 4He 5-
323
I
(cm/s 1
I
I
T
=
I
170K
4LJ
1 200-
1
[PAPI
3-
t
T =
170K 2-
l0 0
-I 1
z
Fig 11. The absolute value of ence versus the absolute value pure superflmd flow and pure I = 41 cm, ~8 = 87 67 Hz. The
3
1 1U
the chemical-potential dlfferof the relative velocity for both normal-fluld flow. d = 620 Mm, slope of the full hne IS 3.1.
6. Concluding remarks Many drfferent flow srtuatrons m a capillary with an mner drameter of 620 fl have been investigated by measurmg the four mdependant quantities &, AT, cr’and CX”.Each quantity appears to be a complicated function of the two average velocities u, and us, that could be adjusted mdependently. Choosing any two hnear combinations of u,, and us to descnbe the results cannot essentrally simplify the results. In spite of this it is found that the ratios of Ap and (Y’,and of on and o’ can be described wrth only one velocity, namely the relative velocity u,, - us. Thrs srgruficant result, which does not allow for a trrvral explanation, might rndicate m what way an analysis of the flow phenomena should be pursued. It also should be remembered that an analogous result, obtained with measurements of & and AT on a 294 pm capillary, 1s reported m ref. 2. The data for the two ratios &la’ and (Y”/(Y’are
00
0
(cm/, 1
0
0’0
’
Fig. 12. gradient (un) for d = 620
I
I
I
I
1
2
3
4
Ratio of the absolute value of the chemical-potentlal and the extra (total) attenuation as a function of us pure superfluid (normal fluid) flow at 1.70 K. pm, 2 = 41 cm, ~8 = 87.67 Hz.
located on three separate branches m such a way that for most values of un - us, there are two possible values for A/.&Y’and &“/a’. There seems to be a tendency toward a unique relation at higher relative velocities. Investrgatrons at other temperatures seem to indicate that this can happen at lower un - us when the temperature 1s hrgher. The way rn whrch the zero attenuation 1s rnfluenced by the flow seems to be different m the various branches. Possibly a deeper analysis of the dispersion data can bring more light in thrs matter. At some velocity combmatrons, devratrons from the three branches were seen. Thrs was in the region rn which oscillatrons occurred. In a capillary of 216 /.un (ref. 4), two types of oscrllations were found m measurements of A/J and AT, together wrth an extra region in whrch stable situations were found. Thrs may indicate that in narrow capillaries, more branches are found rn plots of A&Y’ and (~“/a’ m the nerghbourhood of the ongm. To check on these speculations experiments like the present one should be done in other geometries of the flow caprllary, and
324
R R IJsselstem et al /Flow of superfluid 4He
of the data for the drspersron, and of the combination of the varrous measured quantities, might give more interesting results. Thrs mvestrgatron ISpart of the research program of the “Strchtmg voor Fundamenteel Onderzoek der Materie (F.O.M.)“, whrch is financrally supported by the “Nederlandse Orgamsatie voor Zumer-Wetenschappehjk Onderzoek (Z.W.O.)“.
References ill T. M. Wrarda, Thesis, Leaden (1967). VI G. van der Heyden, A. G. M van der Boog and H. C. 131 I41
Vs( Vn) cm/S
o.o,1
2
2
5
10
Frg. 13. Ratio of extra (extra plus zero) drspersron and extra (total) attenuatron as a function of us (un) for pure superflmd (normal fluid) flow. d = 620 pm, 1= 41 cm, ~8 = 87.67 Hz The slope of the full hne IS 1.8.
wrth larger velocities, A further study of the large amount of information obtained from the present experiment IS strll m progress. Especially an analysis
ISI 161
[71 181 PI
Kramers, Physica 77 (1974) 487. (Commun. Kamerhngh Onnes Lab., Leaden No. 411~). W. de Haas and H. van Beelen, Physica 83B (1976) 129 (Commun. Kamerhngh Onnes Lab., Leaden No. 422b). R. P Slegtenhorst and H. van Beelen, Physrca 90B (1977) 245 (Commun. Kamerhngh Onnes Lab., Leaden No 4293). W. F Vmen, Proc. Roy. Sot A 240 (1957) 114 G van der Heqden, W J. P. de Voogt and H. C. Kramers, Physrca 59 (1972) 473 (Commun. Kamerlmgh Omres Lab., Leaden No 392a). H. C Kramers, J. Phys. Radium 23 (1962) 326. D. R. Ladner and J. T. Tough, Phys. Rev. B17 (1978) 1455. H. C. Kramers, T. M. Wrarda and G. van der Hegden, Physma 69 (1973) 245 (Commun,Kamerhngh Onnes Lab., Leaden, suppl. No. 129).