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The influence of pressure on the viscosity of liquid Helium I
Physica XVIII,
no 11
N o v e m b e r 1952
THE INFLUENCE OF PRESSURE ON T H E V I S C O S I T Y O F L I Q U I D H E L I U M
I
by H. H. T J E R K S T R A Communication No. 290a from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis T h e v i s c o s i t y of l i q u i d 4He h a s b e e n m e a s u r e d a t p r e s s u r e s f r o m 5 t o 50 k g / c m 2 a n d t e m p e r a t u r e s b e t w e e n t h e ),-point a n d 4 . 2 0 ° K b y a c a p i l l a r y method.
I. Introduction. In liquid helium, the internal friction seems to be closely connected with the remarkable phenomena occurring at low temperatures. The purpose of the experiments, described below, was to contribute to the understanding of the transport of m o m e n t u m in a liquid which is strongly influenced by quantum effects, without showing such large deviations as occur below the k-point. Because of its large zero-point motion, helium, under its vapour pressure, has a low density and shows in connection with this certain aspects of a gas. A comparison of the internal friction at higher densities with that of other monoatomic liquids might therefore be of interest. Some of these liquids have been investigated in considerable detail, theoretically as well as experimentally. The viscosities of liquids at higher pressures have been measured frequently, but in the case of liquid helium the usual methods do not apply. The viscosity is so small that a falling body method is not appropriate and at pressures higher than 2.5 kg/cm 2 the meniscus has disappeared. 2. Description o/the method. The apparatus (see fig. I) consisted of a capillary C immersed in the helium bath at the bottom of the Dewar vessel, an U-shaped differential manometer U filled with - -
8 5 3
- -
854
H.H.
TJERKSTRA
mercury connected to the ends of C b y means of the tubes T, and a high-pressure gauge M. IM
tL
Z,. . . . .
;;
',
~i
:I
II
I: J~N
u
I
o--
I
T
-1-
o
~
I
I s° ' \
) jc Fig. 1
Before taking a run, the system was filled with helium, by opening the stopcocks S 1 and S 2, to a pressure between 3 kg/cm 2 and 50 kg/cm 2. As the critical pressure of helium is exceeded everywhere along the connecting tubes T, the variation of the density will be gradual. The stopcock $2, connecting both ends of the differential gauge U is then closed and a pressure difference is created b y blowing off some gas by means of stopcock $3. The capillary C now forms the only connection between the two halves of the apparatus and the liquid flows through it under the driving force of the pressure difference which can be read from the U-gauge as a function of time. For the calculation of the viscosity from these data it is important that the following conditions are fulfilled. 1°. The two halves of the apparatus must be symmetrical. The tubes T are therefore of similar construction and good thermal contact is obtained b y soldering them together over their whole length (So). 2 °. The variation of the average pressure in the apparatus in the
PRESSURE AND VISCOSITY OF LIQUID HELIUM I
855
course of a measurement should be small. These variations are caused by the lowering of the bath level and b3~ temperature fluctuations in the Dewar vessel, especially in the vapour. Their influence on h, the deviation of a mercury level from its zero position o - o is due to the difference in pressure between both halves; for h = 0 and stopcock S 2 closed, no effect is observed if condition 1) is met, because the temperature distributions vary in the same fashion. 3 °. During a run there is a flow of gas through the tubes T; consequently the temperature distributions are not the same as in the case of equilibrium. In order to check the influence of this difference on the measurements, which must be small, runs were made with tubes and capillaries of widely different diameters as well as with widely varying pressure differences, especially small ones. No effect due to the gas flow was observed and an estimate of the occurring temperature differences also supports the assumption that no systematic error is involved. A favourable circumstance in this respect is the heat exchange at So between the descending and ascending gas streams. 4 ° . The flow resistance of the connecting tubes must be small compared to the resistance of C. Proceeding now to the calculation we write for the mass of helium in the system, using the subscripts 1 and 2 for the right and left half respectively: C 1 --~
fo~(T, Pl) dvi - - Pl ~r2h/RT*,
G2 ----f e(T, P2) dv2 + P2 ~r2h/RT*, where o is the density in g/cm 3, p is the pressure in dyne/cm 2, R is the gas constant in dyne cm/g deg.K, T* is room temperature in deg.K and r is the radius of the U-tube. The integration is performed over the volume in T and in U, including the volume in each leg up to o - o. In this notation we have taken into account that the densit'y Q is a single-valued function of the temperature and pressure if either one or the other exceeds its critical value. We define the mean pressure p = ½ (Pl + P2) and shall use the subscript 0 when the deviation h is zero. In order to find the variation in the mean pressure associated
856
H . H . TJERKSTRA
with h, we write, assuming a stationary temperature distribution:
G l = fo~(T, Po)dv I + [ p - h/a --Po] (O/OPo)fQ(T, P0)dvt - - - (p - - h/a) =r2h/RT * G2 = f o(T, Po)d% + [P + h / a - Po] (o/OPo)f ~(T, Po) dr2 + + (p + h/a) rtr2h/RT * where 1/a = ~,,,e,c,,,y * g, g being the gravity constant and the asterisk denoting room temperature. Introducing
Gto = f e(T, p0) dv
G2o = f e(T, Po) dv
and observing that G 1+ G2 = Gi0 -[- G2o, because the total mass does not depend on h, and that OGlo/ePo = OG2o/cOpo which we shall abbreviate to aGo/OPo we obtain:
P = P o - ~r2h2/(aRT* OGo/OPo) so:
GI
=
Glo
h OGo
a apo
Po
~r2h
~
+
:r2r4h 3
aR2T .2 OGo/OPo
(I)
One obtains for the flow, by differentiating (1) with respect to time : 3~2r4h2 ~ dh dG, rpo~zr2 OGo/c~po ¥
= -- L-RY* +
aR2T .20Go/OPo
-a
dt
hence, for laminar flow,
~o,,q,~z~s42h r po.~r2 OGo~OPoldh 8rlLa -- L R ~ - + dt
(2)
where s is the radius of the capillary C, L its length, and ~ the viscosity coefficient of the liquid. Deriving (2) we have neglected the difference in hydrostatic pressure of the gas, the flow resistance of T and on the right the term 3~2r%2[aR2T*2aGo/aPo]-t which is about one thousandth of the two remaining terms. The deviation h is given by: log h ----log h i
t --
l i ~Otiq,,i d
e,*,
[ ~r
2 +
RT*OG°I-' 2~s41°ge (2bi,s) __
p0a
g oJ
8L
where h i and t i a r e integration constants and 9~** = po/RT* is the density of the gas in the U-gauge.
P R E S S U R E AND VISCOSITY OF L I Q U I D H E L I U M I
857
In this derivation it is assumed t h a t the temperature distribution remains stationary; actually, however, it is varying owing to 1°. the lowering of the bath level 2 ° . the variation of the gas flow in the connecting tubes. A d 1°. Varying equation (l) we find for a slight variation of the temperature distribution, keeping G t constant: 8~
r ~-p~+ ~f~j +~P°L~
Rr*J.
80 o=f~am~dv~+a,,[~oG ° p0=,21
F°c° =r~hI ~-f;o-+ ~ - J + 6PoL~-~o+ ~-*J
(3)
and hence, because aT 1 = aT 2, we obtain by subtraction, 6h = - - @o ~r2h [(RT*/a) OGo/@o + Po~r2]- l
(4)
Keeping h zero one obtains, varying G I + G2 ----- G~o + G2o 0 = f (Oe/OT) aT~ dv~ + f (Oe/OT) 6T~ dr2 + 2@o OGo/@o
(5)
SO
f (O0/ST) aT dv ~Po =
(SGo/~Po)
which is the relation between the variations of the temperature and the pressure. A d 2 °. As a result of the gas flow the average temperature in a given cross section of a connecting tube will differ from its equilibrium value and to a first approximation is: T'l = T 1 + bT 1, T'2= T u + O T 2 , ~Ti = - - $ T 2 , for the gas, while at the wall: T'l = T t --~ T'2 = T u. Therefore, as is seen from (5), bP0 = 0 and using (3) we obtain: ~h ---- [RT* f (OQ/OT) ~T dvl.[(RT*/a) (SGo/OPo) + p0zw2]- '
(6)
If there were no heat flow to the wall, we would find for the variation with time of the temperature in a cross section: ~T/Ot = wOT/Sx, where w is the gas velocity and x is the distance along the tube. This corresponds with a heat source along the axis H = z~2wocpOT/ax, where a is the radius of the tube and cp the specific heat of the gas. Actually H is compensated by the conduction to the wall. If we assume that for the latter process the time to reach equilibrium is small in comparison with the time
858
H . H . TJERKSTRA
constant of the U-gauge, the resulting temperature difference 6T will be proportional to H and therefore to w, or 6h = / h from (2) and (6); / is a proportionality factor not dependent on h. So h' = = (1 + [)h, which shows t h a t the log h, t-curve is displaced parallel to itself. Consequently no correction for the gasflow is required. The situation in a connecting tube m a y approximatively be represented by a cylindrical ring of helium with the outer surface (radius a) at the temperature T and the inner surface (radius a/2 ~) at the temperature T ' = T + 6T. The coefficient of heat conductivity is 4. A heat current H is flowing radially through the ring. The temperature distribution along the tube is approximated by taking ~TIO~ = constant. We then find, for a = 0.035 cm, = 6 . 1 0 -s c a l / d e g c m sec, c p = 0 . 5 cal/g deg, q - - 0 . 1 5 g / c m , considering the heat capacity and thermal resistance of the ring, that the time in which the heat conduction to the wall becomes stationary is of the order of a few seconds. The time constant of the U-gauge was, as found experimentally, about four minutes and consequently the assumption, mentioned before, seems to be justified. Estimating dT for a run at p = 5 kg/cm 2 and a temperature of the helium bath T = 4°K we obtain 6T = 0.02°K. The variation ~h is then approximatively bh = 0,01 h.
3. Experimental results. In order to determine OGo/~Po the calibrated volume Ve of the high-pressure gauge with its connections and the apparatus (the stopcock S 2 remains open during the experiment) are brought at the slightly different pressures p' respectively Po, When stopcock S l is opened these pressures become equal to p", so that: It
It
(p'--p")vc/RT* = (o/op) [Gio(Po) + G10(P0)]-[P'--P"I. The values of RT*(O/~po ) [Gl0 + G2o ~ -~- 2RT* OGo/~Po thus obtained have been plotted as a function of the pressure in fig. 2. The uncertainty of the determination is about ten per cent, which is not serious, because it is only a correction. No dependence on the temperature of the bath was found. If the helium in the system could be treated as an ideal gas, then RT* OGo/OPowould represent the volume of G1,2 reduced at the uniform temperature T* at the original pressure of p kg/cm 2. This volume would be independent of p. Actually the gas is non-ideal which explains the rapid increase
PRESSURE AND VISCOSITY OF LIQUID HELIUM I 30cm =
\
20
\
2RT. d
T
I0
20
3 0 kglcm~'
~p
Fig. 2
14o ~ P
,~o 12o
~ "~
,,o
~
\
8O
70
60
50
40
7/
3o j~ I J
T 20
iI II
I0
4~
Fig. 3
859
860
H.H. TJERKSTRA
of RT* gGo/OPo with the lowering of the pressure. In the part of the connecting tube, where T ~ Tc,,ica~ high values of OGo/gPo occur when p ~ Pc,,ca~, which give an appreciable contribution to
gGo/gPo = f (go(T, po)/Opo) dV. In order to calibrate the capillary C a run was taken at the boiling point of liquid oxygen, at a pressure of 10 kg/cm 2. The viscosity of liquid helium was then obtained by calculating ~tiquia/~?cwhere ~7~= 91 # P (see table I and fig. 3; in fig. 4 we have plotted the viscosity as a function of the temperature at constant density). TABLE I The v i s c o s i t y as a function of the t e m p e r a t u r e at c o n s t a n t pressure. P k g c m -~-
t ]
T °K
[ t
r/ /~P
p k g ~cm-2
T °K
r/ /zP
29.8 29.9 30.0
2.27 3.07 4.07
94 89 81
34.9 35.0 34.9
2.00 2.43 2.97
109 103 95
40.0 39.8 39.8 39.6
2.19 2.50 3.50 4.03
115 96 88
50.0 50.0 50.0
2.45 3.33 4.03
131 112 98
10.2 10.5 10.0 10.0 10.5 9.8
2.09 2.19 2.27 2.99 3.08 4.07
44 48 54 57 58 55
20.2 20.0 20.0 19.8 19.8 19.8 19.9
1.89 2.01 2.09 2.19 3.0B 4.07 4.07
70 73 77 74 70 70
30.0 30.0 30.0 30.0
1.78 2.00 2.09 2.19
90 98 97 97
* N.B.
122
N o definite value can be given o w i n g to the a n o m a l o u s b e h a v i o u r of the flow.
The Reynolds number was about 42 at the lowest pressures and largest pressure differences so that the flow was laminar in all cases. The capillary that was finally used had an effective radius of about 5.8/z. The pressure variation due to the lowering of the bath level was found in most cases to be less than 0.07 kg/cm 2 and the correction is therefore very small. The uncertainty in the experimental values of the viscosity is about four percent, which is due to 1°) the value taken for the mean pressure,
PRESSURE
ON T H E V I S C O S I T Y O F L I Q U I D
HELIUM
I
861
2 °) sudden pressure variations as a consequence of temperature fluctuations in the cryostat, which could not always be eliminated, t h o u g h m u c h care was taken, and 3 ° ) the small diameter of the U-tube, in which the shape of the meniscus sometimes changed. A theoretical discussion of these results will soon be forthcoming. 120
=~
Iio
tOO 90
p=0180 80
S
~0=O.171
70 o
0161
60
5O
40
/
30
2O
t.6
a
3
4~
Fig 4 The writer is greatly indebted to Prof. C. J. G o r t e r for his interest and valuable discussions as well as to Dr. K. W. T ac o n i s for his helpful suggestions. Received 31-7-52.
REFERENCES 1) B o w e r s, R. a n d M e n d e 2) D e T r o y e r , A., v a n C o m m u n . No. 284¢, P h y s i c a , 3) K e e s o m, W. H., Helium
1 s s o h n, K., Proe. phys. Soe., L o n d o n A 6 2 ( 1 9 4 9 ) 394. Itterbeek, A. and v a n den Berg, G. J., 17 (1951) 50. (Elsevier, A m s t e r d a m ) 1942.
no 11
N o v e m b e r 1952
THE INFLUENCE OF PRESSURE ON T H E V I S C O S I T Y O F L I Q U I D H E L I U M
I
by H. H. T J E R K S T R A Communication No. 290a from the Kamerlingh Onnes Laboratorium, Leiden, Nederland
Synopsis T h e v i s c o s i t y of l i q u i d 4He h a s b e e n m e a s u r e d a t p r e s s u r e s f r o m 5 t o 50 k g / c m 2 a n d t e m p e r a t u r e s b e t w e e n t h e ),-point a n d 4 . 2 0 ° K b y a c a p i l l a r y method.
I. Introduction. In liquid helium, the internal friction seems to be closely connected with the remarkable phenomena occurring at low temperatures. The purpose of the experiments, described below, was to contribute to the understanding of the transport of m o m e n t u m in a liquid which is strongly influenced by quantum effects, without showing such large deviations as occur below the k-point. Because of its large zero-point motion, helium, under its vapour pressure, has a low density and shows in connection with this certain aspects of a gas. A comparison of the internal friction at higher densities with that of other monoatomic liquids might therefore be of interest. Some of these liquids have been investigated in considerable detail, theoretically as well as experimentally. The viscosities of liquids at higher pressures have been measured frequently, but in the case of liquid helium the usual methods do not apply. The viscosity is so small that a falling body method is not appropriate and at pressures higher than 2.5 kg/cm 2 the meniscus has disappeared. 2. Description o/the method. The apparatus (see fig. I) consisted of a capillary C immersed in the helium bath at the bottom of the Dewar vessel, an U-shaped differential manometer U filled with - -
8 5 3
- -
854
H.H.
TJERKSTRA
mercury connected to the ends of C b y means of the tubes T, and a high-pressure gauge M. IM
tL
Z,. . . . .
;;
',
~i
:I
II
I: J~N
u
I
o--
I
T
-1-
o
~
I
I s° ' \
) jc Fig. 1
Before taking a run, the system was filled with helium, by opening the stopcocks S 1 and S 2, to a pressure between 3 kg/cm 2 and 50 kg/cm 2. As the critical pressure of helium is exceeded everywhere along the connecting tubes T, the variation of the density will be gradual. The stopcock $2, connecting both ends of the differential gauge U is then closed and a pressure difference is created b y blowing off some gas by means of stopcock $3. The capillary C now forms the only connection between the two halves of the apparatus and the liquid flows through it under the driving force of the pressure difference which can be read from the U-gauge as a function of time. For the calculation of the viscosity from these data it is important that the following conditions are fulfilled. 1°. The two halves of the apparatus must be symmetrical. The tubes T are therefore of similar construction and good thermal contact is obtained b y soldering them together over their whole length (So). 2 °. The variation of the average pressure in the apparatus in the
PRESSURE AND VISCOSITY OF LIQUID HELIUM I
855
course of a measurement should be small. These variations are caused by the lowering of the bath level and b3~ temperature fluctuations in the Dewar vessel, especially in the vapour. Their influence on h, the deviation of a mercury level from its zero position o - o is due to the difference in pressure between both halves; for h = 0 and stopcock S 2 closed, no effect is observed if condition 1) is met, because the temperature distributions vary in the same fashion. 3 °. During a run there is a flow of gas through the tubes T; consequently the temperature distributions are not the same as in the case of equilibrium. In order to check the influence of this difference on the measurements, which must be small, runs were made with tubes and capillaries of widely different diameters as well as with widely varying pressure differences, especially small ones. No effect due to the gas flow was observed and an estimate of the occurring temperature differences also supports the assumption that no systematic error is involved. A favourable circumstance in this respect is the heat exchange at So between the descending and ascending gas streams. 4 ° . The flow resistance of the connecting tubes must be small compared to the resistance of C. Proceeding now to the calculation we write for the mass of helium in the system, using the subscripts 1 and 2 for the right and left half respectively: C 1 --~
fo~(T, Pl) dvi - - Pl ~r2h/RT*,
G2 ----f e(T, P2) dv2 + P2 ~r2h/RT*, where o is the density in g/cm 3, p is the pressure in dyne/cm 2, R is the gas constant in dyne cm/g deg.K, T* is room temperature in deg.K and r is the radius of the U-tube. The integration is performed over the volume in T and in U, including the volume in each leg up to o - o. In this notation we have taken into account that the densit'y Q is a single-valued function of the temperature and pressure if either one or the other exceeds its critical value. We define the mean pressure p = ½ (Pl + P2) and shall use the subscript 0 when the deviation h is zero. In order to find the variation in the mean pressure associated
856
H . H . TJERKSTRA
with h, we write, assuming a stationary temperature distribution:
G l = fo~(T, Po)dv I + [ p - h/a --Po] (O/OPo)fQ(T, P0)dvt - - - (p - - h/a) =r2h/RT * G2 = f o(T, Po)d% + [P + h / a - Po] (o/OPo)f ~(T, Po) dr2 + + (p + h/a) rtr2h/RT * where 1/a = ~,,,e,c,,,y * g, g being the gravity constant and the asterisk denoting room temperature. Introducing
Gto = f e(T, p0) dv
G2o = f e(T, Po) dv
and observing that G 1+ G2 = Gi0 -[- G2o, because the total mass does not depend on h, and that OGlo/ePo = OG2o/cOpo which we shall abbreviate to aGo/OPo we obtain:
P = P o - ~r2h2/(aRT* OGo/OPo) so:
GI
=
Glo
h OGo
a apo
Po
~r2h
~
+
:r2r4h 3
aR2T .2 OGo/OPo
(I)
One obtains for the flow, by differentiating (1) with respect to time : 3~2r4h2 ~ dh dG, rpo~zr2 OGo/c~po ¥
= -- L-RY* +
aR2T .20Go/OPo
-a
dt
hence, for laminar flow,
~o,,q,~z~s42h r po.~r2 OGo~OPoldh 8rlLa -- L R ~ - + dt
(2)
where s is the radius of the capillary C, L its length, and ~ the viscosity coefficient of the liquid. Deriving (2) we have neglected the difference in hydrostatic pressure of the gas, the flow resistance of T and on the right the term 3~2r%2[aR2T*2aGo/aPo]-t which is about one thousandth of the two remaining terms. The deviation h is given by: log h ----log h i
t --
l i ~Otiq,,i d
e,*,
[ ~r
2 +
RT*OG°I-' 2~s41°ge (2bi,s) __
p0a
g oJ
8L
where h i and t i a r e integration constants and 9~** = po/RT* is the density of the gas in the U-gauge.
P R E S S U R E AND VISCOSITY OF L I Q U I D H E L I U M I
857
In this derivation it is assumed t h a t the temperature distribution remains stationary; actually, however, it is varying owing to 1°. the lowering of the bath level 2 ° . the variation of the gas flow in the connecting tubes. A d 1°. Varying equation (l) we find for a slight variation of the temperature distribution, keeping G t constant: 8~
r ~-p~+ ~f~j +~P°L~
Rr*J.
80 o=f~am~dv~+a,,[~oG ° p0=,21
F°c° =r~hI ~-f;o-+ ~ - J + 6PoL~-~o+ ~-*J
(3)
and hence, because aT 1 = aT 2, we obtain by subtraction, 6h = - - @o ~r2h [(RT*/a) OGo/@o + Po~r2]- l
(4)
Keeping h zero one obtains, varying G I + G2 ----- G~o + G2o 0 = f (Oe/OT) aT~ dv~ + f (Oe/OT) 6T~ dr2 + 2@o OGo/@o
(5)
SO
f (O0/ST) aT dv ~Po =
(SGo/~Po)
which is the relation between the variations of the temperature and the pressure. A d 2 °. As a result of the gas flow the average temperature in a given cross section of a connecting tube will differ from its equilibrium value and to a first approximation is: T'l = T 1 + bT 1, T'2= T u + O T 2 , ~Ti = - - $ T 2 , for the gas, while at the wall: T'l = T t --~ T'2 = T u. Therefore, as is seen from (5), bP0 = 0 and using (3) we obtain: ~h ---- [RT* f (OQ/OT) ~T dvl.[(RT*/a) (SGo/OPo) + p0zw2]- '
(6)
If there were no heat flow to the wall, we would find for the variation with time of the temperature in a cross section: ~T/Ot = wOT/Sx, where w is the gas velocity and x is the distance along the tube. This corresponds with a heat source along the axis H = z~2wocpOT/ax, where a is the radius of the tube and cp the specific heat of the gas. Actually H is compensated by the conduction to the wall. If we assume that for the latter process the time to reach equilibrium is small in comparison with the time
858
H . H . TJERKSTRA
constant of the U-gauge, the resulting temperature difference 6T will be proportional to H and therefore to w, or 6h = / h from (2) and (6); / is a proportionality factor not dependent on h. So h' = = (1 + [)h, which shows t h a t the log h, t-curve is displaced parallel to itself. Consequently no correction for the gasflow is required. The situation in a connecting tube m a y approximatively be represented by a cylindrical ring of helium with the outer surface (radius a) at the temperature T and the inner surface (radius a/2 ~) at the temperature T ' = T + 6T. The coefficient of heat conductivity is 4. A heat current H is flowing radially through the ring. The temperature distribution along the tube is approximated by taking ~TIO~ = constant. We then find, for a = 0.035 cm, = 6 . 1 0 -s c a l / d e g c m sec, c p = 0 . 5 cal/g deg, q - - 0 . 1 5 g / c m , considering the heat capacity and thermal resistance of the ring, that the time in which the heat conduction to the wall becomes stationary is of the order of a few seconds. The time constant of the U-gauge was, as found experimentally, about four minutes and consequently the assumption, mentioned before, seems to be justified. Estimating dT for a run at p = 5 kg/cm 2 and a temperature of the helium bath T = 4°K we obtain 6T = 0.02°K. The variation ~h is then approximatively bh = 0,01 h.
3. Experimental results. In order to determine OGo/~Po the calibrated volume Ve of the high-pressure gauge with its connections and the apparatus (the stopcock S 2 remains open during the experiment) are brought at the slightly different pressures p' respectively Po, When stopcock S l is opened these pressures become equal to p", so that: It
It
(p'--p")vc/RT* = (o/op) [Gio(Po) + G10(P0)]-[P'--P"I. The values of RT*(O/~po ) [Gl0 + G2o ~ -~- 2RT* OGo/~Po thus obtained have been plotted as a function of the pressure in fig. 2. The uncertainty of the determination is about ten per cent, which is not serious, because it is only a correction. No dependence on the temperature of the bath was found. If the helium in the system could be treated as an ideal gas, then RT* OGo/OPowould represent the volume of G1,2 reduced at the uniform temperature T* at the original pressure of p kg/cm 2. This volume would be independent of p. Actually the gas is non-ideal which explains the rapid increase
PRESSURE AND VISCOSITY OF LIQUID HELIUM I 30cm =
\
20
\
2RT. d
T
I0
20
3 0 kglcm~'
~p
Fig. 2
14o ~ P
,~o 12o
~ "~
,,o
~
\
8O
70
60
50
40
7/
3o j~ I J
T 20
iI II
I0
4~
Fig. 3
859
860
H.H. TJERKSTRA
of RT* gGo/OPo with the lowering of the pressure. In the part of the connecting tube, where T ~ Tc,,ica~ high values of OGo/gPo occur when p ~ Pc,,ca~, which give an appreciable contribution to
gGo/gPo = f (go(T, po)/Opo) dV. In order to calibrate the capillary C a run was taken at the boiling point of liquid oxygen, at a pressure of 10 kg/cm 2. The viscosity of liquid helium was then obtained by calculating ~tiquia/~?cwhere ~7~= 91 # P (see table I and fig. 3; in fig. 4 we have plotted the viscosity as a function of the temperature at constant density). TABLE I The v i s c o s i t y as a function of the t e m p e r a t u r e at c o n s t a n t pressure. P k g c m -~-
t ]
T °K
[ t
r/ /~P
p k g ~cm-2
T °K
r/ /zP
29.8 29.9 30.0
2.27 3.07 4.07
94 89 81
34.9 35.0 34.9
2.00 2.43 2.97
109 103 95
40.0 39.8 39.8 39.6
2.19 2.50 3.50 4.03
115 96 88
50.0 50.0 50.0
2.45 3.33 4.03
131 112 98
10.2 10.5 10.0 10.0 10.5 9.8
2.09 2.19 2.27 2.99 3.08 4.07
44 48 54 57 58 55
20.2 20.0 20.0 19.8 19.8 19.8 19.9
1.89 2.01 2.09 2.19 3.0B 4.07 4.07
70 73 77 74 70 70
30.0 30.0 30.0 30.0
1.78 2.00 2.09 2.19
90 98 97 97
* N.B.
122
N o definite value can be given o w i n g to the a n o m a l o u s b e h a v i o u r of the flow.
The Reynolds number was about 42 at the lowest pressures and largest pressure differences so that the flow was laminar in all cases. The capillary that was finally used had an effective radius of about 5.8/z. The pressure variation due to the lowering of the bath level was found in most cases to be less than 0.07 kg/cm 2 and the correction is therefore very small. The uncertainty in the experimental values of the viscosity is about four percent, which is due to 1°) the value taken for the mean pressure,
PRESSURE
ON T H E V I S C O S I T Y O F L I Q U I D
HELIUM
I
861
2 °) sudden pressure variations as a consequence of temperature fluctuations in the cryostat, which could not always be eliminated, t h o u g h m u c h care was taken, and 3 ° ) the small diameter of the U-tube, in which the shape of the meniscus sometimes changed. A theoretical discussion of these results will soon be forthcoming. 120
=~
Iio
tOO 90
p=0180 80
S
~0=O.171
70 o
0161
60
5O
40
/
30
2O
t.6
a
3
4~
Fig 4 The writer is greatly indebted to Prof. C. J. G o r t e r for his interest and valuable discussions as well as to Dr. K. W. T ac o n i s for his helpful suggestions. Received 31-7-52.
REFERENCES 1) B o w e r s, R. a n d M e n d e 2) D e T r o y e r , A., v a n C o m m u n . No. 284¢, P h y s i c a , 3) K e e s o m, W. H., Helium
1 s s o h n, K., Proe. phys. Soe., L o n d o n A 6 2 ( 1 9 4 9 ) 394. Itterbeek, A. and v a n den Berg, G. J., 17 (1951) 50. (Elsevier, A m s t e r d a m ) 1942.