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The viscosity of liquid helium
P h y s i c a V, n o 8
A u g u s t u s 1938
THE VISCOSITY OF
LIQUID H E L I U M
b y W. H. KEESOM and G. E. MACWOOD *) Commun. No. 254a from the Kamerlingh Onnes Laboratory at Leiden.
Summary A n a p p a r a t u s f o r t h e m e a s u r e m e n t of t h e v i s c o s i t y of l i q u i d s a n d gases, i n t e r c h a n g e a b l y , is d e s c r i b e d . M e a s u r e m e n t s of t h e v i s c o s i t y of l i q u i d h e l i u m m a d e w i t h t h i s a p p a r a t u s a r e r e p o r t e d . T h e m e a s u r e m e n t s w e r e m a d e o v e r t h e t e m p e r a t u r e r a n g e f r o m 1.2 t o 4.0oK.
§ 1. Introduction. It was decided to continue the measurements that have been made in this laboratory on the viscosity of gases and liquefied gases which should be completed in the region of very low temperatures. The first measurements of the viscosity of liquid helium made at Toronto 1) indicated that a more intensive investigation of this property would be of interest. Furthermore, it seemed possible that helium II might show the phenomenon of slip. With a view to measuring the viscosity and, if possible, the slip, we decided to use a modified oscillating disc apparatus. The theory of the method has been extended by one of us 2) 3) to take care of a fluid of high density and low viscosity, and exhibiting slip. § 2. Apparatus. The apparatus used is a modification of that used by v a n I t t e r b e e k and K e e s o m l ) . It is shown in Fig. 1. The oscillation system consisted of a copper disc (a), 50 mm in diameter and 0.56 mm thick. This was fastened to a thin steel rod (b), 100 mm long and 1 mm in diameter, care being taken that it was perpendicular to the plane of the disc. The rod b was fastened with Dekothinsky in turn to a glass rod (c), 500 mm long and 1 mm in diameter. Into a small opening in the rod c, the suspension wire (d) of *) National Research Fellow, Cutting Traveling Fellow of Columbia University. --
Physica V
737
--
47
738
W . I-I.K E E S O M
and
G. E. M A C W O O D
phosphorbronze, 250 mm long and 0.05 mm in diameter, was sealed. The other end was seMed into a heavy metal rod (e). The latter allowed for the raising and lowering of the suspension system. The disc was enclosed in an open brass box (h) in which at the top and b o t t o m are plane copper plates (k) parallel to the disc a. The mechanism l allowed for changing the distance between the plates b y 6 cm. A fiat mirror (m) was fixed to the rod c for the purpose of observing the oscillations. These Were observed b y means of the standard reading arrangement used in the Kamerlingh Onnes Laboratory. It consisted of a telescope and a porcelain scale, which could be read to a 0.1 mm, mounted at about 2.5 meters from the apparatus on a separate pillar. To set the system in motion, a small bar of soft iron (8 mm × 5 ram) was rigidly attacked to rod c. Deviations were than produced b y carefully bringing up a magnet. To the large tube of german silver (A) a thick copper plate (B) was soldered. B y means of the three screws n, o, p, the apparatus rested on a heavy brass plate (C) which was fastened rigidly to a stone column standing on a pillar oi the laboratory. With the aid of the screws n, o, p, the apparatus could be made vertical. This done, the fourth screw q served to lock the apparatus in position. The remainder of the apparatus shown in Fig. 1 is self-explanatory.
§ 3. Determination o/ the logarithmic decrement and period. The values of the logarithmic decrement are average values, each being calculated from about 15 complete oscillations. The period of oscillation was determined b y measuring the time of 15 oscillations with a stop-watch. The watch could be read to 0.02 sec and was checked against the Leiden Observatory clock. § 4. Calibration o~ the apparatus. In the case of liquid helium, as was pointed out in the paper mentioned above 2), the formulae to be used for the reduction of experiments are for the viscosity and slip respectively: %/~ = ~ 4 / ] / - - ~A- (m2A ) ~ .¢Ao) _ (1 p __
,
(1)
and I X----~
I
[ 1 - - ( 1 - - 2A) m2 - 2 m A o ] ,
(2)
THE VISCOSITY
Fig.
1. Oscillating
OF LIQUID
Disc Apparatus
HELIUM
for Measuring
739
Viscosity.
740
W. H. K E E S O M A N D G. E. MACWOOD
where the symbols have the same meaning as in the paper referred to. However, before they m a y be used, I/~R 4 and A0 must first be determined. This was done by measuring the viscosity of helium gas in the liquid hydrogen range using tbe formula 2): 21 17 (.2~__ "~8o),
(3)
where D is the average separation of the disc and was measured with the aid of a cathetometer. Using the measured values of D, T, • and S and the viscosities of helium gas as given by v a n Itterbeek and K e e s o m 4 ) a n d H . K a m e r l i n g h Onn e s and W e b e r 5), I/rcR 4 and A0 were obtained by a least square solution. The measurements with helium gas ar~ given below in Table I. TABLE I Calibration of a p p a r a t u s Temp. °K
Period see.
Log. Decrement x 10~
Known Viscosity g. poise
Cale. Viscosity tz poise
Difference %
15.76 17..59 18.43 19.14
24.624 24.624 24.631 24.624
266.2 286.6 294.0 304.0
29.82 32.05 33.03 33.82
29.78 32.06 32.89 34.01
+ 0.2 -- 0.05 + 0.8 0.8
19.62
24.634
306.9
34.55
34.33
+ O. 1
From these measurements we obtained: D = 0.0605cm, I/~R 4 = 0.2293 where as Ao proved to be negligible.
§ 5. Measurements o~ liquid helium. In these measurements, we observed from 10 to 15 complete oscillations for each point from which the period and decrement were obtained. On various experimental days, the disc separation was always different, varying from 1 to 1.3 cm. Considering Eq. (1), it is seen that one more datum is needed, the density. The table given by W. H. K e e s o m and Miss A. P. K e e s o m 6 ) were used. Table n gives a summary of the experimental results, together with the values of the viscosity according to Eq. (1). The last column gives the values of the viscosity corrected for the exterior liquid according to the calculations of one of us 3). Fig. 2 gives the temperature dependence of the viscosity.
THE VISCOSITY
OF LIQUID
741
HELIUM
TABLE II M e a s u r e m e n t s of v i s c o s i t y of l i q u i d h e l i u m
Date
Temp. °K
Period sec
Log. Decrement x 10 ~
Uncorrected Viscosity ~ poise
Corrected Viscosity p. poise
15 Dec.
1937
1.335 1.586 1.762 1.907 1.973 2.111 21178
24.62 24.65 24.66 24.67 24.69 24.71 24.97
100.5 114.6 141.8 196.0 215.0 269.4 367.8
1.67 2.17 3.33 6.36 7.66 12.0 22.3
1.82 2.33 3.56 6.75 8.11 12.6 23.0
14 J a n .
1938
2.258 2.702
24.81 24.78
318.4 404. I
16.6 27.4
17.3 28.2
21 Jan~
1938
2.08 2.14 2.15
24.71 24.75 24.74.
321.4 317.6 294.2
17.1 16.7 14.3
17.8 17.4 14.8
11 F e b .
1938
1.731 1.988 2.086 2.116 2.145 2.159 2.174 2.642 3.81 3.97
24.60 24.62 24.63 24.64 24.65 24.64 24.65 24.66 24.66 24.64
151.2 208.7 256.7 278.3 299.4 319.7 336.0 367.2 390.6 386.8
3.80 7.22 10.9 12.9 14.9 17.0 18.8 22.7 28.0 28.2
4.10 7.60 11.5 13.5 15.5 17.7 19.5 23.5 28.7 28.8
18Feb.
1938
2.155 2.156 2.171 2.290 2.315 2.934 3.738
24.57 24.61 24.58 24.58 24.58 24.69 24.68
320.2 317.9 342.0 371.4 377. I 406.1 416.8
17.1 16.8 19.5 18.1 18.5 23.7 2618
17.8 17.5 20.2 18.7 19.7 24.0 27.5
25 F e b .
1938
2.203 3.069 3.548 3.922 4.021
24.76 24.79 24.81 24.77 24.79
349.0 395.3 417.6 412.4 423.3
17.5 23.2 26.8 27.4 29.2
18.2 23.9 27.5 28.1 29.8
1 8 M a r c h 1938
1.304 1.282 1.335 1.379
24.77 24.76 24.76 24.77
83.8 90.8 I00.0 119.5
1.14 1.43 1.66 2.43
1.24 1.54 1.79 2.61
§ 6. Discussion. Several points of interest are immediately evident from a consideration of Fig. 2. As for most properties of liquid helium, the neighbourhood of the X-point is abnormal for the viscosity. From the present results the course of the viscosity of He I, quite near to the ~-point, is somewhat uncertain, so that it is impossible to say whether, for the viscosity, the X~point is merely a
742
W. H. KEESOM A N D G. E. MACWOOD
singular point or whether there exists a finite discontinuity there. The temperature dependence of the viscosity of both helium I and II is unusual, ~/OT being positive, while for normal liquids it is negative.
20
Y Fig. 2. Viscosity of liquid helium. Though the theoretical basis of T i s z a's considerations 7) needs further discussion, the picture given b y this author is in excellent qualitative agreement with the results given here and m a y be used to account for the exceedingly large differences between the values reported b y M i s e n e r and A l l e n s ) and K a p i t z a g ) on t h e one hand, and those reported here, as was indicated by Tisza. / Furthermore, in agreement with the results of T i s z a, it appears that the viscosity goes to zero as the temperature approaches zero. Up to now, we have not mentioned slip. It had been hoped that we'would be able to decide whether or not helium II possesses slip. However, our results were not sufficiently precise to decide; the most we m a y safely say is that, if there is a slip, it is smaller than l0 -3. All the calculations of the viscosity given here, therefore, were on the assumption of no slip.
THE VISCOSITY OF LIOUID HELIUM
743
Finally, it m a y be well to observe t h a t the amplitude of oscillation was at all times so small that the linear velocity of the edge of the disc was never larger than 0.3 mm/sec. In Fig. 3 is shown the dependence of the amplitude as a function of time in two typical cases. On the basis of the fact, t h a t the logarithmic decrement appears to be independent of the amplitude, we are quite sure that the motion of the fluid at all times was laminar. 40C ~m
8C
6C
~'-"-'-'--~--~-=-'~" '--"--"~-t335°K
2.178% ~
2O
t 10~)~ periods
4
8
12
Fig. 3. Two examples of the dependence of t h e a m p l i t u d e on t h e time.
§ 7. Acknowledgements. We wish to record our thanks to J. H. Schweers, phil. nat. drs., and P. H. K e e s o m , phil. nat. cand., for their help during the measurements and with the calculations. Our thanks are also due to the Smithsonian Institution for a grant-in-aid. Received June 28, 1.938.
744
THE VISCOSITY OF LIQUID HELIUM REFERENCES
1) J . O . W i l h e l m , A. D. M i s e n e r and A. R. C l a r k , Proc. roy. Soc. London A 151, 342, 1935. 2) G. E. M a e W o o d, C o m m u n . Kamerlingh Onnes Lab., Leiden. Suppl. No. 84b; Physica 5, 374, 1938. 3) G. E. M a c W o o d, C o m m u n . Kamerlingh Onnes Lab., Leiden Suppl. No. 84c; Physica 5, 763, 1938. 4) A. v a n Itterbeek and W . H . Keesom, C o m m u n . Kamerlingh Onnes Lab., Leiden No. 252s; Physica, 's-Grav. 5, 257, 1938. 5) H. K a m e r l i n g h Onnes and S. W e b e r , C o m m u n . Kamerlingh Onnes Lab., Leiden No. 134b; Proc. I(on. Akad. Amsterdam 15, 1396, 1913. 6) W. H. K e e s o m and Miss A. P. K e e s o m , C o m m u n . Kamerlingh Onnes Lab., Leiden Suppl. No. 76b; Physi~a, 's-Grav. I, 128, 1933. 7) L. T i s z a , Nature, Londen141,913, 1938. 8) J. F. A l l e n and A.D. M i s e n e r , Nature, London 141, 75, 1938. 9) P. K a p i t z a , Nature, London 141, 74, 1938.
A u g u s t u s 1938
THE VISCOSITY OF
LIQUID H E L I U M
b y W. H. KEESOM and G. E. MACWOOD *) Commun. No. 254a from the Kamerlingh Onnes Laboratory at Leiden.
Summary A n a p p a r a t u s f o r t h e m e a s u r e m e n t of t h e v i s c o s i t y of l i q u i d s a n d gases, i n t e r c h a n g e a b l y , is d e s c r i b e d . M e a s u r e m e n t s of t h e v i s c o s i t y of l i q u i d h e l i u m m a d e w i t h t h i s a p p a r a t u s a r e r e p o r t e d . T h e m e a s u r e m e n t s w e r e m a d e o v e r t h e t e m p e r a t u r e r a n g e f r o m 1.2 t o 4.0oK.
§ 1. Introduction. It was decided to continue the measurements that have been made in this laboratory on the viscosity of gases and liquefied gases which should be completed in the region of very low temperatures. The first measurements of the viscosity of liquid helium made at Toronto 1) indicated that a more intensive investigation of this property would be of interest. Furthermore, it seemed possible that helium II might show the phenomenon of slip. With a view to measuring the viscosity and, if possible, the slip, we decided to use a modified oscillating disc apparatus. The theory of the method has been extended by one of us 2) 3) to take care of a fluid of high density and low viscosity, and exhibiting slip. § 2. Apparatus. The apparatus used is a modification of that used by v a n I t t e r b e e k and K e e s o m l ) . It is shown in Fig. 1. The oscillation system consisted of a copper disc (a), 50 mm in diameter and 0.56 mm thick. This was fastened to a thin steel rod (b), 100 mm long and 1 mm in diameter, care being taken that it was perpendicular to the plane of the disc. The rod b was fastened with Dekothinsky in turn to a glass rod (c), 500 mm long and 1 mm in diameter. Into a small opening in the rod c, the suspension wire (d) of *) National Research Fellow, Cutting Traveling Fellow of Columbia University. --
Physica V
737
--
47
738
W . I-I.K E E S O M
and
G. E. M A C W O O D
phosphorbronze, 250 mm long and 0.05 mm in diameter, was sealed. The other end was seMed into a heavy metal rod (e). The latter allowed for the raising and lowering of the suspension system. The disc was enclosed in an open brass box (h) in which at the top and b o t t o m are plane copper plates (k) parallel to the disc a. The mechanism l allowed for changing the distance between the plates b y 6 cm. A fiat mirror (m) was fixed to the rod c for the purpose of observing the oscillations. These Were observed b y means of the standard reading arrangement used in the Kamerlingh Onnes Laboratory. It consisted of a telescope and a porcelain scale, which could be read to a 0.1 mm, mounted at about 2.5 meters from the apparatus on a separate pillar. To set the system in motion, a small bar of soft iron (8 mm × 5 ram) was rigidly attacked to rod c. Deviations were than produced b y carefully bringing up a magnet. To the large tube of german silver (A) a thick copper plate (B) was soldered. B y means of the three screws n, o, p, the apparatus rested on a heavy brass plate (C) which was fastened rigidly to a stone column standing on a pillar oi the laboratory. With the aid of the screws n, o, p, the apparatus could be made vertical. This done, the fourth screw q served to lock the apparatus in position. The remainder of the apparatus shown in Fig. 1 is self-explanatory.
§ 3. Determination o/ the logarithmic decrement and period. The values of the logarithmic decrement are average values, each being calculated from about 15 complete oscillations. The period of oscillation was determined b y measuring the time of 15 oscillations with a stop-watch. The watch could be read to 0.02 sec and was checked against the Leiden Observatory clock. § 4. Calibration o~ the apparatus. In the case of liquid helium, as was pointed out in the paper mentioned above 2), the formulae to be used for the reduction of experiments are for the viscosity and slip respectively: %/~ = ~ 4 / ] / - - ~A- (m2A ) ~ .¢Ao) _ (1 p __
,
(1)
and I X----~
I
[ 1 - - ( 1 - - 2A) m2 - 2 m A o ] ,
(2)
THE VISCOSITY
Fig.
1. Oscillating
OF LIQUID
Disc Apparatus
HELIUM
for Measuring
739
Viscosity.
740
W. H. K E E S O M A N D G. E. MACWOOD
where the symbols have the same meaning as in the paper referred to. However, before they m a y be used, I/~R 4 and A0 must first be determined. This was done by measuring the viscosity of helium gas in the liquid hydrogen range using tbe formula 2): 21 17 (.2~__ "~8o),
(3)
where D is the average separation of the disc and was measured with the aid of a cathetometer. Using the measured values of D, T, • and S and the viscosities of helium gas as given by v a n Itterbeek and K e e s o m 4 ) a n d H . K a m e r l i n g h Onn e s and W e b e r 5), I/rcR 4 and A0 were obtained by a least square solution. The measurements with helium gas ar~ given below in Table I. TABLE I Calibration of a p p a r a t u s Temp. °K
Period see.
Log. Decrement x 10~
Known Viscosity g. poise
Cale. Viscosity tz poise
Difference %
15.76 17..59 18.43 19.14
24.624 24.624 24.631 24.624
266.2 286.6 294.0 304.0
29.82 32.05 33.03 33.82
29.78 32.06 32.89 34.01
+ 0.2 -- 0.05 + 0.8 0.8
19.62
24.634
306.9
34.55
34.33
+ O. 1
From these measurements we obtained: D = 0.0605cm, I/~R 4 = 0.2293 where as Ao proved to be negligible.
§ 5. Measurements o~ liquid helium. In these measurements, we observed from 10 to 15 complete oscillations for each point from which the period and decrement were obtained. On various experimental days, the disc separation was always different, varying from 1 to 1.3 cm. Considering Eq. (1), it is seen that one more datum is needed, the density. The table given by W. H. K e e s o m and Miss A. P. K e e s o m 6 ) were used. Table n gives a summary of the experimental results, together with the values of the viscosity according to Eq. (1). The last column gives the values of the viscosity corrected for the exterior liquid according to the calculations of one of us 3). Fig. 2 gives the temperature dependence of the viscosity.
THE VISCOSITY
OF LIQUID
741
HELIUM
TABLE II M e a s u r e m e n t s of v i s c o s i t y of l i q u i d h e l i u m
Date
Temp. °K
Period sec
Log. Decrement x 10 ~
Uncorrected Viscosity ~ poise
Corrected Viscosity p. poise
15 Dec.
1937
1.335 1.586 1.762 1.907 1.973 2.111 21178
24.62 24.65 24.66 24.67 24.69 24.71 24.97
100.5 114.6 141.8 196.0 215.0 269.4 367.8
1.67 2.17 3.33 6.36 7.66 12.0 22.3
1.82 2.33 3.56 6.75 8.11 12.6 23.0
14 J a n .
1938
2.258 2.702
24.81 24.78
318.4 404. I
16.6 27.4
17.3 28.2
21 Jan~
1938
2.08 2.14 2.15
24.71 24.75 24.74.
321.4 317.6 294.2
17.1 16.7 14.3
17.8 17.4 14.8
11 F e b .
1938
1.731 1.988 2.086 2.116 2.145 2.159 2.174 2.642 3.81 3.97
24.60 24.62 24.63 24.64 24.65 24.64 24.65 24.66 24.66 24.64
151.2 208.7 256.7 278.3 299.4 319.7 336.0 367.2 390.6 386.8
3.80 7.22 10.9 12.9 14.9 17.0 18.8 22.7 28.0 28.2
4.10 7.60 11.5 13.5 15.5 17.7 19.5 23.5 28.7 28.8
18Feb.
1938
2.155 2.156 2.171 2.290 2.315 2.934 3.738
24.57 24.61 24.58 24.58 24.58 24.69 24.68
320.2 317.9 342.0 371.4 377. I 406.1 416.8
17.1 16.8 19.5 18.1 18.5 23.7 2618
17.8 17.5 20.2 18.7 19.7 24.0 27.5
25 F e b .
1938
2.203 3.069 3.548 3.922 4.021
24.76 24.79 24.81 24.77 24.79
349.0 395.3 417.6 412.4 423.3
17.5 23.2 26.8 27.4 29.2
18.2 23.9 27.5 28.1 29.8
1 8 M a r c h 1938
1.304 1.282 1.335 1.379
24.77 24.76 24.76 24.77
83.8 90.8 I00.0 119.5
1.14 1.43 1.66 2.43
1.24 1.54 1.79 2.61
§ 6. Discussion. Several points of interest are immediately evident from a consideration of Fig. 2. As for most properties of liquid helium, the neighbourhood of the X-point is abnormal for the viscosity. From the present results the course of the viscosity of He I, quite near to the ~-point, is somewhat uncertain, so that it is impossible to say whether, for the viscosity, the X~point is merely a
742
W. H. KEESOM A N D G. E. MACWOOD
singular point or whether there exists a finite discontinuity there. The temperature dependence of the viscosity of both helium I and II is unusual, ~/OT being positive, while for normal liquids it is negative.
20
Y Fig. 2. Viscosity of liquid helium. Though the theoretical basis of T i s z a's considerations 7) needs further discussion, the picture given b y this author is in excellent qualitative agreement with the results given here and m a y be used to account for the exceedingly large differences between the values reported b y M i s e n e r and A l l e n s ) and K a p i t z a g ) on t h e one hand, and those reported here, as was indicated by Tisza. / Furthermore, in agreement with the results of T i s z a, it appears that the viscosity goes to zero as the temperature approaches zero. Up to now, we have not mentioned slip. It had been hoped that we'would be able to decide whether or not helium II possesses slip. However, our results were not sufficiently precise to decide; the most we m a y safely say is that, if there is a slip, it is smaller than l0 -3. All the calculations of the viscosity given here, therefore, were on the assumption of no slip.
THE VISCOSITY OF LIOUID HELIUM
743
Finally, it m a y be well to observe t h a t the amplitude of oscillation was at all times so small that the linear velocity of the edge of the disc was never larger than 0.3 mm/sec. In Fig. 3 is shown the dependence of the amplitude as a function of time in two typical cases. On the basis of the fact, t h a t the logarithmic decrement appears to be independent of the amplitude, we are quite sure that the motion of the fluid at all times was laminar. 40C ~m
8C
6C
~'-"-'-'--~--~-=-'~" '--"--"~-t335°K
2.178% ~
2O
t 10~)~ periods
4
8
12
Fig. 3. Two examples of the dependence of t h e a m p l i t u d e on t h e time.
§ 7. Acknowledgements. We wish to record our thanks to J. H. Schweers, phil. nat. drs., and P. H. K e e s o m , phil. nat. cand., for their help during the measurements and with the calculations. Our thanks are also due to the Smithsonian Institution for a grant-in-aid. Received June 28, 1.938.
744
THE VISCOSITY OF LIQUID HELIUM REFERENCES
1) J . O . W i l h e l m , A. D. M i s e n e r and A. R. C l a r k , Proc. roy. Soc. London A 151, 342, 1935. 2) G. E. M a e W o o d, C o m m u n . Kamerlingh Onnes Lab., Leiden. Suppl. No. 84b; Physica 5, 374, 1938. 3) G. E. M a c W o o d, C o m m u n . Kamerlingh Onnes Lab., Leiden Suppl. No. 84c; Physica 5, 763, 1938. 4) A. v a n Itterbeek and W . H . Keesom, C o m m u n . Kamerlingh Onnes Lab., Leiden No. 252s; Physica, 's-Grav. 5, 257, 1938. 5) H. K a m e r l i n g h Onnes and S. W e b e r , C o m m u n . Kamerlingh Onnes Lab., Leiden No. 134b; Proc. I(on. Akad. Amsterdam 15, 1396, 1913. 6) W. H. K e e s o m and Miss A. P. K e e s o m , C o m m u n . Kamerlingh Onnes Lab., Leiden Suppl. No. 76b; Physi~a, 's-Grav. I, 128, 1933. 7) L. T i s z a , Nature, Londen141,913, 1938. 8) J. F. A l l e n and A.D. M i s e n e r , Nature, London 141, 75, 1938. 9) P. K a p i t z a , Nature, London 141, 74, 1938.