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Some further experiments with an oscillating disc in liquid helium
P h y s i c a X V I , n o 10
O c t o b e r 1950
SOME FURTHER EXPERIMENTS WITH AN OSCILLATING DISC IN LIQUID HELIUM by P. L. SMITH Comnmnication No. 282c from the Kamerlingh Onnes l.aboratory, l.eiden Synopsis
A n a t t e m p t was m a d e t o s h o w t h e e x i s t e n c e of m u t u a l friction b e t w e e n liquid h e l i u m I a n d liquid h e l i u m II. S u c h a f r i c t i o n s h o u l d be o b s e r v e d as an i n c r e a s e in t h e d a m p i n g c o e f f i c i e n t of a n o s c i l l a t i n g disc w h e n large a m p l i t u d e s w e r e e m p l o y e d . N o s u c h increase, h o w e v e r , was f o u n d . Preliminary experiments above the lambda-point roughly confirmed M e n d e 1 s s o h n's viscosity measurements made with a capillary tube.
1. Introduction. T h e discussion b y G o r t e r and M e 11 i n k 1) of the c o n d u c t i o n of h e a t and of the fountain effect in liquid helium 11 suggests t h e existence of some f o r m of friction b e t w e e n the t w o c o m p o n e n t s of liquid helium which are considered to exist below the )l-point. Accordingly, in an a t t e m p t to m a k e direct m e a s u r e m e n t s of this friction, the e x p e r i m e n t s of K e e s o m and M a c W o o d 2) were modified to include large a m p l i t u d e oscillations. T h e period was also a l t e r e d to less t h a n half its f o r m e r value. I n their p a p e r G o r t e r a n d M e 11 i n k suggest t h a t the force due to the m u t u a l friction is p r o p o r t i o n a l to the cube of the relative velocities of the two fluids, and t h a t it m a y be w r i t t e n
F = Ag.es ( v s -
vn) 3,
where Q., Qs are the densities and vs and v. the velocities of the norm a l and superfluid c o m p o n e n t s respectively, and A a c o n s t a n t w i t h dimensions c m . s e c / g r a m . Supposing t h a t a l a y e r of n o r m a l fluid is carried along w i t h the disc while the m o t i o n of the superfluid is m u c h smaller t h a n t h a t of the n o r m a l fluid, it is possible to predict the order of m a g n i t u d e of the increase of the d a m p i n g which is due to the m u t u a l friction. - -
8 0 8
- -
EXPERIMENTS WITH OSCILLATING DISC IN LIQUID HELIUM 8 0 9
According to a calculation b y G. C. J. Z w a n i k k e n 3), which is given in a separate note, the d a m p i n g is a p p r o x i m a t e l y proportional to I -~- l / l 0 y - -
• ....
where y = ~o~a2A a ~ . H e r e a is the radius of the disc, ~0 is the a m p l i t u d e in radians and a~ the angular f r e q u e n c y of the oscillation. W h e n a p p r o p r i a t e values are inserted, (~0 = 0.2 radians, ~ = 0.1, ~o=2:~/10 and a = 2 cm), it is seen that, if A is of the order 50; y is of the order 0.5. Thus a 5 % increase in the d a m p i n g should be expected at the largest amplitudes, but falling off rapidly with the square of the amplitude. An effect of this m a g n i t u d e should be detectable.
2. Experimental details. The a p p a r a t u s used was t h a t used b y K e e s o m and M a c W o o d, in their 1938 viscosity d e t e r m i n a tions. It was, however, modified to give a disc spacing of one centim e t r e (i.e. the distance from the moving disc to the fixed discs above and below it was I cm in each case) and the period reduced to l0 seconds. The damping, or logarithmic d e c r e m e n t , was measured using a telescope and illuminated scale. The telescope was focussed onto the image of the scale in a small m i r r o r a t t a c h e d to the rod s u p p o r t i n g the disc. The logarithmic d e c r e m e n t was obtained from about ten complete oscillations, the a r i t h m e t i c a l m e t h o d of calculation e m p l o y e d being t h a t used and described b y L i g n a c ~). The amplitude e m p l o y e d ranged from 0.2 radians to 0.05 radians. To measure such large amplitudes it was necessary to have the scale one m e t r e from the mirror; hence the correction for the scale distance was quite considerable. The torsion wire was of p h o s p h o r b r o n z e and about 6 cm long. Its d i a m e t e r was of the order of 70 t,, and its n a t u r a l d a m p i n g was v e r y small. T e m p e r a t u r e control was effected b y the usual regulating device.
3. Experimental results. Values of the logarithmic d e c r e m e n t and its square are given below for various amplitudes and t e m p e r a t u r e s . T h e s t a n d a r d deviations are also given.
810
P . L . SMITH TABLE I T oK
A m p l i t u d e . 10 z
D e c r e m e n t . 10s
Standard deviation
2.15
19.61 8.60 4.80
19.62 19.44 19.45
0.319
1.88
20.90 11.04 5.34
I0.64 10.35 10.19
0.309
1.69
20.90 11.86 6.65
9.51 9.29 9.46
0.203
19.68 11.49
7.85 7.88
21.48 12.42 7.98
6.34 6.50 6.23
1.54 1.35
0.165
0.300
4. Discussion o~ results. It is seen t h a t the effect s o u g h t a f t e r does not a p p e a r to a n y significant extent. T h e d a m p i n g was found to r e m a i n a p p r e c i a b l y constant x)ver a wide range of amplitudes. This leads to the conclusion t h a t m u t u a l friction, if it exists, is not of a sufficiently large m a g n i t u d e to be d e t e c t e d b y an e x p e r i m e n t of this degree of accuracy. Thus, if Z w a n i k k e n's calculation is valid, the conclusion m u s t be t h a t A is smaller t h a n was e s t i m a t e d on the basis of the m e a s u r e m e n t s on heat c u r r e n t s in wide capillaries i). According to "C o c h r a n's s) c r i t e r i u m no t u r b u l e n c e should have occurred. A few e x p e r i m e n t s have been carried out over the lambdat e m p e r a t u r e in order to check K e e s o m and M a c W o o d ' s m e a s u r e m e n t s in this region. It was found however, in a g r e e m e n t with K a p i t z a e ) and M e n d e l s s o h n T ) , t h a t the viscosity varies only slightly between 2.4 and 3.7°K. This is being tested with g r e a t e r a c c u r a c y b y V a n I t t e r b e e k and D e T r o y e r. My t h a n k s are due to Prof. C. J. G o r t e r, b o t h for his interest, suggestions and criticism, and for the generous h o s p i t a l i t y of his l a b o r a t o r y . Also to Mr. L. N e u t e b o o m for his invaluable help with the cryogenic a r r a n g e m e n t s , and to the N e t h e r l a n d s - B r i t i s h C u l t u r a l Relations D e p a r t m e n t of the N e t h e r l a n d s Ministry of E d u c a t i o n , whose grant enabled me to c a r r y out the above work. Received 7-9-50
E X P E R I M E N T S W I T H OSCILLATING DISC IN L I Q U I D H E L I U M
81 1
REFERENCES 1) G o r t e r , C.J.,and Mellink, J. H . , C o m m u n . Kamerlingh Onnes Laboratory, Suppl. No. 99a; Physiea, 's-Gray. 15 (1949) 523. 2) K e e s o m , W.H.,and MacWood, G . E . , C o m m u n . No. 254a;Physiea,'s-Grav. 5 (1938) 737. 3) Z w a n i k k e n, G. C. J., Commun. Suppl. No. 1O3a; Physiea, 's-Grav. 16 (1950), 805. 4) L i g n a c , W. P. J.r Thesis 1949, Leiden. 5) C o e h r a n, W. G., Proc. Cambridge phil. Soc. 3@ (1934) 365. 6) Cf. A l l e n, J. F., and M i s e n e r , A. D., Nature, London 141 (1938) 75. 7) B o w e r s , R.,and M e n d e l s s o h n , K.,Proe. phys. Soc., London 62 (1949) 394.
O c t o b e r 1950
SOME FURTHER EXPERIMENTS WITH AN OSCILLATING DISC IN LIQUID HELIUM by P. L. SMITH Comnmnication No. 282c from the Kamerlingh Onnes l.aboratory, l.eiden Synopsis
A n a t t e m p t was m a d e t o s h o w t h e e x i s t e n c e of m u t u a l friction b e t w e e n liquid h e l i u m I a n d liquid h e l i u m II. S u c h a f r i c t i o n s h o u l d be o b s e r v e d as an i n c r e a s e in t h e d a m p i n g c o e f f i c i e n t of a n o s c i l l a t i n g disc w h e n large a m p l i t u d e s w e r e e m p l o y e d . N o s u c h increase, h o w e v e r , was f o u n d . Preliminary experiments above the lambda-point roughly confirmed M e n d e 1 s s o h n's viscosity measurements made with a capillary tube.
1. Introduction. T h e discussion b y G o r t e r and M e 11 i n k 1) of the c o n d u c t i o n of h e a t and of the fountain effect in liquid helium 11 suggests t h e existence of some f o r m of friction b e t w e e n the t w o c o m p o n e n t s of liquid helium which are considered to exist below the )l-point. Accordingly, in an a t t e m p t to m a k e direct m e a s u r e m e n t s of this friction, the e x p e r i m e n t s of K e e s o m and M a c W o o d 2) were modified to include large a m p l i t u d e oscillations. T h e period was also a l t e r e d to less t h a n half its f o r m e r value. I n their p a p e r G o r t e r a n d M e 11 i n k suggest t h a t the force due to the m u t u a l friction is p r o p o r t i o n a l to the cube of the relative velocities of the two fluids, and t h a t it m a y be w r i t t e n
F = Ag.es ( v s -
vn) 3,
where Q., Qs are the densities and vs and v. the velocities of the norm a l and superfluid c o m p o n e n t s respectively, and A a c o n s t a n t w i t h dimensions c m . s e c / g r a m . Supposing t h a t a l a y e r of n o r m a l fluid is carried along w i t h the disc while the m o t i o n of the superfluid is m u c h smaller t h a n t h a t of the n o r m a l fluid, it is possible to predict the order of m a g n i t u d e of the increase of the d a m p i n g which is due to the m u t u a l friction. - -
8 0 8
- -
EXPERIMENTS WITH OSCILLATING DISC IN LIQUID HELIUM 8 0 9
According to a calculation b y G. C. J. Z w a n i k k e n 3), which is given in a separate note, the d a m p i n g is a p p r o x i m a t e l y proportional to I -~- l / l 0 y - -
• ....
where y = ~o~a2A a ~ . H e r e a is the radius of the disc, ~0 is the a m p l i t u d e in radians and a~ the angular f r e q u e n c y of the oscillation. W h e n a p p r o p r i a t e values are inserted, (~0 = 0.2 radians, ~ = 0.1, ~o=2:~/10 and a = 2 cm), it is seen that, if A is of the order 50; y is of the order 0.5. Thus a 5 % increase in the d a m p i n g should be expected at the largest amplitudes, but falling off rapidly with the square of the amplitude. An effect of this m a g n i t u d e should be detectable.
2. Experimental details. The a p p a r a t u s used was t h a t used b y K e e s o m and M a c W o o d, in their 1938 viscosity d e t e r m i n a tions. It was, however, modified to give a disc spacing of one centim e t r e (i.e. the distance from the moving disc to the fixed discs above and below it was I cm in each case) and the period reduced to l0 seconds. The damping, or logarithmic d e c r e m e n t , was measured using a telescope and illuminated scale. The telescope was focussed onto the image of the scale in a small m i r r o r a t t a c h e d to the rod s u p p o r t i n g the disc. The logarithmic d e c r e m e n t was obtained from about ten complete oscillations, the a r i t h m e t i c a l m e t h o d of calculation e m p l o y e d being t h a t used and described b y L i g n a c ~). The amplitude e m p l o y e d ranged from 0.2 radians to 0.05 radians. To measure such large amplitudes it was necessary to have the scale one m e t r e from the mirror; hence the correction for the scale distance was quite considerable. The torsion wire was of p h o s p h o r b r o n z e and about 6 cm long. Its d i a m e t e r was of the order of 70 t,, and its n a t u r a l d a m p i n g was v e r y small. T e m p e r a t u r e control was effected b y the usual regulating device.
3. Experimental results. Values of the logarithmic d e c r e m e n t and its square are given below for various amplitudes and t e m p e r a t u r e s . T h e s t a n d a r d deviations are also given.
810
P . L . SMITH TABLE I T oK
A m p l i t u d e . 10 z
D e c r e m e n t . 10s
Standard deviation
2.15
19.61 8.60 4.80
19.62 19.44 19.45
0.319
1.88
20.90 11.04 5.34
I0.64 10.35 10.19
0.309
1.69
20.90 11.86 6.65
9.51 9.29 9.46
0.203
19.68 11.49
7.85 7.88
21.48 12.42 7.98
6.34 6.50 6.23
1.54 1.35
0.165
0.300
4. Discussion o~ results. It is seen t h a t the effect s o u g h t a f t e r does not a p p e a r to a n y significant extent. T h e d a m p i n g was found to r e m a i n a p p r e c i a b l y constant x)ver a wide range of amplitudes. This leads to the conclusion t h a t m u t u a l friction, if it exists, is not of a sufficiently large m a g n i t u d e to be d e t e c t e d b y an e x p e r i m e n t of this degree of accuracy. Thus, if Z w a n i k k e n's calculation is valid, the conclusion m u s t be t h a t A is smaller t h a n was e s t i m a t e d on the basis of the m e a s u r e m e n t s on heat c u r r e n t s in wide capillaries i). According to "C o c h r a n's s) c r i t e r i u m no t u r b u l e n c e should have occurred. A few e x p e r i m e n t s have been carried out over the lambdat e m p e r a t u r e in order to check K e e s o m and M a c W o o d ' s m e a s u r e m e n t s in this region. It was found however, in a g r e e m e n t with K a p i t z a e ) and M e n d e l s s o h n T ) , t h a t the viscosity varies only slightly between 2.4 and 3.7°K. This is being tested with g r e a t e r a c c u r a c y b y V a n I t t e r b e e k and D e T r o y e r. My t h a n k s are due to Prof. C. J. G o r t e r, b o t h for his interest, suggestions and criticism, and for the generous h o s p i t a l i t y of his l a b o r a t o r y . Also to Mr. L. N e u t e b o o m for his invaluable help with the cryogenic a r r a n g e m e n t s , and to the N e t h e r l a n d s - B r i t i s h C u l t u r a l Relations D e p a r t m e n t of the N e t h e r l a n d s Ministry of E d u c a t i o n , whose grant enabled me to c a r r y out the above work. Received 7-9-50
E X P E R I M E N T S W I T H OSCILLATING DISC IN L I Q U I D H E L I U M
81 1
REFERENCES 1) G o r t e r , C.J.,and Mellink, J. H . , C o m m u n . Kamerlingh Onnes Laboratory, Suppl. No. 99a; Physiea, 's-Gray. 15 (1949) 523. 2) K e e s o m , W.H.,and MacWood, G . E . , C o m m u n . No. 254a;Physiea,'s-Grav. 5 (1938) 737. 3) Z w a n i k k e n, G. C. J., Commun. Suppl. No. 1O3a; Physiea, 's-Grav. 16 (1950), 805. 4) L i g n a c , W. P. J.r Thesis 1949, Leiden. 5) C o e h r a n, W. G., Proc. Cambridge phil. Soc. 3@ (1934) 365. 6) Cf. A l l e n, J. F., and M i s e n e r , A. D., Nature, London 141 (1938) 75. 7) B o w e r s , R.,and M e n d e l s s o h n , K.,Proe. phys. Soc., London 62 (1949) 394.