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The influence of mutual friction between the two fluids in liquid helium II on the energy dissipation by an oscillating disc
P h y s i c a X V I , n o 10
O c t o b e r 1950
THE INFLUENCE OF MUTUAL FRICTION BETWEEN THE TWO FLUIDS I N LIQUID HELIUM II ON THE ENERGY DISSIPATION BY AN OSCILLATING DISC b y G. C. J. Z W A N I K K E N Bisschoppelijk College, Roermond, Nederland Communications from the Kamerlingh Onnes Laboratory, Suppl. No. 103a
Synopsis T h e d i s s i p a t i o n s b y m u t u a l f r i c t i o n b e t w e e n t h e t w o fluids in l i q u i d h e l i u m I I a n d b y v i s c o u s f r i c t i o n are c o m p u t e d for t h e case t h a t oscillat i o n s a r e c a u s e d b y a n o s c i l l a t i n g disc.
The dissipations b y m u t u a l friction and b y viscous friction are c o m p u t e d for the case t h a t the two fluids describe oscillations about a vertical z-axis, caused b y an" oscillating disc with radius a. In absence of gradients of t e m p e r a t u r e or c o n c e n t r a t i o n the equations of m o t i o n of the two fluids are 1) ~,1 A vn - - ~ grad div v . + Ae.~s [ v S- - vn [2 (v s __ v.) = Qn~;n, -- Ae.es
[ v s - - v. 12 (vs - - Vn) = esL,
with ~/n the viscosity of the n o r m a l fluid, en and 9, the densities of the n o r m a l fluid and the superfluid respectively, v . and v s the velocities and A a constant for the m u t u a l friction. I n t r o d u c i n g cylindrical coordinates and a n g u l a r velocities b y v= :
--yu,
vy :
XU, v= :
O,
X2 +
y2 :
and assuming v -----0 for r > a, the equations read ~1. /1 u. + e.es Ar2 (us - --
e . e s A r 2 (us - --
805
--
u.) 3 =
enU,,,
u,,) 3 ~
e,us.
r 2,
806
G.C.J. ZWANIKKEN
The motion is in first a p p r o x i m a t i o n determined b y t h a t of the disc
cos (o~t
u. ~ C e-~'--~ 'Us
-
6z),
-
~ 0 .
Substitution gives
(6 2 _ --
+
7 2) ~]n
2 y6q. +
aen •
o~e.
N o w , since i n t h e e x p e r i m e n t s
0, 0.
=
a "~ 10 --3 sec - l
a n d a~ -~ 1 sec - 1 ,
we shall take a = 0, and thus 2 76 = 2 62 = °~e"
(1)
The second approximation then reads u.
-
-
~3 ~esrZA - C3 e__~8s cos (oJt - - 6z) +
Ce -~" c o s (o,t - - 6z) +
3 Qsr2A 1 ~sr2A C3e-38s sin (o~t--~z), (2) + -- -C3e- ° ~ sin (or - - 6z) + - - - 40 oJ 24 oJ us=-~3 gnr2A C3e--3~" sin (wt--6z) + l e"r2~A C3e-33s sin 3 (oJt--6z). (3) w 12 oJ The m e a n energy dissipation in a ring-shaped volume of radius r caused b y m u t u a l friction is given b y 2~r dr dz. A q.qsr 4 (u s u,) 4. -
N°w (us--u")4
-
3 C4e_~,21- 9 qsr2A_AC6e___6~"+ .....
= g
40
o,
I n t e g r a t i o n over space, i.e. 2 × over z from 0 to co and over r from 0 to a, gives the total dissipation b y m u t u a l friction
~a : z-(-~ ~a6 AOnOs C 4 (I + 103esaZAo9 C2 + . . . . )"
(4)
The m e a n dissipation b y viscous friction of the normal fluid in the
(OUn~2,
same volume is 2~r dr dz.~nr 2 \-~z /
(Ou.~ 2 -~z/
62C2e-2~, ----
9 + 20
with
osr2A62 C4e--4~, +
....
co
I n t e g r a t i n g over space we find the dissipation b y viscous friction
~o,~-
7~a4cgqn~2 ( 3Osa2ac2-[) ~ c 1+2-6-L-7 ....
(5) ,.
ENERGY DISSIPATION BY MUTUAL FRICTION IN HELIUM II
807
The above approximations are permitted only if 3 Q~az$ C2<~" 1. (6) 20 ~o The velocity u, of the normal component will have its maximum value when, according to (2) and (6), cos ( o J t - Oz)~ + 1, and so this m a x i m u m value will be the sum of the amplitudes of the cosinus terms, i.e. for z = 0 Unm,x.=C
( 1 +20
)
o~
Putting r = a, and neglecting the slip between disc and normal fluid, we m a y put
"3 osr2A
C ( I + ~ - 0 -~
C2)=O~o,
(7)
if the oscillation of the disc is given approximately by ~v ---- ~o0 sin o~t. The ratio Wa/%j of the dissipations is the quotient of (4) and (5). Making use of (1) and (6), this ratio reads
YJ_a= ¼ o#2A
°JCpo
1 -- -~ o~a2A oxp~ + . . . .
(8)
The total energy dissipation is
m4~"C ~( 1 + -2- -~sa2A c2 + . . . ) V'a + V",l- -
~
5
(9)
o~
and making use of (7) I
This latter formula, in which the dissipation is given as a function of the amplitude ~0o is suited for comparison with measurements on the damping of the oscillating disc. Received 7-9-50 REFERENCES 1) G o l ] t e r , C. J . , a n d M e l l i n k , J.H.,Commun. KamerlinghOnnesLab.,Leiden Suppl. No. 99a; Physiea, 's-Gray. 15 (1949) 523. G o r t e r , C.J., K a s t e l e y n , P . W . , a n d M e l l i n k , J. H.,Commun. Suppl. No. 100b; Physiea, 's-Gray. 16 (1950) 113. 2) S m i t h, P. I.., Commun. No. 282c; Physiea 16 (1950) 808.
O c t o b e r 1950
THE INFLUENCE OF MUTUAL FRICTION BETWEEN THE TWO FLUIDS I N LIQUID HELIUM II ON THE ENERGY DISSIPATION BY AN OSCILLATING DISC b y G. C. J. Z W A N I K K E N Bisschoppelijk College, Roermond, Nederland Communications from the Kamerlingh Onnes Laboratory, Suppl. No. 103a
Synopsis T h e d i s s i p a t i o n s b y m u t u a l f r i c t i o n b e t w e e n t h e t w o fluids in l i q u i d h e l i u m I I a n d b y v i s c o u s f r i c t i o n are c o m p u t e d for t h e case t h a t oscillat i o n s a r e c a u s e d b y a n o s c i l l a t i n g disc.
The dissipations b y m u t u a l friction and b y viscous friction are c o m p u t e d for the case t h a t the two fluids describe oscillations about a vertical z-axis, caused b y an" oscillating disc with radius a. In absence of gradients of t e m p e r a t u r e or c o n c e n t r a t i o n the equations of m o t i o n of the two fluids are 1) ~,1 A vn - - ~ grad div v . + Ae.~s [ v S- - vn [2 (v s __ v.) = Qn~;n, -- Ae.es
[ v s - - v. 12 (vs - - Vn) = esL,
with ~/n the viscosity of the n o r m a l fluid, en and 9, the densities of the n o r m a l fluid and the superfluid respectively, v . and v s the velocities and A a constant for the m u t u a l friction. I n t r o d u c i n g cylindrical coordinates and a n g u l a r velocities b y v= :
--yu,
vy :
XU, v= :
O,
X2 +
y2 :
and assuming v -----0 for r > a, the equations read ~1. /1 u. + e.es Ar2 (us - --
e . e s A r 2 (us - --
805
--
u.) 3 =
enU,,,
u,,) 3 ~
e,us.
r 2,
806
G.C.J. ZWANIKKEN
The motion is in first a p p r o x i m a t i o n determined b y t h a t of the disc
cos (o~t
u. ~ C e-~'--~ 'Us
-
6z),
-
~ 0 .
Substitution gives
(6 2 _ --
+
7 2) ~]n
2 y6q. +
aen •
o~e.
N o w , since i n t h e e x p e r i m e n t s
0, 0.
=
a "~ 10 --3 sec - l
a n d a~ -~ 1 sec - 1 ,
we shall take a = 0, and thus 2 76 = 2 62 = °~e"
(1)
The second approximation then reads u.
-
-
~3 ~esrZA - C3 e__~8s cos (oJt - - 6z) +
Ce -~" c o s (o,t - - 6z) +
3 Qsr2A 1 ~sr2A C3e-38s sin (o~t--~z), (2) + -- -C3e- ° ~ sin (or - - 6z) + - - - 40 oJ 24 oJ us=-~3 gnr2A C3e--3~" sin (wt--6z) + l e"r2~A C3e-33s sin 3 (oJt--6z). (3) w 12 oJ The m e a n energy dissipation in a ring-shaped volume of radius r caused b y m u t u a l friction is given b y 2~r dr dz. A q.qsr 4 (u s u,) 4. -
N°w (us--u")4
-
3 C4e_~,21- 9 qsr2A_AC6e___6~"+ .....
= g
40
o,
I n t e g r a t i o n over space, i.e. 2 × over z from 0 to co and over r from 0 to a, gives the total dissipation b y m u t u a l friction
~a : z-(-~ ~a6 AOnOs C 4 (I + 103esaZAo9 C2 + . . . . )"
(4)
The m e a n dissipation b y viscous friction of the normal fluid in the
(OUn~2,
same volume is 2~r dr dz.~nr 2 \-~z /
(Ou.~ 2 -~z/
62C2e-2~, ----
9 + 20
with
osr2A62 C4e--4~, +
....
co
I n t e g r a t i n g over space we find the dissipation b y viscous friction
~o,~-
7~a4cgqn~2 ( 3Osa2ac2-[) ~ c 1+2-6-L-7 ....
(5) ,.
ENERGY DISSIPATION BY MUTUAL FRICTION IN HELIUM II
807
The above approximations are permitted only if 3 Q~az$ C2<~" 1. (6) 20 ~o The velocity u, of the normal component will have its maximum value when, according to (2) and (6), cos ( o J t - Oz)~ + 1, and so this m a x i m u m value will be the sum of the amplitudes of the cosinus terms, i.e. for z = 0 Unm,x.=C
( 1 +20
)
o~
Putting r = a, and neglecting the slip between disc and normal fluid, we m a y put
"3 osr2A
C ( I + ~ - 0 -~
C2)=O~o,
(7)
if the oscillation of the disc is given approximately by ~v ---- ~o0 sin o~t. The ratio Wa/%j of the dissipations is the quotient of (4) and (5). Making use of (1) and (6), this ratio reads
YJ_a= ¼ o#2A
°JCpo
1 -- -~ o~a2A oxp~ + . . . .
(8)
The total energy dissipation is
m4~"C ~( 1 + -2- -~sa2A c2 + . . . ) V'a + V",l- -
~
5
(9)
o~
and making use of (7) I
This latter formula, in which the dissipation is given as a function of the amplitude ~0o is suited for comparison with measurements on the damping of the oscillating disc. Received 7-9-50 REFERENCES 1) G o l ] t e r , C. J . , a n d M e l l i n k , J.H.,Commun. KamerlinghOnnesLab.,Leiden Suppl. No. 99a; Physiea, 's-Gray. 15 (1949) 523. G o r t e r , C.J., K a s t e l e y n , P . W . , a n d M e l l i n k , J. H.,Commun. Suppl. No. 100b; Physiea, 's-Gray. 16 (1950) 113. 2) S m i t h, P. I.., Commun. No. 282c; Physiea 16 (1950) 808.