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The viscous penetration depth problem in 3He-A
RF 7
Physica 108B (1981) 1145-1146 North-Holland Publishing Company
THE VlSCOUS PENETRATION DEPTH PROBLEM IN +He-A
Physics
J.R. Hook Department, Manchester University Manchester MI3 9PL, U.K.
W. Zimmermann, Jr. Tate Laboratory of Physics, U n i v e r s i t y of Minnesota Minneapolis, Minnesota, U.S.A. 55455 We have solved approximately the hydrodynamic equations for a s e m i - i n f i n i t e volume of 3He-A bounded by an o s c i l l a t i n g plane wall for the case where the d vector is locked perpendicular to the wall by a magnetic f i e l d . The amplitude of o s c i l l a t i o n of the o r b i t a l axis ~ is in general small and the normal f l u i d motion is dominated by the shear v i s c o s i t y , - b u t other c o e f f i c i e n t s in the hydrodynamic equations r e s u l t in the appearance of a tangential stress on the boundary perpendicular to the d i r e c t i o n of motion, the measurement of which would provide a check of both the v a l i d i t y of the hydrodynamic equations and the values obtained from microscopic theory of some of the c o e f f i c i e n t s in these equations. There is now a generally agreed form for the equations which should describe the hydrodynamics of 3He-A [ I ] , and estimates from microscopic theory of many of the hydrodynamic c o e f f i c i e n t s have now been made. We present here a preliminary account of an attempt to solve these equations f o r the experimentally important viscous penetration depth problem. The f l u i d occupies the region x > 0 and is bounded at x = 0 by a plane w a l l , o s c i l l a t i n g in the y d i r e c t i o n at v e l o c i t y w = WoeXp(-imt ).
~-
,l)
.)
where
We assume that in the undisturbed state (w ÷ O) the o r b i t a l axis m is p a r a l l e l to the x axis. We take d to be locked perpendicular to the wall by a magnetic f i e l d in the plane of the wall (H >> 30 gauss), d then influences the motion of z through the-nuclear d i p o l e - d i p o l e i n t e r a c t i o~ -g (d. J~)~ = -g Zx2. To a good approximation the motion of the f l u i d should be governed by the f o l l o w i n g equations.
V~)= 0
~~
=0
(n)
- i . ~ v z = ~, ~_~_~)-
0378-4363/81/0000-0000/$02.50
+
© North-HollandPublishingCompany
, and we have used the n o t a t i o n of Hu and Saslow [1], e x c e p t t h a t we use the o r b i t a l v i s c o s i t y ~ ~nstead of t h e i r n(=(~/2m)a/u) and we use the symbol x f o r the s h e a r v i s c o s i t y f o r s h e a r flow in a plane c o n t a i n i n g ~.
In deriving these equations we have made the f o l l o w i n g assumptions and approximations: (a) Spatial v a r i a t i o n only in the x d i r e c t i o n . (b) Uniform temperature and pressure. (c) Only terms up to f i r s t order in v(n) need be considered (the d r i v i n g v e l o c i t y w-is w i t h i n the control of the experimenter and can, at least in p r i n c i p l e , be made s u f f i c i e n t l y small that t h i s approximation is v a l i d ) . (d) Departures of ~ from ~ : ~ are small so that we may take +z--: (0, -Zy, +z) and consider only terms up to f i r s t order in c and Zz. (Note Y that Zy and ~z are f i r s t order in ~ ( n ) . )
1145
1146
(e) Cy, ~z' yV(n) and v (n)z haveatime dependence exp(-iwt). (This approximation obviously becomes i n v a l i d when the two previous approximations break down.) ( f ) v (s) = 0 (the d r i v i n g terms in the superf l u i d - a c c e l e r a t i o n equation are second order in v(n)~ Tg) In accordance with microscopic theory we put ~ = O. Approximations (a) to ( f ) are s e l f cons i s t e n t and they can be checked using the solution obtained below. T h e o r e t i c a l l y the problem becomes much less tractable i f these approximations are not made; experimentally t h e i r breakdown could be detected by the appearance of stresses on the boundary which were non l i n e a r in w. Relaxing assumption (g) on the other hand complicates s l i g h t l y the solution of the above equations but causes no q u a l i t a t i v e differences. The order of magnitude of the hydrodynamic c o e f f i c i e n t s is given by
a l t h o u g h t h e d i v e r g e n c e o f a 2 and ~ c a u s e s t h e
In order that the s o l u t i o n should be well behaved as x ÷ ~, i t is necessary to choose values of k+, k and ~ with p o s i t i v e imaginary parts. The approximation used to obtain the above solution depends on the fact that the natural length scale for spatial v a r i a t i o n of is much less than that for spatial v a r i a t i o n of
v
i.e.
]~-I I << ] k ; l [ ~ Ik - I ]
I t is
possible to use t h i s solution to check some of the assumptions made in deriving the equations of motion ( I ) to (4). A s u f f i c i e n t condition f o r the neglect of terms of second order and above in ~(n) appears to be wo <<~]~n/2m;
the
minimum value of~Im]/2m (at low frequencies) is about 1 mm.s-I so that t h i s condition is not very r e s t r i c t i v e . The amplitude of o s c i l l a t i o n of ~ is very small except very close to T -
c
where the divergence of C causes the amplitude to become large. The above s o l u t i o n may be used to evaluate the tangential stresses on the o s c i l l a t i n g w a l l . Because of (5) the component a of the stress tensor is very close to that yx obtained i f a l l the hydrodynamic c o e f f i c i e n t s except × are neglected, namely
i n e q u a l i t y t o break down very c l o s e t o T . c E x p l o i t i n g (5) we are a b l e t o f i n d t h e f o l l o w i n g a p p r o x i m a t i o n s o l u t i o n o f the above e q u a t i o n s .
9
B
but the existence of the other c o e f f i c i e n t s results in a f i n i t e value for the stress tensor component azx. To lowest non-zero order in k/a we f i n d
_
-
;
where The detection and measurement of o
-
"o(i_ t!l- ' o(l t
b=- k+A(i- r)j E=-k_g(l+
would zx provide a check on the v a l i d i t y of the hydrodynamic equations and the values of the hydro dynamic c o e f f i c i e n t s as given by e.g. Nagai [2]. We are in the process of conside?ing the case where d is not locked by a magnetic f i e l d and s i t u a t i o n s where the assumptions and approximations used above are no longer v a l i d . Full d e t a i l s of these investigations and of the above calculations w i l l be published elsewhere. [ I ] See e.g. Hu C.R. and Saslow W.M., Phys. Rev. Letts. 38 (1977) 605-609. [2] Nagai K-7~,J. Low Temp. Phys. 3__8(1980) 677-705.
Physica 108B (1981) 1145-1146 North-Holland Publishing Company
THE VlSCOUS PENETRATION DEPTH PROBLEM IN +He-A
Physics
J.R. Hook Department, Manchester University Manchester MI3 9PL, U.K.
W. Zimmermann, Jr. Tate Laboratory of Physics, U n i v e r s i t y of Minnesota Minneapolis, Minnesota, U.S.A. 55455 We have solved approximately the hydrodynamic equations for a s e m i - i n f i n i t e volume of 3He-A bounded by an o s c i l l a t i n g plane wall for the case where the d vector is locked perpendicular to the wall by a magnetic f i e l d . The amplitude of o s c i l l a t i o n of the o r b i t a l axis ~ is in general small and the normal f l u i d motion is dominated by the shear v i s c o s i t y , - b u t other c o e f f i c i e n t s in the hydrodynamic equations r e s u l t in the appearance of a tangential stress on the boundary perpendicular to the d i r e c t i o n of motion, the measurement of which would provide a check of both the v a l i d i t y of the hydrodynamic equations and the values obtained from microscopic theory of some of the c o e f f i c i e n t s in these equations. There is now a generally agreed form for the equations which should describe the hydrodynamics of 3He-A [ I ] , and estimates from microscopic theory of many of the hydrodynamic c o e f f i c i e n t s have now been made. We present here a preliminary account of an attempt to solve these equations f o r the experimentally important viscous penetration depth problem. The f l u i d occupies the region x > 0 and is bounded at x = 0 by a plane w a l l , o s c i l l a t i n g in the y d i r e c t i o n at v e l o c i t y w = WoeXp(-imt ).
~-
,l)
.)
where
We assume that in the undisturbed state (w ÷ O) the o r b i t a l axis m is p a r a l l e l to the x axis. We take d to be locked perpendicular to the wall by a magnetic f i e l d in the plane of the wall (H >> 30 gauss), d then influences the motion of z through the-nuclear d i p o l e - d i p o l e i n t e r a c t i o~ -g (d. J~)~ = -g Zx2. To a good approximation the motion of the f l u i d should be governed by the f o l l o w i n g equations.
V~)= 0
~~
=0
(n)
- i . ~ v z = ~, ~_~_~)-
0378-4363/81/0000-0000/$02.50
+
© North-HollandPublishingCompany
, and we have used the n o t a t i o n of Hu and Saslow [1], e x c e p t t h a t we use the o r b i t a l v i s c o s i t y ~ ~nstead of t h e i r n(=(~/2m)a/u) and we use the symbol x f o r the s h e a r v i s c o s i t y f o r s h e a r flow in a plane c o n t a i n i n g ~.
In deriving these equations we have made the f o l l o w i n g assumptions and approximations: (a) Spatial v a r i a t i o n only in the x d i r e c t i o n . (b) Uniform temperature and pressure. (c) Only terms up to f i r s t order in v(n) need be considered (the d r i v i n g v e l o c i t y w-is w i t h i n the control of the experimenter and can, at least in p r i n c i p l e , be made s u f f i c i e n t l y small that t h i s approximation is v a l i d ) . (d) Departures of ~ from ~ : ~ are small so that we may take +z--: (0, -Zy, +z) and consider only terms up to f i r s t order in c and Zz. (Note Y that Zy and ~z are f i r s t order in ~ ( n ) . )
1145
1146
(e) Cy, ~z' yV(n) and v (n)z haveatime dependence exp(-iwt). (This approximation obviously becomes i n v a l i d when the two previous approximations break down.) ( f ) v (s) = 0 (the d r i v i n g terms in the superf l u i d - a c c e l e r a t i o n equation are second order in v(n)~ Tg) In accordance with microscopic theory we put ~ = O. Approximations (a) to ( f ) are s e l f cons i s t e n t and they can be checked using the solution obtained below. T h e o r e t i c a l l y the problem becomes much less tractable i f these approximations are not made; experimentally t h e i r breakdown could be detected by the appearance of stresses on the boundary which were non l i n e a r in w. Relaxing assumption (g) on the other hand complicates s l i g h t l y the solution of the above equations but causes no q u a l i t a t i v e differences. The order of magnitude of the hydrodynamic c o e f f i c i e n t s is given by
a l t h o u g h t h e d i v e r g e n c e o f a 2 and ~ c a u s e s t h e
In order that the s o l u t i o n should be well behaved as x ÷ ~, i t is necessary to choose values of k+, k and ~ with p o s i t i v e imaginary parts. The approximation used to obtain the above solution depends on the fact that the natural length scale for spatial v a r i a t i o n of is much less than that for spatial v a r i a t i o n of
v
i.e.
]~-I I << ] k ; l [ ~ Ik - I ]
I t is
possible to use t h i s solution to check some of the assumptions made in deriving the equations of motion ( I ) to (4). A s u f f i c i e n t condition f o r the neglect of terms of second order and above in ~(n) appears to be wo <<~]~n/2m;
the
minimum value of~Im]/2m (at low frequencies) is about 1 mm.s-I so that t h i s condition is not very r e s t r i c t i v e . The amplitude of o s c i l l a t i o n of ~ is very small except very close to T -
c
where the divergence of C causes the amplitude to become large. The above s o l u t i o n may be used to evaluate the tangential stresses on the o s c i l l a t i n g w a l l . Because of (5) the component a of the stress tensor is very close to that yx obtained i f a l l the hydrodynamic c o e f f i c i e n t s except × are neglected, namely
i n e q u a l i t y t o break down very c l o s e t o T . c E x p l o i t i n g (5) we are a b l e t o f i n d t h e f o l l o w i n g a p p r o x i m a t i o n s o l u t i o n o f the above e q u a t i o n s .
9
B
but the existence of the other c o e f f i c i e n t s results in a f i n i t e value for the stress tensor component azx. To lowest non-zero order in k/a we f i n d
_
-
;
where The detection and measurement of o
-
"o(i_ t!l- ' o(l t
b=- k+A(i- r)j E=-k_g(l+
would zx provide a check on the v a l i d i t y of the hydrodynamic equations and the values of the hydro dynamic c o e f f i c i e n t s as given by e.g. Nagai [2]. We are in the process of conside?ing the case where d is not locked by a magnetic f i e l d and s i t u a t i o n s where the assumptions and approximations used above are no longer v a l i d . Full d e t a i l s of these investigations and of the above calculations w i l l be published elsewhere. [ I ] See e.g. Hu C.R. and Saslow W.M., Phys. Rev. Letts. 38 (1977) 605-609. [2] Nagai K-7~,J. Low Temp. Phys. 3__8(1980) 677-705.