- Email: [email protected]
Energy transfer and phase slip by vortex motion in superfluid4He
PHYSICA
Physica B 194-196 (1994) 585-586 North-Holland
Energy Transfer and Phase Slip by Vortex Motion in Superfluid 4He W. Z i m m e r m a n n , J r . Tate L a b o r a t o r y of Physics, University of Minnesota, Minneapolis, M i n n e s o t a 55455, USA A d i s c u s s i o n is given of the i m p o r t a n c e of ideas developed b y H u g g i n s for u n d e r s t a n d i n g t h e energy t r a n s f e r w h i c h t a k e s place d u r i n g p h a s e slip by vortex m o t i o n in s u p e r f l u i d 4He, a n d s i m p l e e x a m p l e s are given. In addition, a g e n e r a l i z a t i o n of t h e e x p r e s s i o n for t h e effective e n e r g y of a n e l e m e n t a r y e x c i t a t i o n in flowing s u p e r f l u i d is p r e s e n t e d t h a t is applicable to a vortex s y s t e m . 1. INTRODUCTION
d T p / d t = I (lal -~2) - d E / d t ( p - ~ v } ,
R e c e n t e x p e r i m e n t s on t h e flow of superfluid 4He t h r o u g h small apertures have given evidence of the o c c u r r e n c e of p h a s e slip by v o r t e x m o t i o n . In o r d e r to u n d e r s t a n d t h e e n e r g y t r a n s f e r s in t h i s process, it is helpful to m a k e u s e of i d e a s developed for a n ideal i n c o m p r e s s i b l e fluid by H u g g i n s . [1] In p a r t i c u l a r , t h e s e ideas s h o w t h a t e n e r g y c a n be t r a n s f e r r e d from p o t e n t i a l flow to a vortex s y s t e m w i t h or without the concurrent action of d i s s i p a t i v e forces on the vortices. E s p e c i a l l y in t h e l a t t e r case, t h e vortex s y s t e m c a n act as a reservoir for the energy lost from p o t e n t i a l flow in the p h a s e - s l i p event, p o s s i b l y delivering t h a t e n e r g y to s o m e r e m o t e location after t h e p h a s e slip is essentially complete.
dTvldt=dEIdt(p-~v}
2. ENERGY TRANSFER Imagine t h a t t h e fluid is confined to a relatively long flow c h a n n e l w i t h p i s t o n s at b o t h e n d s t h a t move with t h e fluid. The velocity field v of a n i n c o m p r e s s i b l e fluid in t h e c h a n n e l c a n be w r i t t e n as the s u m of a p o t e n t i a l flow field Vp a n d a vortex field Vv. Basic to the development is the idea t h a t if t h e vortices are far from the pistons, so t h a t Vv = 0 there, the total kinetic energy of the liquid c a n be written simply as the s u m of kinetic energies Tp of potential flow a n d Tv of vortex flow, w i t h o u t a n y i n t e r a c t i o n term. The time rates of change of Tp a n d Tv c a n be w r i t t e n as follows:
+ p J r " gedV,
(1) {2)
w h e r e I is the total m a s s c u r r e n t in t h e c h a n n e l , P-I a n d kt2 are effective c h e m i c a l p o t e n t i a l s at the e n d s of the c h a n n e l , p is the m a s s d e n s i t y of the fluid, ge is s o m e n o n c o n s e r v a t i v e e x t e r n a l force field p e r u n i t m a s s acting on the fluid at the cores of v o r t i c e s , a n d d E / d t ( p ~ v ) is t h e r a t e of e n e r g y t r a n s f e r f r o m p o t e n t i a l to v o r t e x flow. For s u p e r f l u i d vortices, we c a n write dE/dt(p--~v)=
- pK J V p ' V l x d l ,
(3)
w h e r e ~cis the q u a n t u m of circulation, Vl is t h e velocity of a n e l e m e n t of v o r t e x line, a n d the integral is t a k e n along the cores of t h e vortices in t h e system. This e x p r e s s i o n r e p r e s e n t s *: t i m e s t h e negative of t h e n e t r a t e at w h i c h t h e m a s s c u r r e n t d u e to p o t e n t i a l flow t h r o u g h s u r f a c e s s p a n n i n g the vortex cores c h a n g e s with time d u e to v o r t e x m o t i o n a c r o s s t h e s t r e a m l i n e s of p o t e n t i a l flow. B e c a u s e Vl c o n t a i n s a convective p a r t a s well a s a p a r t d u e to e x t e r n a l force ge, d E / d t ( p - ~ v } c a n be n o n z e r o even w i t h o u t the action of ge. G u i d e d by e q u a t i o n s (1) - (31, we m a y i m a g i n e a p r o c e s s in w h i c h w o r k d o n e on the fluid by the p i s t o n s a c t s to increase the e n e r g y of p o t e n t i a l flow. T h i s e n e r g y in t u r n is t r a n s f o r m e d i n t o t h e e n e r g y of vortex flow by t h e m o t i o n of vortex lines. The latter energy is d i s s i p a t e d t h r o u g h the action of some external force ge.
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0873-F
586 3. ILLUSTRATIONS d I / d t = p*:~v I x d l , (a) C o n s i d e r first the s p o n t a n e o u s growth of a p l a n e c i r c u l a r vortex ring w h o s e selfi n d u c e d velocity relative to t h e s u p e r f l u i d is opposite to a n d less t h a n t h e u n i f o r m velocity of t h e s u p e r f l u i d itself. T h i s is a s i t u a t i o n i n v o k e d in t h e ILF t h e o r y of critical velocities. [2,3] Here the growth of the ring d e p e n d s on the c o n c u r r e n t action of a d r a g force ge d u e to n o r m a l fluid at rest. A l t h o u g h t h e w o r k d o n e by t h i s force is negative a n d vortex e n e r g y is being lost to it, t h i s e n e r g y loss is o v e r b a l a n c e d by the t r a n s f e r of e n e r g y from p o t e n t i a l flow to t h e v o r t e x ring, a n d t h e r i n g g a i n s energy as it expands. (b) C o n s i d e r n e x t a p l a n e s e m i c i r c u l a r vortex loop w h o s e axis lies in a plane wall a n d p a s s e s t h r o u g h a tiny circular a p e r t u r e in t h e wall from w h i c h s u p e r f l u i d flow is diverging. In the limit t h a t the r a d i u s of the a p e r t u r e is m u c h less t h a n t h a t of the loop, t h e loop will m o v e in a d i r e c t i o n p e r p e n d i c u l a r to its p l a n e a n d e x p a n d while r e t a i n i n g its s h a p e . [4] The e n d s of t h e loop are a s s u m e d to move freely over the wall. Here the vortex loop g a i n s energy f r o m p o t e n t i a l flow a s it c u t s l i n e s of p o t e n t i a l flow, w i t h o u t the a c t i o n of a n y e x t e r n a l force. On topological g r o u n d s , either of t h e s e two p r o c e s s e s c o u l d f o r m p a r t of a 2n p h a s e - s l i p in s u p e r f l u i d 4 H e . C a s e (b), i n v o l v i n g d i v e r g i n g flow w i t h o u t a n y e x t e r n a l force, is i l l u s t r a t i v e of w h a t in principle m a y h a p p e n a t low t e m p e r a t u r e s in t h e a b s e n c e of t h e n o r m a l f l u i d component. A more realistic case e q u i v a l e n t to c a s e (b) h a s r e c e n t l y b e e n p r o p o s e d for vortex growth in a n orifice. [5]
(4)
w h e r e the i n t e g r a l r u n s a l o n g t h e v o r t e x cores. Like d E / d t (p~v), d I / d t h a s b o t h a convective part, w h i c h m a y be p r e s e n t even in the absence of ge, a n d a part due to ge. As a result, I will n o t in g e n e r a l be conserved, even in the absence of ge. 5. EFFECTIVE ENERGY Use is sometimes m a d e of the expression F= E +vs'p
(5)
to give the effective e n e r g y of a v o r t e x s y s t e m in t h e l a b o r a t o r y f r a m e in t h e p r e s e n c e of u n i f o r m s u p e r f l u i d flow w i t h velocity v s. Here, E is the e n e r g y of t h e vortex s y s t e m in the m o v i n g frame, a n d p is t a k e n to be the impulse, even t h o u g h no vortex m o m e n t u m is involved. [2,3] A j u s t i f i c a t i o n for this expression a n d a g e n e r a l i z a t i o n to the c a s e of n o n u n i f o r m s u p e r f l u i d flow is p r o v i d e d , a t l e a s t in derivative form, by Eqs. (2) - (4) c o m b i n e d a n d rewritten a s follows: dWe/dt = dTv/dt + I vp.d(dI/dt).
(6)
Here, d W e / d t is the rate a t w h i c h force ge d o e s w o r k on t h e v o r t e x s y s t e m , to be identified with the r a t e of c h a n g e of t h e e f f e c t i v e e n e r g y of t h e s y s t e m . T h e derivative d T v / d t gives the rate of c h a n g e of vortex energy, the energy being a s s u m e d to be e n t i r e l y kinetic in t h i s t r e a t m e n t . The i n t e g r a l along t h e v o r t e x c o r e s gives a n appropriate average of Vp • d I / d t . Work s u p p o r t e d by the NSF u n d e r g r a n t DMR 90-02890.
4. IMPULSE OF THE VORTEX SYSTEM A l t h o u g h the vortex s y s t e m carries no m o m e n t u m , it is u s e f u l to d e f i n e its "impulse" I equal to the i m p u l s e n e e d e d to be delivered "impulsively" to the fluid by s o m e local e x t e r n a l force to c r e a t e the vortex s y s t e m . Despite the a m b i g u i t y in I so defined, its time rate of c h a n g e is given u n a m b i g u o u s l y by the expression
1. E. 1~ Huggins, Plays Rev A 1,327, 332 (1970). 2. S.V. Iordanskii, Sov Phys JETP 2 1 , 4 6 7 (1965). 3. J. S. Langer a n d M. E. Fisher, Phys Rev Lett 19, 560 (1967). 4. O. Avenel a n d E. Varoquaux, private communication. 5. K.W. Schwarz, preprmt.
Physica B 194-196 (1994) 585-586 North-Holland
Energy Transfer and Phase Slip by Vortex Motion in Superfluid 4He W. Z i m m e r m a n n , J r . Tate L a b o r a t o r y of Physics, University of Minnesota, Minneapolis, M i n n e s o t a 55455, USA A d i s c u s s i o n is given of the i m p o r t a n c e of ideas developed b y H u g g i n s for u n d e r s t a n d i n g t h e energy t r a n s f e r w h i c h t a k e s place d u r i n g p h a s e slip by vortex m o t i o n in s u p e r f l u i d 4He, a n d s i m p l e e x a m p l e s are given. In addition, a g e n e r a l i z a t i o n of t h e e x p r e s s i o n for t h e effective e n e r g y of a n e l e m e n t a r y e x c i t a t i o n in flowing s u p e r f l u i d is p r e s e n t e d t h a t is applicable to a vortex s y s t e m . 1. INTRODUCTION
d T p / d t = I (lal -~2) - d E / d t ( p - ~ v } ,
R e c e n t e x p e r i m e n t s on t h e flow of superfluid 4He t h r o u g h small apertures have given evidence of the o c c u r r e n c e of p h a s e slip by v o r t e x m o t i o n . In o r d e r to u n d e r s t a n d t h e e n e r g y t r a n s f e r s in t h i s process, it is helpful to m a k e u s e of i d e a s developed for a n ideal i n c o m p r e s s i b l e fluid by H u g g i n s . [1] In p a r t i c u l a r , t h e s e ideas s h o w t h a t e n e r g y c a n be t r a n s f e r r e d from p o t e n t i a l flow to a vortex s y s t e m w i t h or without the concurrent action of d i s s i p a t i v e forces on the vortices. E s p e c i a l l y in t h e l a t t e r case, t h e vortex s y s t e m c a n act as a reservoir for the energy lost from p o t e n t i a l flow in the p h a s e - s l i p event, p o s s i b l y delivering t h a t e n e r g y to s o m e r e m o t e location after t h e p h a s e slip is essentially complete.
dTvldt=dEIdt(p-~v}
2. ENERGY TRANSFER Imagine t h a t t h e fluid is confined to a relatively long flow c h a n n e l w i t h p i s t o n s at b o t h e n d s t h a t move with t h e fluid. The velocity field v of a n i n c o m p r e s s i b l e fluid in t h e c h a n n e l c a n be w r i t t e n as the s u m of a p o t e n t i a l flow field Vp a n d a vortex field Vv. Basic to the development is the idea t h a t if t h e vortices are far from the pistons, so t h a t Vv = 0 there, the total kinetic energy of the liquid c a n be written simply as the s u m of kinetic energies Tp of potential flow a n d Tv of vortex flow, w i t h o u t a n y i n t e r a c t i o n term. The time rates of change of Tp a n d Tv c a n be w r i t t e n as follows:
+ p J r " gedV,
(1) {2)
w h e r e I is the total m a s s c u r r e n t in t h e c h a n n e l , P-I a n d kt2 are effective c h e m i c a l p o t e n t i a l s at the e n d s of the c h a n n e l , p is the m a s s d e n s i t y of the fluid, ge is s o m e n o n c o n s e r v a t i v e e x t e r n a l force field p e r u n i t m a s s acting on the fluid at the cores of v o r t i c e s , a n d d E / d t ( p ~ v ) is t h e r a t e of e n e r g y t r a n s f e r f r o m p o t e n t i a l to v o r t e x flow. For s u p e r f l u i d vortices, we c a n write dE/dt(p--~v)=
- pK J V p ' V l x d l ,
(3)
w h e r e ~cis the q u a n t u m of circulation, Vl is t h e velocity of a n e l e m e n t of v o r t e x line, a n d the integral is t a k e n along the cores of t h e vortices in t h e system. This e x p r e s s i o n r e p r e s e n t s *: t i m e s t h e negative of t h e n e t r a t e at w h i c h t h e m a s s c u r r e n t d u e to p o t e n t i a l flow t h r o u g h s u r f a c e s s p a n n i n g the vortex cores c h a n g e s with time d u e to v o r t e x m o t i o n a c r o s s t h e s t r e a m l i n e s of p o t e n t i a l flow. B e c a u s e Vl c o n t a i n s a convective p a r t a s well a s a p a r t d u e to e x t e r n a l force ge, d E / d t ( p - ~ v } c a n be n o n z e r o even w i t h o u t the action of ge. G u i d e d by e q u a t i o n s (1) - (31, we m a y i m a g i n e a p r o c e s s in w h i c h w o r k d o n e on the fluid by the p i s t o n s a c t s to increase the e n e r g y of p o t e n t i a l flow. T h i s e n e r g y in t u r n is t r a n s f o r m e d i n t o t h e e n e r g y of vortex flow by t h e m o t i o n of vortex lines. The latter energy is d i s s i p a t e d t h r o u g h the action of some external force ge.
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0873-F
586 3. ILLUSTRATIONS d I / d t = p*:~v I x d l , (a) C o n s i d e r first the s p o n t a n e o u s growth of a p l a n e c i r c u l a r vortex ring w h o s e selfi n d u c e d velocity relative to t h e s u p e r f l u i d is opposite to a n d less t h a n t h e u n i f o r m velocity of t h e s u p e r f l u i d itself. T h i s is a s i t u a t i o n i n v o k e d in t h e ILF t h e o r y of critical velocities. [2,3] Here the growth of the ring d e p e n d s on the c o n c u r r e n t action of a d r a g force ge d u e to n o r m a l fluid at rest. A l t h o u g h t h e w o r k d o n e by t h i s force is negative a n d vortex e n e r g y is being lost to it, t h i s e n e r g y loss is o v e r b a l a n c e d by the t r a n s f e r of e n e r g y from p o t e n t i a l flow to t h e v o r t e x ring, a n d t h e r i n g g a i n s energy as it expands. (b) C o n s i d e r n e x t a p l a n e s e m i c i r c u l a r vortex loop w h o s e axis lies in a plane wall a n d p a s s e s t h r o u g h a tiny circular a p e r t u r e in t h e wall from w h i c h s u p e r f l u i d flow is diverging. In the limit t h a t the r a d i u s of the a p e r t u r e is m u c h less t h a n t h a t of the loop, t h e loop will m o v e in a d i r e c t i o n p e r p e n d i c u l a r to its p l a n e a n d e x p a n d while r e t a i n i n g its s h a p e . [4] The e n d s of t h e loop are a s s u m e d to move freely over the wall. Here the vortex loop g a i n s energy f r o m p o t e n t i a l flow a s it c u t s l i n e s of p o t e n t i a l flow, w i t h o u t the a c t i o n of a n y e x t e r n a l force. On topological g r o u n d s , either of t h e s e two p r o c e s s e s c o u l d f o r m p a r t of a 2n p h a s e - s l i p in s u p e r f l u i d 4 H e . C a s e (b), i n v o l v i n g d i v e r g i n g flow w i t h o u t a n y e x t e r n a l force, is i l l u s t r a t i v e of w h a t in principle m a y h a p p e n a t low t e m p e r a t u r e s in t h e a b s e n c e of t h e n o r m a l f l u i d component. A more realistic case e q u i v a l e n t to c a s e (b) h a s r e c e n t l y b e e n p r o p o s e d for vortex growth in a n orifice. [5]
(4)
w h e r e the i n t e g r a l r u n s a l o n g t h e v o r t e x cores. Like d E / d t (p~v), d I / d t h a s b o t h a convective part, w h i c h m a y be p r e s e n t even in the absence of ge, a n d a part due to ge. As a result, I will n o t in g e n e r a l be conserved, even in the absence of ge. 5. EFFECTIVE ENERGY Use is sometimes m a d e of the expression F= E +vs'p
(5)
to give the effective e n e r g y of a v o r t e x s y s t e m in t h e l a b o r a t o r y f r a m e in t h e p r e s e n c e of u n i f o r m s u p e r f l u i d flow w i t h velocity v s. Here, E is the e n e r g y of t h e vortex s y s t e m in the m o v i n g frame, a n d p is t a k e n to be the impulse, even t h o u g h no vortex m o m e n t u m is involved. [2,3] A j u s t i f i c a t i o n for this expression a n d a g e n e r a l i z a t i o n to the c a s e of n o n u n i f o r m s u p e r f l u i d flow is p r o v i d e d , a t l e a s t in derivative form, by Eqs. (2) - (4) c o m b i n e d a n d rewritten a s follows: dWe/dt = dTv/dt + I vp.d(dI/dt).
(6)
Here, d W e / d t is the rate a t w h i c h force ge d o e s w o r k on t h e v o r t e x s y s t e m , to be identified with the r a t e of c h a n g e of t h e e f f e c t i v e e n e r g y of t h e s y s t e m . T h e derivative d T v / d t gives the rate of c h a n g e of vortex energy, the energy being a s s u m e d to be e n t i r e l y kinetic in t h i s t r e a t m e n t . The i n t e g r a l along t h e v o r t e x c o r e s gives a n appropriate average of Vp • d I / d t . Work s u p p o r t e d by the NSF u n d e r g r a n t DMR 90-02890.
4. IMPULSE OF THE VORTEX SYSTEM A l t h o u g h the vortex s y s t e m carries no m o m e n t u m , it is u s e f u l to d e f i n e its "impulse" I equal to the i m p u l s e n e e d e d to be delivered "impulsively" to the fluid by s o m e local e x t e r n a l force to c r e a t e the vortex s y s t e m . Despite the a m b i g u i t y in I so defined, its time rate of c h a n g e is given u n a m b i g u o u s l y by the expression
1. E. 1~ Huggins, Plays Rev A 1,327, 332 (1970). 2. S.V. Iordanskii, Sov Phys JETP 2 1 , 4 6 7 (1965). 3. J. S. Langer a n d M. E. Fisher, Phys Rev Lett 19, 560 (1967). 4. O. Avenel a n d E. Varoquaux, private communication. 5. K.W. Schwarz, preprmt.