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Onset of convection in dilute superfluid 3He4He mixtures
Physica 107B (1981) 149-130 North-Holland Publishing Company
CD 2
ONSET OF CONVECTION IN DILUTE SUPERFLUID 3He-4He MIXTURES A l e x a n d e r L. F e t t e r Institute of Theoretical Physics, Department of Physics Stanford University, Stanford, CA 94505 The onset of convection in a thin slab (thickness d) of dilute superfluid 3He-4He mixture is analyzed with the two-fluid equations. The nonzero hydrodynamic flow in the conducting state introduces several new dimensionless ratios, the largest being (Pn/P)dv~/~. Although this parameter is small (~ 10 "2) for current experiments, a reduced thickness should render its presence observable. The two-fluid hydrodynamic equations of superfluid 3He-4He mixtures predict many unusual phenomena. 1 In the particular case of a fluid slab of thickness d with surface temperatures T and T+AT, the uniform conducting state involves steady hydrodynamic transport, with nonzero values of both normal and superfluid velocities and v~0, but with zero total mass flow. The resulting heat transport is accompanied by a flow p c ~ of 5He atoms, where c is the corresponding mass concentration. In the steady state, there is an opposing impurity f l u x ~ 0 associated with the gradients in c and T (ignoring gravitational pressure gradients, whose effects are generally negligible). Since ~ / a t vanishes, the chemical potential ~4 per unit mass must be a spatial constant, and the gradients in c and T are related by ~c 0 = -(cy/T)~T 0, where y = -(~£nc/~£nT)p~ 4 ~ 1 + s40c/msk B in the limit 3 of dilute He~ and s40 is the entropy per unit mass of pure ~He. Although the concentration g r a d i e n t opposes t h e t e m p e r a t u r e g r a d i e n t , t h e i n c r e a s e d volume p e r 5He atom i m p l i e s t h a t t h e d e n s i t y g r a d i e n t g e n e r a l l y has t h e same s i g n as ~r. The p r e c e d i n g r e l a t i o n s a r e r e a d i l y combined to g i v e t h e e x p r e s s i o n 0
Y~n = - (D/T) ( Y - k T / c ) ~
el)
where D is the diffusion coefficient and kT is the thermal diffusion ratio. The dimensionless f a c t o r in p a r e n t h e s e s i s t y p i c a l l y of o r d e r 1, a l t h o u g h d e t a i l e d measurements of k T have only been c a r r i e d out i n t h e normal phase n e a r t h e lambda l i n e . 2 C o r r e s p o n d i n g l y , t h e conducting s t a t e may be c h a r a c t e r i z e d by an e f f e c t i v e t h e r mal conductivity 1
t r a n s v e r s e wavenumber, j u s t as f o r t h e c l a s s i c a l Raylei~h-B~nard i n s t a b i l i t y in a one-component f l u i d . ° The t w o - f l u i d hydrodynamics i n t r o d u c e s one i m p o r t a n t new f e a t u r e , however, owing to the f i n i t e ~0 in t h e conducting s t a t e . Previous t h e o r e t i c a l s t u d i e s 4,5 have dropped Vn0 from t h e t w o - f l u i d e q u a t i o n s . With t h i s a p p r o x i m a t i o n , t h e e q u a t i o n s f o r an unbounded s l a b of t h i c k n e s s d can be r e c a s t in t h e form of t h o s e f o r a onecomponent f l u i d 6 w i t h a R a y l e i g h number Ra -= gO.p~4 d 3 A T / X e f f V ,
where ep~ 4 = - p - l ( a p / a T ) p p 4 , v = n / p i s the k i n ematic viscosity, Xeff =
cT[a(s/c)/aT]pu4.The heat flush of 3He
makes e / negative, so that Ra is positive when P4 the fluid is heated from above. Comparison with known results 3 then predicts the critical value Ra c = 1708 for the onset of convection with a critical transverse wavenumber 5.12 d "I. To include the effect of finite ~ , the full two-fluid equations have been linearized about the initial conducting state. Apart from corrections of relative order [gd/(~4/~C)TDl gdmS/kBT ~ 5x10- and other similar sma~l quantities, the perturbation T, wholly fixes c' ~-cyT'/T. Elimination of the irrotational superfluid velocity ~s' eventually yields two coupled equations for Vn_' and T' • representing Z the balance of vertical momentum and of entropy. In suitable dimensionless units, v 0 enters these equations in the dimensionless ratxos Pn dvO
T \
0¢
-
(2)
where T ' I ( ~ A / ~ C ) _ ~ -kn/m~ f o r small c, and < is the int~insiciPtherm~l ~onductivity. When t h e f l u i d i s h e a t e d from below, t h e negative density gradient yields a stable gravit a t i o n a l c o n f i g u r a t i o n , w i t h °He accumulating n e a r t h e upper s u r f a c e . In c o n t r a s t , h e a t i n g from above can r e n d e r t h e f l u i d g r a v i t a t i o n a l l y u n s t a b l e with r e s p e c t t o c o n v e c t i o n a t a f i n i t e
03784363/81/0000-0000/$02.50
@ North4lolland Publtehin8 Company
(3)
On XeffD Ra =
p
y2
v
-
-
c~_~) (y
-
-
,
(4a)
P gl~pp4 Id3T
c/akt4.)dvOn
Cpla4 ~ l - g ~ ] T p ×ef£
y2CkB
vDRa
m3Cpla4 gl ~pl~41 d3T
(y kTc.~, (4b)
multiplying the d e r i v a t i v e ~/~z. These terms i n t r o d u c e a p r e f e r r e d d i r e c t i o n in t h e c o n d u c t ing s t a t e , so t h a t t h e p e r t u r b a t i o n s o l u t i o n s 149
150
j u s t above t h r e s h o l d no l o n g e r have w e l l - d e f i n e d parity. For r i g i d upper and lower w a l l s , t h e boundary c o n d i t i o n s r e d u c e to t h e c l a s s i c a l ones (T' = v ~ , ' = 8vn , / 3 z = 0), a p a r t from c o r r e c t i o n s o ~ ' o r d e r ~ / g J ~ p ~ 4 l d 3 T ~ 10-8. Recent e x p e r i m e n t s 6 f o r c = 0.00555 and 0.8K ~ T ~ 0.95K in a unit aspect-ratio cylinder with d = 2.07 cm ha~e found a critical Rayleigh number Ra c = 1.7x10 ~ for the onset of convection with rigid bounding walls. At T = 0.8K, for example, ~he measured values 7 yield ~ -7x10 -3 and ~ I0- , respectively, for the quantities in Eq. (4). Thus the specific two-fluid aspects of the hydrodynamics are unimportant in these observations. Since Eq. (4) s c a l e s with d -Z, however, a change to smaller thickness would increase these dimensionless ratios, thus enhancing the effect on the nonzero v 0. (They have the approximate values ~ -0.5 a~d ~ 0.01 for d = 0.5 cm but AT c is then so large that the temperature dependence of the other parameters may become important, requiring non-Boussinesq corrections.) More generally, if Ra c remained unchanged, a reduction in d by a factor f increases AT c and v~ by f-3. In practice, the introduction of additional parameters will also affect Rac, and a numerical analysis of these effects is in progress. The resulting s h i f t in Ra c from the theoretical value 1708 for an unbounded slab with rigid walls might provide experimental evidence for the occurrence of twofluid effects. The present analysis leaves several questions unanswered. In addition to the effect of finite vS, it would be interesting to understand why the measured 6 Ra c is so close to the value for an unbounded slab. Previous studies of classical one-component fluids indicate that the presence of lateral boundaries increases Ra c. especially for aspect ratios of order unity. 8 What is the corresponding situation for superfluid mixtures? Another important problem is the nonlinear behavior beyond threshold, which can conveniently be characterized with an amplitude equation.9, 10 Its detailed form will doubtless differ from that for a classical onecomponent system, 9 however, because the eigenvalues and eigenvalues of the present linearized equations have a more complicated structure. These matters will require considerable further study.
Acknowledgments I am g r a t e f u l t o J . C. N h e a t l e y f o r sending me t h e u n p u b l i s h e d m a t e r i a l in Ref. 6. This work i s supported in p a r t by t h e NSF, g r a n t no. DMR 78-25258.
References 1. I . M. K h a l a t n i k o v , I n t r o d u c t i o n to t h e Theory of S u p e r f l u i d i t y (Benjamin, N.Y., I955), P a r t IV. 2. G. Lee, P. Lucas, and A. T y l e r , Phys. L e t t . 75A, 81 (19793. 3. S. Chandrasekhar, HTdredznamic and Hydromagn e t i c S t a b i l i t y (Oxford, London, 1961}, Chap. I I 4. A. Y. P a r s h i n , Soy. Phys.-JETP L e t t . 10, 362 (1969). 5. V. S t e i n b e r g , Phys. Rev. L e t t . 45, 2050 (1980). 6. P. A. Warkentin, H. J . Haucke, and J . C. Wheatley, Phys. Rev. L e t t . 45, 918 (1980); P. A. Warkentin, H. J . Hauck-~, P. Lucas, and J . C. Wheatley, Proc. N a t l . Acad. S c i . USA 77, 6983 (1980); p. A. Warkentin, Ph.D. t h e s i s , - UCSD, (1980). 7. In a d d i t i o n to t h e d a t a from Ref. 6, D was e x t r a p o l a t e d from C. Ebner, Phys. Rev. 156, 222 (1967). 8, E. L. Koschmieder, Adv. Chem. Phys. XXVI, 177 (1974). 9. M. C. Cross, Phys. F l u i d s 2_~3, 1727 (1980). 10. A. L, F e t t e r , in Recent P r o g r e s s in ManyBody T h e o r i e s ( S p r i n g e g t o be p u b l l s h e d ) .
CD 2
ONSET OF CONVECTION IN DILUTE SUPERFLUID 3He-4He MIXTURES A l e x a n d e r L. F e t t e r Institute of Theoretical Physics, Department of Physics Stanford University, Stanford, CA 94505 The onset of convection in a thin slab (thickness d) of dilute superfluid 3He-4He mixture is analyzed with the two-fluid equations. The nonzero hydrodynamic flow in the conducting state introduces several new dimensionless ratios, the largest being (Pn/P)dv~/~. Although this parameter is small (~ 10 "2) for current experiments, a reduced thickness should render its presence observable. The two-fluid hydrodynamic equations of superfluid 3He-4He mixtures predict many unusual phenomena. 1 In the particular case of a fluid slab of thickness d with surface temperatures T and T+AT, the uniform conducting state involves steady hydrodynamic transport, with nonzero values of both normal and superfluid velocities and v~0, but with zero total mass flow. The resulting heat transport is accompanied by a flow p c ~ of 5He atoms, where c is the corresponding mass concentration. In the steady state, there is an opposing impurity f l u x ~ 0 associated with the gradients in c and T (ignoring gravitational pressure gradients, whose effects are generally negligible). Since ~ / a t vanishes, the chemical potential ~4 per unit mass must be a spatial constant, and the gradients in c and T are related by ~c 0 = -(cy/T)~T 0, where y = -(~£nc/~£nT)p~ 4 ~ 1 + s40c/msk B in the limit 3 of dilute He~ and s40 is the entropy per unit mass of pure ~He. Although the concentration g r a d i e n t opposes t h e t e m p e r a t u r e g r a d i e n t , t h e i n c r e a s e d volume p e r 5He atom i m p l i e s t h a t t h e d e n s i t y g r a d i e n t g e n e r a l l y has t h e same s i g n as ~r. The p r e c e d i n g r e l a t i o n s a r e r e a d i l y combined to g i v e t h e e x p r e s s i o n 0
Y~n = - (D/T) ( Y - k T / c ) ~
el)
where D is the diffusion coefficient and kT is the thermal diffusion ratio. The dimensionless f a c t o r in p a r e n t h e s e s i s t y p i c a l l y of o r d e r 1, a l t h o u g h d e t a i l e d measurements of k T have only been c a r r i e d out i n t h e normal phase n e a r t h e lambda l i n e . 2 C o r r e s p o n d i n g l y , t h e conducting s t a t e may be c h a r a c t e r i z e d by an e f f e c t i v e t h e r mal conductivity 1
t r a n s v e r s e wavenumber, j u s t as f o r t h e c l a s s i c a l Raylei~h-B~nard i n s t a b i l i t y in a one-component f l u i d . ° The t w o - f l u i d hydrodynamics i n t r o d u c e s one i m p o r t a n t new f e a t u r e , however, owing to the f i n i t e ~0 in t h e conducting s t a t e . Previous t h e o r e t i c a l s t u d i e s 4,5 have dropped Vn0 from t h e t w o - f l u i d e q u a t i o n s . With t h i s a p p r o x i m a t i o n , t h e e q u a t i o n s f o r an unbounded s l a b of t h i c k n e s s d can be r e c a s t in t h e form of t h o s e f o r a onecomponent f l u i d 6 w i t h a R a y l e i g h number Ra -= gO.p~4 d 3 A T / X e f f V ,
where ep~ 4 = - p - l ( a p / a T ) p p 4 , v = n / p i s the k i n ematic viscosity, Xeff =
cT[a(s/c)/aT]pu4.The heat flush of 3He
makes e / negative, so that Ra is positive when P4 the fluid is heated from above. Comparison with known results 3 then predicts the critical value Ra c = 1708 for the onset of convection with a critical transverse wavenumber 5.12 d "I. To include the effect of finite ~ , the full two-fluid equations have been linearized about the initial conducting state. Apart from corrections of relative order [gd/(~4/~C)TDl gdmS/kBT ~ 5x10- and other similar sma~l quantities, the perturbation T, wholly fixes c' ~-cyT'/T. Elimination of the irrotational superfluid velocity ~s' eventually yields two coupled equations for Vn_' and T' • representing Z the balance of vertical momentum and of entropy. In suitable dimensionless units, v 0 enters these equations in the dimensionless ratxos Pn dvO
T \
0¢
-
(2)
where T ' I ( ~ A / ~ C ) _ ~ -kn/m~ f o r small c, and < is the int~insiciPtherm~l ~onductivity. When t h e f l u i d i s h e a t e d from below, t h e negative density gradient yields a stable gravit a t i o n a l c o n f i g u r a t i o n , w i t h °He accumulating n e a r t h e upper s u r f a c e . In c o n t r a s t , h e a t i n g from above can r e n d e r t h e f l u i d g r a v i t a t i o n a l l y u n s t a b l e with r e s p e c t t o c o n v e c t i o n a t a f i n i t e
03784363/81/0000-0000/$02.50
@ North4lolland Publtehin8 Company
(3)
On XeffD Ra =
p
y2
v
-
-
c~_~) (y
-
-
,
(4a)
P gl~pp4 Id3T
c/akt4.)dvOn
Cpla4 ~ l - g ~ ] T p ×ef£
y2CkB
vDRa
m3Cpla4 gl ~pl~41 d3T
(y kTc.~, (4b)
multiplying the d e r i v a t i v e ~/~z. These terms i n t r o d u c e a p r e f e r r e d d i r e c t i o n in t h e c o n d u c t ing s t a t e , so t h a t t h e p e r t u r b a t i o n s o l u t i o n s 149
150
j u s t above t h r e s h o l d no l o n g e r have w e l l - d e f i n e d parity. For r i g i d upper and lower w a l l s , t h e boundary c o n d i t i o n s r e d u c e to t h e c l a s s i c a l ones (T' = v ~ , ' = 8vn , / 3 z = 0), a p a r t from c o r r e c t i o n s o ~ ' o r d e r ~ / g J ~ p ~ 4 l d 3 T ~ 10-8. Recent e x p e r i m e n t s 6 f o r c = 0.00555 and 0.8K ~ T ~ 0.95K in a unit aspect-ratio cylinder with d = 2.07 cm ha~e found a critical Rayleigh number Ra c = 1.7x10 ~ for the onset of convection with rigid bounding walls. At T = 0.8K, for example, ~he measured values 7 yield ~ -7x10 -3 and ~ I0- , respectively, for the quantities in Eq. (4). Thus the specific two-fluid aspects of the hydrodynamics are unimportant in these observations. Since Eq. (4) s c a l e s with d -Z, however, a change to smaller thickness would increase these dimensionless ratios, thus enhancing the effect on the nonzero v 0. (They have the approximate values ~ -0.5 a~d ~ 0.01 for d = 0.5 cm but AT c is then so large that the temperature dependence of the other parameters may become important, requiring non-Boussinesq corrections.) More generally, if Ra c remained unchanged, a reduction in d by a factor f increases AT c and v~ by f-3. In practice, the introduction of additional parameters will also affect Rac, and a numerical analysis of these effects is in progress. The resulting s h i f t in Ra c from the theoretical value 1708 for an unbounded slab with rigid walls might provide experimental evidence for the occurrence of twofluid effects. The present analysis leaves several questions unanswered. In addition to the effect of finite vS, it would be interesting to understand why the measured 6 Ra c is so close to the value for an unbounded slab. Previous studies of classical one-component fluids indicate that the presence of lateral boundaries increases Ra c. especially for aspect ratios of order unity. 8 What is the corresponding situation for superfluid mixtures? Another important problem is the nonlinear behavior beyond threshold, which can conveniently be characterized with an amplitude equation.9, 10 Its detailed form will doubtless differ from that for a classical onecomponent system, 9 however, because the eigenvalues and eigenvalues of the present linearized equations have a more complicated structure. These matters will require considerable further study.
Acknowledgments I am g r a t e f u l t o J . C. N h e a t l e y f o r sending me t h e u n p u b l i s h e d m a t e r i a l in Ref. 6. This work i s supported in p a r t by t h e NSF, g r a n t no. DMR 78-25258.
References 1. I . M. K h a l a t n i k o v , I n t r o d u c t i o n to t h e Theory of S u p e r f l u i d i t y (Benjamin, N.Y., I955), P a r t IV. 2. G. Lee, P. Lucas, and A. T y l e r , Phys. L e t t . 75A, 81 (19793. 3. S. Chandrasekhar, HTdredznamic and Hydromagn e t i c S t a b i l i t y (Oxford, London, 1961}, Chap. I I 4. A. Y. P a r s h i n , Soy. Phys.-JETP L e t t . 10, 362 (1969). 5. V. S t e i n b e r g , Phys. Rev. L e t t . 45, 2050 (1980). 6. P. A. Warkentin, H. J . Haucke, and J . C. Wheatley, Phys. Rev. L e t t . 45, 918 (1980); P. A. Warkentin, H. J . Hauck-~, P. Lucas, and J . C. Wheatley, Proc. N a t l . Acad. S c i . USA 77, 6983 (1980); p. A. Warkentin, Ph.D. t h e s i s , - UCSD, (1980). 7. In a d d i t i o n to t h e d a t a from Ref. 6, D was e x t r a p o l a t e d from C. Ebner, Phys. Rev. 156, 222 (1967). 8, E. L. Koschmieder, Adv. Chem. Phys. XXVI, 177 (1974). 9. M. C. Cross, Phys. F l u i d s 2_~3, 1727 (1980). 10. A. L, F e t t e r , in Recent P r o g r e s s in ManyBody T h e o r i e s ( S p r i n g e g t o be p u b l l s h e d ) .