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Boundary effects in second sound near Tλ
Physica B 1658~166 (1990) 561-562 North-Holland
BOUNDARY EFFECTS IN SECOND SOUND NEAR TX D.R. Swanson, T.C.P. Chui, K.W. Rigby*, and J.A. Lipa Department of Physics, Stanford University, Stanford, California 94305 U.S.A. We investigate the effect of non-ideal boundary conditions applied to a second sound resonator near the lambda point. Appropriate boundary conditions for He-II are applied to a second sound resonator with a heater and detector at either end. We find the analytic boundary value solution for the homogeneous (zero gravity) differential equation. We show that the standing wave approximation is accurate to a few tenths of a percent at 10e8 K from the transition. 1. INTRODUCTION Recent second sound experiments (1) near TX
s(z,t) = Re[s (z)e’y ,
show large deviations from theory in the reduced temperature range 10m7< E E 1-T/Th < 10s6. The theoretical model used in (1) made several simplifying assumptions. The gravitational effect was approximated with a time of flight method. In addition the boundaries of the resonator were assumed to be ideal insulators. For materials ordinarily used in the resonator, this is a good approximation. However we find that this is not necessarily accurate for the porous salt pill detector used in (1). In this paper we derive the boundary conditions for the resonator and explore their effect for the simple homogeneous case (zero gravity), very close to the transition. We are currently investigating the more complete solution of numerically integrating the inhomogeneous differential equation for second sound and matching to the boundary conditions presented here.
where the bold style indicates a complex number. To finds we apply appropriate boundary conditions at the heater and detector ends of the cell. The required boundary conditions for He-II at a solid wall are given by London (3): (3)
T=T’ -Rq,
(4)
where p is the He-II density, v, is the velocity of the normal fluid, K is the thermal conductivity of the normal fluid, T’ is the temperature of the wall and R is the Kapitza boundary resistance. Expressing v, in terms of VT gives (2): (- pS2Tp,/iw pn - K)~VT~ = ql.
a,VT, = uI~V%(z,t),
(pSv, - KVT/~)~ = q,/T
(5)
With a harmonically driven heater, the boundary equation [S] at the heater end of the cell becomes:
2. BOUNDARY VALUE PROBLEM In the absence of dissipation the linearized equation for second sound is usually written (2): &(z,t)/&2
(2)
= q. - tc’,VT’, g qo,
(6)
(1) where the term K’~VT’~, denoting the heater
where t.11~~= p,TS2/p,Cp is the velocity of second sound with the parameters defined in the usual way. Second sound is transmitted through He-II in the steady state with: *Current address: General Electric R&D, P.O. Box 8, Schenectady, NY 12603. 0921.4526/90/$03.50
backing material, is negligible compared to q, and a=a~+ia2=
- K + ipS2Tp,/o
pn. The helium
thermal conductivity is small compared to a2 and we also neglect it. At the detector end of the cell the boundary condition [5] is similar to [6] but without
@ 1990 - Elsevier Science Publishers B.V. (North-Holland)
D.R. Swanson, T.C.P. Chui, K. W. Rigby, J.A. Lipa
562
the driving term: a,VTL
0.0
= -K’~VT’~,
(7)
where the primed symbols represent quantities in the wall and the subscript L indicates quantities to be evaluated at z=L, the interface between the fluid and the detector. Similarly [4] becomes:
-0.6 -0.8 -1.0
! -10
T, = T’, + Rtc’$‘T’,.
(8)
In a solid wall heat travels as a thermal diffusive wave which is given by (3): T’ = Re[T, e-z’(i-l)/~e~~,
(9)
where Th is the temperature at the boundary (z’=O), and the thermal penetration depth 6 is given by: 62 = 2 lc’L/p’C&&
(10)
where K’~, p ’and C,’ are properties of the detector wall material. Substituting [9] into [7] and [8] and simpliiing we obtain: sflsL
= -aL (6(1+i)/2K’~+ R) = CL
(11)
3. EXACT SOLUTION IN ZERO GRAVITY In a gravitational potential, ps (thus uII and a) vary with height in the resonator. The resulting non-linear problem requires numerical techniques which we will report on elsewhere. Here we estimate the frequency shift from the boundary effects in zero gravity. This may be useful in designing a future space based experiment. The general solution to [l] is the sum of two waves traveling in opposite directions with wave vectors +k. :
I
I
-8
-6
-4
log (1 - T/-Q>
FIGURE 1 This result scales with qo, as expected for small_excitations, and displays a resonance as a function of o (=uIIk from [ 11, [2] and [ 121). Figure 1 shows the deviation of the resonant frequency obtained from [ 131 from the ideal standing wave approximation (UJJ= wL&) for two different boundary materials, Cu and the porous salt pill used in (1). For the salt pill we approximate the parameters in [lo] as those of He-I. For both materials we use R = 0.5 cm2K/W. The frequency shift for glass was several orders of magnitude smaller and is negligible on this scale. 4. CONCLUSION It is clear from these results that the boundary effects alone are unable to explain the experimental discrepancies. The deviations shown in Fig. 1 are about on third are large as the experiment. Analysis of the more complete model mentioned above may produce a closer agreement with experiment. ACKNOWLEDGMENTS We thank U.S. National Aeronautics and Space administration for its support with contract JPL 957448. REFERENCES
(1) D. Marek, J.A. Lipa, and D. Philips, Phys. Rev. B 38 (1988) 4465.
s(z) = AeikZ + B emikz.
(12)
The complex coefftcients A and B contain the amplitude and phase of each wave. Solving for s(L) using [6], [ll] and [12] we obtain: s(L) = qoCLCp / a,T(cos kL + kC sin kL).
(13)
(2) I.M.Khalatnikov, An Introduction to the Theory Superfluidity (Benjamin, New York, 1965). (3) FLondon, Superfluids (John Wiley & Sons, London, 1954). (4) H.S.Carslaw and J.C.Jaeger, Conduction of Heat in Solids, Second Edition (Clarendon, Oxford, 1959).
BOUNDARY EFFECTS IN SECOND SOUND NEAR TX D.R. Swanson, T.C.P. Chui, K.W. Rigby*, and J.A. Lipa Department of Physics, Stanford University, Stanford, California 94305 U.S.A. We investigate the effect of non-ideal boundary conditions applied to a second sound resonator near the lambda point. Appropriate boundary conditions for He-II are applied to a second sound resonator with a heater and detector at either end. We find the analytic boundary value solution for the homogeneous (zero gravity) differential equation. We show that the standing wave approximation is accurate to a few tenths of a percent at 10e8 K from the transition. 1. INTRODUCTION Recent second sound experiments (1) near TX
s(z,t) = Re[s (z)e’y ,
show large deviations from theory in the reduced temperature range 10m7< E E 1-T/Th < 10s6. The theoretical model used in (1) made several simplifying assumptions. The gravitational effect was approximated with a time of flight method. In addition the boundaries of the resonator were assumed to be ideal insulators. For materials ordinarily used in the resonator, this is a good approximation. However we find that this is not necessarily accurate for the porous salt pill detector used in (1). In this paper we derive the boundary conditions for the resonator and explore their effect for the simple homogeneous case (zero gravity), very close to the transition. We are currently investigating the more complete solution of numerically integrating the inhomogeneous differential equation for second sound and matching to the boundary conditions presented here.
where the bold style indicates a complex number. To finds we apply appropriate boundary conditions at the heater and detector ends of the cell. The required boundary conditions for He-II at a solid wall are given by London (3): (3)
T=T’ -Rq,
(4)
where p is the He-II density, v, is the velocity of the normal fluid, K is the thermal conductivity of the normal fluid, T’ is the temperature of the wall and R is the Kapitza boundary resistance. Expressing v, in terms of VT gives (2): (- pS2Tp,/iw pn - K)~VT~ = ql.
a,VT, = uI~V%(z,t),
(pSv, - KVT/~)~ = q,/T
(5)
With a harmonically driven heater, the boundary equation [S] at the heater end of the cell becomes:
2. BOUNDARY VALUE PROBLEM In the absence of dissipation the linearized equation for second sound is usually written (2): &(z,t)/&2
(2)
= q. - tc’,VT’, g qo,
(6)
(1) where the term K’~VT’~, denoting the heater
where t.11~~= p,TS2/p,Cp is the velocity of second sound with the parameters defined in the usual way. Second sound is transmitted through He-II in the steady state with: *Current address: General Electric R&D, P.O. Box 8, Schenectady, NY 12603. 0921.4526/90/$03.50
backing material, is negligible compared to q, and a=a~+ia2=
- K + ipS2Tp,/o
pn. The helium
thermal conductivity is small compared to a2 and we also neglect it. At the detector end of the cell the boundary condition [5] is similar to [6] but without
@ 1990 - Elsevier Science Publishers B.V. (North-Holland)
D.R. Swanson, T.C.P. Chui, K. W. Rigby, J.A. Lipa
562
the driving term: a,VTL
0.0
= -K’~VT’~,
(7)
where the primed symbols represent quantities in the wall and the subscript L indicates quantities to be evaluated at z=L, the interface between the fluid and the detector. Similarly [4] becomes:
-0.6 -0.8 -1.0
! -10
T, = T’, + Rtc’$‘T’,.
(8)
In a solid wall heat travels as a thermal diffusive wave which is given by (3): T’ = Re[T, e-z’(i-l)/~e~~,
(9)
where Th is the temperature at the boundary (z’=O), and the thermal penetration depth 6 is given by: 62 = 2 lc’L/p’C&&
(10)
where K’~, p ’and C,’ are properties of the detector wall material. Substituting [9] into [7] and [8] and simpliiing we obtain: sflsL
= -aL (6(1+i)/2K’~+ R) = CL
(11)
3. EXACT SOLUTION IN ZERO GRAVITY In a gravitational potential, ps (thus uII and a) vary with height in the resonator. The resulting non-linear problem requires numerical techniques which we will report on elsewhere. Here we estimate the frequency shift from the boundary effects in zero gravity. This may be useful in designing a future space based experiment. The general solution to [l] is the sum of two waves traveling in opposite directions with wave vectors +k. :
I
I
-8
-6
-4
log (1 - T/-Q>
FIGURE 1 This result scales with qo, as expected for small_excitations, and displays a resonance as a function of o (=uIIk from [ 11, [2] and [ 121). Figure 1 shows the deviation of the resonant frequency obtained from [ 131 from the ideal standing wave approximation (UJJ= wL&) for two different boundary materials, Cu and the porous salt pill used in (1). For the salt pill we approximate the parameters in [lo] as those of He-I. For both materials we use R = 0.5 cm2K/W. The frequency shift for glass was several orders of magnitude smaller and is negligible on this scale. 4. CONCLUSION It is clear from these results that the boundary effects alone are unable to explain the experimental discrepancies. The deviations shown in Fig. 1 are about on third are large as the experiment. Analysis of the more complete model mentioned above may produce a closer agreement with experiment. ACKNOWLEDGMENTS We thank U.S. National Aeronautics and Space administration for its support with contract JPL 957448. REFERENCES
(1) D. Marek, J.A. Lipa, and D. Philips, Phys. Rev. B 38 (1988) 4465.
s(z) = AeikZ + B emikz.
(12)
The complex coefftcients A and B contain the amplitude and phase of each wave. Solving for s(L) using [6], [ll] and [12] we obtain: s(L) = qoCLCp / a,T(cos kL + kC sin kL).
(13)
(2) I.M.Khalatnikov, An Introduction to the Theory Superfluidity (Benjamin, New York, 1965). (3) FLondon, Superfluids (John Wiley & Sons, London, 1954). (4) H.S.Carslaw and J.C.Jaeger, Conduction of Heat in Solids, Second Edition (Clarendon, Oxford, 1959).