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Saturated dissipation in a He II superflow
Volume 40A, number 1
PHYSICS LETTERS
19 June 1972
SATURATED DISSIPATION IN A He II SUPERFLOW M. FRANCOIS and D. LHUILLIER Luboratoire de Mkanique
desjluides.
Campus Universitaire, Batimenr 502, 91 -Orsay, France
Received 24 April 1972
We present experimental results showing that the superfluid dissipation is limited to a maximum value when the superflow is subjected to the presence of a constriction.
The experimental apparatus consits in a parrallelepidedic cavity made with Epibond (8 X 5 X 80 mm3), and separated in two parts by a constricted superleak (jeweller rouge; 0.4 /J) with diameter d (fig. 1) . A superflow is thermomecanically driven by a heater through a second sound resonator (S.S.R.) with emitter E and receiver R and through the constricted superleak; it then goes as normal fluid flow, through the fountain (inner diameter 3 mm) to the vapor. To analyse the dissipation in the S.S.R. with and without constriction, we have measured the excess attenuation cr’= (Y- cu, of the second sound amplitude; at the same time we have controlled the temperature along the flow. Fig. 2 shows simultaneously 1) the square-root of cr’versus V,, the superflow velocity in the S.S.R. 2) the temperature TN in the heater chamber versus Vs. We verify from the two curves C (superleak without constriction) that beyond a critical velocity Vslof order 1 mm/s, 0~’is proportional to V.$. Moreover AT = TN - To (To is the bulk temperature) is found to nearly correspond (few percent) to the hydrostatic fountain pressure AP ( a small pressure difference Ap due to the viscous flow of normal fluid in the fountain pipe, calculated from Poiseuille law, is always smaller than 1% AP in our experiments). Submitting now the superflow to constrictions of diameter d, we see (curves A and B), that 1) the results concerning a’ and TN are absolutely identical to those obtained without constriction, up to a second “critical velocity” Vsz in the S.S.R., to which corresponds Vi2 in the superleak constriction; 2) above Vs2,TN. rises abruptly and Q’ (measured by the quality factor of a resonant second sound wave) reaches a constant value 4’m, thus indicating the existence of an upper
R Id Second sound resonator
Constricted ruparlcak
Heater chamber
Fig. 1. Experimental apparatus. limiting value of the dissipation in the part of the apparatus in which stands the S.S.R. Before giving more details about this surprising result, let us emphasize that: (i) the velocity of the superflow created by the thermomechanical effect is by no means blocked to a limiting value (this would be an immediate explanation of the saturation): Indeed, let W be the heat generated by the heater, S and &!$be respectively the cross section of the S.S.R. and the fountain pipe; in stationary situations heat evacuated must compensate heat generated, and when applying this to our parrallelepidedic cavity, we get
W=p*s(T~)*T~*V,,.SF,
all terms have then usual meaning; using the mass conservation law:
PHYSICS LETTERS
Volume 40A, number 1
1/‘,2. d = cste
Fig. 2. Full line: Square root of extra attenuation of second sound versus the superfluid velocity Vs; &shed line: temperature TN in the heater chamber versus Vs. The cross-section of the second sound resonator is S = 40 mm’. Curve A: with a constriction of diameter: d = 1.3 mm. Curve B: with a constriction of diameter: d = 2.3 mm. Curve C: without constriction.
and defining the specific entropy s, of the normal fluid, by ps = p,, . s, , we get : W=Sps(To)*s,(TN)~TN
* vs.
The product s,T is a slowly increasing function in our temperature range and if IVincreases with V, fixed, TN ought to rise much more than what is observed experimentally. (ii) Pressure and temperature-induced back flow in the constriction are quite negligible since the ratio of normal fluid flux in the fountain pipe and in capillaries of the superleak is readily shown to be around 1OS. So a possible normal fluid flux through the S.S.R. would be only due to the thermal conductivity of the powder, and for our highest value of AT= TN - To, this is perfectly negligible. (iii) Precise measurements of the temperature at points L and M, show that for V, > Vs2, a temperature gradient begins to develop between L and M (precisely at the same time as TN begins to rise). Using the same powder compressed in the same way but different diameter d of the constriction we find to a few percent that
90
19 June 1972
$9
which is equivalent to Pa/d = cste,for a given temperature. Moreover, using different resonators with the same ratio of constriction S/S, (Sd = 7n.f2/4),we obtain the same VdUe for oh and for S/S, large enOUgh to get Vs2G PSI we have observed that C& = 0; in fact with S = 130 mm2 and Sd = 0.2 mm2, we had, at T = 1.54”K, PSI = 1.8 mm/s and Vs2 m 1.5 mm/s (obtained from the curve AT versus V,) and it has been impossible to detect any attenuation of the second sound amplitude for V, up to 1 cm/s (beyong which AT becomes prohibitively large). Notice that from (iii) we may conclude that there is an (unsolved) link between saturation of dissipation in the S.S.R. and the appearance of a temperature gradient along the constriction. The most natural idea is that this temperature is due to dissipation in the constriction and thus that Vi2 is the superfluid critical velocity in the powder. But in this case we ought to find that whatever, d, I’; = cste, since we have used the same powder. The relation (2) shows a quite different behaviour and suggests that the flow properties in the constriction are determined not by one but by two characteristic lenghts [l] : the mean pore diameter and the diameter of the constriction. In conclusion, if S/S, is of order lo2 or more, it looks as if the critical velocity of the flow above the constriction was displaced to a very high value, as in [2]. And even for smaller value of s/Sd the presence of a constriction considerably affects the superflow and may play a dominant role in various experiments involving it, as for instance in [3] . We wish to thank M. Le Ray and F. Vidal for helpful discussions and B. Allegri for technical assistance. References
[lj H. Kojima et al., Phys. Rev. Letters 27 (1971) 714; A. L. Fetter, Phys. Rev. 153 (1967) 285. [2] G. B. Hess, Phys. Rev. Letters 27 (1971) 15,977. [ 3) H. C. Kramers, in: Superfluid helium, ed. J. F. Allen (Academic Press, London and New York, 1966) p. 206; R. De Bruyn Ouboter et al., in: Progress in Low Temperature Physics, ed. C. J. Gorter, Vol. 5 (North-Holland, Amsterdam, 1967) p. 60.
PHYSICS LETTERS
19 June 1972
SATURATED DISSIPATION IN A He II SUPERFLOW M. FRANCOIS and D. LHUILLIER Luboratoire de Mkanique
desjluides.
Campus Universitaire, Batimenr 502, 91 -Orsay, France
Received 24 April 1972
We present experimental results showing that the superfluid dissipation is limited to a maximum value when the superflow is subjected to the presence of a constriction.
The experimental apparatus consits in a parrallelepidedic cavity made with Epibond (8 X 5 X 80 mm3), and separated in two parts by a constricted superleak (jeweller rouge; 0.4 /J) with diameter d (fig. 1) . A superflow is thermomecanically driven by a heater through a second sound resonator (S.S.R.) with emitter E and receiver R and through the constricted superleak; it then goes as normal fluid flow, through the fountain (inner diameter 3 mm) to the vapor. To analyse the dissipation in the S.S.R. with and without constriction, we have measured the excess attenuation cr’= (Y- cu, of the second sound amplitude; at the same time we have controlled the temperature along the flow. Fig. 2 shows simultaneously 1) the square-root of cr’versus V,, the superflow velocity in the S.S.R. 2) the temperature TN in the heater chamber versus Vs. We verify from the two curves C (superleak without constriction) that beyond a critical velocity Vslof order 1 mm/s, 0~’is proportional to V.$. Moreover AT = TN - To (To is the bulk temperature) is found to nearly correspond (few percent) to the hydrostatic fountain pressure AP ( a small pressure difference Ap due to the viscous flow of normal fluid in the fountain pipe, calculated from Poiseuille law, is always smaller than 1% AP in our experiments). Submitting now the superflow to constrictions of diameter d, we see (curves A and B), that 1) the results concerning a’ and TN are absolutely identical to those obtained without constriction, up to a second “critical velocity” Vsz in the S.S.R., to which corresponds Vi2 in the superleak constriction; 2) above Vs2,TN. rises abruptly and Q’ (measured by the quality factor of a resonant second sound wave) reaches a constant value 4’m, thus indicating the existence of an upper
R Id Second sound resonator
Constricted ruparlcak
Heater chamber
Fig. 1. Experimental apparatus. limiting value of the dissipation in the part of the apparatus in which stands the S.S.R. Before giving more details about this surprising result, let us emphasize that: (i) the velocity of the superflow created by the thermomechanical effect is by no means blocked to a limiting value (this would be an immediate explanation of the saturation): Indeed, let W be the heat generated by the heater, S and &!$be respectively the cross section of the S.S.R. and the fountain pipe; in stationary situations heat evacuated must compensate heat generated, and when applying this to our parrallelepidedic cavity, we get
W=p*s(T~)*T~*V,,.SF,
all terms have then usual meaning; using the mass conservation law:
PHYSICS LETTERS
Volume 40A, number 1
1/‘,2. d = cste
Fig. 2. Full line: Square root of extra attenuation of second sound versus the superfluid velocity Vs; &shed line: temperature TN in the heater chamber versus Vs. The cross-section of the second sound resonator is S = 40 mm’. Curve A: with a constriction of diameter: d = 1.3 mm. Curve B: with a constriction of diameter: d = 2.3 mm. Curve C: without constriction.
and defining the specific entropy s, of the normal fluid, by ps = p,, . s, , we get : W=Sps(To)*s,(TN)~TN
* vs.
The product s,T is a slowly increasing function in our temperature range and if IVincreases with V, fixed, TN ought to rise much more than what is observed experimentally. (ii) Pressure and temperature-induced back flow in the constriction are quite negligible since the ratio of normal fluid flux in the fountain pipe and in capillaries of the superleak is readily shown to be around 1OS. So a possible normal fluid flux through the S.S.R. would be only due to the thermal conductivity of the powder, and for our highest value of AT= TN - To, this is perfectly negligible. (iii) Precise measurements of the temperature at points L and M, show that for V, > Vs2, a temperature gradient begins to develop between L and M (precisely at the same time as TN begins to rise). Using the same powder compressed in the same way but different diameter d of the constriction we find to a few percent that
90
19 June 1972
$9
which is equivalent to Pa/d = cste,for a given temperature. Moreover, using different resonators with the same ratio of constriction S/S, (Sd = 7n.f2/4),we obtain the same VdUe for oh and for S/S, large enOUgh to get Vs2G PSI we have observed that C& = 0; in fact with S = 130 mm2 and Sd = 0.2 mm2, we had, at T = 1.54”K, PSI = 1.8 mm/s and Vs2 m 1.5 mm/s (obtained from the curve AT versus V,) and it has been impossible to detect any attenuation of the second sound amplitude for V, up to 1 cm/s (beyong which AT becomes prohibitively large). Notice that from (iii) we may conclude that there is an (unsolved) link between saturation of dissipation in the S.S.R. and the appearance of a temperature gradient along the constriction. The most natural idea is that this temperature is due to dissipation in the constriction and thus that Vi2 is the superfluid critical velocity in the powder. But in this case we ought to find that whatever, d, I’; = cste, since we have used the same powder. The relation (2) shows a quite different behaviour and suggests that the flow properties in the constriction are determined not by one but by two characteristic lenghts [l] : the mean pore diameter and the diameter of the constriction. In conclusion, if S/S, is of order lo2 or more, it looks as if the critical velocity of the flow above the constriction was displaced to a very high value, as in [2]. And even for smaller value of s/Sd the presence of a constriction considerably affects the superflow and may play a dominant role in various experiments involving it, as for instance in [3] . We wish to thank M. Le Ray and F. Vidal for helpful discussions and B. Allegri for technical assistance. References
[lj H. Kojima et al., Phys. Rev. Letters 27 (1971) 714; A. L. Fetter, Phys. Rev. 153 (1967) 285. [2] G. B. Hess, Phys. Rev. Letters 27 (1971) 15,977. [ 3) H. C. Kramers, in: Superfluid helium, ed. J. F. Allen (Academic Press, London and New York, 1966) p. 206; R. De Bruyn Ouboter et al., in: Progress in Low Temperature Physics, ed. C. J. Gorter, Vol. 5 (North-Holland, Amsterdam, 1967) p. 60.