Effective thermal conductivity of superfluid helium: laminar, turbulent and ballistic regimes

Michele Sciacca; Luca Galantucci

doi:10.1515/caim-2016-0009

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Effective thermal conductivity of superfluid helium: laminar, turbulent and ballistic regimes

Communications in Applied and Industrial Mathematics

Volume 7 (2016): Issue 2 (June 2016)

By: Michele Sciacca and Luca Galantucci

Open Access

|May 2016

Abstract

In this paper we extend previous results on the effective thermal conductivity of liquid helium II in cylindrical channels to rectangular channels with high aspect ratio. The aim is to compare the results in the laminar regime, the turbulent regime and the ballistic regime, all of them obtained within a single mesoscopic formalism of heat transport, with heat flux as an independent variable.

References

1. V. Arp, Heat transfer to superfluid and supercritical helium, Journal of Applied Physics, vol. 40, p. 2010, 1969.10.1063/1.1657900
Search in Google Scholar
2. K. Mendelsohn, Liquid helium, in Encyclopedia of Physics, vol. XV, Berlin: Springer, 1956.10.1007/978-3-642-45838-5_5
Search in Google Scholar
3. S. W. Van Sciver, Helium Cryogenics. Berlin: Springer, second ed., 2012.10.1007/978-1-4419-9979-5
Search in Google Scholar
4. B. Bertman and T. A. Kitchens, Heat transport in superfluid filled capillaries, Cryogenics, vol. 8, pp. 36–41, 1968.10.1016/S0011-2275(68)80051-7
Search in Google Scholar
5. D. Benin and H. J. Maris, Phonon heat transport and Knudsen”s minimum in liquid helium at low temperatures, Physical Review B, vol. 18, pp. 3112–3125, 1978.
Search in Google Scholar
6. D. S. Greywall, Thermal-conductivity measurements in liquid ⁴He below 0.7k, Physical Review B, vol. 23, pp. 2152–2168, 1981.
Search in Google Scholar
7. H. J. Maris, Dissipative coefficients of superfluid helium, Physical Review A, vol. 7, pp. 2074–2081, 1973.
Search in Google Scholar
8. L. D. Landau, Theory of the superfluidity of helium II, Physical Review, vol. 60, no. 4, p. 356, 1941.10.1103/PhysRev.60.356
Search in Google Scholar
9. M. Sciacca, A. Sellitto, and D. Jou, Transition to ballistic regime for heat transport in helium II, Physics Letters A, vol. 378, pp. 2471–2477, 2014.
Search in Google Scholar
10. M. Sciacca, D. Jou, and M. S. Mongioví, Effective thermal conductivity of helium ii: from landau to gorter–mellink regimes, Zeitschrift für angewandte Mathematik und Physik, pp. 1–17, 2013.
Search in Google Scholar
11. P. Critchlow and R. Hemstreet, Heat transport in superfluid helium in wide tubes, Journal of Applied Physics, vol. 40, p. 2675, 1969.10.1063/1.1658056
Search in Google Scholar
12. L. D. Landau and E. M. Lishitz, Mechanics of fluids. Oxford: Pergamon, 1985.
Search in Google Scholar
13. R. J. Donnelly, Quantized vortices in helium II. Cambridge, UK: Cambridge University Press, 1991.
Search in Google Scholar
14. C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, Quantized Vortex Dynamics and Superfluid Turbulence. Berlin: Springer, 2001.10.1007/3-540-45542-6
Search in Google Scholar
15. M. Tsubota, M. Kobayashi, and H. Takeuchi, Quantum hydrodynamics, Physics Report, vol. 522, pp. 191–238, 2013.10.1016/j.physrep.2012.09.007
Search in Google Scholar
16. S. K. Nemirovskii, Quantum turbulence: Theoretical and numerical problems, Physics Reports, vol. 524, pp. 85–202, 2013.10.1016/j.physrep.2012.10.005
Search in Google Scholar
17. D. R. Ladner and J. T. Tough, Temperature and velocity dependence of superfluid turbulence, Physical Review B, vol. 20, pp. 2690–2702, 1979.
Search in Google Scholar
18. A. Sato and et al., Temperature dependence of the Gorter-Mellink exponent measured in a channel containing He II, in ADVANCES IN CRYOGENIC ENGINEERING: Transactions of the Cryogenic Engineering Conference-CEC, vol. 823, pp. 387–392, AIP Publishing, 2006.10.1063/1.2202439
Search in Google Scholar
19. R. Whitworth, Experiments on the flow of heat in liquid helium below 0.7 degrees K, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 246, no. 1246, pp. 390–405, 1958.
Search in Google Scholar
20. S. Putterman, Superfluid hydrodynamics. Amsterdam: North-Holland Publishing Co., 1974.
Search in Google Scholar
21. S. Nemirovskii, V. Koren’kov, and V. Krupitskii, Heat flux through a phase interface in superfluid helium, Journal of Engineering Physics, vol. 47, no. 6, pp. 1413–1419, 1984.
Search in Google Scholar
22. D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics. Berlin: Springer-Verlag, fourth ed., 2010.10.1007/978-90-481-3074-0_2
Search in Google Scholar
23. I. Müller and T. Ruggeri, Rational Extended Thermodynamics. New York: Springer-Verlag, 1998.10.1007/978-1-4612-2210-1
Search in Google Scholar
24. M. S. Mongioví, Extended irreversible thermodynamics of liquid helium II, Physical Review B, vol. 48, pp. 6276–6283, 1993.
Search in Google Scholar
25. D. Jou, J. Casas-Vázquez, and M. Criado-Sancho, Thermodynamics of Fluids Under Flow. Berlin: Springer, second ed., 2011.10.1007/978-94-007-0199-1
Search in Google Scholar
26. D. Jou, G. Lebon, and M. S. Mongiovì, Second sound, superfluid turbulence, and intermittent effects in liquid helium II, Physical Review B, vol. 66, p. 224509 (9 pages), 2002.
Search in Google Scholar
27. D. Jou and M. Mongiovì, Description and evolution of anisotropy in superfluid vortex tangles with counterflow and rotation, Physical Review B, vol. 74, p. 054509 (11 pages), 2006.
Search in Google Scholar
28. H. Bruus, Theoretical microfluidics. Oxford: Oxford University Press, 2007.
Search in Google Scholar
29. D. Jou, G. Lebon, and M. Criado-Sancho, Variational principles for thermal transport in nanosystems with heat slip flow, Physical Review E, vol. 82, no. 3, p. 031128, 2010.
Search in Google Scholar
30. G. Lebon and P. Dauby, Heat transport in dielectric crystals at low temperature: A variational formulation based on extended irreversible thermodynamics, Physical Review A, vol. 42, p. 4710, 1990.10.1103/PhysRevA.42.4710
Search in Google Scholar
31. D. Jou and M. Sciacca, Quantum Reynolds number for superfluid counterflow turbulence, in Bollettino di Matematica Pura e Applicata (M. S. Mongioví, M. Sciacca, and S. Triolo, eds.), vol. VI, pp. 95–103, Aracne editrice, 2013.
Search in Google Scholar
32. J. Tough, Superfluid turbulence, in Progress of Low Temperature Physics (D. Brewer, ed.), vol. VIII, pp. 133–219, North Holland, 1982.10.1016/S0079-6417(08)60006-2
Search in Google Scholar
33. M. S. Mongiovì and D. Jou, Generalization of Vinen’s equation including transition to superfluid turbulence, Journal of Physics: Condensed Matter, vol. 17, pp. 4423–4440, 2005.
Search in Google Scholar
34. K. P. Martin and J. T. Tough, Evolution of superfluid turbulence in thermal counterflow, Physical Review B, vol. 27, pp. 2788–2799, 1983.
Search in Google Scholar
35. E. Yarmchuk and W. I. Glaberson, Thermorotation Effects in Superfluid Helium, Physical Review Letters, vol. 41, no. 8, pp. 564–568, 1978.10.1103/PhysRevLett.41.564
Search in Google Scholar
36. L. Galantucci, C.F. Barenghi, M. Sciacca, M. Quadrio, and P. Luchini, Turbulent Superfluid Profiles in a Counterflow Channel, Journal of Low Temperature Physics, vol. 162, pp. 354–360, 2011.10.1007/s10909-010-0266-4
Search in Google Scholar
37. L. Galantucci and M. Sciacca, Non-classical velocity statistics in counterflow quantum turbulence, Acta Applicandae Mathematicae, vol. 132, no. 1, pp. 273–281, 2014.10.1007/s10440-014-9902-3
Search in Google Scholar
38. L. D. Landau, Theory of the Superfluidity of Helium II, Journal of Physics, vol. 5, pp. 71–90, 1941.
Search in Google Scholar
39. L. Tisza, Transport phenomena in Helium II, Nature, vol. 141, p. 913, 1938.10.1038/141913a0
Search in Google Scholar
40. I. K. I.L. Bekarevich, Phenomenological derivation of the equation of vortex motion in he ii, Soviet Physics JETP, vol. 13, pp. 643–646, 1961.
Search in Google Scholar
41. H. Hall and W. Vinen, The rotation of liquid helium ii. ii. the theory of mutual friction in uniformly rotating helium II, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 238, no. 1213, pp. 215–234, 1956.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.1515/caim-2016-0009 | Journal eISSN: 2038-0909

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Language: English

Page range: 111 - 129

Submitted on: Dec 31, 2014

Accepted on: Aug 4, 2015

Published on: May 20, 2016

Published by: Italian Society for Applied and Industrial Mathemathics

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Thermal conductivity,

Related subjects:

Numerical and computational mathematics,

Applied mathematics

© 2016 Michele Sciacca, Luca Galantucci, published by Italian Society for Applied and Industrial Mathemathics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 7 (2016): Issue 2 (June 2016)