Skip to main content
SpringerLink
Log in

Contribution of the normal component to the thermal resistance of turbulent liquid helium

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

Previous results for the velocity profile of the normal component of helium II in counterflow are used to evaluate the viscous contribution to the effective thermal resistance. It turns out that such a contribution becomes considerably higher than the usual Landau estimate, because in the presence of vortices, the velocity profile is appreciably different from the Poiseuille parabolic profile. Thus, a marked increase in the contribution of the normal component to the thermal resistance with respect to the viscous Landau estimate does not necessarily imply that the normal component is turbulent. Furthermore, we examine the influence of a possible slip flow along the walls when the radius of the tube becomes comparable with the phonon mean free path; this implies a reduction of the thermal resistance with respect to that obtained for nonslip boundary conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mendelsohn K.: Liquid Helium, vol. XV. Springer, Berlin (1956)

    Google Scholar 

  2. van Sciver S.: Helium Cryogenics. 2nd edn. Springer, Berlin (2012)

    Book  Google Scholar 

  3. Donnelly R.J.: Quantized Vortices in Helium II. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  4. Barenghi, C.F., Sergeev, Y.A.: Vortices and turbulence at very low temperatures, CISM International Centre for Mechanical Sciences, vol. 501. Springer, Berlin, p 280 (2008)

  5. Nemirovskii S.K., Fiszdon W.: Chaotic quantized vortices and hydrodynamic processes in superfluid helium. Rev. Mod. Phys. 67, 37 (1995). doi:10.1103/RevModPhys.67.37

    Article  Google Scholar 

  6. Nemirovskii S.K.: Quantum turbulence: theoretical and numerical problems. Phys. Rep. 524, 85 (2013). doi:10.1016/j.physrep.2012.10.005

    Article  MathSciNet  Google Scholar 

  7. Tsubota M., Kobayashi M., Takeuchi H.: Quantum hydrodynamics. Phys. Rep. 522, 191 (2012). doi:10.1016/j.physrep.2012.09.007

    Article  MathSciNet  Google Scholar 

  8. Barenghi C.F.: Laminar, turbulent, or doubly turbulent?. Physics 3, 60 (2010). doi:10.1103/Physics.3.60

    Article  Google Scholar 

  9. Guo W., Cahn S.B., Nikkel J.A., Vinen W.F., McKinsey D.N.: Visualization study of counterflow in superfluid Helium-4 using metastable helium molecules. Phys. Rev. Lett. 105, 045301 (2010). doi:10.1103/PhysRevLett.105.045301

    Article  Google Scholar 

  10. Galantucci L., Barenghi C.F., Sciacca M., Quadrio M., Luchini P.: Turbulent superfluid profiles in a counterflow channel. J. Low Temp. Phys. 162, 354 (2011). doi:10.1007/s10909-010-0266-4

    Article  Google Scholar 

  11. Galantucci, L., Sciacca, M.: Turbulent superfluid profiles and vortex density waves in a counterflow channel. Acta Appl. Math. 122, 407–418 (2012). doi:10.1007/s10440-012-9752-9

  12. Galantucci, L., Sciacca, M.: Non-classical velocity statistics in counterflow quantum turbulence. Acta Appl. Math. 132, 273–281 (2014). doi:10.1007/s10440-014-9902-3

  13. Hanninen R., Baggaley A.W.: Vortex filament method as a tool for computational visualization of quantum turbulence. Proc Natl Acad Sci USA 111(Sup.1), 4667–4674 (2014). doi:10.1073/pnas.1312535111

    Article  MathSciNet  Google Scholar 

  14. Guo W., McKinsey D.N., Marakov A., Thompson K.J., Ihas G.G., Vinen W.F.: Visualization technique for determining the structure functions of normal-fluid turbulence in superfluid helium-4. J. Low Temp. Phys. 171, 497–503 (2013). doi:10.1007/s10909-012-0708-2

    Article  Google Scholar 

  15. Landau L.D.: The theory of superfluidity of He II. J. Phys. 5, 71 (1941)

    Google Scholar 

  16. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Pergamon Press, Oxford (1987)

  17. Saluto L., Mongiovì M.S., Jou D.: Longitudinal counterflow in turbulent liquid helium: velocity profile of the normal component. Z. Angew. Math. Phys. 65, 531–548 (2014). doi:10.1007/s00033-013-0372-7

    Article  MathSciNet  MATH  Google Scholar 

  18. Martin K.P., Tough J.T.: Evolution of superfluid turbulence in thermal counterflow. Phys. Rev. B 27, 2788 (1983). doi:10.1103/PhysRevB.27.2788

    Article  Google Scholar 

  19. Jou D., Casas-Vàzquez J., Criado-Sancho M.: Thermodynamics of Fluids Under Flow. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  20. Muller I., Ruggeri T.: Rational Extended Thermodynamics. Springer, New York (1998)

    Book  Google Scholar 

  21. Mongiovì M.S.: Extended irreversible thermodynamics of liquid helium II: boundary condition and propagation of fourth sound. Phys. A 292, 55 (2001). doi:10.1016/S0378-4371(00)00537-9

    Article  MATH  Google Scholar 

  22. Mongiovì M.S., Jou D.: Thermodynamical derivation of a hydrodynamical model of inhomogeneous superfluid turbulence. Phys. Rev. B 75, 024507 (2007). doi:10.1103/PhysRevB.75.024507

    Article  Google Scholar 

  23. Mongiovì M.S.: Extended irreversible thermodynamics of liquid helium II. Phys. Rev. B 48, 6276 (1993). doi:10.1103/PhysRevB.48.6276

    Article  Google Scholar 

  24. Hall H.E., Vinen W.F.: The rotation of liquid helium II. I. the theory of mutual friction in uniformly rotating helium II. Proc. R. Soc. A 238, 204 (1956). doi:10.1098/rspa.1956.0214

    Article  Google Scholar 

  25. Ardizzone L., Gaeta G., Mongiovì M.S.: Wave propagation in anisotropic turbulent superfluids. Z. Angew. Math. Phys. 64, 1571–1586 (2013). doi:10.1007/s00033-013-0308-2

    Article  MathSciNet  MATH  Google Scholar 

  26. Jou D., Mongiovì M.S., Sciacca M.: Hydrodynamic equations of anisotropic, polarized and inhomogeneous superfluid vortex tangles. Phys. D 240, 249 (2011). doi:10.1016/j.physd.2010.09.001

    Article  MathSciNet  MATH  Google Scholar 

  27. Donnelly R.J., Barenghi C.F.: The observed properties of liquid helium at the saturated vapor pressure. J. Phys. Chem. 27, 1217 (1998). doi:10.1063/1.556028

    Google Scholar 

  28. Struchtrup H.: Macroscopic transport equations for rarefied gas flows. Springer, Berlin (2005)

    MATH  Google Scholar 

  29. Tabeling P.: Introduction to Microfluidics. Oxford University Press, Oxford (2005)

    Google Scholar 

  30. Sellitto A., Alvarex F.X., Jou D.: Second law of thermodynamics and phonon-boundary conditions in nanowires. J. Appl. Phys. 107, 064302 (2010). doi:10.1063/1.3309477

    Article  Google Scholar 

  31. Alvarex F.X., Jou D., Sellitto A.: Pore-size dependence of the thermal conductivity of porous silicon: a phonon hydrodynamic approach. Appl. Phys. Lett. 97, 033103 (2010). doi:10.1063/1.3462936

    Article  Google Scholar 

  32. Dong Y., Cao B.-Y., Guo Z.-Y.: Size dependent thermal conductivity of Si nanosystems based on phonon gas dynamics. Phys. E 56, 256–262 (2014). doi:10.1016/j.physe.2013.10.006

    Article  Google Scholar 

  33. Greywall D.S.: Thermal-conductivity measurement in liquid 4 He below 0.7 K. Phys. Rev. B 23, 2152–2168 (1981)

    Article  Google Scholar 

  34. Childers R.K., Tough J.T.: Helium II thermal counterflow: temperature and pressure-difference data and analysis in terms of the Vinen theory. Phys. Rev. B 13(3), 1040 (1976). doi:10.1103/PhysRevB.13.1040

    Article  Google Scholar 

  35. Geurst J.A.: Hydrodynamics of quantum turbulence in He II: Vinen’s equation derived from energy and impulse of vortex tangle. Phys. B 154, 327–343 (1989). doi:10.1016/0921-4526(89)90167-1

    Article  Google Scholar 

  36. Sciacca, M., Sellitto, A., Jou, D.: Transition to ballistic regime for heat transport in helium II. Phys. Lett. A 378, 2471–2477 (2014)

  37. Sciacca, M., Jou, D., Mongiovì, M.S.: Effective thermal conductivity of helium II: from Landau to Gorter-Mellink regimes. Z. Angew. Math. Phys. (2014). doi:10.1007/s00033-014-0479-5

  38. Tsubota M., Araki T., Vinen F.: Diffusion of an inhomogeneous vortex tangle. Phys. B 224, 329 (2003)

    Google Scholar 

  39. Saluto L., Mongiovì M.S., Jou D.: Vortex diffusion and vortex-line hysteresis in radial quantum turbulence. Phys. B 440C, 99–103 (2014). doi:10.1016/j.physb.2014.01.041

    Article  Google Scholar 

  40. Saluto L., Jou D., Mongiovì M.S.: Thermodynamic approach to vortex production and diffusion in inhomogeneous superfluid turbulence. Phys. A 406, 272–280 (2014). doi:10.1016/j.physa.2014.03.062

    Article  MathSciNet  Google Scholar 

  41. Jou D., Sciacca M., Mongiovì M.S.: Vortex dynamics in rotating counterflow and plane couette and poiseuille turbulence in superfluid helium. Phys. Rev. B 78, 024524 (2008). doi:10.1103/PhysRevB.78.024524

    Article  Google Scholar 

  42. Nemirovskii S.K.: Diffusion of inhomogeneous vortex tangle and decay of superfluid turbulence. Phys. Rev. B 81, 064512 (2010). doi:10.1103/PhysRevB.81.064512

    Article  Google Scholar 

  43. Mongiovì M.S., Jou D.: Generalization of Vinen’s equation including transition to superfluid turbulence. J. Phys. Condens. Matter 17, 4423–4440 (2005). doi:10.1088/0953-8984/17/28/003

    Article  Google Scholar 

  44. Mongiovì M.S., Jou D., Sciacca M.: Energy and temperature of superfluid turbulent vortex tangles. Phys. Rev. B 75, 214514 (2007). doi:10.1103/PhysRevB.75.214514

    Article  Google Scholar 

  45. Khalatnikov I.M.: An Introduction to the Theory of Superfluidity. Benjamin, New York (1965)

    Google Scholar 

  46. Sciacca, M.: Non-equilibrium thermodynamics analysis of rotating counterflow superfluid turbulence. Math. Comput. Model. 51(2), 91–99 (2010). doi:10.1016/j.mcm.2009.09.002

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici (DEIM), Università degli Studi di Palermo, Viale delle Scienze, 90128, Palermo, Italy

    L. Saluto

  2. Departament de Física, Universitat Autònoma de Barcelona, 08193, Bellaterra, Catalonia, Spain

    D. Jou

  3. Dipartimento di Ingegneria Chimica, Gestionale, Informatica, Meccanica, Università degli Studi di Palermo, Viale delle Scienze, 90128, Palermo, Italy

    M. S. Mongiovì

Authors
  1. L. Saluto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Jou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. S. Mongiovì
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. Saluto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saluto, L., Jou, D. & Mongiovì, M.S. Contribution of the normal component to the thermal resistance of turbulent liquid helium. Z. Angew. Math. Phys. 66, 1853–1870 (2015). https://doi.org/10.1007/s00033-015-0493-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-015-0493-2

Mathematics Subject Classification

Keywords

Search

Navigation