We suggest a “minimal model” for the 3D turbulent energy spectra in superfluids, based on their two-fluid description. We start from the Navier–Stokes equation for the normal fluid and from the coarse-grained hydrodynamic equation for the superfluid component (obtained from the Euler equation for the superfluid velocity after averaging over the vortex lines) and introduce a mutual friction coupling term, proportional to the counterflow velocity, the average superfluid vorticity and to the temperature dependent parameter q = α/(1 + α′), where α and α′ denote the dimensionless parameters characterizing the mutual friction between quantized vortices and the normal component of the liquid. We then derive the energy balance equations, taking into account the cross-velocity correlations. We obtain all asymptotical solutions for normal and superfluid energy spectra for limiting cases of small/big normal to superfluid density ratio and coupling. We discuss the applicability of our model to superfluid He II and to 3He-B.
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References
Vinen W.F., Niemela J J., (2002) . J. Low Temp. Phys. 128, 167
Vinen W F., Proc. Royal Soc. A240 114, 128 (1957); A242 489 (1957); A242, 493 (1957).
Skrbek L., Niemela J.J., Sreenivasan K.R., (2001) . Phys. Rev. E 64: 067301
Bradley D.I., Clubb D.O., Fisher S.N., Guénault A.M., Haley R.P., Matthews C.J., Pickett G.R., Tsepelin V., Zaki K., (2005) . Phys. Rev. Lett. 95, 035302
Araki T., Tsubota M., Nemirovskii S.K., (2002) . Phys. Rev. Lett. 89: 145301
Nore C., Abid M., Brachet M.E., (1997) . Phys. Fluids 9: 2644
Kobayashi M., Tsubota M., (2005) . Phys. Rev. Lett. 94: 065302
Maurer J., Tabeling P., (1998) . Europhys. Lett. 43, 29
Volovik G E., JETP Letters 78, 533 (2003); J. Low Temp. Phys. 136, 309 (2004).
Vinen W.F., (2005) . Phys. Rev. B 71: 024513
L’vov V.S., Nazarenko S.V., Volovik G.E., (2004) . JETP Lett. 80, 479
Skrbek L., (2006) . JETP Lett. 83, 127
Donnelly R.J., Barenghi C.F.,(1998) . J. Phys. Chem. Data 27: 1217
Vollhardt D. and P.Völfle, The Superfluid Phases of Helium 3, Taylor and Francis, London (1990).
Donnelly R J., Quantized Vortices in Hellium II, Cambridge University Press (1991).
Kovasznay L., (1947) . J. Aeronaut. Sci. 15, 745
Nazarenko S., (2005) . JETP Lett 83, 198
Kozik E.V., Svistunov B.V., (2004) . Phys. Rev. Lett. 92: 035301
T.D.C. Bevan et al., Nature, 386, 689 (1997); J. Low Temp. Phys, 109, 423 (1997).
Stalp S.R., Niemela J.J., Vinen W.F., Donnelly R.J., (2002) . Phys. Fluids 14: 1377
Vinen W F., private communication.
Tsubota M., Barenghi C.F., Araki T., Mitani A., (2004) . Phys. Rev. B 69: 134515
Barenghi C.F., Donnelly R.J., Vinen W.F., (1983) . J. Low Temp. Phys. 52, 189
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L’vov, V.S., Nazarenko, S.V. & Skrbek, L. Energy Spectra of Developed Turbulence in Helium Superfluids. J Low Temp Phys 145, 125–142 (2006). https://doi.org/10.1007/s10909-006-9230-8
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DOI: https://doi.org/10.1007/s10909-006-9230-8