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The propagation of small amplitude long waves on the surface of superfluid helium

Published online by Cambridge University Press:  17 February 2009

R. S. Johnson
R. S. Johnson
Affiliation:
School of Mathematics, The University, Newcastle upon Tyne, NEI 7RU, U.K.
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Abstract

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Various problems related to the propagation of small amplitude long waves on the surface of superfluid helium (helium II), usually called third sound, are studied on the basis of the appropriate governing equations. The two-fluid continuum model due to Landau is considered, with the effects of healing and relaxation incorporated, and viscosity, heat conduction and compressibility terms retained. The helium vapour is treated as a classical (Newtonian) compressible gas and the exact jump conditions across the liquid/vapour interface are employed. These liquid, vapour and jump equations constitute the exact problem although, in an effort to reduce the complexity of the equations, a simplified set of ‘model’ surface boundaiy conditions is also introduced. This full set of equations is non-dimensionalised taking care that all physical parameters are defined using only the undisturbed depth of the layer as the appropriate length scale. The ratio depth/wave-lenght (δ) is then a separate parameter as is the wave amplitude/depth ratio (ɛ). The limit which corresponds to the wave under discussion is then ɛ, δ → 0 with all the other parameters fixed.

A number of analyses are presented, four of which describe various aspects of the linearised theory and two examine the nature of the far-field nonlinear problem. Using the simplified surface boundary conditions we discuss in turn: the wave motion in the absence of healing: the rôle of a second wave speed leading to a wave hierarchy; and the effects of healing. The final linearised problem makes use of the full vapour model, but again the healing terms are ignored. This latter analysis suggests that if the upper boundary of the vapour is sufficiently close to the liquid surface then third sound is suppressed.

The complexity of the equations, particularly when the nonlinear terms are to be examined, is such that the incompressible limit is now taken in the absence of both healing and relaxation. Imposing the physically realistic limiting process (ɛ, δ → 0) we show that the only equation valid in the far-field is the Burgers equation. However, we also demonstrate that allowing the other parameters to be functions of e (which is not physically realisable in practice) it is easy to derive, for example, the Korteweg-de Vries equation.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 3 , January 1984 , pp. 311 - 348
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Atkins, K. R., “Third and fourth sound in liquid helium II”, Phys. Rev. 113 (1959), 962–965.CrossRefGoogle Scholar
[2]Atkins, K. R. and Rudnick, I., “Third sound” in Progress in low temperature physics, Vol. 7 (North Holland, Amsterdam, 1970).Google Scholar
[3]Bergman, D., “Hydrodynamics and third sound in thin Hc II films”, Phys. Rev. 188 (1969), 370–384.Google Scholar
[4]Bergman, D., “Third sound in superfluid helium films of arbitrary thickness”, Phys. Rev. A 3 (1971), 2058–2066.CrossRefGoogle Scholar
[5]Chester, M. and Yang, L. C., “Superfluid fraction in thin helium films”, Phys. Rev. Lett. 31 (1973), 1377–1380.CrossRefGoogle Scholar
[6]Donnelly, R. J., Experimental superfluidity (Univ. of Chicago Press, Chicago, 1967).Google Scholar
[7]Everitt, C. W. F., Atkins, K. R. and Denenstein, A., “Detection of third sound in liquid helium films”, Phys. Rev. Len. 8 (1962), 161–165.Google Scholar
[8]Ginzburg, V. L. and Sobaynin, A. A., “Superfluidity of helium II near the λ-point”, Soviet Phys. Uspekhi 19 (1976), 773–812.Google Scholar
[9]Hills, R. N. and Roberts, P. H., “On Landau's two-fluid model for helium II”, J. Inst. Math. Appl. 9 (1972), 56–67.Google Scholar
[10]Hills, R. N. and Roberts, P. H., “Healing and relaxation in flows of helium II-I Generalization of Landau's equations”, Internat. J. Engrg. Sci. 15 (1977), 305–316.CrossRefGoogle Scholar
[11]Hills, R. N. and Roberts, P. H., “On the interface conditions between a liquid and its vapour with application to helium II”, J. Non-Equilib. Thermodyn. 4 (1979), 131–142.CrossRefGoogle Scholar
[12]Huberman, B. A., “Superfluid solitons in helium films”, Phys. Rev. Len. 41 (1978), 1389–1393.Google Scholar
[13]Khalatnikov, I. M., “Absorption and dispersion of sound in a superfluid liquid near the λ-point”, Soviet Phys. JETP 30 (1970), 268–272.Google Scholar
[14]Johnson, R. S., “Shallow water waves on a viscous fluid—the undular bore”, Phys. Fluids 15 (1972), 1693–1699.CrossRefGoogle Scholar
[15]Johnson, R. S., “On the propagation of long waves on the surface of helium II: linearised theory”, Proc. Roy Soc. London Ser. A 362 (1978), 97–111.Google Scholar
[16]Johnson, R. S., “A note on the effect of healing and relaxation in helium II due to heat transfer at a wall”, Roy. Soc. London Ser. A 362 (1978), 375–382.Google Scholar
[17]Johnson, R. S., in preparation, 1983.CrossRefGoogle Scholar
[18]Landau, L. D., “On a two-fluid model of superfluid helium”, J. Phys. 5 (1941), 71–90 (in Russian).Google Scholar
[19]Landau, L. D. and Lifshitz, E. M., Fluid mechanics (Pergamon, Oxford, 1959).Google Scholar
[20]Nakajima, S., Kurihara, S. and Tohdoh, K., “Soliton in superfluid4 He films”, Preprint (University of Toyko, 1980).Google Scholar
[21]Pickar, K. A. and Atkins, K. R., “Critical velocity of a superflowing liquid-helium film using third sound”, Phys. Rev. 178 (1969), 389–399.CrossRefGoogle Scholar
[22]Putterman, S. J., Superfluid hydrodynamics (North-Holland, Amsterdam, 1974).Google Scholar
[23]Roberts, P. H. and Donnelly, R. J., “Superfluid mechanics”, Annual Rev. Fluid. Mech. 6 (1974), 179–225.Google Scholar
[24]Sobaynin, A. A., “Surface tension of helium II near the λ-point”, Soviet Phys. JETP 34 (1972), 229–232.Google Scholar
[25]Whitham, G. B., “Some comments on wave propagation and shock wave structure with application to magnetohydrodynamics”, Comm. Pure Appl. Math. 12 (1959), 113–135.CrossRefGoogle Scholar
[26]Whitham, G. B., Linear and nonlinear waves (Wiley-Interscience, New York, 1974).Google Scholar