Journal of Low Temperature Physics

, Volume 90, Issue 1–2, pp 95–117 | Cite as

Superfluid effects at the onset of convection in3He-superfluid-4He mixtures

  • Guy Metcalfe
  • R. P. Behringer


The equations of motion for convection in dilute3He-superfluid-4He mixtures are the same as those for convection in a conventional pure fluid with the addition of several correction terms. Fetter has considered, for a horizontally infinite layer with realistic boundary conditions, the effect of these corrections on the critical Rayleigh number,R c . The results are a perturbation expansion forR c to lowest order in three perturbation terms, ε 1 , ε 2 , ε 3 . In order to make a comparison with recent precise experiments which have yieldedRc as a function of the layer heightd, we have carried out several calculations. First we show that the analysis can be recast as an expansion in inverse powers ofd2. We then carry out a complete expansion toO(d−6). Up toO(d−4), the expansion involves only the ratio (λ 0 /d) where λ 0 is a length scale which is intrinsic to superfluid mixtures. We consider the effect of the superfluid perturbations on both the critical Rayleigh numbers and wavevectors. These are shifted very little as long as λ 0 /d is small; the crossover from large to small occurs for λ 0 /d∼0.1. We also solve a simplified version of the stability problem which contains the dominant superfluid effect. The simplified problem is Hermitian, and is therefore amenable to an exact solution. A comparison with experimental data forR c and the simplified model shows excellent agreement with the calculations.


Boundary Condition Convection Exact Solution Magnetic Material Excellent Agreement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    See for instance R. P. Behringer,Rev. Mod. Phys. 57, 657 (1985).Google Scholar
  2. 2.
    G. P. Metcalfe and R. P. Behringer,Phys. Rev. A 41, 5735 (1990); andPhysica D 51, 520 (1991); also Guy Metcalfe, Ph.D. dissertation (1991), unpublished.Google Scholar
  3. 3.
    A. L. Fetter,Phys. Rev. A 26, 1164 (1982).Google Scholar
  4. 4.
    A. L. Fetter,Phys. Rev. 26, 1174 (1982).Google Scholar
  5. 5.
    A. Ya. Parshin,Zh. Eksp. Teor. Fiz. Pis'ma Red. 10, 567 (1969) [JETP Lett. 10, 362 (1969)].Google Scholar
  6. 6.
    V. Steinberg,Phys. Rev. A 24, 975 (1981).Google Scholar
  7. 7.
    V. Steinberg,Phys. Rev. A 24, 988 (1981).Google Scholar
  8. 8.
    V. Steinberg and H. R. Brand,Phys. Rev. B 28, 1618 (1983).Google Scholar
  9. 9.
    I. M. Khalatnikov,Introduction to the Theory of Superfluidity (Benjamin, New York, 1965).Google Scholar
  10. 10.
    G. Metcalfe and R. P. Behringer, to be published.Google Scholar
  11. 11.
    S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981).Google Scholar
  12. 12.
    W. Press, B. Flannery, S. Teukolsky, and W. Vetterling,Numerical Recipes (Cambridge University Press, 1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Guy Metcalfe
    • 1
  • R. P. Behringer
    • 1
  1. 1.Department of Physics and Center for Nonlinear and Complex SystemsDuke UniversityDurham

Personalised recommendations