Abstract
The Couette configuration of a fluid contained between two rotating concentric cylinders has proved useful to test and validate the HVBK equations which govern the motion of superfluid helium II. We critically review the current understanding of the superfluid Couette problem and compare theory and experiment, distinguishing between the results obtained with infinitely long cylinders and those obtained at small aspect ratio. After discussing some issues which are still unsolved, we point to what should be fruitful directions of further investigation which can be pursued in the Couette configuration.
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References
London, F.: Superfluids. John Wiley (1954)
Khalatnikov, I.M.: An Introduction to the Theory Superfluidity. Benjamin (1965)
Donnelly, R.J.: Quantized Vortex Lines in Helium II. Cambridge University Press (1991)
Donnelly, R.J., Barenghi, C.F.: The observed properties of liquid helium at the saturated vapour pressure. J. Physical and Chemical Reference Data 27, 1217–1274 (1998)
Smith, M.R., Donnelly, R.J., Vinen, W.F.: Decay of vorticity in homogeneous turbulence. Physical Review Letters 71, 2583–2586 (1993)
Barenghi, C.F., Swanson, C.J., Donnelly, R.J.: Emerging issues in helium turbulence. Journal of Low Temperature Physics 100, 385–413 (1995)
Maurer, J., Tabeling, P.: Local investigation of superfluid turbulence. Europhys. Lett. 43, 29 (1998)
Finne, A.P., Araki, T., Blaauwgeers, R., Eltsov, V.B., Kopnin, N.B., Krusius, M., Skrbek, L., Tsubota, M., Volovik, G.E.: Transition to superfluid turbulence governed by an intrinsic parameter. Nature 424, 1022–1025 (2003)
Barenghi, C.F., Hulton, S., Samuels, D.C.: Polarisation of superfluid turbulence. Physical Review Letters 89, 275301 (2002)
Landau, L.D., Lifshitz, E.M.: Fluid Mechanics. Pergamon (1987)
Barenghi, C.F., Donnelly, R.J., Vinen, W.F. (eds.): Quantized Vortex Dynamics And Superfluid Turbulence, Springer’s Lecture Notes in Physics 571 (2001)
Hall, H.E., Vinen, W.F.: The rotation of liquid He II. II the theory of mutual friction in uniformly rotating He II. Proceedings of the Royal Society of London A 238, 215–234 (1956)
Hall, H.E.: The rotation of liquid helium II. Advances in Physics 9, 89–146 (1960)
Bekharevich, I.L., Khalatnikov, I.M.: Phenomenological derivation of the equations of vortex motion in He II. Soviet Phys. JETP 13, 643–646 (1961)
Hills, R.N., Roberts, P.H.: Superfluid mechanics for a high density of vortex lines. Arch Rat Mech Anal 66, 43–71 (1977)
Barenghi, C.F., Donnelly, R.J., Vinen, W.F.: Friction on quantized vortices in He II. A review. Journal of Low Temperature Physics 52, 189–247 (1983)
Taylor, G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. Lond. A 223, 289–343 (1923)
Poiseuille, J.L.: Comptes rendus 11, 961 (1840)
Kapitza, P.L.: Journal of Physics, USSR 4, 181 (1941)
Donnelly, R.J., Lamar, M.M.: Flow and stability of helium II between rotating cylinders. Journal of Fluid Mechanics 186, 163–198 (1988)
Swanson, C.J., Donnelly, R.J.: Instability of Taylor–Couette flow of helium II. Physical Review Letters 67, 1578–1581 (1991)
Chandrasekhar, S., Donnelly, R.J.: The hydrodynamic stability of He II between rotating cylinders. I. Proceedings of the Royal Society of London A 241, 9–28 (1957)
Barenghi, C.F., Jones, C.A.: The stability of the Couette flow of helium II. Journal of Fluid Mechanics 197, 551–569 (1988)
Barenghi, C.F.: Vortices and the Couette flow of helium II. Physical Review B 45, 2290–2293 (1992)
Henderson, K.L., Barenghi, C.F., Jones, C.A.: Nonlinear Taylor–Couette flow of helium II. Journal of Fluid Mechanics 283, 329–340 (1995)
Henderson, K.L., Barenghi, C.F.: Numerical methods for helium’s two fluid model. Journal of Low Temperature Physics 98, 351–381 (1995)
Henderson, K.L., Barenghi, C.F.: Calculation of the torque in nonlinear Taylor vortex flow of helium II. Physics Letters A 191, 438–442 (1994)
Donnelly, R.J.: Experiments on the hydrodynamic stability of helium II between rotating cylinders. Physical Review Letters 3, 507–508 (1959)
Henderson, K.L., Barenghi, C.F.: The anomalous motion of superfluid helium in a rotating cavity. Journal of Fluid Mechanics 406, 199–219 (2000)
Cliffe, K.A., Mullin, T.: Numerical and experimental study of anomalous modes in the Taylor experiment. Journal of Fluid Mechanics 153, 243–258 (1985)
Schaeffer, D.G.: Qualitative analysis of a model for boundary effects in the Taylor problem. Math. Proc. Camb. Phil. Soc. 87, 307–337 (1980)
Pfister, G., Schmidt, H., Cliffe, K.A., Mullin, T.: Bifurcation phenomena in Taylor–Couette flow in a very short annulus. Journal of Fluid Mechanics 191, 1–18 (1988)
Cliffe, K.A., Kobine, J.J., Mullin, T.: The role of anomalous modes in Taylor–Couette flow. Proceedings of the Royal Society A 439, 341–357 (1992)
Henderson, K.L., Barenghi, C.F.: Transition from Ekman flow to Taylor vortex flow in superfluid helium. Journal of Fluid Mechanics 508, 319–331 (2004)
Bielert, F., Stamm, G.: Visualisation of Taylor–Couette flow in superfluid helium. Cryogenics 33, 938–940 (1993)
Donnelly, R.J., Karpetis, A.N., Niemela, J.J., Sreenivasan, K.R., Vinen, W.F., White, C.M.: The use of particle image velocimetry in the study of turbulence in liquid helium. Journal of Low Temperature Physics 126, 327–332 (2002)
Van Sciver, S.W.: Private Communication (2003)
Zhang, T., Celik, D., Van Sciver, S.W.: Tracer particles for application to PIV studeies of liquid helium. Journal of Low Temperature Physics 134, 985–1000 (2004)
Adronikashvili, E.L., Mamaladze, Y.G.: Rotation of helium II. In Progress in Low Temperature Physics, volume 5, pages 79–161. C.J. Gorter, North Holland, 1967
Hollerbach, R.: Private Communication (2002)
Heikkila, W.J., HollisHallett, C.: The viscosity of liquid helium II. Canadian J. Phys. 33, 420–435 (1955)
Barenghi, C.F., Jones, C.A.: The stability of superfluid helium between rotating concentric cylinders. Physics Letters A 122, 425–430 (1987)
Richards, D., Zhang, J.P.: Turbulence in differentially rotating flows: what can be learned from the Taylor–Couette experiment. Astronomy and Astrophysics 347, 734–738 (1999)
Barenghi, C.F., Samuels, D.C., Bauer, G.H., Donnelly, R.J.: Superfluid vortex lines in a model of turbulent flow. Physics of Fluids 9, 2631–2643 (1997)
Samuels, D.C., Kivotides, D.: A damping length scale for superfluid turbulence. Physical Review Letters, 83, 5306–5309 (1999)
Cheng, D.K., Cromar, M.W., Donnelly, R.J.: Influence of an axial heat current on negative ions trapping in rotating helium. Physical Review Letters 31, 433–436 (1973)
Ostermeier, R.M., Glaberson, G.I.: Instability of vortex lines in the presence of axial normal fluid. Journal of Low Temperature Physics 21, 191–196 (1975)
Infeld, E., Lenkowska-Czerwinska, T.: Growth of the normal-flow instability of a vortex array in 2-component He II. Physical Review B 57, 13390–13392, (1998)
Tsubota, M., Araki, T., Barenghi, C.F.: Rotating superfluid turbulence. Physical Review Letters 90, 205301 (2003)
Chandrasekhar, S.: Hydrodynamic and hydro-magnetic stability. Dover publications (1961)
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H.J.S. Fernando
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Henderson, K., Barenghi, C. Superfluid Couette flow in an enclosed annulus. Theor. Comput. Fluid Dyn. 18, 183–196 (2004). https://doi.org/10.1007/s00162-004-0124-6
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DOI: https://doi.org/10.1007/s00162-004-0124-6