Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-18T10:31:43.284Z Has data issue: false hasContentIssue false

A continuum theory of the isothermal flow of liquid helium II

Published online by Cambridge University Press:  28 March 2006

A. A. Townsend
Affiliation:
Emmanuel College, Cambridge

Abstract

Recent work by Hall and Vinen has established that mutual friction between the normal and superfluid components of liquid helium II is caused by interactions between quantized vortex-lines and the normal fluid. If the mean separation of the vortex-lines is small compared with the channel width, the general character of the flow may not depend on the discrete nature of the lines except in so far as this is the cause of the mutual friction. Equations of motion are developed which refer to components of the velocity field with a scale large compared with the line separation, and these are used to discuss the nature of possible turbulent motions. Reasons are given for believing that isothermal flow is very similar to that of a Newtonian fluid, and the theory is developed for turbulent pressure flow along a channel and a circular pipe. The predicted variation of flow rate with pressure gradient is in good agreement with experimental measurements for Reynolds numbers (based on tube diameter and normal fluid viscosity) above 1400, and it is likely that turbulent flow can exist only above this critical Reynolds number. For Reynolds numbers which are not too small, the equations of motion apply to steady ’laminar’ flow and these lead to a relation between flow rate and pressure gradient in reasonable agreement with experiment.

Type
Research Article
Copyright
© 1961 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atkins, K. R. 1951 Proc. Phys. Soc. A, 64, 833.
Atkins, K. R. 1959 Liquid Helium. Cambridge University Press.
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Bhagat, S. M. 1960 Proc. Phys. Soc. A, 75, 303.
Feynman, R. P. 1955 Progr. Low Temp. Phys., vol. I, chap. II. North Holland Publishing Co.
Goldstein, S. (ed.) 1938 Modern Development in Fluid Dynamics, vol. II. Oxford University Press.
Hall, H. E. 1960 Phil. Mag. Suppl. 9, 89.
Hall, H. E. & Vinen, W. F. 1956a Proc. Roy. Soc. A, 238, 204.
Hall, H. E. & Vinen, W. F. 1956b Proc. Roy. Soc. A, 238, 215.
Laufer, J. 1955 N.A.C.A. Rep. no. 1174.
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.
Vinen, W. F. 1957a Proc. Roy. Soc. A, 240, 114.
Vinen, W. F. 1957b Proc. Roy. Soc. A, 240, 128.
Vinen, W. F. 1957c Proc. Roy. Soc. A, 242, 493.
Vinen, W. F. 1957d Proc. Roy. Soc. A, 243, 400.