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Secondary flow in a Hele-Shaw cell

Published online by Cambridge University Press:  28 March 2006

B. W. Thompson
B. W. Thompson
Affiliation:
Mathematics Department, University College London
Now at Mathematics Department, University of Melbourne.
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Abstract

Riegels (1938) investigated the breakdown of Hele-Shaw flow in a Hele-Shaw cell with unusually large separation distance 2h* between the walls. A theoretical outer expansion for the velocity was constructed in the case where the obstacle is a circular cylinder, using an intuitive inner boundary condition that seems to be correct in the limit h* → 0, but without explicit matching with the inner expansion.

An inner expansion has now been found, and it shows that the solution in the inner layer forces terms into the outer expansion that are larger than those found by Riegels whenever h* is finite and not zero.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 31 , Issue 2 , 29 January 1968 , pp. 379 - 395
Copyright
© 1968 Cambridge University Press

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References

Gaydon, F. A. & Shepherd, W. M. 1964 Proc. Roy. Soc. A, 281, 184.
Harpy, G. H. 1902 Mess. of Math. xxxi, 161.
Hele-Shaw, H. S. 1897 Trans. Roy. Inst. Nav. Arch. 41, 21.
Hele-Shaw, H. S. 1898a Trans. Roy. Inst. Nav. Arch. 42, 49.
Hele-Shaw, H. S. 1898b Rept. 68th Mtg. Brit. Ass. 136.
Hillman, A. P. & Salzer, H. E. 1943 Phil. Mag. 34, 575.
Riegels, F. 1938 ZAMM, 18, 95.
Stokes, G. G. 1898 appendix to Hele-Shaw (1898b).
Thompson, B. W. 1964 M.Sc. thesis, Melbourne.
Thompson, B. W. 1967 Ph.D. Thesis, London.