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On rotationally symmetric flow above an infinite rotating disk

Published online by Cambridge University Press:  29 March 2006

R. J. Bodonyi
Affiliation:
Department of Aeronautical and Astronautical Engineering, The Ohio State University, Columbust

Abstract

The similarity equations for rotationally symmetric flow above an infinite counter–rotating disk are investigated both numerically and analytically. Numerical solutions are found when α, the ratio of the disk's angular speed to that of the rigidly rotating fluid far from it, is greater than −0.68795. It is deduced that there exists a critical value αcr, of α above which finite solutions are possible. The value of α and the limiting structure as α → αcr are found using the method of matched asymptotic expansions. The flow structure is found to consist of a thin viscous wall region above which lies a thick inviscid layer and yet another viscous transition layer. Furthermore, this structure is not unique: there can be any number of thick inviscid layers, each separated from the next by a viscous transition layer, before the outer boundary conditions on the solution are satisfied. However, comparison with the numerical solutions indicates that a single inviscid layer is preferred.

Type
Research Article
Copyright
© 1975 Cambridge University Press

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References

Batchelor, G. K. 1951 Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow. Quart. J. Mech. Appl. Math., 4, 29.Google Scholar
Bödewadt, V. T. 1940 Die Drehstromung uber festem Grunde. 2. angew. Math. Mech., 20, 241.Google Scholar
Bodonyi, R. J. 1973 The laminar boundary layer on a finite rotating disc. Ph.D. dissertation, The Ohio State University.
Evans, D. J. 1969 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disc with uniform suction. Quart. J. Mech. Appl. Math., 22, 467.Google Scholar
Kármán, T. Von 1921 Über laminare und turbulente Reibung. 2. angew. Math. Mech., 1, 233.Google Scholar
Kuiken, H. K. 1971 The effect of normal blowing on the flow near a rotating disk of infinite extent. J. Fluid Mech., 47, 789.Google Scholar
Mcleod, J. B. 1970 A note on rotationally symmetric flow above an infinite rotating disc. Mathematika, 17, 243.Google Scholar
Ockendon, H. 1972 An asymptotic solution for the steady flow above an infinite rotating disc with suction. Quart. J. Mech. Appl. Math., 25, 291.Google Scholar
Rogers, M. H. & Lance, G. N. 1960 The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. J. Fluid Mech., 7, 617.Google Scholar