Summary
In this paper it is shown numerically that axially-symmetric solutions of the Navier-Stokes equations, which describe the rotating flow above a disk which is itself rotating, are non-unique. The numerical techniques designed to calculate such solutions with a high power of resolution are given. Especially the behaviour in and around the first branching point is considered. It is found that fors=−0.16054 two branches coincide. The second branch has been almost completely calculated. It ranges back to positive values ofs.
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References
M. H. Rogers and G. N. Lance, The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk.Journal of Fluid Mechanics, 7 (1960) 617–631.
D. J. Evans, The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk with uniform suction.Quarterly Journal of Mechanics and Applied Mathematics, 22 (1969) 467–485.
H. Ockendon, An asymptotic solution for steady flow above an infinite rotating disk with suction.Quarterly Journal of Mechanics and Applied Mathematics, 25 (1972) 291–301.
R. J. Bodonyi, On rotationally symmetric flow above an infinite rotating disk.Journal of Fluid Mechanics, 67 (1975) 657–666.
P. D. Weidman and L. G. Redekopp,On the motion of a rotating fluid in the presence of an infinite rotating disk. Lecture presented at the 12th Biennial Fluid Dynamics Symposium, september 1975, Bialowieza, Poland.
S. P. Hastings, An existence theorem for some problems from boundary layer theory.Archive for Rational Mechanics and Analysis, 38 (1970) 308–316.
J. B. McLeod, The asymptotic form of solutions of Von Karman's swirling flow problem.Quarterly Journal of Mathematics (Oxford), 20 (1969) 483–496.
J. B. McLeod, The existence of axially symmetric flow above a rotating disk.Proceedings of the Royal Society, A324 (1971) 391–414.
J. B. McLeod, A note on rotationally symmetric flow above an infinite rotating disk.Mathematika, 17 (1970) 243–249.
P. J. Bushell, On Von Karman's equations of swirling flow.Journal of the London Mathematical Society (2), 4 (1972) 701–710.
P. Hartman, On the swirling flow problem.Indiana University Mathematical Journal, 21 (1972) 849–855.
C. C. Lan, On functional differential equations and some laminar boundary layer problems.Archive for Rational Mechanics and Analysis, 42 (1971) 24–39.
H. Schlichting,Boundary-layer theory. McGraw-Hill, New York (1968).
R. Bulirsch and S. Stoer, Numerical treatment of ordinary differential equations by extrapolation methods.Numerische Mathematik, 8 (1966) 1–13, 93–104.
R. J. Bodonyi,The laminar boundary layer on a finite rotating disc. Dissertation, Dept. of Aeronautical and Astronautical Engineering, The Ohio State University (1973).
T. S. Cham, The laminar boundary layer of a source and vortex flow.The Aeronautical Quarterly, May 1971, 196–206.
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Zandbergen, P.J., Dijkstra, D. Non-unique solutions of the Navier-Stokes equations for the Karman swirling flow. J Eng Math 11, 167–188 (1977). https://doi.org/10.1007/BF01535696
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DOI: https://doi.org/10.1007/BF01535696