Hostname: page-component-5447f9dfdb-lr6j9 Total loading time: 0 Render date: 2025-07-30T08:59:48.006Z Has data issue: false hasContentIssue false
  • English
  • Français

On the swirling flow between rotating coaxial disks, asymptotic behaviour, I

Published online by Cambridge University Press:  14 November 2011

Heinz Otto Kreiss  and
Seymour V. Parter
Heinz Otto Kreiss
Affiliation:
California Institute of Technology, Pasadena, California, U.S.A
Seymour V. Parter
Affiliation:
University of Wisconsin-Madison, Madison, Wisconsin, U.S.A
Get access Rights & Permissions [Opens in a new window]

Synopsis

Consider solutions 〈H(x, ε), G(x, ε)〉 of the von Kármán equations for the swirling flow between two rotating coaxial disks

and

We also assume that |H(x, ε)|≦B√(ε) while |G(x, ε)|≦B. This work considers the shapes and asymptotic behaviour as ε→0+. We consider the kind of limit functions that are permissible. The only possible limits (interior) for G(x, ε) are constants. If that limit constant is not zero, then ε−½H(x, ε) will also tend to a constant.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 90 , Issue 3-4 , 1981 , pp. 293 - 316
Copyright
Copyright © Royal Society of Edinburgh 1981

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.)

Article purchase

Temporarily unavailable

References

1Batchelor, G. K.. Note on a class of solutions of the Navier–Stokes equations representing steady rotationally-symmetric flow. Quart. J. Mech. Appl. Math. 4 (1951), 29–41.CrossRefGoogle Scholar
2Elcrat, A. R.. On the swirling flow between rotating coaxial disks. J. Differential Equations 18 (1975), 423–430.CrossRefGoogle Scholar
3Greenspan, D.. Numerical studies of flow between rotating coaxial disks. J. Inst. Math. Appl. 9 (1972), 370–377.CrossRefGoogle Scholar
4Hastings, S. P.. On existence theorems for some problems from boundary layer theory. Arch. Rational Mech. Anal. 38 (1970), 308–316.CrossRefGoogle Scholar
5von Kármán, T.. Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1 (1921), 232–252.Google Scholar
6Landau, E.. Einige Ungleichungen für zweimal differenzbare Funktionen. Proc. London Math. Soc. 13 (1913), 43–49.Google Scholar
7Lance, G. N. and Rogers, M. H.. The axially symmetric flow of a viscous fluid between two infinite rotating disks. Proc. Roy. Soc. London Ser. A 266 (1962), 109–121.Google Scholar
8McLeod, J. B.. The asymptotic form of solutions of von Kármán's swirling flow problem. Quart. J. Math. Oxford 20 (1969), 483–496.CrossRefGoogle Scholar
9McLeod, J. B.. A note on rotationally symmetric flow above an infinite rotating disc. Mathematiku 17 (1970), 243–249.CrossRefGoogle Scholar
10McLeod, J. B. and Parter, S. V.. On the flow between two counter-rotating infinite plane disks. Arch. Rational Mech. Anal. 54 (1974), 301–327.CrossRefGoogle Scholar
11McLeod, J. B. and Parter, S. V.. The non-monotonicity of solutions in swirling flow. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 161–182.CrossRefGoogle Scholar
12Matkowsky, B. J. and Siegmann, W. L.. The flow between counter-rotating disks at high Reynolds numbers. SIAM J. Appl. Math. 30 (1976), 720–727.CrossRefGoogle Scholar
13Mellor, G. L., Chapple, P. J. and Stokes, V. K.. On the flow between a rotating and a stationary disk. J. Fluid Mech. 31 (1968), 95–112.CrossRefGoogle Scholar
14Nguyen, N. D., Ribault, J. P. and Florent, P.. Multiple solutions for flow between coaxial disks. J. Fluid Mech. 68 (1975), 369–388.CrossRefGoogle Scholar
15Pearson, C. E.. Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks. J. Fluid Mech. 21 (1965), 623–633.CrossRefGoogle Scholar
16Rasmussen, H.. High Reynolds number flow between two infinite rotating disks. J. Austral. Math. Soc. 12 (1971), 483–501.CrossRefGoogle Scholar
17Roberts, S. M. and Shipman, J. S.. Computation of the flow between a rotating and a stationary disk. J. Fluid Mech. 73 (1976), 53–63.CrossRefGoogle Scholar
18Rogers, M. H. and Lance, G. N.. The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk. J. Fluid Mech. 7 (1960), 617–631.CrossRefGoogle Scholar
19Schultz, D..and Greenspan, D.. Simplification and improvement of a numerical method for Navier–Stokes problems. Proc. of the Colloquium on Differential Equations, Kesthaly, Hungary; Sept. 2–6, 1974, pp. 201–222.Google Scholar
20Stewartson, K.. On the flow between two rotating coaxial disks. Proc. Cambridge Philos. Soc. 49 (1953), 333–341.CrossRefGoogle Scholar
21Tam, K. K.. A note on the asymptotic solution of the flow between two oppositely rotating infinite plane disks. SIAM J. Appl. Math. 17 (1969), 1305–1310.CrossRefGoogle Scholar
22Watts, A. M.. On the von Kármán equations for axi-symmetric flow. Appl. Math.—Preprint No. 74, (1974), University of Queensland.Google Scholar
23Wilson, L. O. and Schryer, N. L.. Flow between a stationary and a rotating disk with suction. J. Fluid Mech. 85 (1978), 479–496.CrossRefGoogle Scholar