Summary
An investigation is made of the instability of stationary flow between two rotating cylinders for any gap distance. Previous work on this problem, viz. the Taylor theory, has been done with the restriction of small gaps. Moreover, in contrast to the recent work ofS. Chandrasekhar [2], who has also analysed this problem, the method used is independent of the basic flow and is therefore valid for all flows between two coaxial cylinders. In fact, it can even be used for the analysis of the flow in a curved channel.
The major item of physical interst is the critical Reynoldsnumber, i. e. the value of the Reynoldsnumber where small disturbances are amplified for the first time. The value of this parameter is determined by the smallest positive eigen value of the boundary value problem. An existence proof is made for this eigenvalue for any wavenumber and for all possible cases of cylindrical flow, with the exception of the case where the cylinders rotate in opposite directions. The results are depicted in a convenient form, where the critical Reynoldsnumber is the dependent variable and the gap distance and the angular velocity ratio of the two cylinders are the independent variables respectively.
A comparison with experiment is made for the case where the ratio of the two radii is 2: 1 and the outer cylinder is at rest. The agreement with the theory is good; the noticeable error being approximately 1%.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Literaturverzeichinis
G. I. Taylor,Stability of a Viscous Liquid Contained Between two Rotating Cylinders, Phil. Trans. Roy. Soc. London, [A]223, 289 (1923).
S. Chandrasekhar,The Stability of Viscous Flow Between Rotating Cylinders, Proc. Roy. Soc. [A],246, 301 (1958).
H. Görtler,Über eine dreidimensionale Instabilität laminarer Grenzschichten an konkaven Wänden, Nachr. Ges. Wiss. Göttingen, Fachgruppe I, Neue Folge2, 1 (1940).
G. Hämmerlin,Über das Eigenwertproblem der dreidimensionalen Instabilität laminarer Grenzschichten an konkaven Wänden, J. Rat. Mech. Anal.4, 279 (1955).
R. J. Donelly,Experiments on the Stability of Viscuous Flow Between Rotating Cylinders, Proc. Roy. Soc. [A],246, 312 (1958).
H. Schlichting,Grenzschichttheorie, 3. Auflage (Verlag G. Braun, Karlsruhe, 1958) S. 69.
H. Witting,Über den Einfluss der Stromlinienkrümmung auf die Stabilität laminarer Strömungen, Arch. Rat. Mech. Anal.2, 243 (1958).
R. Jentzsch,Über Integralgleichungen mit positivem Kern, J. Math.141, 235 (1912).
J. L. Synge,On the Stability of a Viscous Liquid Between Rotating Coaxial Cylinders, Proc. Roy. Soc. [A],167, 250 (1938).
H. Wielandt,Das Iterationsverfahren bei nicht selbstadjungierten Eigenwertaufgaben, Math. Z.50, 93 (1944).
H. Bückner,Die praktische Behandlung von Integralgleichungen, Ergebnisse der Angew. Math., Bd. I, (Springer-Verlag, Berlin-Göttingen-Heidelberg 1952), S. 81.
G. Hämmerlin,Zur Theorie der dreidimensionalen Instabilität laminarer Grenzschichten, ZAMP7, 156 (1956).
Author information
Authors and Affiliations
Additional information
Diese Untersuchung wurde vom Wirtschafsministerium des Landes Baden/Württemberg gefördert.
Rights and permissions
About this article
Cite this article
Kirchgässner, K. Die Instabilität der Strömung zwischen zwei rotierenden Zylindern gegenüber Taylor-Wirbeln für beliebige Spaltbreiten. Journal of Applied Mathematics and Physics (ZAMP) 12, 14–30 (1961). https://doi.org/10.1007/BF01601104
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01601104