Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T21:14:17.906Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of viscosity on the stability of a uniformly rotating liquid column in zero gravity

Published online by Cambridge University Press:  23 January 2007

J. P. KUBITSCHEK  and
P. D. WEIDMAN
J. P. KUBITSCHEK*
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, USA
P. D. WEIDMAN
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309-0427, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

An investigation of the linear temporal stability of a uniformly rotating viscous liquid column in the absence of gravity is presented. The governing parameters are the rotational Reynolds number Re and the Hocking parameter L, defined as the ratio of surface tension to centrifugal forces. Though the viscosity-independent condition L≥(k2 + n2-1)−1 for stability to disturbances of axial wavenumber k and azimuthal mode number n has been known for some time, the preferred modes, growth rates and frequencies at onset of instability have not been reported. We compute these results over a wide range of L–Re space and determine certain asymptotic behaviours in the limits of L→0, L→∞ and Re→∞. The computations exhibit a continuous evolution toward known inviscid stability results in the large-Re limit and their ultimate transition to an n = 1 spiral mode at small Re. While viscosity is shown to reduce growth rates for axisymmetric disturbances, it also produces a destabilizing effect for n = 2 planar and n = 1 spiral disturbances in certain regions of parameter space. A special feature is the appearance of a tricritical point in L–Re space at which maximum growth rates of the axisymmetric, n = 1 spiral, and n = 2 planar disturbances are equal.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 572 , February 2007 , pp. 261 - 286
Copyright
Copyright © Cambridge University Press 2007

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.)

References

REFERENCES

Ali, M. E. 1988 The stability of Taylor-Couette flow with radial heating. PhD thesis, University of Colorado, Boulder, CO.Google Scholar
Ashmore, J. & Stone, H. A. 2004 Instability of a rotating thread in a second immiscible liquid. Phys. Fluids 16, 29–38.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Dávalos-Orozco, L. A. & Vázquez-Luis, E. 2003 Instability of the interface between two inviscid fluids inside a rotating annulus in the absence of gravity. Phys. Fluids 15, 2728–2739.CrossRefGoogle Scholar
Donnelly, R. J. & Glaberson, W. 1965 Experiments on the capillary instability of a liquid jet. Proc. R. Soc. Lond. A 290, 547–556.Google Scholar
Eggers, J. & Brenner, M. P. 1999 Spinning jets. Proc. IUTAM Symp. on Nonlinear Waves in Multiphase Flow (ed. Chang, H. C.). Kluwer.Google Scholar
Gillis, J. 1961 Stability of a column of rotating viscous liquid. Proc. Camb. Phil. Soc. 57, 152–159.CrossRefGoogle Scholar
Gillis, J. & Kaufman, B. 1962 Stability of a rotating viscous jet. Q. Appl. Maths 19, 301–308.CrossRefGoogle Scholar
Gillis, J. & Suh, K. S. 1962 Stability of a rotating liquid column. Phys. Fluids. 5, 149–155.CrossRefGoogle Scholar
Goedde, E. F. & Yuen, M. C. 1970 Experiments on liquid jet instability. J. Fluid Mech. 40, 495–511.CrossRefGoogle Scholar
Hocking, L. M. 1960 The stability of a rigidly rotating column of liquid. Mathematika 7, 1–9.CrossRefGoogle Scholar
Hocking, L. M. & Michael, D. H. 1959 The stability of a column of rotating liquid. Mathematika 6, 25–32.CrossRefGoogle Scholar
Kubitschek, J. P. 2006 The effect of viscosity on the stability of a uniformly rotating viscous liquid column. PhD Thesis, University of Colorado, Boulder, CO.CrossRefGoogle Scholar
Rayleigh, Lord 1879 On the stability of jets. Proc. Lond. Math. Soc. 9, 4–13.Google Scholar
Rayleigh, Lord 1892 On the instability of a cylinder of viscous liquid under capillary force. Phil. Mag. 34, 145–154.CrossRefGoogle Scholar
Rayleigh, Lord 1945 The Theory of Sound, vol. II, 2nd edn. Dover.Google Scholar
Rutland, D. F. & Jameson, G. J. 1970 Theoretical prediction of the sizes of drops formed in the breakup of capillary jets. Chem. Engng Sci. 25, 1689–1698.CrossRefGoogle Scholar
Sanmiguel-Rojas, E. & Fernandez-Feria, R. 2006 Nonlinear instabilities in a vertical pipe flow discharging from a cylindrical container. Phys. fluids 18, 024101.CrossRefGoogle Scholar
Weidman, P. 1994 Stability criteria for two immiscible fluids rotating in zero gravity. Mécanique Appliquée 39, 481–496.Google Scholar
Weidman, P. D., Goto, M. & Fridberg, A. 1997 On the stability of inviscid, rotating immiscible fluids in zero gravity. Z. Agnew Math. Phys. 48, 921–950.CrossRefGoogle Scholar
Wolfram, S. 1999 Mathematica 4.0. Wolfram Research, Inc.Google Scholar
Yih, C. S. 1960 Instability of a rotating liquid film with a free surface. Proc. R. Soc. Lond. A 258, 63–89.Google Scholar