Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-18T13:41:58.525Z Has data issue: false hasContentIssue false
  • English
  • Français

The influence of viscosity on the frozen wave instability: theory and experiment

Published online by Cambridge University Press:  25 July 2007

EMMA TALIB ,
SHREYAS V. JALIKOP  and
ANNE JUEL
EMMA TALIB
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, The University of Manchester, Manchester M13 9PL, UK, anne.juel@manchester.ac.uk.
SHREYAS V. JALIKOP
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, The University of Manchester, Manchester M13 9PL, UK, anne.juel@manchester.ac.uk.
ANNE JUEL
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, The University of Manchester, Manchester M13 9PL, UK, anne.juel@manchester.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the results of an experimental and linear stability study of the influence of viscosity on the frozen wave (FW) instability, which arises when a vessel containing stably stratified layers of immiscible liquids is oscillated horizontally. Our linear stability model consists of two superposed fluid layers of arbitrary viscosities and infinite lateral extent, subject to horizontal oscillation. The effect of the endwalls of the experimental vessel is simulated by enforcing the conservation of horizontal volume flux, so that the base flow consists of counterflowing layers.

We perform experiments with four pairs of fluids, keeping the viscosity of the lower layer (ν1) constant, and increasing the viscosity of the upper layer (ν2), so that 1.02 × 102 ≤ N1 = ν2/ν1 ≤ 1.21 × 104. We find excellent quantitative agreement between theory and experiment despite the simple model geometry, for both the critical onset parameter and wavenumber of the FW. We show that the model of lyubimov:1987 (Fluid Dyn. vol. 86, 1987, p. 849), which is valid in the limit of inviscid fluids, consistently underestimates the instability threshold for fluids of equal viscosity, but generally overestimates the threshold for fluids of unequal viscosity. We extend the experimental parameter range numerically to viscosity contrasts 1 ≤ N1 ≤ 6 × 104 and identify four regions of N1 where qualitatively different dynamics occur, which are reflected in the non-monotonic dependence of the most unstable wavenumber and the critical amplitude on N1. In particular, we find that increasing the viscosity contrast between the layers leads to destabilization over a wide range of N1, 10 ≤ N1 ≤ 8 × 103. The intricate dependence of the instability on viscosity contrast is due to considerable changes in the time-averaged perturbation vorticity distribution near the interface.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 584 , 10 August 2007 , pp. 45 - 68
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

Barnea, D. & Taitel, Y. 1993 Kelvin–Helmholtz stability criteria for stratified flow: viscous versus non-viscous (inviscid) approaches. Intl J. Multiphase Flow 19, 639–649.CrossRefGoogle Scholar
Beckwith, T. G., Marangoni, R. D. & Lienhard, V. J. H. 1993 Mechanical Measurements, 5th edn. Prentice Hall.Google Scholar
Berman, A. S., Bradford, J. & Lundgren, T. S. 1978 Two-fluid spin-up in a centrifuge. J. Fluid Mech. 84, 411–431.CrossRefGoogle Scholar
Burgess, J. M., Juel, A., McCormick, W. D., Swift, J. B. & Swinney, H. L. 2001 Suppression of dripping from a ceiling. Phys. Rev. Lett. 86, 1203–1206.CrossRefGoogle ScholarPubMed
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fermigier, M., Limat, L., Westfreid, J. E., Boudinet, P. & Quilliet, C. 1992 Two-dimensional patterns in Rayleigh–Taylor instability of a thin layer. J. Fluid Mech. 236, 349–383.CrossRefGoogle Scholar
González-Viñas, W. & Salán, G. 1994 Surface waves periodically excited in a CO2 tube. Europhys. Lett. 26, 665–670.CrossRefGoogle Scholar
Hesla, T. I., Pranckh, F. R. & Preziosi, L. 1986 Squire's theorem for two stratified fluids. Phys. Fluids 29, 2808–2811.CrossRefGoogle Scholar
Hinch, E. J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463–465.CrossRefGoogle Scholar
Hogan, J. M. & Ayyaswamy, P. S. 1985 Linear stability of a viscous-inviscid interface. Phys. Fluids 28, 2709–2715.CrossRefGoogle Scholar
Hooper, A. P. & Boyd, W. G. C. 1983 Shear-flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507–528.CrossRefGoogle Scholar
Ivanova, A. A., Kozlov, V. G. & Evesque, P. 2001 Interface dynamics of immiscible fluids under horizontal vibration. Fluid Dyn. 36, 362–368.CrossRefGoogle Scholar
Kelly, R. E. 1965 The stability of an unsteady Kelvin–Helmholtz flow. J. Fluid Mech. 22, 547–560.CrossRefGoogle Scholar
Khenner, M. V., Lyubimov, D. V., Belozerova, T. S. & Roux, B. 1999 Stability of plane-parallel vibrational flow in a two-layer system. Eur. J. Mech. B/Fluids 18, 1085–1101.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Lanczos, C. 1956 Applied Analysis. Prentice Hall.Google Scholar
Legendre, M., Petitjeans, P. & Kurowski, P. 2003 Instabilité à l'interface entre fluides miscibles par forçage oscillant horizontal. C. R. Méc. 331, 617–622.CrossRefGoogle Scholar
Lindsay, K. A. 1984 The Kelvin–Helmholtz instability for a viscous interface. Acta Mech. 52, 51–61.CrossRefGoogle Scholar
Lyubimov, D. V. & Cherepanov, A. 1987 Development of a steady relief at the interface of fluids in a vibrational field. Fluid Dyn. 86, 849–854.Google Scholar
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley.Google Scholar
Or, A. C. 1997 Finite wavelength instability in a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 335, 213–232.CrossRefGoogle Scholar
Roberts, R. M., Ye, Y., Demekhin, E. A. & Chang, H. C. 2000 Wave dynamics in two-layer Couette flow. Chem. Engng Sci. 55, 345–362.CrossRefGoogle Scholar
Shyh, C. K. & Munson, B. R. 1986 Interfacial instability of an oscillating shear layer. J. Fluids Engng 108, 89–92.CrossRefGoogle Scholar
Talib, E. & Juel, A. 2006 Instability of a viscous interface under horizontal oscillation. Phys. Fluids. (submitted).CrossRefGoogle Scholar
Wolf, G. H. 1969 The dynamic stabilization of the Rayleigh–Taylor instability and the corresponding dynamic equilibrium. Z. Physik 227, 291–300.CrossRefGoogle Scholar
Wolf, G. H. 1970 Dynamic stabilization of the interchange instability of a liquid-gas interface. Phys. Rev. Lett. 24, 444–446.CrossRefGoogle Scholar
Wunenburger, R., Evesque, P., Chabot, C., Garrabos, Y., Fauve, S. & Beysens, D. 1999 Frozen wave instability by high frequency horizontal vibrations on a CO2 liquid-gas interface near the critical point. Phys. Rev. E 59, 5440–5445.Google ScholarPubMed
Yoshikawa, H. 2006 Instabilités des interfaces sous oscillations. PhD thesis, Université Paris 6.Google Scholar