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Moving contact lines and rivulet instabilities. Part 1. The static rivulet

Published online by Cambridge University Press:  19 April 2006

Stephen H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60201

Abstract

A rivulet is a narrow stream of liquid located on a solid surface and sharing a curved interface with the surrounding gas. Capillary instabilities are investigated by a linearized stability theory. The formulation is for small, static rivulets whose contact (common or three-phase) lines (i) are fixed, (ii) move but have fixed contact angles or (iii) move but have contact angles smooth functions of contact-line speeds. The linearized stability equations are converted to a disturbance kinetic-energy balance showing that the disturbance response exactly satisfies a damped linear harmonic-oscillator equation. The ‘damping coefficient’ contains the bulk viscous dissipation, the effect of slip along the solid and all dynamic effects that arise in contact-line condition (iii). The ‘spring constant’, whose sign determines stability or instability in the system, incorporates the interfacial area changes and is identical in cases (ii) and (iii). Thus, for small disturbances changes in contact angle with contact-line speed constitute a purely dissipative process. All the above results are independent of slip model at the liquid–solid interface as long as a certain integral inequality holds. Finally, sufficient conditions for stability are obtained in all cases (i), (ii) and (iii).

Type
Research Article
Copyright
© 1980 Cambridge University Press

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References

Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics. Wiley-Interscience.
Culkin, J. B. 1979 Preliminary experiments on rivulet instabilities. M.S. thesis, Johns Hopkins University, Baltimore, Md.
Dussan, V. E. B. 1975 Arch. Rat. Mech. Anal. 57, 363.
Dussan, V. E. B. 1976 J. Fluid Mech. 77, 665.
Dussan, V. E. B. 1979a Ann. Rev. Fluid Mech. 11, 371.
Dussan, V. E. B. 1979b unpublished.
Dussan, V. E. B. & Davis, S. H. 1974 J. Fluid Mech. 65, 71.
Gibbs, J. W. 1948 Collected Works, vol. 1, p. 55.
Hocking, L. M. 1977 J. Fluid Mech. 79, 209.
Huh, C. & Mason, S. G. 1977 J. Fluid Mech. 81, 401.
Kern, J. 1969 Verfahrenstechnik 3, 425.
Kern, J. 1971 Verfahrenstechnik 5, 289.
Michael, D. H. & Williams, P. G. 1977 Proc. Roy. Soc. A 354, 127.
Rayleigh, Lord 1879 Proc. Roy. Soc. A 10, 4.
Towell, G. D. & Rothfeld, L. B. 1966 A.I.Ch.E. J. 12, 972.