Abstract
Asymptotic solutions for large and small surface tension are developed for the profile of a symmetric sessile drop. The problem for large surface tension (i.e., small Bond number) is a regular perturbation problem, where the solution may be written as a uniformly valid asymptotic expansion. The problem for small surface tension (i.e., large Bond number) is a singular perturbation problem with boundary-layer behaviour in the edge region. The solution is a matched asymptotic expansion, where some care is to be taken for the matching. The respective ranges of validity are established by comparing the asymptotic results with solutions obtained by numerical integration of the full equations.
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References
F. Bashforth and J. C. Adams, An attempt to test the theories of capillary attraction, Cambridge University Press, Cambridge (1883).
J. F. Padday, The profiles of axially symmetric menisci, Philosophical Transactions of the Royal Society of London A269 (1971) 265–293.
J. N. Butler and B. H. Bloom, A curve fitting method for calculating interfacial tension from the shape of a sessile drop, Surface Science 4 (1966) 1–17.
H.K. Kuiken, personal communication; unpublished reports.
A. K. Chesters, An analytical solution for the profile and volume of a small drop or bubble symmetrical about the vertical axis, Journal of Fluid Mechanics 81 (1977) 609–624.
M. E. R. Shanahan, Profile and contact angle of small sessile drops, Journal of the Chemical Society, Faraday Transactions 80 (1984) 37–45.
P. G. Smith and T. G. M. van de Ven, Profiles of slightly deformed axisymmetric drops, Journal of Colloid and Interface Science 97 (1984) 1–8.
P. Concus, Static menisci in a vertical right circular cylinder, Journal of Fluid Mechanics 34 (1968) 481–495.
W. Eckhaus, Asymptotic analysis of singular perturbations, North-Holland Publishing Company, Amsterdam (1979).
M. van Dyke, Perturbation methods in fluid mechanics, Parabolic Press, Stanford (1975).
A. H. Nayfeh, Perturbation methods, John Wiley & Sons, New York (1973).
M. B. Lesser and D. G. Crighton, Physical acoustics and the method of matched asymptotic expansions, Physical Acoustics, Vol. XI edited by W. P. Mason and R. M. N. Thurston, Academic Press, New York (1975).
R. Finn, Equilibrium capillary surfaces, Springer-Verlag, New York (1986).
D. Siegel, The behavior of a capillary surface for small Bond number, Variational methods for free surface interfaces, pp. 109–113, edited by P. Concus and R. Finn, Springer-Verlag, New York (1987).
M. Abramowitz and I.A. Stegun (eds.) Handbook of mathematical functions, National Bureau of Standards (1964).
E. Fehlberg, Low-order classical Runge-Kutta formulas with stepsize control, NASA TR R-315.
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Rienstra, S.W. The shape of a sessile drop for small and large surface tension. J Eng Math 24, 193–202 (1990). https://doi.org/10.1007/BF00058465
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DOI: https://doi.org/10.1007/BF00058465