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Plane Stokes flow driven by capillarity on a free surface

Published online by Cambridge University Press:  26 April 2006

Robert W. Hopper
Robert W. Hopper
Affiliation:
Chemistry and Materials Science Department, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
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Abstract

The free creeping viscous incompressible plane flow of a finite region, bounded by a simple smooth closed curve and driven solely by surface tension, is analysed. The shape evolution is described in terms of a time-dependent mapping function z = Ω (ζ, t) of the unit circle, conformal on |ζ| [les ] 1. An equation giving the time evolution of the Ω (ζ, t) is derived. In practice, it has been necessary to guess a parametric form, i.e. Ω (ζ, t) = Ω[ζ; a1(t), a2(t), …], whose validity must be verified using the shape-evolution equation. Polynomial and proper rational mappings with no repeated factors are apparently always valid in principle. Solutions are given for (i) regions bounded initially by a regular epitrochoid, (ii) the limiting case of a half-plane bounded by a trochoid, and (iii) a class of rosettes whose mapping is rational. The two-lobed rosette gives the exact solution of the coalescence of equal cylinders. All these mappings involve limiting initial shapes having inward-pointing cusps. Useful parameterizations providing regions whose limiting shapes possess corners or outward-pointing cusps have not been found.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 213 , April 1990 , pp. 349 - 375
Copyright
© 1990 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. eds. 1965 Handbook of Mathematical Functions. Dover Publications, New York.
Batchelor, G. K. 1967 An Introduction to Fluid Mechanics, 4.8. Cambridge University Press.
Bolz, R. E. & Tuve, G. L. 1973 CRC Handbook of Tables for Applied Engineering Science, 2nd edn. table 10.51B. CRC Press. Boca Raton, Florida.
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.
Elliot, C. M. & Ockendon, J. R. 1982 Weak and Variational Methods for Moving Boundary Problems. Pitman.
Galin, L. A. 1945 Unsteady filtration with a free surface. C. R. (Dokl.) Acad. Sci. URSS 47, 246–249.Google Scholar
Garabedian, P. R. 1966 Free boundary flows of a viscous liquid. Commun. Pure Appl. Maths 19, 421–434.Google Scholar
Gradshteyn, I. S. & Rhyzhik, I. M. 1980 Table of Integrals, Series and Products (transl. A. Jeffrey), corrected and enlarged edition. Academic.
Henrici, P. 1974 Applied and Computational Complex Analysis, vol. 1. John Wiley.
Henrici, P. 1986 Applied and Computational Complex Analysis, vol. 3. John Wiley.
Hopper, R. W. 1984 Coalescence of two equal cylinders – Exact results for creeping viscous plane flow driven by capillarity. J. Am. Ceram. Soc. (Comm.) 67, C262–264. See also errata, ibid. 68, C138 (1985), and in footnote on p. 367 in this article.Google Scholar
Hopper, R. W. 1990a Articles on coalescence of unequal cylinders and on unbounded regions. To be submitted to J. Fluid Mech.Google Scholar
Hopper, R. W. 1990b Articles on cylinder coalescence and on crack healing. To be submitted to J. Am. Ceram. Soc.Google Scholar
Howison, S. D. 1986a Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Maths 46, 20–26.Google Scholar
Howison, S. D. 1986b Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439–453.Google Scholar
Ince, E. L. 1926 Ordinary Differential Equations, 3.3. Dover (1956).
Keller, J. B. & Miksis, M. J. 1986 Surface tension driven flows. SIAM J. Appl. Maths 43, 268–277.Google Scholar
Lamb, H. 1932 Hydrodynamics (6th edn, 1945).
Muskhelishvili, N. I. 1953a Some Basic Problems in the Mathematical Theory of Elasticity (transl. J. R. M. Radok). Groningen: P. Noordhoff.
Muskhelishvili, N. I. 1953b Singular Integral Equations (transl. J. R. M. Radok). Leyden: Noordhoff International Publishing.
Richardson, S. 1968 Two-dimensional bubbles in slow viscous flow. J. Fluid Mech. 33, 476–493.Google Scholar
Richardson, S. 1972 Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618.Google Scholar
Richardson, S. 1973 Two-dimensional bubbles in slow viscous flow. Part 2. J. Fluid Mech. 58, 115–127.Google Scholar
Richardson, S. 1982 Hele-Shaw flows with time-dependent free boundaries in infinite and semiinfinite strips. Q. J. Mech. Appl. Maths 35, 531–548.Google Scholar
Rudin, W. 1966 Real and Complex Analysis. McGraw-Hill.
Shraiman, B. & Bensimon, D. 1984 Singularities in nonlocal interface dynamics.. Phys. Rev. A 30, 2840–2842.Google Scholar
Sills, L. G. Jr 1986 Coalescence of glassy fibers. BS thesis in Ceramic Engineering, New York State College of Ceramics at Alfred University (L. D. Pye, advisor).
Sokolnikov, I. S. 1956 Mathematical Theory of Elasticity, 2nd edn. McGraw-Hill.
Titchmarsh, E. C. 1939 The Theory of Functions, 2nd edn. Oxford University Press.
Trefethen, L. N. Ed. 1986 Numerical Conformal Mapping. Special issue of J. Comp. Appl. Maths (vol. 14).