Theory of drop formation: Physics of Fluids: Vol 7, No 5

Abstract

The motion of an axisymmetric column of Navier–Stokes fluid with a free surface is considered. Due to surface tension, the thickness of the fluid neck goes to zero in finite time. After the singularity, the fluid consists of two halves, which constitute a unique continuation of the Navier–Stokes equation through the singular point. The asymptotic solutions of the Navier–Stokes equation are calculated, both before and after the singularity. The solutions have scaling form, characterized by universal exponents as well as universal scaling functions, which are computed without adjustable parameters.

REFERENCES

1. P. S. de Laplace, Méchanique Celeste, Supplement au X Libre (Courier, Paris, 1805).
2. F. Savart, “Mémoire sur la Constitution des Veines liquid lancées par des orifices circulaires en mince paroi,” Ann. Chim. 53, 337 (1833).
3. G. Hagen, “Über die Auflösung flüssiger Cylinder in Tropfen,” Verhand-lungen Preuss. Akad. Wissenschaften Berlin, 281 (1849).
4. Lord Rayleigh, “On the instability of jets,” Proc. London Math. Soc. 4, 10 (1878). https://doi.org/PLMTAL,
5. C. Weber, “Zum Zerfall eines Flüssigkeitsstrahles,” Z. Angew. Math. Mech. 11, 136 (1931). https://doi.org/ZAMMAX, Crossref
6. S. Grossmann and A. Müller, “Instabilities and decay rates of charged viscous liquid jets,” Z. Phys. B 57, 161 (1984). https://doi.org/ZPCMDN, Crossref
7. K. C. Chaudhary and L. G. Redekopp, “The nonlinear instability of a liquid jet, Part 1: Theory,” J. Fluid Mech. 96, 257 (1980). https://doi.org/JFLSA7, Crossref
8. J. Eggers, “Universal pinching of 3D axisymmetric free-surface flow,” Phys. Rev. Lett. 71, 3458 (1993). https://doi.org/PRLTAO, Crossref, CAS
9. R. E. Goldstein, A. I. Pesci, and M. J. Shelley, “Topology transitions and singularities in viscous flows,” Phys. Rev. Lett. 70, 3043 (1993). https://doi.org/PRLTAO, Crossref, CAS
10. M. Tjahjadi, H. A. Stone, and J. M. Ottino, “Satellite and subsatellite formation in capillary breakup,” J. Fluid Mech. 243, 297 (1992). https://doi.org/JFLSA7, Crossref, CAS
11. B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski, and G. Zanetti “Modelling merging and fragmentation in multiphase flows with SURFER,” J. Comput. Phys. 113, 134 (1994). https://doi.org/JCTPAH, Crossref
12. P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley, and S. M. Zhou, “Droplet breakup in a model of the Hele-Shaw cell,” Phys. Rev. E 47, 4169 (1993). https://doi.org/PLEEE8, Crossref, CAS
13. S. E. Bechtel, M. G. Forest, and K. J. Lin, “Closure to all orders in 1-D models for slender viscoelastic free jets: An integrated theory for axisymmetric, torsionless flows,” Stability Appl. Anal. Continuous Media 2, 59 (1992).
14. J. Eggers and T. F. Dupont, “Drop formation in a one-dimensional approximation of the Navier-Stokes equation,” J. Fluid Mech. 262, 205 (1994). https://doi.org/JFLSA7, Crossref, CAS
15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1984).
16. M. G. Forest (private communication, 1994).
17. D. B. Bogy, “Drop formation in a circular liquid jet,” Annu. Rev. Fluid Mech. 11, 207 (1979). https://doi.org/ARVFA3, Crossref
18. J. B. Keller and M. J. Miksis, “Surface tension driven flows,” SIAM J. Appl. Math. 43, 268 (1983). https://doi.org/SMJMAP, Crossref
19. A. L. Bertozzi, M. P. Brenner, T. F. Dupont, and L. P. Kadanoff, “Singularities and similarities in interface flow,” in Trends and Perspectives in Applied Mathematics, Applied Mathematics Series, edited by L. Sirovich (Springer-Verlag, New York, 1994), Vol. 100.
20. I am grateful to Jens Hoppe for pointing this out to me.
21. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
22. R. A. London and B. P. Flannery, “Hydrodynamics of x-ray induced stellar winds,” Astrophys. J. 258, 260 (1982). https://doi.org/ASJOAB, Crossref
23. This generalization is due to Peter Constantin.
24. E. Marschall, “Zur Strömungsmechanik während der Tropfenbildung in flüssigen Zweiphasen,” Verh. Dtsch. Ing. Forschungs. 632, 13 (1985).
25. E. F. Goedde and M. C. Yuen, “Experiments on liquid jet instability,” J. Fluid Mech. 40, 495 (1970). https://doi.org/JFLSA7, Crossref
26. D. H. Peregrine, G. Shoker, and A. Symon, “The bifurcation of liquid bridges,” J. Fluid Mech. 212, 25 (1990). https://doi.org/JFLSA7, Crossref, CAS
27. X. D. Shi, M. P. Brenner, and S. R. Nagel, “A cascade structure in a drop falling from a faucet,” Science 265, 157 (1994). Crossref, CAS
28. T. A. Kowalewski, “Experiments on the late stages of drop formation ”(unpublished).
29. E. Becker, W. J. Hiller, and T. A. Kowalewski, “Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets,” J. Fluid Mech. 231, 189 (1991). https://doi.org/JFLSA7, Crossref, CAS
30. F. Shokoohi and H. G. Elrod, “Algorithms for Eulerian treatment of jet breakup induced by surface tension,” J. Comput. Phys. 89, 483 (1990). https://doi.org/JCTPAH, Crossref
31. D. H. Michael and P. G. Williams, “The equilibrium and stability of axisymmetric pendant drops,” Proc. R. Soc. London Ser. A 351, 117 (1976). https://doi.org/PRLAAZ, Crossref
32. M. P. Brenner, X. D. Shi, and S. R. Nagel, “Iterated instabilities during droplet fission,” Phys. Rev. Lett. 73, 3391 (1994). https://doi.org/PRLTAO, Crossref, CAS
© 1995 American Institute of Physics.

Article Metrics

Views

Citations

Crossref 93