Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-08T20:05:19.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Crumpled water bells

Published online by Cambridge University Press:  24 January 2012

H. Lhuissier  and
E. Villermaux
H. Lhuissier
Affiliation:
Aix-Marseille Université, IRPHE, 13384 Marseille CEDEX 13, France
E. Villermaux*
Affiliation:
Aix-Marseille Université, IRPHE, 13384 Marseille CEDEX 13, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Stationary axi-symmetrical pressurized bells formed by the impact of a liquid jet on a solid disc (so-called Savart water bells) can exhibit uncommon sharp and pointed shapes. They are characterized by two successive inflections of the two-dimensional bell generator profile, corresponding to partially concave bells. We show that this shape is incompatible with the usual assumption that the detail of the flow across the liquid sheet constitutive of the bell is unimportant. We consider the equilibrium of a curved liquid sheet of finite thickness sustaining a pressure difference between both sides, and show that several curvatures of the interface may be a solution under given flow conditions. The inflection of the bell profile is then explained in terms of a spontaneous transition from a ‘negative’ to a ‘positive’ curvature which conserves mass flow, linear and angular momenta. That inflection is also a transition from a super to a subcritical flow (with respect to capillary waves), having the status of a capillary hydraulic jump on a freely suspended sheet, a novel object in fluid mechanics. The azimuthal wrinkles forming at the jump result from the inertial destabilization of the sheet due to the centripetal acceleration fluid particles experience as they flow along the highly curved bell profile in the vicinity of the fold. This finding also explains the singular shape at the edge of freely flapping sheets.

JFM classification

Mathematical Foundations: General fluid mechanics Interfacial Flows (free surface): Interfacial Flows (free surface) Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 693 , 25 February 2012 , pp. 508 - 540
Copyright
Copyright © Cambridge University Press 2012

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.)

Footnotes

‡

Present address: Also at Institut Universitaire de France.

References

1. Bohr, T., Dimon, P. & Putkaradze, V. 1993 Shallow-water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635–648.CrossRefGoogle Scholar
2. Bond, W. N. 1935 The surface tension of a moving water sheet. Proc. Phys. Soc. 47, 549–558.Google Scholar
3. Bond, W. N. & Puls, H. O. 1937 The change of surface tension with time. Phil. Mag. 24, 864–888.CrossRefGoogle Scholar
4. Boussinesq, J. 1869a Théories des expériences de Savart, sur la forme que prend une veine liquide après s’être choquée contre un plan circulaire I. C. R. Acad. Sci. Paris 69, 45–48.Google Scholar
5. Boussinesq, J. 1869b Théories des expériences de Savart, sur la forme que prend une veine liquide après s’être choquée contre un plan circulaire II. C. R. Acad. Sci. Paris 69, 128–131.Google Scholar
6. Bremond, N., Clanet, C. & Villermaux, E. 2007 Atomization of undulating liquid sheets. J. Fluid Mech. 585, 421–456.Google Scholar
7. Bremond, N. & Villermaux, E. 2005 Bursting thin liquid films. J. Fluid Mech. 524, 121–130.CrossRefGoogle Scholar
8. Brunet, P., Clanet, C. & Limat, L. 2004 Transonic liquid bells. Phys. Fluids 16 (7), 2668–2678.Google Scholar
9. Buckingham, R. & Bush, J. W. 2001 Fluid polygons. Phys. Fluids 13, S10.CrossRefGoogle Scholar
10. Bush, J. W. & Aristoff, J. 2003 The influence of surface tension on the circular hydraulic jump. J. Fluid Mech. 489, 229–238.Google Scholar
11. Bush, J. W., Aristoff, J. & Hosoi, A. 2006 An experimental investigation of the stability of the circular hydraulic jump. J. Fluid Mech. 558, 33–52.CrossRefGoogle Scholar
12. Clanet, C. 2001 Dynamics and stability of water bells. J. Fluid Mech. 430, 111–147.Google Scholar
13. Clanet, C. 2007 Waterbells and liquid sheets. Annu. Rev. Fluid Mech. 39, 469–496.CrossRefGoogle Scholar
14. Darrigol, O. 2005 Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press.CrossRefGoogle Scholar
15. Ellegaard, C., Hansen, A. E., Haaning, A., Hansen, K., Marcussen, A., Bohr, T., Hannsen, J. L. & Watanabe, S. 1998 Creating corners in kitchen sinks. Nature 392, 767–768.CrossRefGoogle Scholar
16. Hopwood, F. L. 1952 Water bells. Proc. Phys. Soc. B 65, 2–5.CrossRefGoogle Scholar
17. Huang, J. C. P. 1970 The break-up of axisymmetric liquid sheets. J. Fluid Mech. 43, 305–319.CrossRefGoogle Scholar
18. Keller, J. B. & Kolodner, I. 1954 Instability of liquid surfaces and the formation of drops. J. Appl. Phys. 25 (7), 918–921.CrossRefGoogle Scholar
19. Lance, G. N. & Perry, R. L. 1953 Water bells. Proc. Phys. Soc. B 66, 1067–1072.CrossRefGoogle Scholar
20. Laplace, P. S. 1805 Sur l’action capillaire, supplément au livre X du Traité Mécanique Céleste, t. IV. Courcier.Google Scholar
21. Lefebvre, A. H. 1989 Atomization and Sprays. Hemisphere.Google Scholar
22. Lhuissier, H. & Villermaux, E. 2009 Soap films burst like flapping flags. Phys. Rev. Lett. 103, 054501–(4).Google Scholar
23. Lord Rayleigh, J. W. 1914 On the theory of long waves and bore. Proc. R. Soc. Lond. A 90, 324–328.Google Scholar
24. Newton, I. 1687 Philosophiae Naturalis Principia Mathematica. London.Google Scholar
25. Parlanges, J.-Y. 1967 A theory of water-bells. J. Fluid Mech. 29, 361–372.Google Scholar
26. Puls, H. O. 1936 The surface tension of a moving mercury sheet. Phil. Mag. 22.CrossRefGoogle Scholar
27. Rowlinson, J. S. 2002 Cohesion. Cambridge University Press.Google Scholar
28. Savart, F. 1833a Mémoire sur le choc d’une veine liquide lancée contre un plan circulaire. Ann. Chim. 54, 56–87.Google Scholar
29. Savart, F. 1833b Suite du mémoire sur le choc d’une veine liquide lancée contre un plan circulaire. Ann. Chim. 54, 113–145.Google Scholar
30. Squire, H. B. 1953 Investigation of the stability of a moving liquid film. Br. J. Appl. Phys. 4, 167–169.Google Scholar
31. Taylor, G. I. 1959a The dynamics of thin sheets of fluid I. Water bells. Proc. R. Soc. Lond. A 253, 289–295.Google Scholar
32. Taylor, G. I. 1959b The dynamics of thin sheets of fluid III. Desintegration of fluid sheets. Proc. R. Soc. Lond. A 253, 313–321.Google Scholar
33. Villermaux, E. & Clanet, C. 2002 Life of a flapping liquid sheet. J. Fluid Mech. 462, 342–363.CrossRefGoogle Scholar
34. Watson, E. J. 1964 The radial spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20, 481–499.CrossRefGoogle Scholar