Hostname: page-component-54dcc4c588-scsgl Total loading time: 0 Render date: 2025-09-12T01:38:45.400Z Has data issue: false hasContentIssue false
  • English
  • Français

Capillary retraction of the edge of a stretched viscous sheet

Published online by Cambridge University Press:  03 April 2018

James P. Munro Open the ORCID record for James P. Munro [Opens in a new window]  and
John R. Lister
James P. Munro*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, UK
John R. Lister
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, CMS, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: jpm82@cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Surface tension causes the edge of a fluid sheet to retract. If the sheet is also stretched along its edge then the flow and the rate of retraction are modified. A universal similarity solution for the Stokes flow in a stretched edge shows that the scaled shape of the edge is independent of the stretching rate, and that it decays exponentially to its far-field thickness. This solution justifies the use of a stress boundary condition in long-wavelength models of stretched viscous sheets, and gives the detailed shape of the edge of such a sheet, resolving the position of the sheet edge to the order of the thickness.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Low-Reynolds-number flows

Information

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , R1
Copyright
© 2018 Cambridge University Press 

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.)

Article purchase

Temporarily unavailable

References

Billingham, J. 2005 The initial surface tension-driven flow of a wedge of viscous fluid. SIAM J. Appl. Maths 66, 510–532.10.1137/050622419Google Scholar
Bird, J. C., de Ruiter, R., Courbin, L. & Stone, H. A. 2010 Daughter bubble cascades produced by folding of ruptured thin films. Nature 465, 759–762.10.1038/nature09069Google Scholar
Brenner, M. P., Eggers, J., Joseph, K., Nagel, S. R. & Shi, X. D. 1997 Breakdown of scaling in droplet fission at high Reynolds number. Phys. Fluids 9, 1573–1590.10.1063/1.869279Google Scholar
Brenner, M. P. & Gueyffier, D. 1999 On the bursting of viscous films. Phys. Fluids 11, 737–739.10.1063/1.869942Google Scholar
Culick, F. E. C. 1960 Comments on a ruptured soap film. J. Appl. Phys. 31, 1128–1129.10.1063/1.1735765Google Scholar
Cummings, L. J. & Howell, P. D. 1999 On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity. J. Fluid Mech. 389, 361–389.10.1017/S0022112099005030Google Scholar
Dewynne, J. N., Ockendon, J. R. & Wilmott, P. 1992 A systematic derivation of the leading-order equations for extensional flows in slender geometries. J. Fluid Mech. 244, 323–338.10.1017/S0022112092003094Google Scholar
Eggers, J. 2014 Post-breakup solutions of Navier–Stokes and Stokes threads. Phys. Fluids 26, 072104.10.1063/1.4890203Google Scholar
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–221.10.1017/S0022112094000480Google Scholar
Fadle, J. 1940 Die Selbstspannungs-Eigenwertfunktionen der quadratischen Scheibe. Ing.-Arch. 11, 125–148.10.1007/BF02084699Google Scholar
Keller, J. B. 1983 Breaking of liquid films and threads. Phys. Fluids 26, 3451–3453.10.1063/1.864126Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.10.1017/S0022112064000015Google Scholar
Munro, J. P., Anthony, C. R., Basaran, O. A. & Lister, J. R. 2015 Thin-sheet flow between coalescing bubbles. J. Fluid Mech. 773, R3.10.1017/jfm.2015.253Google Scholar
O’Kiely, D., Breward, C. J. W., Griffiths, I. M., Howell, P. D. & Lange, U. 2015 Edge behaviour in the glass sheet redraw process. J. Fluid Mech. 785, 248–269.10.1017/jfm.2015.629Google Scholar
Papkovich, P. F. 1940 On one form of solution of the plane problem of the theory of elasticity for a rectangular strip. Dokl. Akad. Nauk SSSR 27, 335–339.Google Scholar
Paulsen, J. D., Carmigniani, R., Burton, A., Kannanand, J. C. & Nagel, S. R. 2014 Coalescence of bubbles and drops in an outer fluid. Nat. Commun. 5, 3182.10.1038/ncomms4182Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.10.1017/CBO9780511624124Google Scholar
Savva, N. & Bush, J. W. M. 2009 Viscous sheet retraction. J. Fluid Mech. 626, 211–240.10.1017/S0022112009005795Google Scholar
Scheid, B., Quiligotti, S., Tran, B. & Stone, H. A. 2009 Lateral shaping and stability of a stretching viscous sheet. Eur. Phys. J. B 68, 487–494.10.1140/epjb/e2009-00111-xGoogle Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc. R. Soc. Lond. A 253, 313–321.Google Scholar