Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-15T05:27:44.369Z Has data issue: false hasContentIssue false
  • English
  • Français

Models for thin viscous sheets

Published online by Cambridge University Press:  26 September 2008

P. D. Howell
P. D. Howell
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford, UK
Get accessRights & Permissions [Opens in a new window]

Abstract

Leading-order equations governing the dynamics of a two-dimensional thin viscous sheet are derived. The inclusion of inertia effects is found to result in an ill-posed model when the sheet is compressed, and the resulting paradox is resolved by rescaling the equations over new length-and timescales which depend on the Reynolds number of the flow and the aspect ratio of the sheet. Physically this implies a dominant lengthscale for transverse displacements during viscous buckling. The theory is generalized to give new models for fully three-dimensional sheets.

Information

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 4 , August 1996, pp. 321 - 343
Copyright
Copyright © Cambridge University Press 1996

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

Benjamin, T. B. & Mullin, T. (1988) Buckling instabilities in layers of viscous liquid subjected to shearing. J. Fluid Mech. 195, 523–540.CrossRefGoogle Scholar
Buckmaster, J. D., Nachman, A. & Ting, L. (1975) The buckling and stretching of a viscida. J. Fluid Mech. 69, 1–20.CrossRefGoogle Scholar
Dewynne, J. N., Ockendon, J. R. & Wilmott, P. (1989) On a mathematical model for fiber tapering. SIAM J. Appl. Math. 49, 983–990.Google 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.CrossRefGoogle Scholar
Dewynne, J. N., Howell, P. D. & Wilmott, P. (1994) Slender viscous fibres with inertia and gravity. Quart. J. Mech. Appl. Math. 47, 541–555.CrossRefGoogle Scholar
Eggers, J. (1993) Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458–3460.CrossRefGoogle ScholarPubMed
van de Fliert, B. W., Howell, P. D. & Ockendon, J. R. (1995) Pressure-driven flow of a thin viscous sheet. J. Fluid Mech. 292, 359–376.CrossRefGoogle Scholar
Howell, P. D. (1994) Extensional thin layer flows. D.Phil, thesis, University of Oxford.Google Scholar
Ida, M. P. & Miksis, M. J. (1995) Dynamics of a lamella in a capillary tube. SIAM J. Appl. Math. 55, 23–57.CrossRefGoogle Scholar
Kreyszig, E. (1959) Differential Geometry. University of Toronto Press (reprinted Dover, 1991).CrossRefGoogle Scholar
Love, A. E. H. (1927) A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press.Google Scholar
Myers, T. G. (1995) Thin films with high surface tension. OCIAM report, University of Oxford.Google Scholar
Pearson, J. R. A. & Petrie, C. J. S. (1970) The flow of a tubular film. Part 1. Formal mathematical representation. J. Fluid Mech. 40, 1–19.CrossRefGoogle Scholar
Pearson, J. R. A. & Petrie, C. J. S. (1970) The flow of a tubular film. Part 2. Interpretation of the model and discussion of solutions. J. Fluid Mech. 42, 609–625.CrossRefGoogle Scholar
Schultz, W. W. & Davis, S. H. (1982) One-dimensional liquid fibers. J. Rheology 26, 331–345.CrossRefGoogle Scholar
Ting, L. & Keller, J. B. (1990) Slender jets and sheets with surface tension. SIAM J. Appl. Math. 50, 1533–1546.CrossRefGoogle Scholar
Wilmott, P. (1989) The stretching of a thin viscous inclusion and the drawing of glass sheets. Phys. Fluids A 1, 1098–1103.CrossRefGoogle Scholar
Yarin, A. L., Gospodinov, P. & Roussinov, V. I. (1994) Stability loss and sensitivity in hollow-fiber drawing. Phys. Fluids 6, 1454–1463.CrossRefGoogle Scholar
Yarin, A. L. & Tchavdarov, B. (1995) Onset of folding in plane liquid films. J. Fluid Mech. 307, 85–99.CrossRefGoogle Scholar