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Models for thin viscous sheets

  • P. D. Howell (a1)
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.

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Benjamin T. B. & Mullin T. (1988) Buckling instabilities in layers of viscous liquid subjected to shearing. J. Fluid Mech. 195, 523540.
Buckmaster J. D., Nachman A. & Ting L. (1975) The buckling and stretching of a viscida. J. Fluid Mech. 69, 120.
Dewynne J. N., Ockendon J. R. & Wilmott P. (1989) On a mathematical model for fiber tapering. SIAM J. Appl. Math. 49, 983990.
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, 323338.
Dewynne J. N., Howell P. D. & Wilmott P. (1994) Slender viscous fibres with inertia and gravity. Quart. J. Mech. Appl. Math. 47, 541555.
Eggers J. (1993) Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 34583460.
van de Fliert B. W., Howell P. D. & Ockendon J. R. (1995) Pressure-driven flow of a thin viscous sheet. J. Fluid Mech. 292, 359376.
Howell P. D. (1994) Extensional thin layer flows. D.Phil, thesis, University of Oxford.
Ida M. P. & Miksis M. J. (1995) Dynamics of a lamella in a capillary tube. SIAM J. Appl. Math. 55, 2357.
Kreyszig E. (1959) Differential Geometry. University of Toronto Press (reprinted Dover, 1991).
Love A. E. H. (1927) A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press.
Myers T. G. (1995) Thin films with high surface tension. OCIAM report, University of Oxford.
Pearson J. R. A. & Petrie C. J. S. (1970) The flow of a tubular film. Part 1. Formal mathematical representation. J. Fluid Mech. 40, 119.
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, 609625.
Schultz W. W. & Davis S. H. (1982) One-dimensional liquid fibers. J. Rheology 26, 331345.
Ting L. & Keller J. B. (1990) Slender jets and sheets with surface tension. SIAM J. Appl. Math. 50, 15331546.
Wilmott P. (1989) The stretching of a thin viscous inclusion and the drawing of glass sheets. Phys. Fluids A 1, 10981103.
Yarin A. L., Gospodinov P. & Roussinov V. I. (1994) Stability loss and sensitivity in hollow-fiber drawing. Phys. Fluids 6, 14541463.
Yarin A. L. & Tchavdarov B. (1995) Onset of folding in plane liquid films. J. Fluid Mech. 307, 8599.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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