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Slow slumping of a very viscous liquid bridge

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Abstract

A layer of very viscous liquid (e.g. tar, molten glass) spans a chasm between two vertical walls. The slow fall or slump of this initially-rectangular liquid bridge is analysed. A semi-analytical solution is obtained for the initial motion, for arbitrary thickness/width ratios. The formal limits of large and small thickness/width ratios are also investigated. For example, the centre section of a thin bridge of liquid of density ρ and viscosity µ, with width 2w and thickness 2h≪2w falls under gravity g at an initial velocity ρgw4/(32μh2). A finite element technique is then employed to determine the slumping motion at later times, confirming in passing the semi-analytical prediction of the initial slumping velocity.

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References

  1. L. Smith, R.J. Tillen and J. Winthrop, New directions in aspherics: glass and plastic. In: M.J. Riedl (ed), Replication and Molding of Optical Components, Vol. 896, Proceedings of The Society of Photo-Optical Instrumentation Engineers. Washington: Bellingham (1988) pp. 160–166.

  2. E.B. Shand, Glass Engineering Handbook. New York: McGraw-Hill (1958) 484pp.

    Google Scholar 

  3. G.W. Morey, The Properties of Glass. New York: Reinhold (1938) 561pp.

    Google Scholar 

  4. S.T. Gulati, E.H. Fontana and W.A. Plummer, Disc bending viscometry. Phys and Chem. of Glasses 17 (1976) 114–119.

    Google Scholar 

  5. H. Rawson, Properties and Applications of Glass. New York: Elsevier (1980) 318pp.

    Google Scholar 

  6. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge: Cambridge University Press (1967) 615pp.

    Google Scholar 

  7. J.V. Wehausen and E.V. Laitone, Surface Waves. In: S. Flugge (ed), Handbuch der Physik Vol. 9. Berlin: Springer (1960) pp. 446–778.

    Google Scholar 

  8. B.W. van der Fliert, P.D. Howell and J.R. Ockendon, Pressure-driven flow of a thin viscous sheet. J. Fluid Mech. 292 (1995) 359–376.

    Google Scholar 

  9. G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford: Clarendon Press (1978) 304pp.

    Google Scholar 

  10. S. Timoshenko and J. Goodier, Theory of Elasticity, 3rd edn. New York: McGraw-Hill (1970) 567pp.

    Google Scholar 

  11. A.P. Hillman and H.E. Salzer, Roots of sin z = z. Phil. Mag. 34 (1943) 575.

    Google Scholar 

  12. D.D. Joseph, L.D. Sturges and W.H. Warner, Convergence of biorthogonal series of biharmonic eigenfunctions by the method of Titchmarsh. Arch. Rat. Mech. Anal. 78 (1982) 223–274.

    Google Scholar 

  13. H.K. Moffatt, Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18 (1964) 1–18.

    Google Scholar 

  14. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. New York: McGraw-Hill (1959) 580pp.

    Google Scholar 

  15. A.N. Stokes, Fastflo solutions of natural convection problems close to freezing. In: D. Stewart et al. (ed), Computational Techniques and Applications Conference 1993. Singapore: World Scientific (1994) pp. 446–453.

    Google Scholar 

  16. A.N. Stokes, Using conjugate gradient methods in solving the incompressible Navier-Stokes equations. In: A.K. Easton and R.L. May (eds), Computational Techniques and Applications Conference 1995. Singapore: World Scientific (1996) pp. 725–732.

    Google Scholar 

  17. J.T. Pittman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander, Numerical Analysis of Forming Processes. New York: Wiley (1984) 444pp.

    Google Scholar 

  18. A. Kaye, Convected coordinates and elongational flow. J. Non-Newtonian Fluid Mech. 40 (1991) 55–77.

    Google Scholar 

  19. P.D. Howell, Models for thin viscous sheets. Euro. J. Appl. Math. 7 (1996) 321–343.

    Google Scholar 

  20. E.O. Tuck, Mathematics of honey on toast. National Symposium on the Mathematical Sciences, University of NSW, February 1996. In: R.R. Moore and A.J. van der Poorten (eds), Multiplying Australia's Potential. Canberra: Australian Academy of Science (1997).

    Google Scholar 

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Tuck, E., Stokes, Y. & Schwartz, L. Slow slumping of a very viscous liquid bridge. Journal of Engineering Mathematics 32, 27–40 (1997). https://doi.org/10.1023/A:1004200926153

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  • DOI: https://doi.org/10.1023/A:1004200926153

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