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Converging Dies

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Rheological Measurement

Abstract

Although the flow behaviour of non—Newtonian fluids in channels of constant cross—section is well understood, and there are established models to describe it, the same is not yet true of flow in convergences where the extensional strain rate becomes significant compared to the wall shearrate. Various methods of describing the convergent flow behaviour of polymeric liquids in dies and in other regimes have formed the basis of recent reviews1–3 (see also Chapter 8).

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Abbreviations

Subscripts 0 and 1:

refer to die entry and exit respectively.

R :

radial distance in cylindrical coordinates; die or capillary radius

x :

axial distance from die cone apex

L :

capillary length

r, θ, ϕ :

spherical coordinates, based on die cone apex

α :

die semi-angle; α, value for minimum pressure

a :

surface area of spherical shell

b :

radius of spherical shell in exit region

V :

volume of truncated spherical exit region

β :

angle subtended at axis by die exit at distance b

Re :

Reynolds number

ρ :

density

v :

velocity; v r, r-direction component; ῡ, mean value

σ :

elongating stress

τ :

shear stress

ε :

elongational strain rate

γ :

shear strain rate; γA, apparent wall shear rate at the die exit or in the capillary

A, n :

extensional power law constants

B, m :

shear power law constants

λ :

extensional viscosity or anisotropic extensional viscosity; λA, apparent (n = 1) value. Subscripts ‖ and ┴ refer to directions parallel and perpendicular to anisotropic reference direction. λRes refers to resin phase only

η :

shear viscosity or anisotropic shear viscosity; ηRes resin phase shear viscosity; ηA apparent (m = 1) shear viscosity

a ij :

coefficients in the reciprocal viscosity tensor

P :

reservoir hydrostatic pressure

P Ent, P Cap :

contributions of entry flow and capillary flow towards total pressure

PSEnt, PEEnt :

contributions of shear dissipation and extensional dissipation towards PEnt

P’ Ent :

contribution of exit transition region

Ṗ Ent :

minimum value

P*:

dimensionless pressure

p :

hydrostatic pressure

Q :

volume flow rate

V f, l, d :

volume fraction; length and diameter of reinforcement

References

  1. J. W. Hill and J. A. Cucolo, J. Macromol. Sci., Rev. Macromol. Chem., 1976, C14(1), 107.

    Article  Google Scholar 

  2. T. H. Forsyth, Polym. Plast. Technol. Eng., 1976, 6(1), 101.

    Article  CAS  Google Scholar 

  3. F. N. Cogswell, J. Non—Newtonian Fluid Mech., 1978, 4, 23.

    Article  CAS  Google Scholar 

  4. J. L. Sutterby, Appl. Sci. Res., 1965, A15, 241.

    Article  Google Scholar 

  5. C. D. Han, Rheology in Polymer Processing, Academic Press, New York, 1976.

    Google Scholar 

  6. A. G. Gibson and G. A. Williamson, Polym. Eng. Sci., 1985, 25(15), 968.

    Article  CAS  Google Scholar 

  7. A. G. Gibson and G. A. Williamson, Polym. Eng. Sci., 1985, 25(15), 980.

    Article  CAS  Google Scholar 

  8. R. J. Crowson, A. J. Scott and D. W. Saunders, Polym. Eng. Sci., 1981, 21, 748.

    Article  CAS  Google Scholar 

  9. R. R. Lamonte and C. D. Han, J. Appl. Polym. Sci., 1972, 16, 3285.

    Article  CAS  Google Scholar 

  10. G. K. Batchelor, J. Fluid Mech., 1971, 46, 813.

    Article  Google Scholar 

  11. G. A. Williamson and A. G. Gibson, Plast. Rubber Process. Appl., 1984, 4(3), 203.

    Google Scholar 

  12. A. G. Gibson, A. N. McClelland, G. Cuff and C. R. Gore, Developments in the Science and Technology of Composite Materials, Proc. 1st European Conf. on Composite Materials, Bordeaux, France, 24–27 September, 1985, AEMC, Bordeaux, p. 511.

    Google Scholar 

  13. A. G. Gibson and A. N. McClelland, Fibre Reinforced Composites, University of Liverpool, 8–10 April 1986, Inst. Mech. Engrs, London, paper 32, p. 99.

    Google Scholar 

  14. A. G. Gibson and A. N. McClelland, ICCM6/ECCM2 Conference, Imperial College, London, 20–24 July, 1987.

    Google Scholar 

  15. F. N. Cogswell, Polymer Melt Rheology, George Godwin/Plastics and Rubber Institute, London, 1981.

    Google Scholar 

  16. F. N. Cogswell, Polym. Eng. Sci., 1972, 12, 64.

    Article  CAS  Google Scholar 

  17. S. Oka, J. Phys. Soc. Japan, 1964, 19(8), 1481.

    Article  CAS  Google Scholar 

  18. S. Oka and A. Takami, Japan J. Appl. Phys., 1967, 6, 423.

    Article  CAS  Google Scholar 

  19. A. B. Metzner and A. P. Metzner, Rheol. Acta, 1970, 9(2), 174.

    Article  CAS  Google Scholar 

  20. R. M. Jones, Mechanics of Composite Materials, Scripta Book Company, Washington DC, and McGraw—Hill, New York, 1975.

    Google Scholar 

  21. W. J. Harrison, Proc. Cambridge Phil. Soc., 1916, 19, 307.

    Google Scholar 

  22. W. N. Bond, Phil. Mag., 1925, 50, 1958.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Materials Science and Engineering, University of Liverpool, UK

    A. G. Gibson

Authors
  1. A. G. Gibson
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Editor information

Editors and Affiliations

  1. Department of Applied Physics, Sheffield City Polytechnic, Sheffield, UK

    A. A. Collyer

  2. Department of Metallurgy and Materials Science, University of Nottingham, Nottingham, UK

    D. W. Clegg

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© 1993 Springer Science+Business Media Dordrecht

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Gibson, A.G. (1993). Converging Dies. In: Collyer, A.A., Clegg, D.W. (eds) Rheological Measurement. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2898-0_3

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  • DOI: https://doi.org/10.1007/978-94-017-2898-0_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-2900-0

  • Online ISBN: 978-94-017-2898-0

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