Skip to main content
SpringerLink
Log in

Slow viscoelastic radial flow between parallel disks

  • Published:
Applied Scientific Research Aims and scope Submit manuscript

Abstract

A perturbation analysis is presented for the steady-state radial flow of a third-order fluid between two parallel disks. The results include previous perturbation analyses in which various other rheological models were used. The pressure drop needed to maintain the radial flow is less than that for the Newtonian creeping-flow solution because of fluid inertia and shear-thinning viscosity, whereas the normal stresses have the opposite effect. Possible use of the “radial flow viscometer” for experimental evaluation of second-order constants is also discussed. Finally, molecular stretching in the flow system is examined using the elastic dumbbell model for a polymer molecule.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

A2, A11, A3, ...:

Dimensionless constants in the third-order fluid [see equations (30) and (31)]

c mn :

Coefficients in a function of the form C given in equation (34)

d mn :

Coefficients in a function of the form D given in equation (35)

h :

Half distance between the two disks

H :

Spring constant of Hookean dumbbell

i, j, k, m, n :

Indices

K :

Boltzmann's constant

M :

A dimensionless normal stress difference, (τ rr−τθθ)h/η 0 V

Mi, Mij:

Functions in the expansion for M [see equation (39)]

jackie :

Molecular weight of the macromolecules

jackie :

Number density of polymer molecules

N :

A dimensionless normal stress difference, (τ rr−τzz)h/η 0 V

Ni, Nij:

Functions in the expansion for N [see equation (38)]

Ñ :

Avogadro's number

P :

Dimensionless pressure, (p+τ zz)h/η 0 V

Pi, Pij:

Functions in the expansion for P [see equation (36)]

ΔP :

Dimensionless pressure drop, P(R, ±1)−P(R 1, ±1)

ΔP N :

Newtonian creeping-flow result for ΔP

Q :

Volume rate of flow

r :

Radial coordinate

r 0 :

Radius of the orifice on the upper disk

r 1 :

Radius of the disk

R :

Dimensionless radial coordinate, r/h

R 1 :

Dimensionless radius of the disk, r 1/h

〈jackie 2〉:

Average value of the square of the dumbbell length jackie

〈jackie 2〉 eq :

Value of 〈jackie 2〉 at equilibrium

Re :

Reynolds number, ρVh/η 0

t :

Time

T :

Dimensionless shear stress, τ rzh/τ0 V

Ti, Tij:

Functions in the expansion for T [see equation (37)]

jackie :

Absolute temperature

U :

Dimensionless radial velocity, v r/V

Ui, Uij:

Functions in the expansion for U [see equation (32)]

v :

Velocity vector

V :

Characteristic velocity, Q/4πr 1 h

W :

Dimensionless axial velocity, v z/V

W i :

Functions in the expansion of W [see equation (33)]

x, y, z :

Cartesian coordinates

z :

Axial coordinate

Z :

Dimensionless axial coordinate, z/h

α :

Dimensionless parameter in Spriggs four-constant model

α1, α2, α11, ...:

Constants in the third-order fluid

\(\dot \gamma \) :

Shear rate; magnitude of the \(\dot \gamma \) tensor [defined by \(\dot \gamma \) = \(\sqrt {(1/2)(\dot \gamma :\dot \gamma )]} \)

\(\dot \gamma \) :

Rate-of-strain tensor \((\dot \gamma _{ij}= \partial v_j /\partial x_i+ \partial v_i /\partial x_j)\)

ε :

Dimensionless parameter in Spriggs four-constant model

ζ(α):

Riemann zeta function of α

η :

Non-Newtonian viscosity

η 0 :

Zero-shear-rate viscosity

η s :

Solvent viscosity

[η]0 :

Zero-shear-rate intrinsic viscosity

θ :

Tangential coordinate

λ :

Time constant in Spriggs four-constant model

λ H :

Time constant for Hookean dumbbells

ρ :

Density of fluid

τ :

Stress tensor

τ p :

Polymer contribution to τ

τ s :

Solvent contribution to τ

Ψ1, Ψ2:

Primary and secondary normal stress coefficients

ω :

Vorticity tensor \((\omega _{ij}= \partial v_j /\partial x_i- \partial v_i /\partial x_j )\)

D/Dt :

Substantial or material derivative

D/Dt:

Corotational or Jaumann derivative

o:

Derivative with respect to Z

References

  1. Schwarz, W. H. and C. Bruce, Chem. Eng. Sci. 24 (1969) 399.

    Article  Google Scholar 

  2. Laurencena, B. R. and M. C. Williams, Trans. Soc. Rheol. 18 (1974) 331.

    Article  Google Scholar 

  3. Lee, M. C. H. and M. C. Williams, J. Non-Newtonian Fluid Mech. 1 (1976) 323 and 343.

    Article  MATH  Google Scholar 

  4. Woolard, H. W., J. Appl. Mech. 24 (1957) 9 and 644.

    MATH  ADS  Google Scholar 

  5. Barrie, I. T., Plastics & Polymers 38 (1970) 47.

    Google Scholar 

  6. Berger, J. L. and C. G. Gogos, Polymer Eng. & Sci. 13 (1973) 102.

    Article  Google Scholar 

  7. Kamal, M. R. and S. Kenig, Polymer Eng. & Sci. 12 (1972) 294.

    Article  Google Scholar 

  8. Winter, H. H., Polymer Eng. & Sci. 15 (1975) 460.

    Article  Google Scholar 

  9. Jackson, J. D. and G. R. Symmons, Int. J. Mech. Sci. 7 (1965) 239.

    Article  Google Scholar 

  10. Jackson, J. D. and G. R. Symmons, Appl. Sci. Res. A15 (1965) 59.

    Article  Google Scholar 

  11. Hunt, J. B. and I. Torbe, Int. J. Mech. Sci. 4 (1962) 503.

    Article  Google Scholar 

  12. Livesey, J. L., Int. J. Mech. Sci. 1 (1960) 84.

    Article  Google Scholar 

  13. McGinn, J. H., Appl. Sci. Res. A5 (1956) 255.

    Google Scholar 

  14. Moller, P. S., Aeronaut. Quart. 14 (1963) 163.

    Google Scholar 

  15. Morgan, P. G. and A. Saunders, Int. J. Mech. Sci. 2 (1960) 8.

    Article  Google Scholar 

  16. Savage, S. B., Trans. ASME, J. Appl. Mech. 31 (1964) 594.

    MATH  MathSciNet  Google Scholar 

  17. Cogswell, F. N. and P. Lamb, Plastics & Polymers 38 (1970) 331.

    Google Scholar 

  18. Na, T. Y. and A. G. Hansen, Int. J. Non-Linear Mech. 2 (1967) 261.

    Article  MATH  Google Scholar 

  19. Piau, J. M. and M. Piau, Comptes Rendus Acad. Sci. Paris A269 (1969) 1214.

    Google Scholar 

  20. Piau, J. M. and M. Piau, Comptes Rendus Acad. Sci. Paris A270 (1970) 159.

    Google Scholar 

  21. van Es, H. E., Rheo. Acta. 13 (1974) 905.

    Article  Google Scholar 

  22. Spriggs, T. W., Chem. Eng. Sci. 20 (1965) 931.

    Article  Google Scholar 

  23. Co, A., Ph.D. Thesis, University of Wisconsin-Madison, 1978.

  24. Walters, K., Zeits. für angew. Math. und Physik 21 (1970) 592 Sec. 5.

    Article  MATH  ADS  Google Scholar 

  25. Caswell, B., Chem. Eng. Sci 25 (1970) 1167 Appendix.

    Article  Google Scholar 

  26. Bird, R. B., R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1 Fluid Mechanics, Wiley, New York, New York, 1977.

    Google Scholar 

  27. Bird, R. B., O. Hassager, R. C. Armstrong, and C. F. Curtiss, Dynamics of Polymeric Liquids, Vol. 2 Kinetic Theory, Wiley, New York, New York, 1977.

    Google Scholar 

  28. Davies, J. M. and K. Walters, Rheology of Lubricants, T. Davenport (Ed.), Applied Science Publishers. 1973, pp. 65–80.

  29. Ballal, B. Y. and R. S. Rivlin, Trans. Soc. Rheol. 20 (1976) 65.

    Article  MATH  Google Scholar 

  30. Tanner, R. I. and A. C. Pipkin, Trans. Soc. Rheol. 13 (1969) 471.

    Article  Google Scholar 

  31. Walters, K. and N. D. Waters, Rheo. Acta 3 (1964) 312.

    Article  MATH  Google Scholar 

  32. Giesekus, H., Proc. Fourth International Congress on Rheology, E. H. Lee and A. L. Copley (Eds.), Wiley, New York, New York, 1965, Vol. 1, pp. 249–266.

    Google Scholar 

  33. Joseph, D. D. and R. L. Fosdick, Arch. Rational Mech. Anal. 49 (1973) 321.

    MATH  MathSciNet  ADS  Google Scholar 

  34. Joseph, D. D., G. S. Beavers, and R. L. Fosdick, Arch. Rational Mech. Anal. 49 (1973) 381.

    MATH  MathSciNet  ADS  Google Scholar 

  35. Beavers, G. S. and D. D. Joseph, J. Fluid. Mech. 69 (1975) 475.

    Article  MATH  ADS  Google Scholar 

  36. Sturges, L., and D. D. Joseph, Arch. Rational Mech. Anal. 59 (1975) 359.

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Chem. Eng. and Rheology Research Center, Univ. of Wisconsin, 53706, Madison, Wisconsin, USA

    Albert Co & R. Byron Bird

Authors
  1. Albert Co
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Byron Bird
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Co, A., Bird, R.B. Slow viscoelastic radial flow between parallel disks. Appl. Sci. Res. 33, 385–404 (1977). https://doi.org/10.1007/BF00411821

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00411821

Keywords

Search

Navigation