Viscoelastic flow in a curved channel: A similarity solution for the Oldroyd-B fluid

  • N. Phan-Thien
  • R. Zheng
Original Papers


A similarity solution is used to analyse the flow of the Oldroyd fluid B, which includes the Newtonian and Maxwell fluids, in a curved channel modelled by the narrow annular region between two circular concentric cylinders of large radius. The solution is exact, including inertial forces. It is found that the non-Netonian kinematics are very similar to the Newtonian ones, although some stress components can become very large. At high Reynolds number a boundary layer is developed at the inner cylinder. The structure of this boundary layer is asymptotically analysed for the Newtonian fluid. Non-Newtonian stress boundary layers are also developed at the inner cylinder at large Reynolds numbers.


Boundary Layer Reynolds Number Inertial Force Newtonian Fluid High Reynolds Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Armstrong, R. C. & Brown, R. A., Paper presented at the 6th Workshop on Numerical Methods in Non-Newtonian Flows, Denmark, June 1989.Google Scholar
  2. [2]
    Berman, N. S. & Parsch, M. A.,Laser doppler velocity measurements for dilute polymer solutions in the laminar boundary of a rotating disk, J. Rheology30, 441–458 (1986).Google Scholar
  3. [3]
    Bird, R. B., Armstrong, R. C. & Hassager, O.,Dynamics of Polymeric Liquids: Vol. I Fluid Mechanics, 2nd Edition, John Wiley and Sons, New York 1987.Google Scholar
  4. [4]
    Boger, D. V.,Dilute polymer solutions and their use to model polymer processing flows, in J. C. Seferis and P. S. Theocaris,Interrelations between processing structure and properties of polymeric materials, Elsevier, Amsterdam 1985.Google Scholar
  5. [5]
    Burdette, S. R., Coates, P. J., Armstrong, R. C. & Brown, R. A.,Calculations of viscoelastic flow through an axisymmetric corrugated tube using the explicitly elliptic momemtum equation formulation (EEME), J. non-Newt. Fluid Mech.33, 1–23 (1989).Google Scholar
  6. [6]
    Crochet, M. J.,Numerical simulations of highly viscoelastic flows, Proc. 10th Int. Cong. Rheology, Sydney, August 1988.Google Scholar
  7. [7]
    Hamza, E. A. & MacDonald, D. A.,A fluid film squeezed between two parallel plane surfaces, J. Fluid Mech.109, 147–160 (1981).Google Scholar
  8. [8]
    Huilgol, R. R.,Nearly extensional flows, J. non-Newt. Fluid Mech.5, 210–231 (1979).Google Scholar
  9. [9]
    Hull, A. M.,An exact solution for the slow flow of a general linear viscoelastic fluid through a slit, J. non-Newt. Fluid Mech.8, 327–336 (1981).Google Scholar
  10. [10]
    Larson, R. G.,Analytic results for viscoelastic flow in a porous tube, J. non-Newt. Fluid Mech., in press.Google Scholar
  11. [11]
    Larson, R. G.,Viscoelastic inertial flow driven by an axisymmetric accelerated surface, J. Fluid Mech.196, 449–465 (1988).Google Scholar
  12. [12]
    Magda, J. J. & Larson, R. G.,A transition occurring in ideal elastic liquids during shear flow. J. non-Newt. Fluid Mech.30, 1–19 (1988).Google Scholar
  13. [13]
    Menon, R. K., Kim-E, M. E., Armstrong, R. C., Brown, R. A. & Brady, J. F.,Injection and suction of an upper-conceded Maxwell fluid through a porous-walled tube, J. non-Newt. Fluid Mech.27, 265–297 (1988).Google Scholar
  14. [14]
    Phan-Thien, N.,Coaxial-disk flow and flow about a rotating disk of a Maxwellian fluid, J. Fluid Mech.128, 427–442 (1983a).Google Scholar
  15. [15]
    Phan-Thien, N.,Coaxial-disk flow of an Oldroyd-B fluid: exact solution and instability, J. non-Newt. Fluid Mech.13, 325–340 (1983b).Google Scholar
  16. [16]
    Phan-Thien, N. & Tanner, R. I.,Viscoelastic squeeze-film flows—Maxwell fluids, J. Fluid Mech.129, 265–281 (1983).Google Scholar
  17. [17]
    Phan-Thien, N.,Squeezing a viscoelastic fluid from a wedge: an exact solution, J. non-Newt. Fluid Mech.16, 329–345 (1984a).Google Scholar
  18. [18]
    Phan-Thien, N.,Stagnation flows for the Oldroyd-B fluid, Rheol. Acta23, 172–176 (1984b).Google Scholar
  19. [19]
    Phan-Thien, N.,Cone -and-plate flow of the Oldroyd-B fluid is unstable, J. non-Newt. Fluid Mech.17, 37–44 (1985a).Google Scholar
  20. [20]
    Phan-Thien, N.,Squeezing of an Oldroyd-B fluid from a tube: limiting Weissenberg number, Rheol. Acta24, 15–21 (1985b).Google Scholar
  21. [21]
    Phan-Thien, N.,Squeezing a viscoelastic fluid from a cone: an exact solution, Rheol. Acta24, 119–126 (1985c).Google Scholar
  22. [22]
    Phan-Thien, N.,A three-dimensional stretching flow of an Oldroyd fluid, Quart. Appl. Maths.XLV, 23–27 (1987).Google Scholar
  23. [23]
    Pilitsis, S. & Beris, A. N.,Calculations of steady-state viscoelastic flow in an undulating tube, J. non-Newt. Fluid Mech.31, 231–287 (1989).Google Scholar
  24. [24]
    Pipkin, A. C. & Tanner, R. I.,A survey of theory and experiment in viscometric flows of viscoelastic liquids. In S. Nemat-Nasser,Mechanics Today, Vol. I, Pergamon, Oxford 1972.Google Scholar
  25. [25]
    Pipkin, A. C. & Owen, D. R.,Nearly viscometric flows, Phys. Fluids10, 836–843 (1967).Google Scholar
  26. [26]
    Schlichting, H.,Boundary Layer Theory, Translator J. Kestin, 6th Edition, McGraw-Hill, New York 1968.Google Scholar
  27. [27]
    Tanner, R. I.,Engineering Rheology, Oxford University Press, New York 1988.Google Scholar
  28. [28]
    Zheng, R., Phan-Thien, N., Tanner, R. I. & Bush, M. B.,Numerical analysis of viscoelastic flow through a sinusoidally corrugated tube using a boundary element method, J. Rheology34, 79–102 (1990a).Google Scholar
  29. [29]
    Zheng, R., Phan-Thien, N. & Tanner, R. I.,On the flow past a sphere in a cylindrical tube: limiting Weissenberg number, J. non-Newt. Fluid Mech. (1990b) in press.Google Scholar

Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • N. Phan-Thien
    • 1
  • R. Zheng
    • 1
  1. 1.Department of Mechanical EngineeringThe University of SydneyAustralia

Personalised recommendations