Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong, R. C. & Brown, R. A., Paper presented at the 6th Workshop on Numerical Methods in Non-Newtonian Flows, Denmark, June 1989.
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).
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.
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.
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).
Crochet, M. J.,Numerical simulations of highly viscoelastic flows, Proc. 10th Int. Cong. Rheology, Sydney, August 1988.
Hamza, E. A. & MacDonald, D. A.,A fluid film squeezed between two parallel plane surfaces, J. Fluid Mech.109, 147–160 (1981).
Huilgol, R. R.,Nearly extensional flows, J. non-Newt. Fluid Mech.5, 210–231 (1979).
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).
Larson, R. G.,Analytic results for viscoelastic flow in a porous tube, J. non-Newt. Fluid Mech., in press.
Larson, R. G.,Viscoelastic inertial flow driven by an axisymmetric accelerated surface, J. Fluid Mech.196, 449–465 (1988).
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).
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).
Phan-Thien, N.,Coaxial-disk flow and flow about a rotating disk of a Maxwellian fluid, J. Fluid Mech.128, 427–442 (1983a).
Phan-Thien, N.,Coaxial-disk flow of an Oldroyd-B fluid: exact solution and instability, J. non-Newt. Fluid Mech.13, 325–340 (1983b).
Phan-Thien, N. & Tanner, R. I.,Viscoelastic squeeze-film flows—Maxwell fluids, J. Fluid Mech.129, 265–281 (1983).
Phan-Thien, N.,Squeezing a viscoelastic fluid from a wedge: an exact solution, J. non-Newt. Fluid Mech.16, 329–345 (1984a).
Phan-Thien, N.,Stagnation flows for the Oldroyd-B fluid, Rheol. Acta23, 172–176 (1984b).
Phan-Thien, N.,Cone -and-plate flow of the Oldroyd-B fluid is unstable, J. non-Newt. Fluid Mech.17, 37–44 (1985a).
Phan-Thien, N.,Squeezing of an Oldroyd-B fluid from a tube: limiting Weissenberg number, Rheol. Acta24, 15–21 (1985b).
Phan-Thien, N.,Squeezing a viscoelastic fluid from a cone: an exact solution, Rheol. Acta24, 119–126 (1985c).
Phan-Thien, N.,A three-dimensional stretching flow of an Oldroyd fluid, Quart. Appl. Maths.XLV, 23–27 (1987).
Pilitsis, S. & Beris, A. N.,Calculations of steady-state viscoelastic flow in an undulating tube, J. non-Newt. Fluid Mech.31, 231–287 (1989).
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.
Pipkin, A. C. & Owen, D. R.,Nearly viscometric flows, Phys. Fluids10, 836–843 (1967).
Schlichting, H.,Boundary Layer Theory, Translator J. Kestin, 6th Edition, McGraw-Hill, New York 1968.
Tanner, R. I.,Engineering Rheology, Oxford University Press, New York 1988.
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).
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Phan-Thien, N., Zheng, R. Viscoelastic flow in a curved channel: A similarity solution for the Oldroyd-B fluid. Z. angew. Math. Phys. 41, 766–781 (1990). https://doi.org/10.1007/BF00945834
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00945834