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Instability of a viscoelastic fluid between rotating parallel disks: analysis for the Oldroyd-B fluid

Published online by Cambridge University Press:  26 April 2006

Alparslan Öztekin  and
Robert A. Brown
Alparslan Öztekin
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Robert A. Brown
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

The stability of the viscometric motion of a viscoelastic fluid held between rotating parallel disks with large radii to small-amplitude perturbations is studied for the Oldroyd-B constitutive model. The disturbances are assumed to be radially localized and are expressed in Fourier form so that a separable eigenvalue problem results; these disturbances describe either axisymmetric or spiral vortices, depending on whether the most dangerous disturbance has zero or non-zero azimuthal wavenumber, respectively. The critical value of the dimensionless radius R* for the onset of the instability is computed as a function of the Deborah number De, a dimensionless time constant of the fluid, the azimuthal and radial wavenumbers, and the ratio of the viscosities of the solvent to the polymer solution. Calculations meant to match the experiments of McKinley et al. (1991) for a Boger fluid show that the most dangerous instabilities are spiral vortices with positive and negative angle that start at the same critical radius and travel outward and inward toward the centre of the disk; the axisymmetric mode also becomes unstable at only slightly greater values of R*, or De for fixed R*. The predicted dependence of the value of De for a fixed R* on the gap between the disks agrees quantitatively with the measurements of McKinley et al., when the longest relaxation time for the fluid at the shear rate corresponding to the maximum value of R* is used to define the time constant in the Oldroyd-B model.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 255 , October 1993 , pp. 473 - 502
Copyright
© 1993 Cambridge University Press

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References

Beris, A. N. & Avgousti, M. 1992 Viscoelastic flow instabilities: inception and non-linear evolution. In Theoretical and Applied Rheology. Proc. Xth Intl Cong. on Rheology, Belgium (ed. P. Modenaers & R. Keunings), vol. 1, pp. 33–38. Elsevier.
Binnington, R. J. & Boger, D. V. 1986 Remarks on non-shear thinning elastic fluids. Polymer Engng Sci. 26, 133–138.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987a Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics, 2nd edn. Wiley-Interscience.
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O. 1987b Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory, 2nd edn. Wiley-Interscience.
Brown, R. A., Szady, M. J., Northey, P. J. & Armstrong, R. C. 1993 On the numerical stability of mixed finite element methods for viscoelastic flows governed by differential constitutive equations. J. Theor. Comput. Fluid Dyn. (in press.)Google Scholar
Griffiths, D. F., Jones, D. T. & Walters, K. 1969 A flow reversal due to edge effects. J. Fluid Mech. 36, 161–175.Google Scholar
Gorodstov, V. A. & Leonov, A. I. 1967 On a linear stability of a plane parallel Couette flow of viscoelastic fluid. Z. Angew. Math. Mech. 31, 310–319.Google Scholar
Hill, C. T. 1972 Nearly viscometric flow in the disk and cylinder systems II. Experimental. Trans. Soc. Rheol. 16, 213–245.Google Scholar
Jackson, J. P., Walters, K. & Williams, R. W. 1984 A rheometrical study of Boger fluids. J. Non-Newtonian Fluid Mech. 14, 173–188.Google Scholar
Joo, Y. L. & Shaqfeh, E. S. G. 1992 A purely elastic instability in Dean and Taylor–Dean flow. Phys. Fluids A 4, 524–543.Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheologica Acta 31, 213–263.Google Scholar
Larson, R. G., Shaqfeh, E. S. G. & Muller, S. J. 1990 A purely elastic instability in Taylor–Couette flow. J. Fluid Mech. 218, 573–600.Google Scholar
Laun, H. M. & Hingmann, R. 1990 Rheological characterization of the fluid M1 and of its components. J. Non-Newtonian Fluid Mech. 35, 137–157.Google Scholar
Magda, J. J. & Larson, R. G. 1988 A transition in ideal elastic liquids during shear flow. J. Non-Newtonian Fluid Mech. 30, 1–19.Google Scholar
Malik, M. R. 1986 The neutral curve for stationary disturbances in rotating disk flow. J. Fluid Mech. 164, 275–287.Google Scholar
Malik, M. R., Wilkinson, S. P. & Orszag, S. A. 1981 Instability and transition in rotating disk flow. AIAA J. 19, 1131–1138.Google Scholar
McKinley, G. H., Byars, J. A., Brown, R. A. & Armstrong, R. C. 1991 Observations on the elastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid. J. Non-Newtonian Fluid Mech. 40, 201–229.Google Scholar
Northey, P., Armstrong, R. C. & Brown, R. A. 1990 Finite-element calculation of time dependent, two dimensional inertial flows in circular Couette flow. Annual AIChE Meeting, November, 1990, Paper 167j.
Northey, P., Armstrong, R. C. & Brown, R. A. 1992 Finite-amplitude time-periodic states in viscoelastic Taylor–Couette flow described by the UCM model. J. Non-Newtonian Fluid Mech. 42, 117–139.Google Scholar
Öztekin, A., Bornside, D. E. & Brown, R. A. 1993 Stability of forced flow over a rotating disk. Unpublished.
Phan-Thien, N. 1983 Coaxial-disk flow of an Oldroyd-B fluid, exact solution and stability. J. Non-Newtonian Fluid Mech. 13, 325–340.Google Scholar
Phan-Thien, N. 1985 Cone-and-plate flow of the Oldroyd-B fluid is unstable. J. Non-Newtonian Fluid Mech. 13, 325–340.Google Scholar
Quinzani, L. M., McKinley, G. H., Brown, R. A. & Armstrong, R. C. 1991 Modelling the rheology of polyisobutylene solutions. J. Rheol. 34, 705–748.Google Scholar
Shaqfeh, E. S. G., Muller, S. J. & Larson, R. G. 1992 The effects of gap width and dilute solution properties on the viscoelastic Taylor–Couette instability. J. Fluid Mech. 235, 285–317.Google Scholar
Steiert, P. & Wolff, C. 1990 Rheological properties of a polyisobutylene in a kerosene/polybutene mixture in simple shear flow. J. Non-Newtonian Fluid Mech. 35, 189–196.Google Scholar
Wilkinson, S. P. & Malik, M. R. 1983 Stability experiments in the flow over a rotating disk. AIAA Paper 83–1760.
Zebib, A. 1987 Removal of spurious modes encountered in solving stability problems by spectral methods. J. Comput. Phys. 70, 521–525.Google Scholar