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Observations of purely elastic instabilities in the Taylor–Dean flow of a Boger fluid

Published online by Cambridge University Press:  26 April 2006

Yong Lak Joo  and
Eric S. G. Shaqfeh
Yong Lak Joo
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025, USA
Eric S. G. Shaqfeh
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025, USA
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Abstract

An experimental and theoretical investigation of the stability of the viscoelastic flow of a model Boger fluid between rotating cylinders with an applied pressure gradient is presented. In our theoretical study, a linear stability analysis based on the Oldroyd-B fluid model which predicts the critical conditions and the structure of the vortex flow at the onset of instability is developed. Our results reveal that certain non-axisymmetric modes are more unstable than the previously studied axisymmetric mode when the shearing by the cylinder rotation is the dominant flow-driving force. This is consistent with recent results presented by Beris & Avgousti (1992) on the stability of elastic Taylor–Couette flow. On the other hand, the axisymmetric mode is more unstable when the pressure gradient becomes dominant. Furthermore, we investigate the mechanism of purely elastic Taylor–Dean instability with respect to non-axisymmetric disturbances through an examination of the disturbance-energy equation. It is found that the mechanism of the elastic Taylor–Dean instability is associated with the coupling between the disturbance polymeric stresses due to the azimuthal variation of the disturbance flow and the base state velocity gradients. In our experimental study, evidence of non-inertial, cellular instabilities in the Taylor–Dean flow of a well-characterized polyisobutylene/polybutene Boger fluid is presented. A stationary, meridional obstruction is placed between independently rotating, concentric cylinders to generate an azimuthal pressure gradient in opposition to the shearing flow. Flow visualization experiments near the critical conditions show the transition from purely azimuthal flows to secondary vortex flows, and the development of evenly spaced, banded vortex structures. The critical wavenumber obtained from spectral image analysis of the visualizations, and the critical Deborah number are presented for various ratios of the pressure gradient to the shear driving force. Although there is a quantitative discrepancy between data and theory, the qualitative trends in the data are in agreement with our theoretical predictions. In addition, laser-Doppler velocimetry (LDV) measurements show that the instability is a stationary mode when the pressure gradient is the dominant flow-driving force, while it is an oscillatory instability when the shearing is dominant, again as predicted by the theory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 262 , 3 October 1994 , pp. 27 - 73
Copyright
© 1994 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, Proceedings of the X International Congress on Rheology, Belgium (ed. P. Modenaers & R. Keunings,) vol. 1, pp. 33–38. Elsevier.
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn, vol. 2. Wiley-Interscience.
Boger, D. V. 1977/ 1978 A highly elastic constant-viscosity fluid. J. Non-Newtonian Fluid Mech. 3, 87–91.Google Scholar
Brewster, D. B. & Nissan, A. H. 1958 Hydrodynamics of flow between horizontal concentric cylinders. I. Flow due to the rotation of cylinder. Chem. Engng Sci. 7, 215–221.Google Scholar
Chandrasekar, S. 1961 Hydrodynamic Stability. Clarendon Press, Oxford.
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 93, 385–425.Google Scholar
Conte, S. D. 1966 The numerical solutions of linear boundary value problems. SIAM Rev. 8, 309–321.Google Scholar
Dean, W. R. 1928 Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121, 402–420.Google Scholar
DiPrima, R. C. 1959 The stability of viscous flow between rotating concentric cylinders with a pressure gradient acting around the cylinder. J. Fluid Mech. 6, 462–468.Google Scholar
DiPrima, R. C. 1961 Stability of nonrotationally symmetric disturbances for viscous flow between rotating cylinders. Phys. Fluids 4, 751–755.Google Scholar
DiPrima, R. C. & Swinney, H. L. 1981 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the transition to turbulence, 2nd edn (ed. H. L. Swinney & J. P. Gollub), pp. 139–180. Springer.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Gupta, R. K., Sridhar, T. & Ryan, M. E. 1983 Model viscoelastic liquids. J. Non-Newtonian Fluid Mech. 12, 233–241.Google Scholar
Joo, Y. L. 1993 A theoretical and experimental investigation of viscoelastic instabilities in Taylor–Dean flow. PhD thesis, Stanford University.
Joo, Y. L. & Shaqfeh, E. S. G. 1991 Viscoelastic Poiseuille flow through a curved channel: a new elastic instability. Phys. Fluids A 3, 1691–1694.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
Joo, Y. L. & Shaqfeh, E. S. G. 1993 Weakly nonlinear analysis and experimental studies in the viscoelastic Taylor-Dean instability. In preparation.
Keller, H. B. 1961 Numerical Methods for Two-Point Boundary-Value Problems. Cinn-Blaisdell, Waltham, MA.
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
Mackay, M. E. & Boger, D. V. 1987 An explanation of the rheological properties of Boger fluids. J. Non-Newtonian Fluid Mech. 22, 235–243.Google Scholar
McKinley, G. H., Byars, J. A. & Brown, R. A. 1991a 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
McKinley, G. H., Raiford, W. P., Brown, R. A. & Armstrong, R. C. 1991b Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions. J. Fluid Mech. 223, 411–56.Google Scholar
Magda, J. J. & Larson, R. G. 1988 A transition in ideal liquids during the shear flow. J. Non-Newtonian Fluid Mech. 30, 1–19.Google Scholar
Muller, S. J., Shaqfeh, E. S. G. & Larson, R. G. 1993 Experimental studies of the onset of oscillatory instability in viscoelastic Taylor–Couette flow. J. Non-Newtonian Fluid Mech. 46, 315–330.Google Scholar
Mutabazi, I., Normand, C., Peerhossaini, H. & Wesfreid, J. E. 1989 Oscillatory modes in the flow between two horizontal corotating cylinders with a partially filled gap. Phys. Rev. A 39, 763–771.Google Scholar
Mutabazi, I., Hegseth, J. J. & Andereck, C. D. 1990 Spatiotemporal pattern modulations in the Taylor-Dean system. Phys. Rev. Lett. 64, 1729–1732.Google Scholar
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 an Oldroyd-B fluid is unstable. J. Non-Newtonian Fluid Mech. 17, 37–44.Google Scholar
Quinzani, L. M., McKinley, G. H., Brown, R. A. & Armstrong, R. C. 1991 Modeling the rheology of polyisobutylene solutions. J. Rheol. 34, 705.Google Scholar
Raney, D. C. & Chang, T. S. 1971 Oscillatory modes of instability for flow between rotating cylinders with a transverse pressure gradient. Z. angew. Math. Phys. 22, 680–690.Google Scholar
Shaqfeh, E. S. G. & Acrivos, A. 1987 The effects of inertia on the stability of convective flow in inclined particle settlers. Phys. Fluids 30, 960–973.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
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar