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A purely elastic instability in Taylor–Couette flow

Published online by Cambridge University Press:  26 April 2006

R. G. Larson ,
Eric S. G. Shaqfeh  and
S. J. Muller
R. G. Larson
Affiliation:
AT&T Bell Laboratories, Murray Hill, NJ 07974, USA
Eric S. G. Shaqfeh
Affiliation:
AT&T Bell Laboratories, Murray Hill, NJ 07974, USA
S. J. Muller
Affiliation:
AT&T Bell Laboratories, Murray Hill, NJ 07974, USA
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Abstract

A non-inertial (zero Taylor number) viscoelastic instability is discovered for Taylor–Couette flow of dilute polymer solutions. A linear stability analysis of the inertialess flow of an Oldroyd-B fluid (using both approximate Galerkin analysis and numerical solution of the relevant small-gap eigenvalue problem) show the growth of an overstable (oscillating) mode when the Deborah number exceeds f(S) ε−½, where ε is the ratio of the gap to the inner cylinder radius, and f(S) is a function of the ratio of solvent to polymer contributions to the solution viscosity. Experiments with a solution of 1000 p.p.m. high-molecular-weight polyisobutylene in a viscous solvent show an onset of secondary toroidal cells when the Deborah number De reaches 20, for ε of 0.14, and a Taylor number of 10−6, in excellent agreement with the theoretical value of 21. The critical De was observed to increase as ε decreases, in agreement with the theory. At long times after onset of the instability, the cells become small in wavelength compared to those that occur in the inertial instability, again in agreement with our linear analysis. For this fluid, a similar instability occurs in cone-and-plate flow, as reported earlier. The driving force for these instabilities is the interaction between a velocity fluctuation and the first normal stress difference in the base state. Instabilities of the kind that we report here are likely to occur in many rotational shearing flows of viscoelastic fluids.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 218 , September 1990 , pp. 573 - 600
Copyright
© 1990 Cambridge University Press

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References

Beard, D. W., Davies, M. H. & Walters, K., 1966 The stability of elastico-viscous flow between rotating cylinders. Part 3. Overstability in viscous and Maxwell fluids. J. Fluid Mech. 24, 321–334.Google Scholar
Beavers, G. S. & Joseph, D. D., 1974 Tall Taylor cells in polyacrylamide solutions. Phys. Fluids 17, 650–651.Google Scholar
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O., 1987 Dynamics of Polymeric Liquids, vol. 2, 2nd edn. Wiley-Interscience.
Boger, D. V.: 1977/1978 A highly elastic constant-viscosity fluid. J. Non-Newtonian Fluid Mech. 3, 87–91.Google Scholar
Chandrasekhar, S.: 1961 Hydrodynamic Stability. Clarendon.
Coleman, B. D. & Noll, W., 1961 Foundations of linear viscoelasticity. Rev. Mod. Phys. 33, 239–249.Google Scholar
Coles, D.: 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Conte, S. D.: 1966 The numerical solution of linear boundary value problems. SIAM Rev. 8, 309–321.Google Scholar
Datta, S. K.: 1964 Note on the stability of an elasticoviscous liquid in Couette flow. Phys. Fluids 7, 1915–1919.Google Scholar
Denn, M. M. & Roisman, J. J., 1969 Rotational stability and measurement of normal stress functions in dilute polymer solutions. AIChE J. 15, 454–459.Google Scholar
Drazin, P. G. & Reid, W. H., 1981 Hydrodynamic Stability. Cambridge University Press.
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P., 1979 Dynamic instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103–128.Google Scholar
Giesekus, H.: 1966 Zur Stabilität von Strömungen viskoelastisher Flüssigkeiten. Rheol. Acta 5, 239–252.Google Scholar
Giesekus, H.: 1972 On instabilities in Poiseuille and Couette flows of viscoelastic fluids. Prog. Heat Mass Transfer 5, 187–193.Google Scholar
Ginn, R. F. & Denn, M. M., 1969 Rotational stability in viscoelastic liquids. AIChE J. 15, 450–454.Google Scholar
Gollub, J. P. & Swinney, H. L., 1975 Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35, 927–930.Google Scholar
Green, J. & Jones, W. M., 1982 Couette flow of dilute solutions of macromolecules: embryo cells and overstability. J. Fluid Mech. 119, 491–505.Google Scholar
Greenberg, M. D.: 1978 Foundations of Applied Mathematics. Prentice-Hall.
Grosch, C. E. & Salwen, H., 1968 The stability of steady and time-dependent plane Poiseuille flow. J. Fluid Mech. 34, 177–205.Google Scholar
Hayes, J. W. & Hutton, J. F., 1972 The effect of very dilute polymer solutions on the formation of Taylor vortices. Comparison of theory with experiment. Prog. Heat Mass Transfer 5, 195–209.Google Scholar
Jones, W. M., Davies, D. M. & Thomas, M. C., 1973 Taylor vortices and the evaluation of material constants: a critical assessment. J. Fluid Mech. 60, 19–41.Google Scholar
Keentok, M., Georgescu, A. G., Sherwood, A. A. & Tanner, R. I., 1980 The measurement of the second normal stress difference for some polymer solutions. J. Non-Newtonian Fluid Mech. 6, 303–324.Google Scholar
Lockett, F. J. & Rivlin, R. S., 1968 Stability in Couette flow of a viscoelastic fluid. Part I. J. Méc. 7, 475–498.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
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
Magda, J. J., Lou, J., Baek, S.-G. & DeVries, K. L. 1990 The second normal stress difference of a Boyer fluid. Polymer (submitted).Google Scholar
Miller, C. & Goddard, J. D., 1979 Appendix to Goddard, J. D. 1979 Polymer fluid mechanics. Adv. Appl. Mech. 19, 143–219.Google Scholar
Phan-Thien 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. 17, 37–44.Google Scholar
Prilutski, G., Gupta, R. K., Sridhar, T. & Ryan, M. E., 1983 Model viscoelastic liquids. J. Non-Newtonian Fluid Mech. 12, 233–241.Google Scholar
Ramachandran, S., Gao, H. W. & Christiansen, E. B., 1985 Dependence of viscoelastic flow functions on molecular structure for linear and branched polymers. Macromolecules 18, 695–699.Google Scholar
Rubin, H. & Elata, C., 1966 Stability of Couette flow of dilute polymer solutions. Phys. Fluids 9, 1929–1933.Google Scholar
Shaqfeh, E. S. G. & Acrivos, A. 1987 The effects of inertia on the stability of the convective flow in inclined particle settlers. Phys. Fluids 30, 960–973.Google Scholar
Shaqfeh, E. S. G., Larson, R. G. & Fredrickson, G. H., 1989 The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number. J. Non-Newtonian Fluid Mech. 31, 87–113.Google Scholar
Smith, M. M. & Rivlin, R. S., 1972 Stability in Couette flow of a viscoelastic fluid. Part II. J. Méc. 11, 69–94.Google Scholar
Sridhar, T., Gupta, R. K., Boger, D. V. & Binnington, R., 1986 Steady spinning of the Oldroyd fluid B. II: Experimental results. J. Non-Newtonian Fluid Mech. 21, 115–126.Google Scholar
Sun, Z.-S. & Denn, M. M. 1972 Stability of rotational Couette flow of polymer solutions. AIChE J. 18, 1010–1015.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
Thomas, R. H. & Walters, K., 1964 The stability of elastico-viscous flow between rotating cylinders. Part 1. J. Fluid Mech. 18, 33–43.Google Scholar
Weissenberg, K.: 1947 A continuum theory of rheological phenomena. Nature 159, 310–311.Google Scholar