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Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions

Published online by Cambridge University Press:  26 April 2006

Gareth H. McKinley ,
William P. Raiford ,
Robert A. Brown  and
Robert C. Armstrong
Gareth H. McKinley
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
William P. Raiford
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
Robert C. Armstrong
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

The steady-state and time-dependent flow transitions observed in a well-characterized viscoelastic fluid flowing through an abrupt axisymmetric contraction are characterized in terms of the Deborah number and contraction ratio by laser-Doppler velocimetry and flow visualization measurements. A sequence of flow transitions are identified that lead to time-periodic, quasi-periodic and aperiodic dynamics near the lip of the contraction and to the formation of an elastic vortex at the lip entrance. This lip vortex increases in intensity and expands outwards into the upstream tube as the Deborah number is increased, until a further flow instability leads to unsteady oscillations of the large elastic vortex. The values of the critical Deborah number for the onset of each of these transitions depends on the contraction ratio β, defined as the ratio of the radii of the large and small tubes. Time-dependent, three-dimensional flow near the contraction lip is observed only for contraction ratios 2 [les ] β [les ] 5, and the flow remains steady for higher contraction ratios. Rounding the corner of the 4:1 abrupt contraction leads to increased values of Deborah number for the onset of these flow transitions, but does not change the general structure of the transitions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 223 , February 1991 , pp. 411 - 456
Copyright
© 1991 Cambridge University Press

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References

Apelian, M. R., Armstrong, R. C. & Brown, R. A., 1988 Impact of the constitutive equation and singularity on the calculation of stick-slip flow: the modified Maxwell equation (MUCM). J. Non-Newtonian Fluid Mech. 27, 299–321.Google Scholar
Bagley, E. B. & Schreiber, H. P., 1961 Effect of die entry geometry on polymer melt fracture and extrudate distortion. Trans. Soc. Rheol. 5, 341–360.Google Scholar
Ballenger, T. F. & White, J. L., 1971 The development of the velocity field in polymer melts in a reservoir approaching a capillary die. J. Appl. Polymer Sci. 15, 1949–1962.Google Scholar
Bergé, P., Pomeau, Y. & Vidal, C., 1986 Order within Chaos. John Wiley & Sons.
Binding, D. M. & Walters, K., 1988 On the use of flow through a contraction in estimating the extensional viscosity of mobile polymer solutions. J. Non-Newtonian Fluid Mech. 30, 233–250.Google Scholar
Binding, D. M., Walters, K., Dheur, J. & Crochet, M. J., 1987 Interfacial effects in the flow of viscous and elastico-viscous liquids. Phil. Trans. R. Soc. Lond. A 323, 449–469.Google Scholar
Binnington, R. J. & Boger, D. V., 1985 Constant viscosity elastic liquids. J. Rheol. 29, 887–904.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O., 1987a Dynamics of Polymeric Liquids, vol. 1, 2nd edn. Wiley Interscience.
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O., 1987b Dynamics of Polymeric Liquids, vol. 2, 2nd edn. Wiley Interscience.
Bird, R. B. & DeAguiar, J. R., 1983 An encapsulated dumbbell model for concentrated polymer solutions and melts; I theoretical development and constitutive equation. J. Non-Newtonian Fluid Mech. 13, 149–160.Google Scholar
Bisgaard, C.: 1983 Velocity fields around spheres and bubbles investigated by laser-Doppler anemometry. J. Non-Newtonian Fluid Mech. 12, 283–302.Google Scholar
Boger, D. V.: 1977/78 A highly elastic constant-viscosity fluid. J. Non-Newtonian Fluid Mech. 3, 87–91.Google Scholar
Boger, D. V.: 1982 Circular entry flows of inelastic and viscoelastic fluids. In Advances in Transport Processes, vol. 2 (ed. A. S. Mujumdar & R. A. Mashelkar). Wiley Eastern.
Boger, D. V.: 1987 Viscoelastic flows through contractions. Ann. Rev. Fluid Mech. 19, 157–182.Google Scholar
Boger, D. V., Hur, D. U. & Binnington, R. J., 1986 Further observations of elastic effects in tubular entry flows. J. Non-Newtonian Fluid Mech. 20, 31–49.Google Scholar
Boger, D. V. & Nguyěñ, H. 1978 A model viscoelastic fluid. Polymer Engng Sci. 18, 1037–1043.Google Scholar
Boger, D. V. & Murthy, A. V. Rama 1972 Flow of viscoelastic fluids through an abrupt contraction. Rheol. Acta 11, 61–69.Google Scholar
Burdette, S. R.: 1989 Development of the velocity field in transient shear flows of viscoelastic fluids. J. Non-Newtonian Fluid Mech. 32, 269–294.Google Scholar
Cable, P. J. & Boger, D. V., 1978a A comprehensive experimental investigation of tubular entry flow of viscoelastic fluids: part 1, vortex characteristics in stable flow. AIChE J. 4, 869–879.Google Scholar
Cable, P. J. & Boger, D. V., 1978b A comprehensive experimental investigation of tubular entry flow of viscoelastic fluids: part 2, the velocity field in stable flow. AIChE. J. 24, 882–999.Google Scholar
Cable, P. J. & Boger, D. V., 1979 A comprehensive experimental investigation of tubular entry flow of viscoelastic fluids: part 3, unstable flow. AIChE J. 25, 152–159.Google Scholar
Coates, P. J., Armstrong, R. C. & Brown, R. A., 1991 Convergent calculations in the abrupt axisymmetric contraction using modified Maxwell models (in preparation).
Crochet, M. J.: 1988 Numerical simulation of highly viscoelastic flows. Proc. Xth Intl Congr. On Rheology, Sydney, Australia, vol. 1, pp. 19–24.Google Scholar
DeAguiar, J. R.: 1983 An encapsulated dumbbell model for concentrated polymer solutions and melts; II. Calculation of material functions and experimental comparisons. J. Non-Newtonian Fluid Mech. 13, 161–179.Google Scholar
Evans, R. E. & Walters, K., 1986 Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquids. J. Non-Newtonian Fluid Mech. 20, 11–29.Google Scholar
Evans, R. E. & Walters, K., 1989 Further remarks on the lip-vortex mechanism of vortex enhancement in planar-contraction flow. J. Non-Newtonian Fluid Mech. 32, 95–105.Google Scholar
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P., 1979 Dynamical instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103–128.Google Scholar
Giesekus, H.: 1972 On instabilities in Poiseuille and Couette flows of viscoelastic fluids. In Prog. Heat and Mass Transfer, vol. 5 (ed. W. R. Schowalter, A. V. Luikov, W. J. Minkowycz & N. H. Afgan). Pergamon.
Gollub, J. P. & Benson, S. V., 1980 Many routes to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Hassager, O.: 1988 Working group on numerical techniques, fifth international workshop on numerical methods in non-Newtonian flows, Lake Arrowhead, USA. J. Non-Newtonian Fluid Mech. 29, 2–5.Google Scholar
Iooss, G. & Joseph, D. D., 1980 Elementary Stability and Bifurcation Theory. Springer.
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
Jones, D. M., Walters, K. & Williams, P. R., 1987 On the extensional viscosity of mobile polymer solutions. Rheol. Acta 26, 20–30.Google Scholar
Keunings, R.: 1987 Simulation of viscoelastic fluid flow. In Fundamentals of Computer Modeling for Polymer Processing (ed. C. L. Tucker III). Carl Hanser.
Kim-E, M. E., Brown, R. A. & Armstrong, R. C., 1983 The roles of inertia and shear-thinning in flow of an inelastic liquid through an axisymmetric sudden contraction. J. Non-Newtonian Fluid Mech. 13, 341–363.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 Ml and of its components. J. Non-Newtonian Fluid Mech. 35, 137–157.Google Scholar
Lawler, J. V., Muller, S. J., Brown, R. A. & Armstrong, R. C., 1986 Laser Doppler velocimetry measurements of velocity fields and transitions in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 20, 51–92.Google Scholar
Lipscomb, G. G., Keunings, R. & Denn, M. M., 1987 Implications of boundary singularities in complex geometries. J. Non-Newtonian Fluid Mech. 2, 85–96.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 occurring in ideal elastic liquids during shear flow. J. Non-Newtonian Fluid Mech. 30, 1–19.Google Scholar
Marchal, J. M. & Crochet, M. J., 1987 A new mixed finite element for calculating viscoelastic flow. J. Non-Newtonian Fluid Mech. 26, 77–114.Google Scholar
Mendelson, M. A., Yeh, P.-W., Brown, R. A. & Armstrong, R. C., 1982 Approximation error in finite element calculation of viscoelastic fluid flows. J. Non-Newtonian Fluid Mech. 10, 31–54.Google Scholar
Moffatt, H. K.: 1964 Viscous and resistive eddies near sharp corners. J. Fluid Mech. 18, 1–18.Google Scholar
Muller, S. J.: 1986 Experimental analysis of flow through an axisymmetric sudden contraction: Rheological characterization and LDV measurements. Ph.D thesis, Massachusetts Institute of Technology.
Nguyen, H. & Boger, D. V., 1979 The kinematics and stability of die entry flows. J. Non-Newtonian Fluid Mech. 5, 353–368.Google Scholar
Northey, P. J., Armstrong, R. C. & Brown, R. A., 1990 Finite-element calculation of time-dependent two-dimensional viscoelastic flow with the EEME formulation. J. Non-Newtonian Fluid Mech. 36, 109–134.Google Scholar
Oldroyd, J. G.: 1950 On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541.Google Scholar
den Otter, J. L.: 1970 Mechanisms of melt fracture. Plastics Polymers 38, 155–168.Google Scholar
Petrie, C. J. S. & Denn, M. M. 1976 Instabilities in polymer processing. AIChE J. 22, 209–235.Google Scholar
Piau, J. M., El Kissi, N. & Tremblay, B. 1988 Low Reynolds number flow visualization of linear and branched silicones upstream of orifice dies. J. Non-Newtonian Fluid Mech. 30, 197–232.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
Quinzani, L. M., Mckinley, G. H., Brown, R. A. & Armstrong, R. C., 1990 Modeling the rheology of polyisobutylene solutions. J. Rheol. 34, 705–748.Google Scholar
Raiford, W. P.: 1989 Laser Doppler velocimetry measurements of nonlinear viseoelastic flow transitions in the axisymmetric sudden contraction. Ph.D thesis, M. I. T.
Raiford, W. P., Quinzani, L. M., Coates, P. J., Armstrong, R. C. & Brown, R. A., 1989 LDV measurements of viscoelastic flow transitions in abrupt axisymmetric contractions: interaction of inertia and elasticity. J. Non-Newtonian Fluid Mech. 32, 39–68.Google Scholar
Murthy, A. V. Rama 1974 Flow instabilities in a capillary rheometer for an elastic polymer solution. Trans. Soc. Rheol. 18, 431–452.Google Scholar
Viriyayuthakorn, M. & Caswell, B., 1980 Finite element simulation of viscoelastic flow. J. Non-Newtonian Fluid Mech. 6, 245–267.Google Scholar
Vlachopoulos, J. & Alam, M., 1972 Critical stress and recoverable shear for polymer melt fracture. Polymer Engng Sci. 12, 184.Google Scholar
Walters, K.: 1989 Unpublished data on the extensional rheology of fluid M1. Presented at 6th Intl Workshop on Numerical Methods in Non-Newtonian Flows, Hindsgavl, Denmark.Google Scholar
Walters, K. & Rawlinson, D. M., 1982 On some contraction flows for Boger fluids. Rheol. Acta 21, 547–552.Google Scholar
Walters, K. & Webster, M. F., 1982 On dominating elastico-viscous response in some complex flows. Phil. Trans. R. Soc. Lond. A 308, 199–218.Google Scholar
White, J. L.: 1973 Critique on flow patterns in polymer fluids at the entrance of a die and instabilities leading to extrudate distortion. Appl. Polymer Symp. 20, 155–174.Google Scholar
White, J. L. & Kondo, A., 1977/78 Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: measurement and interpretation. J. Non-Newtonian Fluid Mech. 3, 41–64.Google Scholar
White, S. A. & Baird, D. G., 1986 The importance of extensional flow properties on planar entry flow patterns of polymer melts., J. Non-Newtonian Fluid Mech. 20, 93–101.Google Scholar
White, S. A., Gotsis, A. D. & Baird, D. G., 1987 Review of the entry flow problem: experimental and numerical. J. Non-Newtonian Fluid Mech. 24, 121–160.Google Scholar
Williams, P. R. & Williams, R. W., 1985 On the planar extensional viscosity of mobile liquids. J. Non-Newtonian Fluid Mech. 19, 53–80.Google Scholar
Wunderlich, A. M., Brunn, P. O. & Durst, F., 1988 Flow of dilute polyacrylamide solutions through a sudden planar contraction. J. Non-Newtonian Fluid Mech. 28, 267–285.Google Scholar
Yoganathan, A. P. & Yarlagadda, A. P., 1984 Velocity fields of viscoelastic fluids in sudden tubular contractions. Proc. IX Intl Congress on Rheology, Acapulco, Mexico, Advances in Rheology, vol. 2 (ed. B. Mena, A. García-Rejón & C. Rangel-Nafaile), pp. 135–142.