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Mixing of a viscoelastic fluid in a time-periodic flow

Published online by Cambridge University Press:  26 April 2006

T. C. Niederkorn  and
J. M. Ottino
T. C. Niederkorn
Affiliation:
Department of Chemical Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208, USA
J. M. Ottino
Affiliation:
Department of Chemical Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208, USA
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Abstract

We present an experimental and computational investigation of mixing of a viscoelastic fluid in two-dimensional time-periodic flows generated in an eccentric cylindrical geometry. The objective of the study is to investigate the impact of fluid elasticity on the morphological structures produced by the advection of passive tracers in chaotic flows. The relevant dimensionless numbers that quantify the rheological differences with respect to the Newtonian fluid are the Deborah number (De), defined as the ratio of the fluid timescale to the flow timescale, and the Weissenberg number (We), defined as the product of the fluid timescale and the mean shear rate. The effects of elasticity are investigated in the limit of slow flows, De ≈ 0 and We < 0.1. The experimental window of We is limited to Newtonian behaviour on the low end and the transition to three-dimensional flow on the high end; experiments show that this window is small, 0.02 < We < 0.1. Typical values of the Reynolds number and the Strouhal number are O(0.001) and O(0.1), respectively.

Results from experiments with a constant-viscosity elastic fluid and computations using the upper-convected Maxwell constitutive equation are presented. Even though the streamlines for the elastic flow are nearly indistinguishable from the Newtonian flow, small deviations in the velocity field produce large effects on chaotically advected patterns. Elasticity affects both the asymptotic coverage of a dyed passive tracer and the rate at which the tracer is stretched. In all cases the tracer undergoes exponential stretching, but on a longer timescale as the elasticity increases. According to flow conditions, elasticity might increase or decrease the degree of regularity; however, island symmetry does not seem to be affected. Similar phenomena are observed in both the experiments and computations; therefore, an analysis of the chaotic dynamics of the periodic flow using numerical techniques is possible.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 256 , November 1993 , pp. 243 - 268
Copyright
© 1993 Cambridge University Press

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References

Anderson, D. A., Tannehill, J. C. & Pletcher, R. H. 1984 Computational Fluid Mechanics and Heat Transfer. McGraw-Hill.
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Astarita, G. & Marucci, G. 1974 Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill.
Ballal, B. Y. & Rivlin, R. S. 1976 Flow of a Newtonian fluid between eccentric rotating cylinders: inertial effects. Arch. Rat. Mech. Anal. 62, 237–294.Google Scholar
Beris, A. N., Armstrong, R. C. & Brown, R. A. 1983 Perturbation theory for viscoelastic fluids between eccentric rotating cylinders. J. Non-Newtonian Fluid Mech. 13, 109–148.Google Scholar
Beris, A. N., Armstrong, R. C. & Brown, R. A. 1987 Spectral/finite-element calculations of the flow of a Maxwell fluid between eccentric rotating cylinders. J. Non-Newtonian Fluid Mech. 22, 129–167.Google Scholar
Berry, M. V. & MacKley, M. R. 1977 The six roll mill: unfolding an unstable persistently extensional flow. Phil. Trans. R. Soc. Lond. A 287, 1–16.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. 1987 Fluid Mechanics: Vol. 1, Dynamics of Polymeric Liquids. 2nd edn. John Wiley & Sons.
Boger, D. V. 1977 A highly elastic constant viscosity fluid. J. Non-Newtonian Fluid Mech. 3, 87–91.Google Scholar
Burdette, S. R., Coates, P. J., Armstrong, R. C. & Brown, R. A. 1989 Calculations of viscoelastic flow through an axisymmetric corrugated tube using the explicitly elliptic momentum equation formulation (EEME). J. Non-Newtonian Fluid Mech. 33, 1–23.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. 1986 Experimental study of Lagrangian turbulence in a Stokes flow. Proc. R. Soc. Lond. A 408, 165–174.Google Scholar
Chien, W.-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377.Google Scholar
Crochet, M. J., Davies, A. R. & Walters, K. 1984 Numerical Simulation of Non-Newtonian Flow. Elsevier.
Debbaut, B., Marchal, J. M. & Crochet, M. J. 1988 Numerical simulation of highly viscoelastic flows through an abrupt contraction. J. Non-Newtonian Fluid Mech. 29, 119–146.Google Scholar
Dupret, F. & Marchal, J. M. 1986 Sur le signe des valeurs propres du tenseur des extra-contraintes dans ecoulement de fluide de Maxwell. J. Mec. Theor. Appl. 5, 403–427.Google Scholar
Elemans, P. H. M. 1989 Modeling of the processing of incompatible polymer blends. PhD dissertation, Eindhoven University of Technology.
Franjione, J. G. & Ottino, J. M. 1987 Feasibility of numerical tracking of material lines and surfaces in chaotic flows. Phys. Fluids 30, 3641–3643.Google Scholar
Giesekus, H. 1982 A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newtonian Fluid Mech. 11, 69–109.Google Scholar
Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer.
Joseph, D. D., Renardy, M. & Saut, J. C. 1985 Hyperbolicity and change of type in the flow of viscoelastic fluid. Arch. Rat. Mech. Anal. 87, 213–251.Google Scholar
Keunings, R. 1989 Simulation of viscoelastic fluid flow. In Fundamentals of Computer Modeling for Polymer Processing (ed. Tucker C. L.), pp. 403–469. Hanser.
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
Leong, C.-W. & Ottino, J. M. 1989 Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209, 463–499.Google Scholar
Leong, C.-W. & Ottino, J. M. 1990 Increase in regularity by polymer addition during chaotic mixing in two dimensional flows. Phys. Rev. Lett. 64, 874–877.Google Scholar
Middleman, S. 1977 Fundamentals of Polymer Processing. McGraw-Hill.
Muzzio, F. J., Swanson, P. D. & Ottino, J. M. 1991 The statistics of stretching and stirring in chaotic flows. Phys. Fluids A 3, 822–834.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
Phan-Thien, N. & Tanner, R. I. 1977 A new constitutive equation derived from network theory. J. Non-Newtonian Fluid Mech. 2, 353–365.Google Scholar
Phelan, F. R. 1989 A finite difference scheme for unsteady, nearly incompressible viscoelastic flow. PhD dissertation, University of Massachusetts.
Phelan, F. R., Malone, M. F. & Winter, H. H. 1989 A purely hyperbolic model for unsteady viscoelastic flow. J. Non-Newtonian Fluid Mech. 32, 197–224.Google Scholar
Pilitsis, S. & Beris, A. N. 1989 Calculations of steady-state viscoelastic flow in an undulating tube. J. Non-Newtonian Fluid Mech. 31, 231–287.Google Scholar
Rauwendaal, C. 1991 Mixing in Polymer Processing. Marcel Dekker.
Rutkevich, I. M. 1969 Some general properties of the equations of viscoelastic incompressible fluid dynamics. J. Appl. Math. Mech. (Prikl. Mat. Mech.) 33, 42–51.Google Scholar
Swanson, P. D. 1991 Regular and chaotic mixing of viscous fluids in eccentric rotating cylinders. PhD dissertation, University of Massachusetts.
Swanson, P. D. & Ottino, J. M. 1990 A comparative computational and experimental study of chaotic mixing of viscous fluids. J. Fluid Mech. 213, 227–249.Google Scholar
Tadmor, Z. & Gogos, C. G. 1979 Principles of Polymer Processing. John Wiley & Sons.
Tanner, R. I. 1966 Plane creeping flows of incompressible second-order fluids. Phys. Fluids 9, 1246–1247.Google Scholar
Tjahjadi, M. & Ottino, J. M. 1991 Stretching and breakup of droplets in chaotic flows. J. Fluid Mech. 232, 191–219.Google Scholar