Skip to main content Accessibility help

Mixing of a viscoelastic fluid in a time-periodic flow

  • T. C. Niederkorn (a1) and J. M. Ottino (a1)


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.



Hide All
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, 121.
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, 237294.
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, 109148.
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, 129167.
Berry, M. V. & MacKley, M. R. 1977 The six roll mill: unfolding an unstable persistently extensional flow. Phil. Trans. R. Soc. Lond. A 287, 116.
Binnington, R. J. & Boger, D. V. 1985 Constant viscosity elastic liquids. J. Rheol. 29, 887904.
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, 8791.
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, 123.
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, 165174.
Chien, W.-L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355377.
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, 119146.
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, 403427.
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, 36413643.
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, 69109.
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, 213251.
Keunings, R. 1989 Simulation of viscoelastic fluid flow. In Fundamentals of Computer Modeling for Polymer Processing (ed. Tucker C. L.), pp. 403469. 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, 5192.
Leong, C.-W. & Ottino, J. M. 1989 Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209, 463499.
Leong, C.-W. & Ottino, J. M. 1990 Increase in regularity by polymer addition during chaotic mixing in two dimensional flows. Phys. Rev. Lett. 64, 874877.
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, 822834.
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, 353365.
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, 197224.
Pilitsis, S. & Beris, A. N. 1989 Calculations of steady-state viscoelastic flow in an undulating tube. J. Non-Newtonian Fluid Mech. 31, 231287.
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, 4251.
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, 227249.
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, 12461247.
Tjahjadi, M. & Ottino, J. M. 1991 Stretching and breakup of droplets in chaotic flows. J. Fluid Mech. 232, 191219.
MathJax is a JavaScript display engine for mathematics. For more information see


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed