Skip to main content Accesibility Help

A mechanism for oscillatory instability in viscoelastic cross-slot flow

  • LI XI (a1) and MICHAEL D. GRAHAM (a1)

Interior stagnation-point flows of viscoelastic liquids arise in a wide variety of applications including extensional viscometry, polymer processing and microfluidics. Experimentally, these flows have long been known to exhibit instabilities, but the mechanisms underlying them have not previously been elucidated. We computationally demonstrate the existence of a supercritical oscillatory instability of low-Reynolds-number viscoelastic flow in a two-dimensional cross-slot geometry. The fluctuations are closely associated with the ‘birefringent strand’ of highly stretched polymer chains associated with the outflow from the stagnation point at high Weissenberg number. Additionally, we describe the mechanism of instability, which arises from the coupling of flow with extensional stresses and their steep gradients in the stagnation-point region.

Corresponding author
Email address for correspondence:
Hide All
Arratia, P. E., Thomas, C. C., Diario, J. & Gollub, J. P. 2006 Elastic instabilities of polymer solutions in cross-channel flow. Phys. Rev. Lett. 96, 144502.
Baaijens, F. P. T. 1998 Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newtonian Fluid Mech. 79, 361385.
Baaijens, F. P. T., Selen, S. H. A., Baaijens, H. P. W., Peters, G. W. M. & Meijer, H. E. H. 1997 Viscoelastic flow past a confined cylinder of a low density polyethylene melt. J. Non-Newtonian Fluid Mech. 68, 173203.
Bird, R. B., Curtis, C. F., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn, vol. 2. Wiley Interscience.
Bisgaard, C. 1983 Velocity fields around spheres and bubbles investigated by laser-Doppler anemometry. J. Non-Newtonian Fluid Mech. 12, 283302.
Bisgaard, C. & Hassager, O. 1982 An experimental investigation of velocity fields around spheres and bubbles moving in non-Newtonian fluids. Rheol. Acta. 21, 537539.
Broadbent, J. M., Pountney, D. C. & Walters, K. 1978 Experimental and theoretical aspects of the two-roll mill problem. J. Non-Newtonian Fluid Mech. 3, 359378.
Brooks, A. N. & Hughes, T. J. R. 1982 Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Math. Appl. Mech. Eng. 32, 199259.
Chow, A., Keller, A., Müller, A. J. & Odell, J. A. 1988 Entanglements in polymer solutions under elongational flow: a combined study of chain stretching, flow velocimetry, and elongational viscosity. Macromolecules 21, 250256.
Graham, M. D. 1998 Effect of axial flow on viscoelastic Taylor–Couette instability. J. Fluid Mech. 360, 341374.
Graham, M. D. 2003 Interfacial hoop stress and instability of viscoelastic free surface flows. Phys. Fluid 15, 17021710.
Groisman, A. & Steinberg, V. 2000 Elastic turbulence in a polymer solution flow. Nature 405, 5355.
Groisman, A. & Steinberg, V. 2001 Efficient mixing at low Reynolds numbers using polymer additives. Nature 410, 905908.
Groisman, A. & Quake, S. R. 2004 A microfluidic rectifier: anisotropic flow resistance at low Reynolds number. Phys. Rev. Lett. 92, 094501.
Groisman, A., Enzelberger, M. & Quake, S. R. 2003 Microfluidic memory and control devices. Science 300, 955958.
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
Harlen, O. G. 2002 The negative wake behind a sphere sedimenting through a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 108, 411430.
Harlen, O. G., Rallison, J. M. & Chilcott, M. D. 1990 High-Deborah-number flows of dilute polymer solutions. J. Non-Newtonian Fluid Mech. 34, 319349.
Harris, O. J. & Rallison, J. M. 1993 Start-up of a strongly extensional flow of a dilute polymer solution. J. Non-Newtonian Fluid Mech. 50, 89124.
Harris, O. J. & Rallison, J. M. 1994 Instabilities of a stagnation point flow of a dilute polymer solution. J. Non-Newtonian Fluid Mech. 55, 5990.
Hassager, O. 1979 Negative wake behind bubbles in non-Newtonian liquids. Nature 279, 402403.
Joo, Y. L. & Shaqfeh, E. S. G. 1994 Observations of purely elastic instabilities in Taylor–Dean flow of a Boger fluid. J. Fluid Mech. 262, 2773.
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta. 31, 213263.
Larson, R. G., Muller, S. J. & Shaqfeh, E. S. G. 1990 A purely elastic instability in Taylor–Couette flow. J. Fluid Mech. 218, 573600.
McKinley, G. H., Armstrong, R. C. & Brown, R. A. 1993 The wake instability in viscoelastic flow past confined circular cylinders. Phil. Trans. R. Soc. Lond. A 344, 265304.
Magda, J. J. & Larson, R. G. 1988 A transition occurring in ideal elastic liquids during shear flow. J. Non-Newtonian Fluid Mech. 30, 119.
Müller, A. J., Odell, J. A. & Keller, A. 1988 Elongational flow and rheology of monodisperse polymers in solution. J. Non-Newtonian Fluid Mech. 30, 99118.
Muller, S. J., Shaqfeh, E. S. G. & Larson, R. G. 1989 A purely elastic transition in Taylor–Couette flow. Rheol. Acta. 28, 499503.
Ng, R. C. Y. & Leal, L. G. 1993 Concentration effects on birefringence and flow modification of semidilute polymer solutions in extensional flows. J. Rheol. 37, 443468.
Pakdel, P. & McKinley, G. H. 1996 Elastic instability and curved streamlines. Phys. Rev. Lett. 77, 24592462.
Poole, R. J., Alves, M. A. & Oliveira, P. J. 2007 Purely elastic flow asymmetries. Phys. Rev. Lett. 99, 164503.
Remmelgas, J., Singh, P. & Leal, L. G. 1999 Computational studies of nonlinear elastic dumbbell models of Boger fluids in a cross-slot flow. J. Non-Newtonian Fluid Mech. 88, 3161.
Renardy, M. 2006 A comment on smoothness of viscoelastic stresses. J. Non-Newtonian Fluid Mech. 138, 204205.
Shaqfeh, E. S. G. 1996 Purely elastic instabilities in viscometric flow. Annu. Rev. Fluid. Mech. 28, 129185.
Spiegelberg, S. H. & McKinley, G. H. 1996 Stress relaxation and elastic decohesion of viscoelastic polymer solutions in extensional flow. J. Non-Newtonian Fluid Mech. 67, 4976.
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at nanoliter scale. Rev. Mod. Phys. 77, 9771026.
Thomases, B. & Shelley, M. 2007 Emergence of singular structures in Oldroyd-B fluids. Phys. Fluid 19, 103103.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


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