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Self-similar spiral instabilities in elastic flows between a cone and a plate

Published online by Cambridge University Press:  26 April 2006

Gareth H. Mckinley ,
Alparslan Öztekin ,
Jeffrey A. Byars  and
Robert A. Brown
Gareth H. Mckinley
Affiliation:
Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Alparslan Öztekin
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Jeffrey A. Byars
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
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Abstract

Experimental observations and linear stability analysis are used to quantitatively describe a purely elastic flow instability in the inertialess motion of a viscoelastic fluid confined between a rotating cone and a stationary circular disk. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the spatially homogeneous azimuthal base flow that is stable in the limit of small Reynolds numbers and small cone angles becomes unstable with respect to non-axisymmetric disturbances in the form of spiral vortices that extend throughout the fluid sample. Digital video-imaging measurements of the spatial and temporal dynamics of the instability in a highly elastic, constant-viscosity fluid show that the resulting secondary flow is composed of logarithmically spaced spiral roll cells that extend across the disk in the self-similar form of a Bernoulli Spiral.

Linear stability analyses are reported for the quasi-linear Oldroyd-B constitutive equation and the nonlinear dumbbell model proposed by Chilcott & Rallison. Introduction of a radial coordinate transformation yields an accurate description of the logarithmic spiral instabilities observed experimentally, and substitution into the linearized disturbance equations leads to a separable eigenvalue problem. Experiments and calculations for two different elastic fluids and for a range of cone angles and Deborah numbers are presented to systematically explore the effects of geometric and rheological variations on the spiral instability. Excellent quantitative agreement is obtained between the predicted and measured wavenumber, wave speed and spiral mode of the elastic instability. The Oldroyd-B model correctly predicts the non-axisymmetric form of the spiral instability; however, incorporation of a shear-rate-dependent first normal stress difference via the nonlinear Chilcott–Rallison model is shown to be essential in describing the variation of the stability boundaries with increasing shear rate.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 285 , 25 February 1995 , pp. 123 - 164
Copyright
© 1995 Cambridge University Press

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References

Bird, R. B., Armstrong, R. C. & Hassager, O. 1987a Dynamics of Polymeric Liquids. Volume 1: Fluid Mechanics, 2nd Edn, Wiley Interscience.Google Scholar
Bird, R. B., Curtiss, C. F., Armstrong, R. C. & Hassager, O. 1987b Dynamics of Polymeric Liquids. Volume 2: Kinetic Theory, 2nd Edn, Wiley Interscience.Google Scholar
Bird, R. B., Stewart, E. S. & Lightfoot, E. N. 1960 Transport Phenomena. Wiley Interscience.Google Scholar
Boger, D. V. 1977/78 A highly elastic constant-viscosity fluid. J. Non-Newtonian Fluid Mech. 3, 87–91.Google Scholar
Byars, J. A., Öztekin, A., Brown, R. A. & McKinley, G. H. 1994 Spiral instabilities in the flow of highly elastic fluids between rotating parallel disks. J. Fluid Mech. 271, 173–218.Google Scholar
Chilcott, M. D. & Rallison, J. M. 1988 Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newtonian Fluid Mech. 29, 381–432.Google Scholar
Evans, A. R., Shaqfeh, E. S. G. & Frattini, P. L. 1994 Observations of polymer conformation during flow through a fixed fibre bed. J. Fluid Mech. 281, 319–356.Google Scholar
Giesekus, H. 1963 Some secondary flow phenomena in general viscoelastic fluids. In Proc. 4th Intl Congr. Rheology, Brown Univ., Vol. 1, pp. 249–266. Wiley.Google Scholar
Griffiths, D. F. & Walters, K. 1970 On edge effects in rheometry. J. Fluid Mech. 42, 379–399.Google Scholar
Heuser, G. & Krause, E. 1979 The flow field of Newtonian fluids in cone-and-plate rheometers with small gap angles. Rheol. Acta 18, 531–564.Google Scholar
Hudson, N. E. & Ferguson, J. 1990 The shear flow properties of M1. J. Non-Newtonian Fluid Mech. 35, 159–168.Google Scholar
Joo, Y. L. & Shaqfeh, E. S. G. 1992 The effects of inertia on the viscoelastic Dean and Taylor–Couette flow instabilities with application to coating flows. Phys. Fluids A 4, 2415–2431.Google Scholar
Joo, Y. L. & Shaqfeh, E. S. G. 1994 Observations of purely elastic instabilities in the Taylor–Dean flow of a Boger fluid. J. Fluid Mech. 262, 27–73.Google Scholar
Kocherov, V. L., Lukach, Y. L., Sporyagin, E. A. & Vinogradov, G. V. 1973 Flow of polymer melts in a disc-type extruder and in rotational devices of the ‘cone-plate’ and ‘plate-plate’ type. Polymer Engng Sci. 13, 194–201.Google Scholar
Kulicke, W. M. & Porter, R. S. 1979 Irregularities in steady flow for non-Newtonian fluids between cone and plate. J. Appl. Polymer Sci. 23, 953–965.Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31, 213–263.Google Scholar
Larson, R. G., Muller, S. J. & Shaqfeh, E. S. G. 1994 The effect of fluid rheology on the elastic Taylor–Couette flow instability. J. Non-Newtonian Fluid Mech. 51, 195–225.Google Scholar
Laun, H. M. & Hingmann, R. 1990 Rheological characterization of the fluid M1 and of its components. J. Non-Newtonian Fluid Mech. 35, 137–157.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
McKinley, G. H. 1991 The nonlinear dynamics of viscoelastic flow in complex geometries, PhD thesis, MIT.CrossRefGoogle Scholar
McKinley, G. H., Byars, J. A., Brown, R. A. & Armstrong, R. C. 1991 Observations on the elastic instability in cone-and-plate and parallel-plate flows of a polyisobutylene Boger fluid. J. Non-Newtonian Fluid Mech. 40, 201–229.Google Scholar
Olagunju, D. O. 1993 Asymptotic analysis of the finite cone-and-plate flow of a non-Newtonian fluid. J. Non-Newtonian Fluid Mech. 50, 289–305.Google Scholar
Olagunju, D. O. & Cook, L. P. 1992 Secondary flows in cone and plate flow of an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 46, 29–47.Google Scholar
Olagunju, D. O. & Cook, L. P. 1993 Linear stability analysis of cone-and-plate flow of an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 47, 93–105.Google Scholar
Öztekin, A. & Brown, R. A. 1993 Instability of a viscoelastic fluid between rotating parallel disks: analysis for the Oldroyd-B fluid. J. Fluid Mech. 255, 473–502.Google Scholar
Öztekin, A., Brown, R. A. & McKinley, G. H. 1994 Quantitative prediction of the viscoelastic instability in cone-and-plate flow of a Boger fluid using a multi-mode Giesekus model. J. Non-Newtonian Fluid Mech. 54, 351–379.Google Scholar
Peitgen, H.-O., Jürgens, H. & Saupe, D. 1992 Chaos and Fractals. Springer.CrossRefGoogle 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
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
Rallison, J. M. & Hinch, E. J. 1988 Do we understand the physics in the constitutive equation? J. Non-Newtonian Fluid Mech. 29, 37–55.Google Scholar
Sdougos, H. P., Bussolari, S. R. & Dewey, C. F. 1984 Secondary flow and turbulence in a cone-and-plate device. J. Fluid Mech. 138, 379–404.Google Scholar
Shaqfeh, E. S. G. 1993 Weakly nonlinear analysis of the elastic Taylor–Couette instability. Presentation at the VIIIth Intl Workshop on Num. Meth. in Viscoelastic Flows, Cape Cod, Oct. 21–24.Google Scholar
Steiert, P. & Wolff, C. 1990 Rheological properties of a polyisobutylene in a kerosene/polybutene mixture in simple shear flow. J. Non-Newtonian Fluid Mech. 35, 189–196.Google Scholar
Sureshkumar, R., Beris, A. N. & Avgousti, M. 1994 Non-axisymmetric subcritical bifurcations in viscoelastic Taylor–Couette flow. Proc. R. Soc. Lond. A (in press).Google Scholar
Tirtaatmadja, V. & Sridhar, T. 1993 A filament stretching device for measurement of extensional viscosity. J. Rheol. 37, 1081–1102.Google Scholar
Turian, R. M. 1972 Perturbation solution of the steady Newtonian flow in the cone-and-plate and parallel-plate systems. Ind. Engng Chem. Fundam. 11, 361–368.Google Scholar
Walters, K. & Waters, N. D. 1968 On the use of a rheogoniometer. Part I–Steady shear. In Polymer Systems: Deformation and Flow (ed. A. D. Wetton & A. Whorlow).Google Scholar