Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2025-01-04T04:46:38.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite amplitude instability of second-order fluids in plane Poiseuille flow

Published online by Cambridge University Press:  29 March 2006

Larry V. McIntire  and
C. H. Lin
Larry V. McIntire
Affiliation:
Department of Chemical Engineering, Rice University
C. H. Lin
Affiliation:
Department of Chemical Engineering, Rice University
Get access Rights & Permissions [Opens in a new window]

Abstract

The hydrodynamic stability of plane Poiseuille flow of second—order fluids to finite amplitude disturbances is examined using the method of Stuart, and Watson as extended by Reynolds & Potter. For slightly non-Newtonian fluids subcritical instabilities are predicted. No supercritical equilibrium states are expected if the entire spectrum of disturbance wavelengths is present. Possible implications with respect to the Toms phenomenon are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 52 , Issue 2 , 28 March 1972 , pp. 273 - 285
Copyright
© 1972 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bernstein, B. & Tlapa, G. 1970 Stability of a relaxation-type viscoelastic fluid with slight elasticity. Phys. Fluids, 13, 565.Google Scholar
Chun, D. H. & Schwarz, W. H. 1968 Stability of a plane Poiseuille flow of a second-order fluid. Phys. Fl∼uids, 11, 5.Google Scholar
Craik, A. D. D. 1968 A note on the static stability of an elastico-viscous fluid. J. Fluid Mech. 33, 33.Google Scholar
Datta, S. D. 1964 Note on the stability of an elasticoviscous liquid in Couette flow. Phys. Fluids, 7, 1915.Google Scholar
Denn, M. M. & Ginn, R. F. 1969 Rotational stability in viscoelastic liquids: theory. A.I.Ch.E. J. 15, 450.Google Scholar
Denn, M. M. & Roisman, J. J. 1969 Rotational stability and measurement of normal stress functions in dilute polymer solutions. A.I.Ch.E. J. 15, 454.Google Scholar
Eckhaus, W. 1965 Studies in Non-Linear Stability Theory. Springer.
Finlayson, B. A. 1968 The Galerkin method applied to convective instability problems. J. Fluid Mech. 33, 201.Google Scholar
Jones, J. R. & Walters, T. S. 1967 Flow of elastico-viscous liquids in channels under the influence of a periodic pressure gradient. Rheol. Acta, 6, 240.Google Scholar
Landahl, M. T. & Kaplan, R. E. 1965 The effect of compliant walls on boundary layer stability and transition. Actardograph, 97, 369.Google Scholar
Lee, L. H. & Reynolds, W. C. 1967 On the approximate and numerical solution of On-Somerfeld problems. Q. J. Mech. Appl. Math. 20, 1.Google Scholar
Mcintire, L. V. & Schowalter, W. R. 1970 Stability of viscoelastic fluids: plane Couette flow with superposed temperature gradient. Trans. Soc. Rheol. 14, 585.Google Scholar
Metzner, A. B. & Park, M. G. 1964 Turbulent flow characteristics of viscoelastic fluids. J. Fluid Mech. 20, 291.Google Scholar
Paterson, R. W. & Abernathy, F. H. 1970 Turbulent flow drag reduction and degradation with dilute polymer solutions. J. Fluid Mech. 43, 689.Google Scholar
Pippin, A. C. 1964a Annular effect in viscoelastic fluids. Phys. Fluids, 7, 1143.Google Scholar
Pippin, A. C. 1964b Alternating flow of non-Newtonian fluids in tubes of arbitrary cross section. Archs. Ration. Mech. Analysis, 15, 1.Google Scholar
Platten, J. & Schechter, R. S. 1970 Stability of the flow of a slightly viscoelastic fluid. Phys. Fluids, 13, 832.Google Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 3.Google Scholar
Savins, J. G. 1969 Contrasts in the solution drag reduction characteristics of polymeric solutions and mioellar systems. In Viscozcs Drag Reduction (ed. C. S. Wells). Plenum.
Seyer, F. A. 1970 Friction reduction in turbulent flow of polymer solution. J. Fluid Mech. 40, 807.Google Scholar
Shen, S. F. 1964 Stability of Laminar Flows. In Theory of Laminar Flows (ed. F. K. Moore). Princeton University Press.
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1.Google Scholar
Stuaet, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fzuid Mech. 9, 353.Google Scholar
Thomas, L. H. 1953 The stability of plane Poiseuille flow. Phys. Rev. 91, 780.Google Scholar
Toms, B. A. 1948 Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. Proc. First Int. Cong. Rheol. 2, 135.Google Scholar
Virk, P. S. & Merrill, E. W. 1969 The onset of dilute polymer solution phenomena. In Viscous Drag Reduction (ed. C. S. Wells). Plenum.
Walters, K. & Thomas, R. H. 1964 The stability of elastico-viscous flow between rotating cylinders. J. Fluid Mech. 19, 557.Google Scholar
Walters, K. & Townsend, P. 1970 The flow of viscous and elastico-viscous liquids in straight pipes under a varying pressure gradient. Proc. Fqth Int. Cong. Rheol. 4, 471.Google Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable flows. Part 2. J. Fluid Mech. 9, 371.Google Scholar
White, D. A. 1970 Correlation of pressure drop data in pipe flow of dilute polymer solutions. Chern. Engng. Sci. 25, 1177.Google Scholar