On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation

Journal of Non-Newtonian Fluid Mechanics

Volumn 127, Issue 2-3, 2005, Pages 169-190

  Kupferman, Raz ^a  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

Couette flow; Generalized functions; Linear stability; Non normal operators; Oldroyd B model; Stress diffusion

Indexed keywords

APPROXIMATION THEORY; DIFFUSION; EIGENVALUES AND EIGENFUNCTIONS; ERRORS; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES;

CROSS-STREAM GRADIENTS; LINEAR STABILITY; NON-MODAL STRESSES; STREAMLINES;

FLOW OF FLUIDS;

COUETTE FLOW;

EID: 20344387190     PISSN: 03770257     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jnnfm.2005.03.002     Document Type: Article

References (28)

1. V. Gorodsov A. Leonov On a linear instability of a plane parallel Couette flow of viscoelastic fluids PPM 31 1966 289-299
2. M. Renardy Y. Renardy Linear stability of plane Couette flow of an upper convected Maxwell fluid J. Non-Newton. Fluid Mech. 22 1986 23-33
3. R. Sureshkumar A. Beris Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm J. Non-Newton. Fluid Mech. 56 1995 151-182
4. M. Graham Effect of axial flow on viscoelastic Taylor-Couette instability J. Fluid Mech. 360 1998 341-374
5. R. Sureshkumar M. Smith R. Armstrong R. Brown Linear stability and dynamics of viscoelastic flows using time dependent numerical simulations J. Non-Newton. Fluid Mech. 82 1999 57-104
6. H. Wilson M. Renardy Y. Renardy Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids J. Non-Newton. Fluid Mech. 80 1999 251-268
7. M. Renardy Location of the continuous spectrum in complex flows of the UCM fluid J. Non-Newton. Fluid Mech. 94 2000 75-85
8. M. Renardy A rigorous stability proof for plane Couette flow of an upper convected Maxwell fluid at zero Reynolds number Eur. J. Mech. B 11 1992 511-516
9. R. Keiller Numerical instability of time-dependent flows J. Non-Newton. Fluid Mech. 43 1992 229-246
10. R. Brown M. Szady P. Northey R. Armstrong On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations Theor. Comput. Fluid Dynam. 5 1993 77-106
11. M. Szady T. Salamon A. Liu D. Bornside R. Armstrong R. Brown A new mixed finite element method for viscoelastic flows governed by differential constitutive equations J. Non-Newton. Fluid Mech. 59 1995 215-243
12. A. Bogaerds G. Peters F. Baaijens Temporal stability of low-order continuous and discontinuous mixed finite element techniques for viscoelastic fluid mechanics Proceedings of the XIIIth International Congress on Rheology Cambridge, UK 2000
13. A. Grillet A. Bogaerds G. Peters F. Baaijens Stability analysis of constitutive equations for polymer melts in viscometric flows J. Non-Newton. Fluid Mech. 103 2002 221-250
14. M. Gerritsma, Time dependent numerical simulations of a viscoelastic fluid on a staggered grid, PhD Thesis, University of Gronigen, 1996.
15. A. El-Kareh L. Leal Existence of solutions for all Deborah numbers for a non-Newtonian model midified to include diffusion J. Non-Newton. Fluid Mech. 33 1989 257-287
16. R. Sureshkumar A. Beris Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows J. Non-Newton. Fluid Mech. 60 1995 53-80
17. T. Vaithianathan L. Collins Numerical approach to simulating turbulent flow of a viscoelastic polymer solution J. Comput. Phys. 187 2003 1-21
18. R. King M. Apelian R. Armstrong R. Brown Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries J. Non-Newton. Fluid Mech. 29 1988 147-216
19. I. Gel'fand G. Shilov Generalized Functions 1964 Academic Press New York
20. J. von Neumann Mathematical Foundations of Quantum Mechanics 1955 Princeton University Press Princeton, NJ
21. M. Abramowitz I. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1971 Dover New York
22. F. Harlow J. Welch Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface Phys. Fluids 8 1965 2182
23. L. Trefethen Pseudospectra of linear operators SIAM Rev. 39 1997 383-406
24. A. Trefethen N. Trefethen P. Schmid Spectra and pseudospectra for pipe Poiseuille flow Comput. Meth. Appl. Mech. Eng. 175 1999 413-420
25. S. Orszag A. Patera Secondary instability of wall-bounded shear flows J. Fluid Mech. 128 1983 347-385
26. K. Atalik R. Keunings Non-linear temporal stability analysis of viscoelastic channel flows using a fully spectral method J. Non-Newton. Fluid Mech. 102 2002 299-319
27. T. Write, Eigtool, a graphical tool for nonsymmetric eigenproblems. http://web.comlab.ox.ac.uk/projects/pseudospectra/eigtool/.
28. M. Embree, L. Trefethen, Generalizing eigenvalue theorems to pseudospectra theorems, SIAM J. Sci. Comput. 23 583-590.