Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T06:37:15.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of the flow of a Bingham fluid in a channel: eigenvalue sensitivity, minimal defects and scaling laws of transition

Published online by Cambridge University Press:  08 December 2009

CHERIF NOUAR  and
ALESSANDRO BOTTARO
CHERIF NOUAR
Affiliation:
LEMTA, UMR 7563 (CNRS-INPL-UHP), 2 Avenue de la Forêt de Haye, BP 160 54504 Vandoeuvre Lès Nancy, France
ALESSANDRO BOTTARO*
Affiliation:
DICAT, Facoltà di Ingegneria, Università di Genova, Via Montallegro 1, 16145 Genova, Italy
*
†Email address for correspondence: cherif.nouar@ensem.inpl-nancy.fr, alessandro.bottaro@unige.it
Get access Rights & Permissions [Opens in a new window]

Abstract

It has been recently shown that the flow of a Bingham fluid in a channel is always linearly stable (Nouar et al., J. Fluid Mech., vol. 577, 2007, p. 211). To identify possible paths of transition we revisit the problem for the case in which the idealized base flow is slightly perturbed. No attempt is made to reproduce or model the perturbations arising in experimental environments – which may be due to the improper alignment of the channel walls or to imperfect inflow conditions – rather a general formulation is given which yields the transfer function (the sensitivity) for each eigenmode of the spectrum to arbitrary defects in the base flow. It is first established that such a function, for the case of the most sensitive eigenmode, displays a very weak selectivity to variations in the spanwise wavenumber of the disturbance mode. This justifies a further look into the class of spanwise homogeneous modes. A variational procedure is set up to identify the base flow defect of minimal norm capable of optimally destabilizing an otherwise stable flow; it is found that very weak defects are indeed capable to excite exponentially amplified streamwise travelling waves. The associated variations in viscosity are situated mostly near the critical layer of the inviscid problem. Neutrally stable conditions are found as function of the Reynolds number and the Bingham number, providing scalings of critical values with the amplitude of the defect consistent with previous experimental and numerical studies. Finally, a structured pseudospectrum analysis is performed; it is argued that such a class of pseudospectra provides information well suited to hydrodynamic stability purposes.

JFM classification

Instability: Instability Instability: Transition to turbulence Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 642 , 10 January 2010 , pp. 349 - 372
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Abbas, M. A. & Crowe, C. T. 1987 Experimental study of the flow properties of homogeneous slurry near transitional Reynolds numbers. Intl J. Multiph. Flow 13, 387–364.CrossRefGoogle Scholar
Balas, G. J., Doyle, J. C., Glover, K., Packard, A. & Smith, R. 2001 μ-Analysis and Synthesis Toolbox. User's Guide. Version 4. The Mathworks.Google Scholar
Balmforth, N. J. & Craster, R. V. 1999 A consistent thin-layer theory for Bingham plastics J. Non-Newton. Fluid Mech. 814, 65–81.CrossRefGoogle Scholar
Barnes, H. A. 1999 The yield stress: a review or ‘παντα ρει’: everything flows? J. Non-Newton. Fluid Mech. 81, 133–178.CrossRefGoogle Scholar
Ben Dov, G. & Cohen, J. 2007 a Critical Reynolds number for a natural transition to turbulence in pipe flows. Phys. Rev. Lett. 98, 064503.CrossRefGoogle ScholarPubMed
Ben Dov, G. & Cohen, J. 2007 b Instability of optimal non-axisymmetric base-flow deviations in pipe Poiseuille flow. J. Fluid Mech. 588, 189–215.CrossRefGoogle Scholar
Bercovier, M. & Engelman, M. 1980 A finite-element method for incompressible non-Newtonian flows J. Comput. Phys. 36, 313–326.CrossRefGoogle Scholar
Bergström, L. B. 2005 Nonmodal growth of three-dimensional disturbances on plane Couette–Poiseuille flows. Phys. Fluids 17, 014105.1–014105.10CrossRefGoogle Scholar
Beverly, C. R. & Tanner, R. I. 1992 Numerical analysis of three-dimensional Bingham plastic flow. J. Non-Newton. Fluid Mech. 42, 85–115.CrossRefGoogle Scholar
Biau, D. & Bottaro, A. 2004 Transient growth and minimal defects: two possible initial paths of transition to turbulence in plane shear flows. Phys. Fluids 16, 3515–3529.CrossRefGoogle Scholar
Biau, D. & Bottaro, A. 2009 An optimal path to transition in a duct. Phil. Trans. R. Soc, A 367, 529–544.CrossRefGoogle Scholar
Biau, D., Soueid, H. & Bottaro, A. 2008 Transition to turbulence in duct flow. J. Fluid Mech. 596, 133–142.CrossRefGoogle Scholar
Bird, R. B., Dai, G. C. & Yarusso, B. J. 1983 The rheology and flow of viscoplastic materials. Rev. Chem. Engng 1, 1–70.CrossRefGoogle Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.CrossRefGoogle Scholar
Chapman, S. J. 2002 Subcritical transition in channel flows. J. Fluid Mech. 451, 35–97.CrossRefGoogle Scholar
Coussot, P. 1999 Saffman–Taylor instability in yield-stress fluids. J. Fluid Mech. 380, 363–376.CrossRefGoogle Scholar
De Kee, D. & Chan Man Fong, C. F. 1993 A true yield stress? J. Rheol. 37, 775–776.CrossRefGoogle Scholar
Dodge, D. W. & Metzner, A. B. 1959 Turbulent Flow of Non-Newtonian Systems. A.I.Ch.E J. 5, 189–204.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Escudier, M. P., Poole, R. J., Presti, F., Dales, C., Nouar, C., Graham, L. & Pullum, L. 2005 Observations of asymmetrical flow behaviour in transitional pipe flow of yield-stress and other shear thinning liquids. J. Non-Newton. Fluid Mech. 127, 143–155.CrossRefGoogle Scholar
Escudier, M. P. & Presti, F. 1996 Pipe flow of thixotropic liquid. J. Non-Newton. Fluid Mech. 62, 291–306.CrossRefGoogle Scholar
Esmael, A. & Nouar, C. 2008 Transitional flow of a yield-stress fluid in a pipe: evidence of a robust coherent structure. Phys. Rev. E 77, 057302.CrossRefGoogle Scholar
Frigaard, I. A., Howison, S. D. & Sobey, I. J. 1994 On the stability of Poiseuille flow of a Bingham fluid. J. Fluid Mech. 263, 133–150.CrossRefGoogle Scholar
Frigaard, I. A. & Nouar, C. 2003 On three-dimensional linear stability of Poiseuille flow of Bingham fluids. Phys. Fluids 15, 2843–2851.CrossRefGoogle Scholar
Frigaard, I. A. & Nouar, C. 2005 On the usage of viscosity regularisation methods for visco-plastic fluid flow computation. J. Non-Newton. Fluid Mech. 127, 1–26.CrossRefGoogle Scholar
Gavarini, I., Bottaro, A. & Nieuwstadt, F. T. M. 2004 The initial stage of transition in cylindrical pipe flow: role of optimal base-flow distortions. J. Fluid Mech. 517, 131–165.CrossRefGoogle Scholar
Georgievskii, D. V. 1993 Stability of two and three-dimensional viscoplastic flows, and generalized Squire theorem. Isv. Akad. Nauk SSR Mekh. Tverd. Tela 28, 117–123.Google Scholar
Govindarajan, R., L'vov, V. S. & Procaccia, I. 2001 Retardation of the onset of turbulence by minor viscosity contrast. Phys. Rev. Lett. 87, 174501.1–174501.4.CrossRefGoogle Scholar
Gupta, G. K. 1999 Hydrodynamic stability analysis of the plane Poiseuille flow of an electrorheological fluid. Intl J. Nonlinear Mech. 34, 589–602.CrossRefGoogle Scholar
Guzel, B., Burghelea, T., Frigaard, I. & Martinez, M. 2009 b Observation of laminar-turbulent transition of a yield stress fluid in Hagen–Poiseuille flow. J. Fluid Mech. 627, 97–128.CrossRefGoogle Scholar
Guzel, B., Frigaard, I. & Martinez, M. 2009 a Predicting laminar-turbulent transition in Poiseuille pipe flow for non-Newtonian fluids. Chem. Engng Sci. 64, 254–264.CrossRefGoogle Scholar
Hanks, R. W. 1963 The laminar turbulent transition for fluids with a yield stress. A.I.Ch.E. J. 9, 306–309.CrossRefGoogle Scholar
Hanks, R. W. & Christiansen, E. B. 1962 The laminar-turbulent transition in nonisothermal flow of pseudoplastic fluids in tubes. A.I.ChE. J. 8, 467–471.Google Scholar
Hanks, R. W. & Pratt, D. R. 1967 On the flow of Bingham plastic slurries in pipes and between parallel plates. Soc. Pet. Eng. J. 87 (4), 342–346.CrossRefGoogle Scholar
Hedström, B. O. A. 1952 Flow of plastic materials in pipes. Ind. Engng Chem. 44, 652–656.Google Scholar
Hwang, Y. & Choi, H. 2006 Control of absolute instability by basic flow modification in a parallel wake at low Reynolds number. J. Fluid Mech. 87 (4), 342–346.Google Scholar
Kreiss, G., Lundbladh, A. & Henningson, D. S. 1994 Bounds for threshold amplitudes in subcritical shear flows. J. Fluid Mech. 270, 175–198.CrossRefGoogle Scholar
Kozicki, W., Chou, C. & Tiou, C. 1966 Non-Newtonian flow in ducts of arbirary cross-sectional shape. Chem. Engng Sci. 21, 665–679.CrossRefGoogle Scholar
Lips, G. C. & Denn, M. M. 1984 Flow of Bingham fluids in complex geometries. J. Non-Newton. Fluid Mech. 14, 337–346.Google Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497–520.CrossRefGoogle Scholar
Metzner, A. B. & Park, M. G. 1964 Turbulent flow characteristics of viscoelastic fluids. J. Fluid Mech. 20 291–303.CrossRefGoogle Scholar
Metzner, A. B. & Reed, J. C. 1955 Flow of non-Newtonian fluids – Correlation of the laminar, transition and turbulent flow regions. A.I.ChE. J. 1, 434–440.Google Scholar
Meyer, W. A. 1966 A correlation of the frictional characteristics for turbulent flow of dilute viscoelastic non-Newtonian fluids in pipes. A.I.ChE. J. 12, 522–525.Google Scholar
Mishra, P. & Tripathi, G. 1971 Transition from laminar to turbulent flow of purely viscous non-Newtonian fluids in tubes. Chem. Engng Sci. 26, 915–921.CrossRefGoogle Scholar
Nguyen, Q. D. & Boger, D. V. 1992 Measuring the flow properties of yield stress fluids. Annu. Rev. Fluid Mech. 24, 47–88.CrossRefGoogle Scholar
Nouar, C. & Frigaard, I. 2001 Nonlinear stability of Poiseuille flow of a Bingham fluid: theoretical results and comparison with phenomenological criteria. J. Non-Newton. Fluid Mech. 100, 127–149.CrossRefGoogle Scholar
Nouar, C., Kabouya, N., Dusek, J. & Mamou, M. 2007 Modal and non-modal linear stability of the plane-Bingham–Poiseuille flow. J. Fluid Mech. 577, 211–239.CrossRefGoogle Scholar
Papanastasiou, T. C. 1987 Flows of materials with yield. J. Rheol. 31, 385–404.CrossRefGoogle Scholar
Park, J. T., Mannheimer, R. J., Grimley, T. A. & Morrow, T. B. 1989 Pipe flow measurements of a transparent non-Newtonian Slurry. ASME J. Fluids Engng 111, 331–336.CrossRefGoogle Scholar
Peixinho, J. & Mullin, T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96, 094501.CrossRefGoogle ScholarPubMed
Peixinho, J., Nouar, C., Desaubry, C. & Théron, B. 2005 Laminar transitional and turbulent flow of yield stress fluid in a pipe. J. Non-Newton. Fluid Mech. 128, 172–184.CrossRefGoogle Scholar
Philip, J., Svizher, A. & Cohen, J. 2007 Scaling law for a subcritical transition in plane Poiseuille flow. Phys. Rev. Lett. 98, 154502.CrossRefGoogle ScholarPubMed
Potter, M. C. 1966 Stability of plane Couette–Poiseuille flow. J. Fluid Mech. 24, 609–619.CrossRefGoogle Scholar
Ryan, N. W. & Johnson, M. M. 1959 Transition from laminar to turbulent flow in pipes. A.I.Ch.E. J. 5, 433–435.CrossRefGoogle Scholar
Shaver, R. G. & Merill, E. W. 1959 Turbulent flow of pseudoplastic polymer solutions in straight cylindrical tubes. A.I.Ch.E. J. 5, 181–188.CrossRefGoogle Scholar
Slatter, P. T. 1999 The laminar turbulent transition prediction for non-Newtonian Slurries. In Proceedings of the International Conference problems in Fluid Mechanics and Hydrology, Prague, Czech Republic, 247–256.Google Scholar
Trefethen, L. N., Chapman, S. J., Henningson, D. S., Meseguer, A., Mullin, T. & Nieuwstadt, F. T. M. 2000 Threshold amplitudes for transition to turbulence in a pipe. Numerical Analysis Report 00/17, Oxford University Computing Laboratory.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Sciences 261, 578–584.CrossRefGoogle ScholarPubMed