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

  • CHERIF NOUAR (a1) and ALESSANDRO BOTTARO (a2)
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.

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Corresponding author
Email address for correspondence: cherif.nouar@ensem.inpl-nancy.fr, alessandro.bottaro@unige.it
References
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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.
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.
Balmforth N. J. & Craster R. V. 1999 A consistent thin-layer theory for Bingham plastics J. Non-Newton. Fluid Mech. 814, 6581.
Barnes H. A. 1999 The yield stress: a review or ‘παντα ρει’: everything flows? J. Non-Newton. Fluid Mech. 81, 133178.
Ben Dov G. & Cohen J. 2007 a Critical Reynolds number for a natural transition to turbulence in pipe flows. Phys. Rev. Lett. 98, 064503.
Ben Dov G. & Cohen J. 2007 b Instability of optimal non-axisymmetric base-flow deviations in pipe Poiseuille flow. J. Fluid Mech. 588, 189215.
Bercovier M. & Engelman M. 1980 A finite-element method for incompressible non-Newtonian flows J. Comput. Phys. 36, 313326.
Bergström L. B. 2005 Nonmodal growth of three-dimensional disturbances on plane Couette–Poiseuille flows. Phys. Fluids 17, 014105.1–014105.10
Beverly C. R. & Tanner R. I. 1992 Numerical analysis of three-dimensional Bingham plastic flow. J. Non-Newton. Fluid Mech. 42, 85115.
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, 35153529.
Biau D. & Bottaro A. 2009 An optimal path to transition in a duct. Phil. Trans. R. Soc, A 367, 529544.
Biau D., Soueid H. & Bottaro A. 2008 Transition to turbulence in duct flow. J. Fluid Mech. 596, 133142.
Bird R. B., Dai G. C. & Yarusso B. J. 1983 The rheology and flow of viscoplastic materials. Rev. Chem. Engng 1, 170.
Bottaro A., Corbett P. & Luchini P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293302.
Chapman S. J. 2002 Subcritical transition in channel flows. J. Fluid Mech. 451, 3597.
Coussot P. 1999 Saffman–Taylor instability in yield-stress fluids. J. Fluid Mech. 380, 363376.
De Kee D. & Chan Man Fong C. F. 1993 A true yield stress? J. Rheol. 37, 775776.
Dodge D. W. & Metzner A. B. 1959 Turbulent Flow of Non-Newtonian Systems. A.I.Ch.E J. 5, 189204.
Eckhardt B., Schneider T. M., Hof B. & Westerweel J. 2007 Turbulence transition in pipe flow Annu. Rev. Fluid Mech. 39, 447468.
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, 143155.
Escudier M. P. & Presti F. 1996 Pipe flow of thixotropic liquid. J. Non-Newton. Fluid Mech. 62, 291306.
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.
Frigaard I. A., Howison S. D. & Sobey I. J. 1994 On the stability of Poiseuille flow of a Bingham fluid. J. Fluid Mech. 263, 133150.
Frigaard I. A. & Nouar C. 2003 On three-dimensional linear stability of Poiseuille flow of Bingham fluids. Phys. Fluids 15, 28432851.
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, 126.
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, 131165.
Georgievskii D. V. 1993 Stability of two and three-dimensional viscoplastic flows, and generalized Squire theorem. Isv. Akad. Nauk SSR Mekh. Tverd. Tela 28, 117123.
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.
Gupta G. K. 1999 Hydrodynamic stability analysis of the plane Poiseuille flow of an electrorheological fluid. Intl J. Nonlinear Mech. 34, 589602.
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, 97128.
Guzel B., Frigaard I. & Martinez M. 2009 a Predicting laminar-turbulent transition in Poiseuille pipe flow for non-Newtonian fluids. Chem. Engng Sci. 64, 254264.
Hanks R. W. 1963 The laminar turbulent transition for fluids with a yield stress. A.I.Ch.E. J. 9, 306309.
Hanks R. W. & Christiansen E. B. 1962 The laminar-turbulent transition in nonisothermal flow of pseudoplastic fluids in tubes. A.I.ChE. J. 8, 467471.
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), 342346.
Hedström B. O. A. 1952 Flow of plastic materials in pipes. Ind. Engng Chem. 44, 652656.
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), 342346.
Kreiss G., Lundbladh A. & Henningson D. S. 1994 Bounds for threshold amplitudes in subcritical shear flows. J. Fluid Mech. 270, 175198.
Kozicki W., Chou C. & Tiou C. 1966 Non-Newtonian flow in ducts of arbirary cross-sectional shape. Chem. Engng Sci. 21, 665679.
Lips G. C. & Denn M. M. 1984 Flow of Bingham fluids in complex geometries. J. Non-Newton. Fluid Mech. 14, 337346.
Mack L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497520.
Metzner A. B. & Park M. G. 1964 Turbulent flow characteristics of viscoelastic fluids. J. Fluid Mech. 20 291303.
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, 434440.
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, 522525.
Mishra P. & Tripathi G. 1971 Transition from laminar to turbulent flow of purely viscous non-Newtonian fluids in tubes. Chem. Engng Sci. 26, 915921.
Nguyen Q. D. & Boger D. V. 1992 Measuring the flow properties of yield stress fluids. Annu. Rev. Fluid Mech. 24, 4788.
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, 127149.
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, 211239.
Papanastasiou T. C. 1987 Flows of materials with yield. J. Rheol. 31, 385404.
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, 331336.
Peixinho J. & Mullin T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96, 094501.
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, 172184.
Philip J., Svizher A. & Cohen J. 2007 Scaling law for a subcritical transition in plane Poiseuille flow. Phys. Rev. Lett. 98, 154502.
Potter M. C. 1966 Stability of plane Couette–Poiseuille flow. J. Fluid Mech. 24, 609619.
Ryan N. W. & Johnson M. M. 1959 Transition from laminar to turbulent flow in pipes. A.I.Ch.E. J. 5, 433435.
Shaver R. G. & Merill E. W. 1959 Turbulent flow of pseudoplastic polymer solutions in straight cylindrical tubes. A.I.Ch.E. J. 5, 181188.
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, 247256.
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.
Trefethen L. N., Trefethen A. E., Reddy S. C. & Driscoll T. A. 1993 Hydrodynamic stability without eigenvalues. Sciences 261, 578584.
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