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  • Journal of Fluid Mechanics, Volume 642
  • January 2010, pp. 349-372

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)
  • DOI: http://dx.doi.org/10.1017/S0022112009991832
  • Published online: 08 December 2009
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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Email address for correspondence: cherif.nouar@ensem.inpl-nancy.fr, alessandro.bottaro@unige.it
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M. A. Abbas & C. T. Crowe 1987 Experimental study of the flow properties of homogeneous slurry near transitional Reynolds numbers. Intl J. Multiph. Flow 13, 387–364.

H. A. Barnes 1999 The yield stress: a review or ‘παντα ρει’: everything flows? J. Non-Newton. Fluid Mech. 81, 133178.

G. Ben Dov & J. Cohen 2007 aCritical Reynolds number for a natural transition to turbulence in pipe flows. Phys. Rev. Lett. 98, 064503.

M. Bercovier & M. Engelman 1980 A finite-element method for incompressible non-Newtonian flows J. Comput. Phys. 36, 313326.

L. B. Bergström 2005 Nonmodal growth of three-dimensional disturbances on plane Couette–Poiseuille flows. Phys. Fluids 17, 014105.1–014105.10

C. R. Beverly & R. I. Tanner 1992 Numerical analysis of three-dimensional Bingham plastic flow. J. Non-Newton. Fluid Mech. 42, 85115.

D. Biau & A. Bottaro 2004 Transient growth and minimal defects: two possible initial paths of transition to turbulence in plane shear flows. Phys. Fluids 16, 35153529.

D. Biau & A. Bottaro 2009 An optimal path to transition in a duct. Phil. Trans. R. Soc, A 367, 529544.





D. De Kee & C. F. Chan Man Fong 1993 A true yield stress? J. Rheol. 37, 775776.

D. W. Dodge & A. B. Metzner 1959 Turbulent Flow of Non-Newtonian Systems. A.I.Ch.E J. 5, 189204.

B. Eckhardt , T. M. Schneider , B. Hof & J. Westerweel 2007 Turbulence transition in pipe flow Annu. Rev. Fluid Mech. 39, 447468.

M. P. Escudier , R. J. Poole , F. Presti , C. Dales , C. Nouar , L. Graham & L. Pullum 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.

M. P. Escudier & F. Presti 1996 Pipe flow of thixotropic liquid. J. Non-Newton. Fluid Mech. 62, 291306.

A. Esmael & C. Nouar 2008 Transitional flow of a yield-stress fluid in a pipe: evidence of a robust coherent structure. Phys. Rev. E 77, 057302.


I. A. Frigaard & C. Nouar 2003 On three-dimensional linear stability of Poiseuille flow of Bingham fluids. Phys. Fluids 15, 28432851.

I. A. Frigaard & C. Nouar 2005 On the usage of viscosity regularisation methods for visco-plastic fluid flow computation. J. Non-Newton. Fluid Mech. 127, 126.


R. Govindarajan , V. S. L'vov & I. Procaccia 2001 Retardation of the onset of turbulence by minor viscosity contrast. Phys. Rev. Lett. 87, 174501.1–174501.4.

G. K. Gupta 1999 Hydrodynamic stability analysis of the plane Poiseuille flow of an electrorheological fluid. Intl J. Nonlinear Mech. 34, 589602.


B. Guzel , I. Frigaard & M. Martinez 2009 aPredicting laminar-turbulent transition in Poiseuille pipe flow for non-Newtonian fluids. Chem. Engng Sci. 64, 254264.

R. W. Hanks 1963 The laminar turbulent transition for fluids with a yield stress. A.I.Ch.E. J. 9, 306309.


W. Kozicki , C. Chou & C. Tiou 1966 Non-Newtonian flow in ducts of arbirary cross-sectional shape. Chem. Engng Sci. 21, 665679.



P. Mishra & G. Tripathi 1971 Transition from laminar to turbulent flow of purely viscous non-Newtonian fluids in tubes. Chem. Engng Sci. 26, 915921.

Q. D. Nguyen & D. V. Boger 1992 Measuring the flow properties of yield stress fluids. Annu. Rev. Fluid Mech. 24, 4788.

C. Nouar & I. Frigaard 2001 Nonlinear stability of Poiseuille flow of a Bingham fluid: theoretical results and comparison with phenomenological criteria. J. Non-Newton. Fluid Mech. 100, 127149.


T. C. Papanastasiou 1987 Flows of materials with yield. J. Rheol. 31, 385404.

J. T. Park , R. J. Mannheimer , T. A. Grimley & T. B. Morrow 1989 Pipe flow measurements of a transparent non-Newtonian Slurry. ASME J. Fluids Engng 111, 331336.

J. Peixinho & T. Mullin 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96, 094501.

J. Peixinho , C. Nouar , C. Desaubry & B. Théron 2005 Laminar transitional and turbulent flow of yield stress fluid in a pipe. J. Non-Newton. Fluid Mech. 128, 172184.

J. Philip , A. Svizher & J. Cohen 2007 Scaling law for a subcritical transition in plane Poiseuille flow. Phys. Rev. Lett. 98, 154502.


N. W. Ryan & M. M. Johnson 1959 Transition from laminar to turbulent flow in pipes. A.I.Ch.E. J. 5, 433435.

R. G. Shaver & E. W. Merill 1959 Turbulent flow of pseudoplastic polymer solutions in straight cylindrical tubes. A.I.Ch.E. J. 5, 181188.

L. N. Trefethen , A. E. Trefethen , S. C. Reddy & T. A. Driscoll 1993 Hydrodynamic stability without eigenvalues. Sciences 261, 578584.

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