Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T19:16:21.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly nonlinear dynamics of thermoconvective instability involving viscoplastic fluids

Published online by Cambridge University Press:  04 August 2010

C. MÉTIVIER ,
C. NOUAR  and
J.-P. BRANCHER
C. MÉTIVIER*
Affiliation:
Laboratoire de Rhéologie, UMR 5520, Domaine universitaire, BP 53, 38041 Grenoble Cedex, France
C. NOUAR
Affiliation:
LEMTA, UMR 7563, 2 avenue de la forêt de Haye, BP 160, 54504 Vandoeuvre Cedex, France
J.-P. BRANCHER
Affiliation:
LEMTA, UMR 7563, 2 avenue de la forêt de Haye, BP 160, 54504 Vandoeuvre Cedex, France
*
†Email address for correspondence: christel.metivier@ujf-grenoble.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, a weakly nonlinear stability of viscoplastic fluid flow is performed. The system consists of a plane Rayleigh–Bénard–Poiseuille (RBP) flow of a Bingham fluid. The basic flow is characterized by a central plug zone, of 2yb width, in which the stresses are smaller than the Bingham number B, the dimensionless yield stress. The Bingham model assumes that inside this zone the material moves as a rigid solid, and that outside this zone it behaves as a viscous fluid. The aim of this study is to investigate the influence of the yield stress on the instability conditions. The linear stability analysis is performed using a modal method and provides critical values of Rayleigh and wavenumbers, from which the system becomes unstable. The critical mode, i.e. the least stable mode, is also determined. This mode, also called the fundamental mode, creates perturbation harmonics which cannot be neglected above criticality. The weakly nonlinear analysis is performed for small-amplitude perturbations. In this study, the quadratic modes of the perturbation are determined. Results indicate that the nonlinear modes perturbation can attain high maximal values, which is the consequence of the high variations of viscosity in the flow. The characterization of the complex Landau equation sheds light on a transition in terms of the bifurcation nature above a critical Péclet number Pec = O(1). Below Pec, it is found that a supercritical equilibrium state could exist, such as in the Newtonian case, while above Pec, the bifurcation becomes subcritical. One observes a sharp transition from supercritical to subcritical bifurcation as the Péclet value is increased. A dependence of Pec on the yield stress is highlighted since the subcritical bifurcation is first observed for weak values of yb (yb < O(10−1)). For this range of values, the transition is mainly due to the presence of the unyielded region via non-homogeneous boundary conditions at the yield surfaces. Then for yb > O(10−1), the change of the bifurcation nature is due to the variations of the effective viscosity in the unyielded regions.

JFM classification

Convection: Convection Instability: Instability Non-Newtonian Flows: Non-Newtonian Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 660 , 10 October 2010 , pp. 316 - 353
Copyright
Copyright © Cambridge University Press 2010

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

Albaalbaki, B. & Khayat, R. E. 2008 Finite-amplitude Rayleigh–Bénard convection for weakly shear thinning fluids. J. Phys.: Conf. Ser. 137, 012024.Google Scholar
Ashrafi, N. & Khayat, R. E. 1999 Shear-thinning-induced chaos in Taylor–Couette flow. Phys. Rev. E 61 (2), 1455–1567.CrossRefGoogle Scholar
Balmforth, N. & Rust, A. 2009 Weakly nonlinear viscoplastic convection. J. Non-Newton. Fluid Mech. 158 (1–3), 36–45.CrossRefGoogle Scholar
Bingham, E. C. 1922 Fluidity and Plasticity. McGraw-Hill.Google Scholar
Carrière, P. & Monkewitz, P. 2001 Transverse-roll global modes in a Rayleigh–Bénard–Poiseuille system with streamwise variable heating. Eur. J. Mech. B Fluids 20, 751–770.Google Scholar
Cheng, P. J. & Lai, H. Y. 2008 Finite-amplitude long-wave instability of Bingham liquid films. Nonlinear Anal.: Real World Appl. 10, 1500–1513.Google Scholar
Fortin, M. & Glowinski, R. 1983 The Augmented Lagrangian Method. Elsevier.Google Scholar
Fujimura, K. 1989 The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows. Proc. R. Soc. Lond. Ser. A 424, 373–392.Google Scholar
Fujimura, K. 1997 Center manifold reduction and the Stuart–Landau equation for fluid motions. Proc. R. Soc. Lond. Ser. A 453, 181–203.CrossRefGoogle Scholar
Gage, K. S. & Reid, W. H. 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33, 21–32.CrossRefGoogle Scholar
Generalis, S. C. & Fujimura, K. 2009 The range of validity of weakly nonlinear theory in the Rayleigh–Bénard problem. J. Phys. Soc. Japan 78 (8), 084401.Google Scholar
Georgievskii, D. V. 1993 Stability of two and three dimensional viscoplastic flows and generalized Squire theorem. Isv. AN SSSR Mekh. Tverdogo Tela. 28, 117–123.Google Scholar
Gross, L. K. 2002 Weakly non-linear dynamics of interface propagation. Stud. Appl. Math. 108, 323–350.Google Scholar
Herbert, T. 1980 Nonlinear stability of parallel flows by high-order amplitude expansions. AIAA J. 18 (3), 243–248.CrossRefGoogle Scholar
Herbert, T. 1983 On perturbation methods in nonlinear stability theory. J. Fluid Mech. 126, 167–186.CrossRefGoogle Scholar
Huilgol, R. R. & Panizza, M. P. 1995 On the determination of the plug flow region in Bingham fluids through the application of variational inequalities. J. Non-Newton. Fluids Mech. 58, 207–217.CrossRefGoogle Scholar
Khayat, R. E. 1996 Chaos in thermal convection of weakly shear-thinning fluids. J. Non-Newton. Fluid Mech. 63, 153–178.CrossRefGoogle Scholar
Kuo, H. L. 1996 Solution of the nonlinear equations of cellular convection and heat transport. J. Fluid Mech. 10, 611–634.CrossRefGoogle Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260.CrossRefGoogle Scholar
Martinand, D., Carrière, P. & Monkewitz, P. 2006 Three-dimensional global instability modes associated with a localized hot spot in Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 551, 275–301.CrossRefGoogle Scholar
Métivier, C., Frigaard, I. A. & Nouar, C. 2009 Non-linear stability of the Bingham–Rayleigh–Bénard Poiseuille flow. J. Non-Newton. Fluid Mech. 158, 127–131.Google Scholar
Métivier, C. & Nouar, C. 2008 On linear stability of Rayleigh–Bénard Poiseuille flow of viscoplastic fluids. Phys. Fluids 20 (10), 104106.1–104106.7.CrossRefGoogle Scholar
Métivier, C., Nouar, C. & Brancher, J. P. 2005 Linear stability involving the Bingham model when the yield stress approaches zero. Phys. Fluids 17 (10), 104101.1–104101.14.CrossRefGoogle Scholar
Müller, H. W., Lücke, M. & Kamps, M. 1989 Convective patterns in horizontal flow. Europhys. Lett. 10 (5), 451.CrossRefGoogle Scholar
Nicolas, X. 2002 Revue bibliographique sur les écoulements de Poiseuille–Rayleigh–Bénard: écoulements de convection mixte en conduites rectangulaires horizontales chauffées par le bas. Intl J. Therm. Sci. 41, 961–1016.CrossRefGoogle Scholar
Oldroyd, J. G. 1947 a Two-dimensional plastic flow of a Bingham solid. Proc. Camb. Phil. Soc. 43, 383–395.Google Scholar
Oldroyd, J. G. 1947 b A rational formulation of the equations of plastic flow for a Bingham solid. Proc. Camb. Phil. Soc. 43, 100–105.Google Scholar
Ouazzani, M. T., Platten, J. K., Müller, H. W. & Lücke, M. 1994 Etude de la convection mixte entre deux plans horizontaux à températures différentes III. Intl J. Heat Mass Transfer 38 (5), 875–886.CrossRefGoogle Scholar
Platten, J. K. 1971 A variational formulation for the stability of flows with temperature gradients. Intl Engng Sci. 9, 855–869.CrossRefGoogle Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 465–492.Google Scholar
Schmid, P. J. & Henningson Dan, S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Vinay, G., Wachs, A. & Agassant, J. F. 2005 Numerical simulation of non-isothermal viscoplastic waxy crude oil flows. J. Non-Newton. Fluid Mech. 128, 144–162.CrossRefGoogle Scholar
Zhang, J., Vola, D. & Frigaard, I. A. 2006 Yield stress effects on Rayleigh–Bénard convection. J. Fluid Mech. 566, 389–419.CrossRefGoogle Scholar