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Stability of hexagonal pattern in Rayleigh–Bénard convection for thermodependent shear-thinning fluids

Published online by Cambridge University Press:  04 November 2020

T. Varé ,
C. Nouar Open the ORCID record for C. Nouar [Opens in a new window] ,
C. Métivier Open the ORCID record for C. Métivier [Opens in a new window]  and
M. Bouteraa
T. Varé
Affiliation:
LEMTA, UMR 7563 CNRS-Université de Lorraine, 2 Avenue de la Forêt de Haye, TSA 60604, 54518Vandoeuvre lès Nancy CEDEX, France
C. Nouar*
Affiliation:
LEMTA, UMR 7563 CNRS-Université de Lorraine, 2 Avenue de la Forêt de Haye, TSA 60604, 54518Vandoeuvre lès Nancy CEDEX, France
C. Métivier
Affiliation:
LEMTA, UMR 7563 CNRS-Université de Lorraine, 2 Avenue de la Forêt de Haye, TSA 60604, 54518Vandoeuvre lès Nancy CEDEX, France
M. Bouteraa
Affiliation:
LEMTA, UMR 7563 CNRS-Université de Lorraine, 2 Avenue de la Forêt de Haye, TSA 60604, 54518Vandoeuvre lès Nancy CEDEX, France
*
†Email address for correspondence: cherif.nouar@univ-lorraine.fr
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Abstract

Stability of hexagonal patterns in Rayleigh–Bénard convection for shear-thinning fluids with temperature-dependent viscosity is studied in the framework of amplitude equations. The rheological behaviour of the fluid is described by the Carreau model and the relationship between the viscosity and the temperature is of exponential type. Ginzburg–Landau equations including non-variational quadratic spatial terms are derived explicitly from the basic hydrodynamic equations using a multiple scale expansion. The stability of hexagonal patterns towards spatially uniform disturbances (amplitude instabilities) and to long wavelength perturbations (phase instabilities) is analysed for different values of the shear-thinning degree $\alpha$ of the fluid and the ratio $r$ of the viscosities between the top and bottom walls. It is shown that the amplitude stability domain shrinks with increasing shear-thinning effects and increases with increasing the viscosity ratio $r$. Concerning the phase stability domain which confines the range of stable wavenumbers, it is shown that it is closed for low values of $r$ and becomes open and asymmetric for moderate values of $r$. With increasing shear-thinning effects, the phase stability domain becomes more decentred towards higher values of the wavenumber. Beyond the stability limits, two different modes go unstable: longitudinal and transverse modes. For the parameters considered here, the longitudinal mode is relevant only in a small region close to the onset. The nonlinear evolution of the transverse phase instability is investigated by numerical integration of amplitude equations. The hexagon–roll transition triggered by the transverse phase instability for sufficiently large reduced Rayleigh number $\epsilon$ is illustrated.

JFM classification

Nonlinear Dynamical Systems: Pattern formation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A33
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Albaalbaki, B. & Khayat, R. 2011 Pattern selection in the thermal convection of non-Newtonian fluids. J. Fluid Mech. 668, 500–550.CrossRefGoogle Scholar
Alloui, Z., Ben-Khelifa, N., Beji, H., Vasseur, P. & Guizani, A. 2013 The onset of convection of power-law fluids in a shallow cavity heated from below by a constant heat flux. J. Non-Newtonian Fluid Mech. 196, 70–82.CrossRefGoogle Scholar
Balmforth, N. J. & Rust, A. C. 2009 Weakly nonlinear viscoplastic convection. J. Non-Newtonian Fluid Mech. 158, 36–45.CrossRefGoogle Scholar
Benouared, O., Mamou, M. & Ait-Messaoudene, N. 2014 Numerical nonlinear analysis of subcritical Rayleigh–Bénard convection in a horizontal confined enclosure filled with non-Newtonian fluids. Phys. Fluids 26 (7), 073101.CrossRefGoogle Scholar
Bird, R. B., Amstrong, R. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Wiley-Interscience.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Bottaro, A., Metzener, P. & Matalon, M. 1992 Onset and two-dimensional patterns of convection with strongly temperature-dependent viscosity. Phys. Fluids 4 (4), 655–663.CrossRefGoogle Scholar
Bouteraa, M. 2016 Convection de Rayleigh–Bénard pour des fluides rhéofluidifiants: approche théorique et expérimentale. PhD thesis, Université de Lorraine.Google Scholar
Bouteraa, M., Nouar, C., Plaut, E., Métivier, C. & Kalck, A. 2015 Weakly nonlinear analysis of Rayleigh–Bénard convection in shear-thinning fluids: nature of the bifurcation and pattern selection. J. Fluid Mech. 767, 696–734.CrossRefGoogle Scholar
Bragard, J. & Velarde, M. G. 1998 Bénard–Marangoni convection: planforms and related theoretical predictions. J. Fluid Mech. 368, 165–194.CrossRefGoogle Scholar
Brand, H. R. 1989 Envelope equations near the onset of a hexagonal pattern. Prog. Theor. Phys. Suppl. 99, 442–449.CrossRefGoogle Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.CrossRefGoogle Scholar
Busse, F. H. 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1930–1967.CrossRefGoogle Scholar
Busse, F. H. & Frick, H. 1985 Square-pattern convection in fluids with strongly temperature-dependent viscosity. J. Fluid Mech. 150, 451–465.CrossRefGoogle Scholar
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47 (2), 305–320.CrossRefGoogle Scholar
Charlson, G. S. & Sani, R. L. 1970 Thermoconvective instability in a bounded cylindrical fluid layer. Intl J. Heat Mass Transfer 13 (9), 1479–1496.CrossRefGoogle Scholar
Ciliberto, S., Coullet, P., Lega, J., Pampaloni, E. & Perez-Garcia, C. 1990 Defects in roll-hexagon competition. Phys. Rev. Lett. 65 (19), 2370.CrossRefGoogle ScholarPubMed
Ciliberto, S., Pampaloni, E. & Perez-Garcia, C. 1988 Competition between different symmetries in convective patterns. Phys. Rev. lett. 61 (10), 1198.CrossRefGoogle ScholarPubMed
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 65 (4), 625–645.CrossRefGoogle Scholar
Cox, S. M. & Matthews, P. C. 2002 Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2), 430–455.CrossRefGoogle Scholar
Darbouli, A., Métivier, C., Leclerc, S., Nouar, C. & Stemmlen, D. 2016 Natural convection in shear-thinning fluids: experimental investigation by MRI. Intl J. Heat Mass Transfer 95, 742–754.CrossRefGoogle Scholar
Davaille, A. & Jaupart, P. 1993 Transient high Rayleigh number thermal convection with large viscosity variation. J. Fluid Mech. 253, 141–166.CrossRefGoogle Scholar
Echebarria, B. & Perez-Garcia, C. 2001 Stability of hexagonal pattern in Bénard–Marangoni convection. Phys. Rev. E 63, 066307.CrossRefGoogle Scholar
Echebarría, B. & Pérez-García, C. 1998 Phase instabilities in hexagonal patterns. Europhys. Lett. 43 (1), 35.CrossRefGoogle Scholar
Getling, A. V. 1988 Rayleigh–Bénard Convection: Structures and Dynamics. World Scientific.Google Scholar
Golubitsky, M., Swift, J. W. & Knoblock, E. 1984 Symmetries and pattern selection in Rayleigh–Bénard convection. Physica D 10, 249–276.CrossRefGoogle Scholar
Hoard, C. Q., Robertson, C. R. & Acrivos, A. 1970 Experiments on the cellular structure in Bénard convection. Intl J. Heat Mass Transfer 13, 849–856.CrossRefGoogle Scholar
Hoyle, R. 1995 Nonlinear phase diffusion equations for the long-wave instabilities of hexagons. Appl. Maths Lett. 8, 81–85.CrossRefGoogle Scholar
Hoyle, R. 2006 Pattern Formation: An Introduction Methods. Cambridge University Press.CrossRefGoogle Scholar
Jenkins, D. R. 1987 Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity. J. Fluid Mech. 178, 491–506.CrossRefGoogle Scholar
Jenny, M., Plaut, E. & Briard, A. 2015 Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids. J. Non-Newtonian Fluid Mech. 219, 19–34.CrossRefGoogle Scholar
Kaddiri, M., Naïmi, M., Raji, A. & Hasnaoui, M. 2012 Rayleigh–Bénard convection of non-Newtonian power-law fluids with temperature-dependent viscosity. ISRN Thermodyn. 2012. ID614712, 1–10.CrossRefGoogle Scholar
Kolodner, P. 1998 Oscillatory convection in viscoelastic DNA suspensions. J. Non-Newtonian Fluid Mech. 75, 167–192.CrossRefGoogle Scholar
Lamsaadi, M., Naimi, M. & Hasnaoui, M. 2005 Natural convection of power law fluids in a shallow horizontal rectangular cavity uniformly heated from below. Heat Mass Transfer 41, 239–249.Google Scholar
Larson, R. G. 1992 Instabilities in viscoelastic flows. Rheol. Acta 31 (3), 213–263.CrossRefGoogle Scholar
Lauzeral, J., Metens, S. & Walgraef, D. 1993 On the phase dynamics of hexagonal patterns. Europhys. Lett. 24 (9), 707.CrossRefGoogle Scholar
Li, Z. & Khayat, R. E. 2005 Finite-amplitude Rayleigh–Bénard convection and pattern selection for viscoelastic fluids. J. Fluid Mech. 529, 221–251.CrossRefGoogle Scholar
Liang, S. F. & Acrivos, A. 1970 Experiments on buoyancy-driven convection in non-Newtonian fluids. Rheol. Acta 9, 447–455.CrossRefGoogle Scholar
McKenzie, D. 1988 The symmetry of convective transitions in space and time. J. Fluid Mech. 191, 287–339.CrossRefGoogle Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.CrossRefGoogle Scholar
Palm, E. 1975 Nonlinear thermal convection. Annu. Rev. Fluid Mech. 7 (1), 39–61.CrossRefGoogle Scholar
Palm, E., Ellingsen, T. & Gjevik, B. 1967 On the occurrence of cellular motion in Bénard convection. J. Fluid Mech. 30, 651–661.CrossRefGoogle Scholar
Pampaloni, E., Perez-Garcia, C., Albavetti, L. & Ciliberto, S. 1992 Transition from hexagons to rolls in convection in fluids under non-Boussinesq conditions. J. Fluid Mech. 234, 393–416.CrossRefGoogle Scholar
Pena, B. & Perez-Garcia, C. 2001 Stability of Turing patterns in the Brusselator model. Phys. Rev. E 64 (5), 056213.CrossRefGoogle ScholarPubMed
Pocheau, A. & Croquette, V. 1984 Dislocation motion: a wavenumber selection mechanism in Rayleigh–Bénard convection. J. Phys. 45 (1), 35–48.CrossRefGoogle Scholar
Richter, F. M. 1978 Experiments on the stability of convection rolls in fluids whose viscosity depends on temperature. J. Fluid Mech. 89 (3), 553–560.CrossRefGoogle Scholar
Segel, L. A. & Stuart, J. T. 1962 On the question of the preferred mode in cellular thermal convection. J. Fluid Mech. 13 (2), 289–306.CrossRefGoogle Scholar
Solomatov, V. S. 1995 Scaling of temperature-and-stress-dependent viscosity convection. Phys. Fluids 7, 266–274.CrossRefGoogle Scholar
Solomatov, V. S. & Barr, A. C. 2006 Onset of convection in fluids with strongly temperature-dependent power-law viscosity. Phys. Earth Planet. Inter. 155 (1–2), 140–145.CrossRefGoogle Scholar
Solomatov, V. S. & Barr, A. C. 2007 Onset of convection in fluids with strongly temperature-dependent, power-law viscosity: 2. Dependence on the initial perturbation. Phys. Earth Plane. Inter. 165 (1–2), 1–13.CrossRefGoogle Scholar
Somerscales, E. F. C. & Dougherty, T. S. 1970 Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below. J. Fluid Mech. 42 (4), 755–768.CrossRefGoogle Scholar
Stengel, K. C., Olivier, D. S. & Booker, J. R. 1982 Onset of convection in a variable-viscosity fluid. J. Fluid Mech. 120, 411–431.CrossRefGoogle Scholar
Sushchik, M. M. & Tsimring, L. S. 1994 The Eckhaus instability in hexagonal patterns. Physica D 74, 90–106.CrossRefGoogle Scholar
Tanner, R. 2000 Engineering Rheology. Oxford University Press.Google Scholar
Van-Den-Berg, J. B., Deschênes, A., Lessard, J. P. & Mireles-James, J. D. 2015 Stationary coexistence of hexagons and rolls via rigorous computations. SIAM J. Appl. Dyn. Syst. 14 (2), 942–979.CrossRefGoogle Scholar
White, D. B. 1988 The planforms and the onset of convection with a temperature dependent-viscosity. J. Fluid Mech. 191, 2247–286.CrossRefGoogle Scholar
Young, Y. N. & Riecke, H. 2002 Mean flow in hexagonal convection: stability and nonlinear dynamics. Physica D 163 (3–4), 166–183.CrossRefGoogle Scholar