Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T00:37:30.710Z Has data issue: false hasContentIssue false
  • English
  • Français

Pattern selection in the thermal convection of non-Newtonian fluids

Published online by Cambridge University Press:  05 January 2011

BASHAR ALBAALBAKI  and
ROGER E. KHAYAT
BASHAR ALBAALBAKI
Affiliation:
Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario N6A 5B9, Canada
ROGER E. KHAYAT*
Affiliation:
Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario N6A 5B9, Canada
*
†Email address for correspondence: rkhayat@eng.uwo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

The thermogravitational instability in a fluid layer of a non-Newtonian medium heated from below is investigated. Linear and weakly nonlinear analyses are successively presented. The fluid is assumed to obey the Carreau–Bird model. Although the critical threshold is the same as for a Newtonian fluid, it is found that non-Newtonian fluids can convect in the form of rolls, squares or hexagons, depending on the shear-thinning level. Similar to Newtonian fluids, shear-thickening fluids convect only in the form of rolls. The stability of the convective steady branches is carried out to determine under which specific conditions a pattern is preferred. The influence of the rheological and physical parameters is examined and discussed in detail.

JFM classification

Non-Newtonian Flows: Non-Newtonian Flows Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 500 - 550
Copyright
Copyright © Cambridge University Press 2011

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. 2010 Thermal convection of non-Newtonian fluids. PhD thesis, The University of Western Ontario, London, Ontario.CrossRefGoogle Scholar
Albaalbaki, B. & Khayat, R. E. 2008 Finite-amplitude Rayleigh–Bénard convection for weakly shear thinning fluids. J. Phys. Conf. Ser. 137, 012024, 1–12.CrossRefGoogle Scholar
Anand, M. & Rajagopal, K. R. 2004 A shear-thinning viscoelastic fluid model for describing the flow of blood. Intl J. Cardiovasc. Med. Sci. 4 (2), 59–68.Google Scholar
Ashurst, W. T. & Hoover, W. G. 1975 Dense fluid shear viscosity via non-equilibrium molecular dynamics. Phys. Rev. A 11 (2), 658–678.CrossRefGoogle Scholar
Assenheimer, M. & Steinberg, V. 1996 Observation of coexisting upflow and downflow in Boussinesq Rayleigh–Bénard convection. Phys. Rev. Lett. 76 (5), 756–759.CrossRefGoogle ScholarPubMed
Balmforth, N. J. & Rust, A. C. 2009 Weakly nonlinear viscoplastic convection. J. Non-Newtonian Fluid Mech. 158, 36–45.CrossRefGoogle Scholar
Behringer, R. P. 1985 Rayleigh–Bénard convection and turbulence in liquid helium. Rev. Mod. Phys. 57 (3), 657–687.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn, vol. 1. John Wiley & Sons.Google Scholar
Boger, D. V. & Walters, K. 1993 Rheological Phenomena in Focus. Elsevier.Google Scholar
Booker, J. R. 1976 Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76 (4), 741–754.CrossRefGoogle Scholar
Booker, J. R. & Stengel, K. C. 1978 Further thoughts on convective heat transport in a variable-viscosity fluid. J. Fluid Mech. 86 (2), 289–291.CrossRefGoogle Scholar
Bruus, H. 2008 Theoretical Microfluidics. Oxford University Press.Google Scholar
Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30 (4), 624–649.CrossRefGoogle Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.CrossRefGoogle Scholar
Busse, F. H. 1985 Transition to turbulence in Rayleigh–Bénard convection. In Topics in Applied Physics, Hydrodynamic Instabilities and Transition to Turbulence (ed. Swinney, H. L. & Gollup, J. P.), p. 97–133. Springer.Google Scholar
Busse, F. H. & Bolton, E. W. 1984 Instabilities of convection rolls with stress-free boundaries near threshold. J. Fluid Mech. 146, 114–125.CrossRefGoogle Scholar
Busse, F. H. & Clever, R. M. 1998 Asymmetric squares as an attracting set in Rayleigh–Bénard convection. Phys. Rev. Lett. 81 (2), 341–344.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
Chandraekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chhabra, R. P. & Richardson, J. F. 1999 Non-Newtoian Flow in the Process Industries: Fundamentals and Engineering Applications. Butterworth-Heinmann.Google Scholar
Chhabra, R. P. & Uhlherr, P. H. T. 1980 Creeping motion of spheres through shear-thinning elastic fluids described by the Carreau viscosity equation. Rheol. Acta. 19, 187–195.CrossRefGoogle Scholar
Christensen, U. 1983 Convection in a variable-viscosity fluid: Newtonian versus power-law rheology. Earth Planet. Sci. Lett. 64, 153–162.CrossRefGoogle Scholar
Ciliberto, S., Coullet, P., Lega, J., Pampaloni, P. & Perez-Garcia, C. 1990 Defects in roll-hexagon competition. Phys. Rev. Lett. 65 (19), 2370–2373.CrossRefGoogle ScholarPubMed
Clever, R. M. & Busse, F. H. 1996 Hexagonal convection cells under conditions of vertical symmetry. Phys. Rev. E. 53 (3), R2037–R2040.CrossRefGoogle ScholarPubMed
Cross, M. C. & Hohenherg, P. C. 1993 Pattern formation outside equilibrium. Rev. Mod. Phys. 65, 851–1112.CrossRefGoogle Scholar
Denier, J. P. & Dabrowski, P. P. 2004 On the boundary-layer equations for power-law fluids. Proc. R. Soc. Lond. A 460, 3143–3158.CrossRefGoogle Scholar
Denier, J. P. & Hewitt, R. E. 2004 Asymptotic matching constraints for a boundary-layer flow of a power-law fluid. J. Fluid Mech. 518, 261–279.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dubois, M. & Berge, P. 1978 Experimental study of velocity field in Rayleigh–Bénard convection. J. Fluid Mech. 85 (4), 641–653.CrossRefGoogle Scholar
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. Springer.CrossRefGoogle Scholar
Fromm, J. E. 1965 Numerical solution of the nonlinear equations for a heated fluid layer. Phys. Fluids 8 (10), 1757–1769.CrossRefGoogle Scholar
Getling, A. V. 1998 Rayleigh–Bénard Convection: Structures and Dynamics. World Scientific.CrossRefGoogle Scholar
Goldstein, R. J. & Graham, D. J. 1969 Stability of a horizontal fluid layer with zero shear boundaries. Phys. Fluids 12 (6), 1133–1137.CrossRefGoogle Scholar
Harder, H. 1991 Numerical simulation of thermal convection with Maxwellian viscoelasticity. J. Non-Newtonian Fluid Mech. 36, 67–88.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
Hur, J. S. & Shaqfeh, E. S. G. 2005 Dynamics of dilute and semidilute DNS solutions in the start-up of shear flow. J. Rheol. 45 (2), 421–451.CrossRefGoogle Scholar
Hwang, H. J., Kim, J. H. & Jeong, K. 2009 Method and apparatus for amplification of nucleic acid sequences by using thermal convection. United States patent 7,628,961.Google Scholar
Inaba, H., Dai, C. & Horibe, A. 2003 Numerical simulation of Rayleigh–Bénard convection in non-Newtonian phase-change-material slurries. Intl J. Therm. Sci. 42, 471–480.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
Jenkins, D. R. 1988 Interpretations of shadowgraph patterns in Rayleigh–Bénard convection. J. Fluid Mech. 190, 451–469.CrossRefGoogle Scholar
Khayat, R. E. 1996 Chaos in the thermal convection of weakly shear-thinning fluids. J. Non-Newtonian Fluid Mech. 63, 153–178.CrossRefGoogle Scholar
Khayat, R. E. 1999 Finite-amplitude Taylor-vortex flow of viscoelastic fluids. J. Fluid Mech. 400, 33–58.CrossRefGoogle Scholar
Kolodner, P. 1998 Oscillatory convection in viscoelastic DNA suspensions. J. Non-Newtonian Fluid Mech. 75, 167–192.CrossRefGoogle Scholar
Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.Google Scholar
Krishnan, M., Uhaz, V. M. & Burns, M. A. 2002 PCR in a Rayleigh–Bénard convection cell. Science 298, 793.CrossRefGoogle Scholar
Lamsaadi, M., Naimi, M. & Hansaoui, M. 2005 Natural convection of non-Newtonian 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, 213–263.CrossRefGoogle Scholar
Li, Z. & Khayat, R. E. 2004 A non-linear dynamical system approach to finite amplitude Taylor–Vortex flow of shear-thinning fluids. Intl J. Numer. Meth Fluids 45, 321–340.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
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4 (3), 225–260.CrossRefGoogle Scholar
Moore, D. R. & Weiss, N. O. 1973 Two-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 58, 289–312.CrossRefGoogle Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 1. McGraw-Hill.Google Scholar
Normand, C., Pomeau, Y. & Velarde, M. G. 1977 Convective instability: a physicist's approach. Rev. Mod. Phys. 49 (3), 581–624.CrossRefGoogle Scholar
Ohta, M., Ohta, M., Akiyoshi, M. & Obata, E. 2002 A numerical study on natural convective heat transfer of pseudoplastic fluids in a square cavity. Numer. Heat Transfer A 41, 357–372.CrossRefGoogle Scholar
Oliver, D. S & Brook, J. R. 1983 Planform of convection with strongly temperature-dependent viscosity. Geophys. Astrophys. Fluid Dyn. 27, 73–85.CrossRefGoogle Scholar
Ozoe, H. & Churchill, S. W. 1972 Hydrodynamic stability and natural convection in Ostwald–de Waele and Ellis fluids: the development of a numerical solution. AICHE J. 18 (6), 1196–1207.CrossRefGoogle Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.CrossRefGoogle Scholar
Palm, E. 1972 A note on a minimum principle in Bénard convection. Intl J. Heat Mass Transfer 15, 2409–2417.CrossRefGoogle Scholar
Palm, E. 1975 Nonlinear thermal convection. Annu. Rev. Fluid Mech. 7, 39–61.CrossRefGoogle Scholar
Park, H. M. & Lee, H. S. 1996 Hopf bifurcation of viscoelastic fluids heated from below. J. Non-Newtonian Fluid Mech. 66, 1–34.CrossRefGoogle Scholar
Parmentier, E. M. 1978 A study of thermal convection in non-Newtonian fluids. J. Fluid Mech. 84 (1), 1–11.CrossRefGoogle Scholar
Parmentier, E. M., Turcotte, D. L. & Torrance, K. E. 1976 Studies of finite amplitude non-Newtonian thermal convection with application to convection in the earth's mantle. J. Geophys. Res. 81 (11), 1839–1846.CrossRefGoogle Scholar
Parmentier, P., Lebon, G. & Reginer, V. 2000 Weakly nonlinear analysis of Bénard–Marangoni instability in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 89, 63–95.CrossRefGoogle Scholar
Parmentier, P. M., Regnier, V. C., Lebon, G. & Legros, J. C. 1996 Nonlinear analysis of coupled gravitational and capillary thermoconvection in thin fluid layers. Phys. Rev. E 54 (1), 411–423.Google Scholar
Peltier, W. R. 1989 Mantle Convection, Plate Tectonics and Global Dynamics. Gordon and Breach.Google Scholar
Pierre, C. St. & Tien, C. 1963 Experimental investigation of natural convection heat transfer in confined space for non-Newtonian fluid. Can. J. Chem. Engng 41, 122–127.CrossRefGoogle Scholar
Pons, M. & Quere, L. P. 2005 An example of entropy balance in natural convection. Part 1. The usual Boussinesq equations. C. R. Mecanique 333, 127–132.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
Schlueter, A., Lortz, D. & Busse, F. 1965 On the stability of steady finite amplitude convection. J. Fluid Mech. 23 (1), 129–144.CrossRefGoogle Scholar
Schubert, G., Turcotte, D. L. & Olson, P. 2001 Mantle Convection in the Earth and Planet. Cambridge University Press.CrossRefGoogle Scholar
Segel, L. A. 1965 The non-linear interaction of a finite number of disturbances to a layer of fluid heated from below. J. Fluid Mech. 21 (2), 359–384.CrossRefGoogle Scholar
Segel, L. A. & Stuart, J. T. 1962 On the question of the preferred mode in cellular thermal convection. J. Fluid Mech. 13, 289–306.CrossRefGoogle Scholar
Shenoy, A. V. & Mashelkar, R. A. 1982 Thermal convection in non-Newtonian fluids. In Advances in heat transfer (ed. Hartnett, J. P. & Irvine, T. F.), vol. 15, 143–225. Jr. Academic.Google Scholar
Solomatov, V. S. 1995 Scaling of temperature- a stress-dependent viscosity convection. Phys. Fluids 7 (2), 266–274.CrossRefGoogle Scholar
Strogatz, S. 1994 Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry and Engineering. Addison Wesley.Google Scholar
Stuart, J. T. 1964 On the cellular patterns in thermal convection. J. Fluid Mech. 18, 481–498.CrossRefGoogle Scholar
Swift, J. W. 1978 Bifurcation and symmetry in convection. PhD thesis, University of California, Berkeley.Google Scholar
Tien, C., Tsuei, H. S. & Sun, Z. S. 1969 Thermal convection of a horizontal layer of non-Newtonian fluid heated from below. Intl J. Heat Mass Transfer 12, 1173–1178.Google Scholar
Tsuei, H. S. 1970 Thermal instability and heat transport of a layer of non-Newtonian fluid. PhD thesis, Syracuse University, New York.Google Scholar
Tsuei, H. S. & Tien, C. 1973 Free convection heat transfer in a horizontal layer of non-Newtonian fluid. Can. J. Chem. Engng 51, 249–251.CrossRefGoogle Scholar
White, D. B. 1988 The planforms and onset of convection with a temperature-dependent viscosity. J. Fluid Mech. 191, 247–286.CrossRefGoogle Scholar
Zhang, J., Vola, D., & Frigaard, I. A. 2006 Yield stress effects on Rayleigh–Bénard convection. J. Fluid Mech. 556, 389–419.CrossRefGoogle Scholar
Zhao, J. & Khayat, R. E. 2008 Spread of a non-Newtonian liquid jet over a horizontal plate. J. Fluid Mech. 613, 411–443.CrossRefGoogle Scholar