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Thermal Instability of a Micropolar Fluid Layer with Temperature-Dependent Viscosity

  • Joginder Singh DhimanEmail author
  • Nivedita Sharma
Research Article

Abstract

In this paper, the effect of temperature-dependent viscosity on the onset of thermal convection in a micropolar fluid layer heated from below for each combination of rigid (the surfaces with non-slip condition) and dynamically free (the surfaces with stress-free condition) boundaries is investigated. It is shown here analytically that the principle of exchange of stabilities is valid for the problem, which means that instability sets in as stationary convection. The expressions for Rayleigh numbers for each combination of rigid and dynamically free boundary conditions are derived using Galerkin method. The effects of micropolar parameters and viscosity variation parameter on critical wave numbers and consequently on the critical Rayleigh numbers are computed numerically.

Keywords

Thermal convection Temperature-dependent viscosity Principle of exchange of stabilities Galerkin method Rayleigh number Microrotation 

References

  1. 1.
    Eringen AC (1964) Simple microfluids. Int J Eng Sci 2:205–217MathSciNetCrossRefGoogle Scholar
  2. 2.
    Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16:1–18MathSciNetGoogle Scholar
  3. 3.
    Askar A (1972) Molecular crystals and the polar theories of the continua experimental values of material coefficients for KNO3. Int J Eng Sci 10:293–300CrossRefGoogle Scholar
  4. 4.
    Bažant ZP, Christensen M (1972) Analogy between micropolar continuum and grid frameworks under initial stress. Int J Solids Struct 8:327–346CrossRefGoogle Scholar
  5. 5.
    Ariman T, Turk MA, Sylvester ND (1973) Microcontinuum fluid mechanics-a review. Int J Eng Sci 11:905–930CrossRefGoogle Scholar
  6. 6.
    Ariman T, Turk MA, Sylvester ND (1974) Review article-applications of microcontinuum fluid mechanics. Int J Eng Sci 12:273–293CrossRefGoogle Scholar
  7. 7.
    Eringen AC (1998) Microcontinuum field theories, II. Fluent Media. Springer, New York InczbMATHGoogle Scholar
  8. 8.
    Lukaszewicz G (1999) Micropolar fluids, theory and applications. Brikhauser, BostonCrossRefGoogle Scholar
  9. 9.
    Bég OA, Bhargava R, Rashidi MM (2011) Numerical simulation in micropolar fluid dynamics: mathematical modelling of nonlinear flows of micropolar fluids. Lambert Academic Publishing, GermanyGoogle Scholar
  10. 10.
    Walzer U (1976) Convective instability of a micropolar fluid layer. Ger Beitr Geophysik Leipzig 85:137–143ADSGoogle Scholar
  11. 11.
    Ahmadi G (1976) Stability of a micropolar fluid layer heated from below. Int J Engng Sci 14:81–89MathSciNetCrossRefGoogle Scholar
  12. 12.
    Datta AB, Sastry VUK (1976) Thermal instability of a horizontal layer of micropolar fluid heated from below. Int J Engng Sci 14:631–637CrossRefGoogle Scholar
  13. 13.
    Dhiman JS, Sharma PK, Singh G (2011) Convective stability analysis of a micropolar fluid layer by variational method. Theor Appl Mech Lett 1:022004CrossRefGoogle Scholar
  14. 14.
    Palm E (1960) On the tendency towards hexagonal cells in steady convection. J Fluid Mech 8:183–192ADSCrossRefGoogle Scholar
  15. 15.
    Stengel KC, Oliver DS, Booker JR (1982) Onset of convection in a variable viscosity fluid. J Fluid Mech 120:411–431ADSCrossRefGoogle Scholar
  16. 16.
    Booker JR, Stengel KC (1978) Further thoughts on convective heat transport in a variable viscosity fluid. J Fluid Mech 86:289–291ADSCrossRefGoogle Scholar
  17. 17.
    Jenkins DR (1987) Rolls versus squares in thermal convection of fluids with temperature dependent viscosity. J Fluid Mech 178:491–506ADSCrossRefGoogle Scholar
  18. 18.
    Selak R, Lebon G (1993) Bénard-Marangoni thermoconvective instability in presence of a temperature dependent viscosity. J De Physique II France 3:1185–1199ADSCrossRefGoogle Scholar
  19. 19.
    Nield DA (1996) The effect of temperature dependent viscosity on the onset of convection in a saturated porous medium. ASME J Heat Trans 118:803–805CrossRefGoogle Scholar
  20. 20.
    Straughan B (2002) Sharp global non-linear stability for temperature dependent viscosity. Proc R Soc Lond 458:1773–1782ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Dhiman JS, Kumar V (2013) On stability analysis of Rayleigh Bénard convection with temperature dependent viscosity for general boundary condition. Int J Emerg Multidiscip Fluid Sci 3:85–98Google Scholar
  22. 22.
    Dhiman JS, Sharma N (2015) Effect of temperature dependent viscosity on the thermal convection of nanofluid layer: steady case. J Thermophys Heat Trans 29:90–101CrossRefGoogle Scholar
  23. 23.
    Pellew A, Southwell RV (1940) On maintained convective motion in a fluid heated from below. Proc R Soc Lond A176:312–343ADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    Chandrasekhar S (1961) Hydrodynamics and hydromagnetic stability. Oxford University Press, Amen HousezbMATHGoogle Scholar
  25. 25.
    Finlayson BA (1972) The method of weighted residuals and variational principles. Academic Press, NYzbMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Department of MathematicsHimachal Pradesh UniversitySummerhill, ShimlaIndia

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