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A Modified Analysis of the Onset of Convection in a Micropolar Liquid Layer Heated From Below


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1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, India
     

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In the present paper, the thermal stability analysis of a micropolar liquid layer heated from below is investigated by utilizing the essential arguments of the modified analysis of Banerjee et al. The principle of exchange of stabilities (PES) is shown to be valid using Pellew and Southwell’s method for the problem, whether the liquid layer is hotter or cooler. A general expression for Rayleigh numbers is derived using Galerkin method valid for all combinations of rigid and dynamically free boundary conditions. The values of critical wave numbers and consequently of critical Rayleigh numbers for each case of boundary combinations are derived and computed numerically, when instability sets in as stationary convection. The effects of microrotation parameters and the coefficient of specific heat variation on critical Rayleigh numbers for each case of boundary conditions are computed numerically. From the obtained results, we conclude that the microrotation viscosity coefficient K and coefficient of specific heat variation for large temperature have stabilizing effects whereas microrotation parameter A has destabilizing effect on the onset of convection.

Keywords

Micropolar Liquid, Modified Boussinesq Approximation, Galerkin Method, Principle Of Exchange Of Stabilities, Stationary Convection, Rayleigh Number.
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  • Ahmadi, G. (1976), Stability of a micropolar fluid layer heated from below, Int. J. Engng. Sci., 14, 81.

  • Banerjee, M.B., Gupta, J.R., Shandil, R.G., Sharma, K.C., Katoch, D.C. (1983), A modified analysis of thermal and thermohaline instability of a liquid layer heated underside, J. Math. Phys. Sci., 17(6), 603.

  • Boussinesq, J. (1903), ‘Théorie analytique de la chaleur’, Gauthier Villas, Paris, 2, 172.

  • Chandrasekhar, S. (1961), ‘Hydrodynamics and Hydromagnetic Stability’, Oxford University Press, Amen. House London, EC. 4.

  • Datta, A.B. and Sastry, V.U.K. (1976), Thermal instability of a horizontal layer of micropolar fluid heated from below, Int. J. Engng. Sci., 14, 631.

  • Dhiman, J. S., Sharma, P. K. and Singh G. (2011), Convective stability analysis of a micropolar fluids Layer by variational method, Theo. Appl. Mech. Lett., 1, 04204-1.

  • Eringen, A.C. (1966), Theory of micropolar fluids, J. Math. Mech., 16, 1.

  • Eringen, A.C. (1998), ‘Microcontinuum Field Theories, II. Fluent Media’, Springer-Verlag, NY, Inc.

  • Finlayson, B.A. (1972), ‘The Method of Weighted Residuals and Variational Principles’, Academic Press, NY.

  • Lukaszewicz, G. (1999), ‘Micropolar Fluids, Theory and Applications’, Brikhauser. Boston, USA.

  • Pellew, A. and Southwell, R. V. (1940), On maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London, E.C.4.

  • Rayleigh, L. (1916): On the convective currents in a horizontal layer of fluid when the higher temperature is on the underside, Phil. Mag., 32, 529.

  • Schmidt, R.J. and Milverton S.W. (1935): On the instability of the fluid when heated from below, Proc. R. Soc. London, A152, 586.


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  • A Modified Analysis of the Onset of Convection in a Micropolar Liquid Layer Heated From Below

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Authors

Joginder Singh Dhiman
Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, India
Nivedita Sharma
Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, India

Abstract


In the present paper, the thermal stability analysis of a micropolar liquid layer heated from below is investigated by utilizing the essential arguments of the modified analysis of Banerjee et al. The principle of exchange of stabilities (PES) is shown to be valid using Pellew and Southwell’s method for the problem, whether the liquid layer is hotter or cooler. A general expression for Rayleigh numbers is derived using Galerkin method valid for all combinations of rigid and dynamically free boundary conditions. The values of critical wave numbers and consequently of critical Rayleigh numbers for each case of boundary combinations are derived and computed numerically, when instability sets in as stationary convection. The effects of microrotation parameters and the coefficient of specific heat variation on critical Rayleigh numbers for each case of boundary conditions are computed numerically. From the obtained results, we conclude that the microrotation viscosity coefficient K and coefficient of specific heat variation for large temperature have stabilizing effects whereas microrotation parameter A has destabilizing effect on the onset of convection.

Keywords


Micropolar Liquid, Modified Boussinesq Approximation, Galerkin Method, Principle Of Exchange Of Stabilities, Stationary Convection, Rayleigh Number.

References