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Dhiman, Joginder Singh
- On the Effects of Magnetic Field and Temperature-Dependent Viscosity on the Onset Magnetoconvection for General Boundary Conditions
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
Source
Research Journal of Science and Technology, Vol 5, No 1 (2013), Pagination: 104-109Abstract
In the present paper, the problem of thermal instability of an electrically conducting fluid layer heated from below and permeated with a uniform vertical magnetic field is studied for all combinations of rigid and dynamically free boundary conditions. The effect of temperature-dependent viscosity on the onset of hydromagnetic thermal convection is investigated both analytically and numerically. The validity of the principle of exchange of stabilities for this general problem has been investigated using the Pellew and Southwell’s method and a sufficient condition for the validity of this principle is also derived. The values of the Rayleigh numbers for each case of boundary combinations are obtained numerically using Galerkin technique. Further, the effect of temperature-dependent viscosity on the onset of stationary convection and consequently on the celebrated ∏2 Q-law of Chandrasekhar for each case of boundary combinations is computed numerically. It is observed that the temperature-dependent viscosity also has the inhibiting effect on the onset of convection as that of magnetic field and the ∏2 Q -law is also valid for this problem.
References
- Banerjee, M. B., Gupta, J. R., Shandil, R. G. and Jamwal, H.S. (1989): Settlement of the long Standing Controversy in Magnetothermoconvection in favour of S. Chandrasekhar, J. Math. Anal. Appl., 144, 356.
- Banerjee, M. B., Shandil, R.G. and Kumar, R. (1995) : On Chandrasekhar’s . –Law, J. Math. Anal. Appl., 191, 460.
- Banerjee, M.B. and Bhowmick, S.K. (1992): Salvaging the Thompson-Chandrasekhar Criterion: A tribute to S. Chandrasekhar, J. Math. Anal. Appl., 167, 57-65.
- Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Amen. House London, E.C.4, 1961.
- Dhiman, J. S. and Kumar Vijay, (2012): On 2 p Q -Law in Magnetoconvection Problem for General Nature of Boundaries Using Galerkin Method, Research J. Engineering and Technology, 3(2), (2012), 186.
- Finlayson, B.A. (1972): ‘The Method of Weighted Residuals and Variational Principles’; Academic Press, New York.
- Gupta, J.R. and Kaushal, M.B. (1988): Rotatory hydromagnetic double-diffusive convection with viscosity variation, J. Math Phys. Sci., 22 (3), 301.
- Jeffreys, H., Some Cases of Instability in Fluids Motions, Proc. Roy. Soc. London, 1928, A118, 195.
- Kumar, Vijay (2012): A Study of Some Convective Stability Problems with Variable Viscosity, Ph.D. thesis (supervised by J.S. Dhiman), submitted to Himachal Pradesh University, Shimla.
- Low, A. R, On the Criterion for Stability of a Layer of Viscous Fluid Heated From Below, Proc. Roy. Soc. London, 1929, A125, 180.
- Pellew, A. and Southwell, R.V., On Maintained Convective Motion in a Fluid Heated From Below, Proc. Roy. Soc. London, 1940, A 176, 312.
- Straughan, B., Sharp Global Non-Linear Stability for Temperature Dependent Viscosity, Proc. Roy. Soc. London, 2002, A 458, 1773.
- A Modified Analysis of the Onset of Convection in a Micropolar Liquid Layer Heated From Below
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
Source
Research Journal of Science and Technology, Vol 5, No 1 (2013), Pagination: 224-231Abstract
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
- 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.