On the Effects of Magnetic Field and Temperature-Dependent Viscosity on the Onset Magnetoconvection for General Boundary Conditions

Joginder Singh Dhiman and Vijay Kumar

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

*Corresponding Author: jsdhiman66@gmail.com

ABSTRACT:

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 -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 -law is also valid for this problem.

1. INTRODUCTION:

The hydrodynamic stability has been recognized as one of the central problems of fluid mechanics. It is concerned with when and how laminar flows break down, their subsequent development and their eventual transition to turbulence. It has many applications in engineering, meteorology, oceanography, astrophysics and geophysics. The hydrodynamic thermal instability or thermal convection in a plane horizontal fluid layer heated from below (also known as Rayleigh–Bénard convection) is a type of convection considered most frequently. In this phenomenon, spatial and temporal effects are largely decomposed because of the lack of intense streams determined by external conditions. For this reason, both the experimental and theoretical treatment of the Rayleigh–Bénard convection proves to be especially fruitful. For broad view of the subject of Rayleigh-Bénard convection, one may be referred to Jeffrey’s [1928], Low [1929], Pellew and Southwell [1940] and Chandrasekhar [1961].

The hydromagnetic stability or magnetoconvection is the study of the effect of an externally impressed magnetic field on the onset of thermal instability in electrically conducting fluids. In broad terms, the subject of hydromagnetics is concerned with the ways in which magnetic fields can affect fluid behavior. The problem of thermal instability in the presence of a magnetic field illustrates many general principles of the subject.

Recent developments in astrophysics and geophysics have widespread interest in magnetoconvection because the magnetic fields are believed to be driven by buoyancy forces. The studies pertaining to magnetoconvection are extensively investigated over the years, and is well established fact that the effect of magnetic field is to make the system more stable.

Chandrasekhar [1961] studied the magnetoconvective instability problems in detail and laid some guiding principles for the onset of magnetoconvection. He studied the magnetoconvection in detail and due to the mathematical complexity involved in solving equations, a very few exact solutions of the problem exist in literature. He proved his famous -law of stationary convection in magnetoconvection problem for the case of both dynamically free boundaries, which expresses the asymptotic dependence on the Chandrasekhar number of the critical Rayleigh number . For the other two cases of boundary conditions, namely; both rigid boundaries and combinations of rigid and dynamically free boundaries he, on the basis of numerical computations (as no exact solutions are obtainable in these cases of boundary conditions), conjectured that the same -law must hold true. Later on many authors including; Banerjee et al. [1989], Banerjee and Bhowmick [1992], Banerjee et al. [1995] and Dhiman and Kumar [2012] studied the magnetohydrodynamic thermal stability for general boundary conditions both analytically and numerically and used the Galerkin method to validate the Chandrasekhar’s conjecture (the -law).

It is well known fact that the viscosity is one of the properties of a fluid which are most sensitive to temperature (c.f. Straughan [2002], Kumar V. [2012]). In the majority of the cases, viscosity becomes the only property which may have considerable effect on the heat transfer, whereas the temperature variation and dependence of other thermo-physical properties to temperature are often negligible. Hence, the variation in viscosity behaviour with temperature may have a pronounced effect on the convective motions of the fluid.

Motivated by the above discussions, the present study is concerned with buoyancy driven magneto convection in an electrically conducting fluid layer heated from below for which the viscosity of the fluid may depend strongly on the local temperature. 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, and 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 (PES) for this general problem has been investigated using the Pellew and Southwell’s method, since the establishment of PES implies the non-occurrence of any slow oscillatory motions which may be neutral or unstable. Further, a sufficient condition for the validity of PES is also derived. The expressions for the Rayleigh numbers for each case of boundary combinations are obtained numerically using Galerkin technique and the values of each expression is computed numerically. The effects of temperature-dependent viscosity and Chandrasekhar number on the onset of stationary convection are computed numerically. Further, the fate of celebrated -law with temperature-dependent viscosity is examined for each case of boundary combinations.

5. REFERENCE:

1. 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.

2. Banerjee, M. B., Shandil, R.G. and Kumar, R. (1995) : On Chandrasekhar’s –Law, J. Math. Anal. Appl., 191, 460.

3. 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.

4. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Amen. House London, E.C.4, 1961.

5. Dhiman, J. S. and Kumar Vijay, (2012): On -Law in Magnetoconvection Problem for General Nature of Boundaries Using Galerkin Method, Research J. Engineering and Technology, 3(2), (2012), 186.

6. Finlayson, B.A. (1972): ‘The Method of Weighted Residuals and Variational Principles’; Academic Press, New York.

7. Gupta, J.R. and Kaushal, M.B. (1988): Rotatory hydromagnetic double-diffusive convection with viscosity variation, J. Math Phys. Sci., 22 (3), 301.

8. Jeffreys, H., Some Cases of Instability in Fluids Motions, Proc. Roy. Soc. London, 1928, A118, 195.

9. 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.

10. Low, A. R, On the Criterion for Stability of a Layer of Viscous Fluid Heated From Below, Proc. Roy. Soc. London, 1929, A125, 180.

11. Pellew, A. and Southwell, R.V., On Maintained Convective Motion in a Fluid Heated From Below, Proc. Roy. Soc. London, 1940, A 176, 312.

12. Straughan, B., Sharp Global Non-Linear Stability for Temperature Dependent Viscosity, Proc. Roy. Soc. London, 2002, A 458, 1773.

Received on 12.01.2013 Accepted on 08.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 104-109