On the Effect of Non-Uniform Temperature Gradients on the Stability of Modified Thermal Convection Problem
Joginder Singh Dhiman1, Vijay Kumar2 and Som Krishan Sharma3
1Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005.
2Govt. Degree College, Sunni, Distt. Shimla (H.P.)-171301.
The aim of the present paper is to study the effect of non-uniform basic temperature gradients on the onset of modified thermal convection in a layer of fluid heated from below for different combinations of rigid and dynamically free boundary conditions. It is shown that the principle of exchange of stabilities (PES) is valid when the temperature gradient is monotonically decreasing upward, which means that the instability sets in as stationary mode. The expressions for the Rayleigh numbers for each combination of rigid and dynamically free boundary conditions for the stationary case of instability are derived using Galerkin method. The effects of non-uniform temperature gradients and the modification factor which arises due to modified theory of Banerjee et al on the instability are studied from the values of the critical Rayleigh numbers calculated numerically for various temperature profiles and the coefficient of specific heat variation due to temperature variation for the given values of other parameters. It is observed from these values that the Cubic temperature profile is more stabilizing than the inverted parabolic temperature distribution profile. Further, it is also found that the critical Rayleigh numbers for thermally insulating boundaries are lower than those for the corresponding isothermal cases.
In the classical buoyancy driven convective instability problem (Bénard Problem), the fluid is driven by maintaining an adverse uniform temperature gradient. Here, the basic temperature distribution is the steady state (conduction state) distribution, i.e. the temperature profile is linear and hence the temperature gradient maintained across the fluid layer is constant. However, Nield  suggested that in many situations, particularly in geophysical contexts, the stability/instability of a fluid in the presence of a non-linear temperature profile is of great practical importance and reported that the non-linearity of the temperature profile is due to rapid heating (or cooling) at a boundary. Malurkar  investigated the stability of a radiating layer of air near the ground with non-linear temperature profile. Graham  and Chandra  observed experimentally that a form convection in thin layers occurs at the values of the overall Rayleigh number lower than the critical value predicted by the classical theory.
Sparrow et al.  and Roberts  studied the onset of convection with a parabolic basic temperature profile. Rogers and Morrison  and Rogers et al.  also studied the onset of convection induced by buoyancy in a saturated porous medium with a non-linear basic temperature distribution. The effect of non-uniform temperature profile on the Rayleigh-Bénard convection was also considered by Nield . Thangaraj  investigated the effects of non-uniform temperature gradient and bounding permeable walls on Rayleigh Bénard convection in a sparsely packed porous medium. Chiang  studied the effect of non-uniform temperature gradient on the onset of stationary and oscillatory Bénard-Marangoni convection. Idris et al.  studied the Bénard-Marangoni convection in micropolar fluid with a cubic basic state temperature profile. Rudraiah et al. , Shivakumara  and Mokhtar et al.  also studied the effect of the non-uniform temperature gradients on the convective stability problems.
Banerjee et al.  presented a modified analysis of thermal instability of a liquid layer heated underside. They remarked that the Rayleigh’s utilization of Boussinesq approximation overlooks a term in the equation of heat conduction, which is on account of the variation in specific heat at constant volume due to variation in temperature, and which is such that in usual circumstances it cannot be ignored if the Boussinesq approximation is to be consistently and more accurately applied throughout the analysis. The essential argument on which this term finds a place in modified theory of Banerjee et al. is that it is the temperature difference which is of moderate amounts but not necessarily the temperature itself. Thus, they asserted that an incorporation of this term into the derivation of heat equation completes the qualitative and quantitative gap in Rayleigh’s theory as pointed out earlier. Subsequently the Boussinesq approximation together with the incorporation of the above term will be referred to as modified Boussinesq approximation. For further detail on the subject, one may be referred to Banerjee and Gupta  and Dhiman .
Motivated by the above discussions and the importance of the non-uniform temperature gradients in controlling the onset of convective motions in fluid layers, in the present paper we have carried out the linear stability analysis of the modified thermal convection problem in order to investigate the effects of various temperature profiles on the onset of instability. The Galerkin method is used to obtain the expressions for the Rayleigh numbers for different cases of boundary conditions.
2. MATHEMATICAL FORMULATION OF THE PROBLEM:
Consider a Bénard layer of a viscous quasi-incompressible (Boussinesq) fluid confined between two horizontal boundaries and , maintained respectively at temperature and ), in the force field of gravity. The temperature gradient so maintained, which shall be uniform or non-uniform, shall be qualified as adverse temperature gradient in order to have interplay in the density of the initially homogeneous fluid. Further, at each boundary the basic temperature assumed to be uniform. The Cartesian coordinates are so chosen that the vertical depth of the fluid is along the z axis.
Following the usual steps of the linear stability analysis and the modified analysis of Banerjee et al.  of thermal convection problem, the non-dimensional linearized perturbation equations governing the above physical configuration are given by (cf. Nield );
5. DISCUSSION AND CONCLUSIONS:
In this chapter we have examined the stability of Modified thermal convection in the presence of non-uniform temperature gradients. Firstly, we have shown the validity of the PES in this general problem. In numerical analysis, we have used the single term Galerkin method to derive the expressions for Rayleigh numbers for different types of boundary conditions. For each of these cases of the boundary conditions, we have computed the values of critical Rayleigh numbers for four different types of temperature profiles.
From the above analysis and obtained data (see Tables 1-3),
(i) We observe that amongst the four temperature profiles, the Cubic 1 profile () is most stabilizing basic temperature distribution profile. Thus, the onset of Modified Rayleigh-Bénard convection for each case of boundary conditions can be delayed by the application this cubic basic state temperature profiles. Further, Cubic 2 temperature profile () is more stabilizing than the inverted parabola temperature distribution profile ().
(ii) We observe from various corollaries that the obtained results are in good agreements with the results obtained earlier.
(iii) We observe that when the value of increases for the decreasing values of , the values of the critical Rayleigh numbers decreases and vice versa.
(iv) Further, it is observed that the obtained values of the critical Rayleigh numbers for the case of thermally insulating boundaries are lower than the corresponding values for the case of isothermal boundary as obtained by .
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Int. J. Tech. 4(1): Jan.-June. 2014; Page 100-105