1 INTRODUCTION:
Double-Diffusive convection is the name given to such convection motions where the density variations are caused by two different components which have different rate of diffusion. Double-diffusive convection generally referred to as Thermohaline convection has been extensively studied in the recent past on account of its interesting complexities as well as its direct relevance in many problems of practical interest.
The striking feature of many systems of interest is that instabilities can develop even when the net density decreases upward. Diffusion, which is generally stabilizing in a fluid containing a single solute, can act so as to allow the release of potential energy in the component that is heavy at top. Because of heat and salt, the process is called thermohaline convection, the name Double-Diffusive Convection has been used to encompass the wider range of phenomenon.
The whole subject of double-diffusive convection was developed from the ‘thought experiments’ of Stommel, Arons and Blanchard [1], who imagined a heat conducting pipe suspended vertically in an ocean with its upper end in warm salty water and its lower end in denser, cooler and fresher water( a temperature and salinity distribution which is common in subtropical ocean). Stern [2] took further decisive step to explain the thought experiment of [1]. He considered a horizontal layer of viscous fluid heated from above and with mass concentrations S0 and S1(S0 <S1) at lower and upper boundary respectively. He showed that even lighter at the top than at the bottom, instability might still occur in this configuration as exchange of stabilities provided the destabilizing concentration gradient is sufficiently large but compatible with the total density field is gravitationally stable.
Banerjee et al ([3], [4]) presented a modified analysis of thermal/thermohaline instability of a liquid layer heated from below. They pointed out that Rayleigh’s utilization of Boussinesq approximation overlooks the terms in the equation of heat conduction which are on account of the variations in the specific heat at constant volume due to variations in temperature/concentration and which is such that in the usual circumstances they cannot be ignored, if the Boussinesq approximation were to be consistently and accurately applied throughout the analysis. The essential argument on which these terms find places in the modified theory is that it is the temperature/concentration differences which are of moderate amounts and not necessarily the temperature/concentration itself. Thus, they asserted that an incorporation of these terms into the calculations completes the qualitative and quantitative gaps as in Rayleigh’s theory as pointed above. Thus, the modified thermal/ thermohaline convection problems originated as a consequence of the proper utilization of Boussinesq approximation in the thermal/Double-Diffusive Convection problems.
The Double-Diffusive fluid systems correspond to those mixtures wherein the concentration and thermal diffusive modes are absent, i.e. the concentration gradient arising from the diffusion induced temperature gradient (Soret effect) and conversely, the diffusion induced temperature gradients due to concentration gradient (Dufour effect) are neglected. When these coupled effects are present in double-diffusive problems the resulting problems are usually termed as Dufour/Soret driven double diffusive convective problems (McDougall [5]).
Critical examination of the derivations of eigenvalue problems describing Stern type double-diffusive convective convection (SDDC), Modified Thermohaline convection (MTHC) and Dufour-Driven Thermohaline convection (DDTHC) reveals that these problems are quite closely related to each other and clearly hints towards the unification. In the present paper, we have constructed of a generalized setup of eigenvalue problems from the perturbations equations governing Modified Thermohaline convection by an appropriate choice of parameters of the fluid. This generalized setup of eigenvalue problem named as; generalized double-diffusive convection problem yields eigenvalue problems for Stern type double-diffusive convection problem and for Dufour-Driven double-diffusive convection problem as a consequence. The stability investigations of this general problem are carried in this paper and some general results concerning the stability or otherwise are derived for the case of both dynamically free boundaries and various consequence of the derived results are also discussed.
2. THE PHYSICAL CONFIGURATION AND EIGENVALUE PROBLEM:
The physical configuration discussed in the present paper is the following;
A viscous and finitely heat conducting
fluid is statically between two horizontal boundaries 
of infinite horizontal extension and
finite vertical depth which are respectively maintained at uniform temperature
T0 and T1(T0 < T1) and uniform
concentration S0 and S1(S0< S1).The
temperature gradient/concentration gradient thus maintained will respectively
be qualified as favorable/ unfavorable because of their tendencies to decrease
the density of the fluid vertically upward.
Following the usual steps of the linear stability theory and the deviation of the eigenvalue problem for Modified Thermohaline convection of Banerjee et. al. [3], we have the following linearized non-dimensional perturbation equations;
