A Semi-Circle Theorem in Triply Diffusive Convection

The paper mathematically establishes that the complex growth rate (P_r, P_i) of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude, in a triply diffusive fluid layer with one of the components as heat with diffusivity k, must lie inside a semicircle in the right- half of the (P_r, P_i)-plane whose centre is origin and radius equals

√(R₁+R₂)σ-27/4π⁴τ²₂

where R₁ and R₂are the Rayleigh numbers for the two concentration components with diffusivities κ₁and κ₂(with no loss of generality, κ > κ₁> κ₂) and σ is the Prandtl number. The bounds obtained herein, in particular, yield a sufficient condition for the validity of 'the principle of the exchange of stability'. Further, it is proved that above result is uniformly valid for quite general nature of the bounding surfaces.