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A Semi-Circle Theorem in Triply Diffusive Convection
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The paper mathematically establishes that the complex growth rate (Pr, Pi) 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 (Pr, Pi)-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.Keywords
Triply Diffusive Convection, Oscillatory Motions, Complex Growth Rate, Principle of the Exchange of Stability.
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