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On the Principle of the Exchange of Stabilities in Rotatory Triply Diffusive Convection

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Abstract

The present paper mathematically establishes that the principle of the exchange of stabilities in a rotatory triply diffusive convection is valid in the regime \( \frac{{R_{1} \sigma }}{{2\tau_{1}^{2} \pi^{4} }} + \frac{{R_{2} \sigma }}{{2\tau_{2}^{2} \pi^{4} }} + \frac{{T_{a} }}{{\pi^{4} }} \le 1, \) where R 1 and R 2 are the Rayleigh numbers for the two concentration components, τ 1 and τ 2 are the Lewis numbers for the two concentration components, T a is the Taylor number and σ is the Prandtl number.

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Abbreviations

a 2 :

Square of the wave number

\( D\, = \,\frac{\text{d}}{{{\text{d}}z}} \) :

Differentiation with respect to z

g :

Acceleration due to gravity

\( p( = p_{r} + i p_{i} ) \) :

Complex growth rate

R :

Thermal Rayleigh number

R 1 :

Concentration Rayleigh number for first concentration component

R 2 :

Concentration Rayleigh number for second concentration component

S 10 :

Concentration of the first component at the lower boundary

S 11 :

Concentration of the first component at the upper boundary

S 20 :

Concentration of the second component at the lower boundary

S 21 :

Concentration of the second component at the upper boundary

T :

Temperature

T 0 :

Temperature at the lower boundary

T 1 :

Temperature at the upper boundary

T a :

Taylor number

w :

Vertical velocity

z :

Vertical coordinate

σ :

Prandtl number

ζ :

Vertical vorticity

θ :

Temperature

ϕ 1 :

Perturbation concentration of first component

ϕ 2 :

Perturbation concentration of second component

\( \tau_{1} \left( { = \frac{{\kappa_{1} }}{\kappa }} \right) \) :

Lewis number for concentration S 1

\( \tau_{2} \left( { = \frac{{\kappa_{2} }}{\kappa }} \right) \) :

Lewis number for concentration S 2

κ:

Thermal diffusivity

κ 1 :

Mass diffusivity of S 1

κ 2 :

Mass diffusivity of S 2

*:

Complex conjugation

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, 171005, India

    Jyoti Prakash, Renu Bala & Kanu Vaid

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  1. Jyoti Prakash
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  2. Renu Bala
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  3. Kanu Vaid
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Correspondence to Jyoti Prakash.

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Prakash, J., Bala, R. & Vaid, K. On the Principle of the Exchange of Stabilities in Rotatory Triply Diffusive Convection. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 84, 433–439 (2014). https://doi.org/10.1007/s40010-014-0155-3

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