International Journal of Technology

Jyoti Prakash ¹, Virender Singh ¹, Shweta Manan ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

Abstract

Condition for characterizing nonoscillatory motions, which may be neutral or unstable, for rotatory hydrodynamic triply diffusive convection in a porous medium is derived. It is analytically proved that the principle of the exchange of stabilities, in rotatory triply diffusive convection in a porous medium, is valid in the regime R₁E₁σ/2τ²₁π⁴ + R₂E₂σ/2τ²₂π⁴ + T_a/π²ΛD_a^-1 ≤ 1, where R₁ and R₂ are the concentration Raleigh numbers, and τ₁ and τ₂ are the Lewis numbers for the two concentration components respectively, T_a is the Taylor number, σ is the Prandtl number, D_a is the Darcy number, E₁ and E₂ are constants.

Keywords

Triply Diffusive Convection, Porous Medium, Darcy-Brinkman Model, the Principle of the Exchange of Stabilities, Taylor Number, Concentration Rayleigh Number.

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Jyoti Prakash ¹, Shweta Manan ¹, Virender Singh ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Shimla – 171005, IN

Abstract

Condition for characterizing nonoscillatory motions, which may be neutral or unstable, for multicomponent convection is derived.

Keywords

Multicomponent Convection, The Principle of the Exchange of Stabilities, Oscillatory Motions, Complex Growth Rate, Concentration Rayleigh Number.

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Jyoti Prakash ¹, Kultaran Kumari ¹, Vinod Kumar ²

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla 171005, IN
2 Department of Physics, M.L.S.M. College, Sunder-Nagar, Distt. Mandi (H.P.), IN

Abstract

Linear stability of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer has been studied analytically. A sufficient condition for the occurrence of stationary convection has been derived in terms of the parameters of the system alone. It is further proved the above result is uniformly valid for any combination of the bounding surfaces.

Keywords

Double-Diffusive Convection, Viscoelastic Fluid, The Principle of the Exchange of Stabilities, Porous Medium.

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Jyoti Prakash ¹, Rajeev Kumar ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Shimla-171005, IN

Abstract

The paper mathematically establishes that magnetohydrodynamic triply diffusive convection, with variable viscosity and with one of the components as heat with diffusivity κ, cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the two concentration Rayleigh numbers R₁ and R₂, the Lewis numbers τ₁ and τ₂ for the two concentrations with diffusivities k₁ and k₂ respectively (with no loss of generality κ > κ₁ > κ₂), μ_min (the minimum value of viscosity μ in the closed interval [0,1]) and the Prandtl number σ satisfy the inequality R₁ + R₂ ≤ 27π⁴/4{μ_min+(τ₁+τ₂)/σ/1+τ₁/τ²₂ provided D²μ is positive everywhere. It is further proved that this result is uniformly valid for any combination of rigid and/or free perfectly conducting boundaries.

Keywords

Triply Diffusive Convection, Variable Viscosity, Concentration Rayleigh Number, Oscillatory Motion, Initially Bottom Heavy Configuration and Chandrasekhar Number.

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Jyoti Prakash ¹, Pankaj Kumar ¹, Shweta Manan ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

Abstract

analogous to magnetohydrodynamic thermohaline convection problem of Veronis (1965) type has been established. It is shown that the total kinetic energy associated with a disturbance exceeds the sum of its total magnetic and concentration energies in the parameter regime Qσ₁/π² + R₁σ/4τ²₁π⁴ + R₂σ/4τ²₂π⁴ + ⋯+ R_n-1σ/τ²_n-1π⁴ ≤ 1, where Q,σ,σ₁,τ₁,τ₂,…,τ_n-1,R₁,R₂,…,R_n-1 represent Chandrasekhar number, Prandtl number, magnetic Prandtl number, Lewis number for first concentration component, Lewis number for second concentration component, Lewis number for (n − 1)th concentration component, concentration Rayleigh number for first component, concentration Rayleigh number for second component, concentration Rayleigh number for (n - 1)th component respectively. Further, this result is uniformly valid for any combination of rigid or free boundaries whether perfectly conducting or insulating.

Keywords

Multicomponent Convection, Chandrasekhar Number, Lewis Number, Prandtl Number, Rayleigh Number.

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Jyoti Prakash ¹, Kanu Vaid ¹, Renu Bala ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

Abstract

The present paper prescribes upper bounds for oscillatory motions of neutral or growing amplitude in magnetorotatory thermohaline configurations of Veronis (Veronis, G., J. Mar. Res., 23(1965)1) and Stern types (Stern, M.E., Tellus 12(1960)172) in porous medium (Darcy model) in such a way that also result in sufficient conditions of stability for an initially bottom-heavy as well as initially top-heavy configuration.

Keywords

Thermohaline Instability, Oscillatory Motions, Initially Bottom-Heavy Configuration, Initially Topheavy Configuration, Porous Medium.

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Jyoti Prakash ¹, Virender Singh ¹, Shweta Manan ¹, Vinod Kumar ²

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN
2 Department of Physics, MLSM College, Sundernagar, H.P., IN

Abstract

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 in porous medium (Darcy Model) with one of the components as heat with diffusivity , 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 where R₁ and R₂ are the Raleigh numbers for the two concentration components with diffusivities k₁ and k₂ (with no loss of generality, k > k₁> k₂ ) and is the Prandtl number. 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, Porous Medium.

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Jyoti Prakash ¹, Rajeev Kumar ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

Abstract

Upper bounds for the complex growth rate of an arbitrary oscillatory perturbation which may be neutral or unstable of thermohaline convection of Veronis (G.Veronis, J.MarineRes., 23, (1965) 1-17) type with the viscosity variation effects included heated from below are obtained which in particular yield sufficient condition for the validity the "principle of the exchange of stabilities" for this configuration. Similar results are also obtained for thermohaline convection of Stern (ME Stern, Tellus, 12,(1960), 171-175) type with the viscosity variation effect included. The results obtained herein are uniformly valid for all combinations of dynamically free and rigid boundaries.

Keywords

Thermohaline Instability, Veronis Type, Stern Type, Oscillatory Motions, Variable Viscosity.

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Jyoti Prakash ¹, Shweta Manan ¹, Virender Singh ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

Abstract

The present paper mathematically establishes that 'the principle of the exchange of stabilities' for triply diffusive convection in porous medium (Darcy model) is valid in the regime (R_1 σ)/(2〖ι_1〗^2 π^4 )+(R_2 σ)/(2〖ι_2〗^2 π^4)≤1, where R_1 and R_2 are the Rayleigh numbers for the two concentration components, ι_1 and ι_2 are the Lewis numbers for the two concentration components and σ is the thermal Prandtl number. It is further proved that the above result is uniformly valid for any combination of rigid and free boundaries.

Keywords

Triply Diffusive Convection, Nonoscillatory Motions, Principle of the Exchange of Stabilities, Concentration Rayleigh Number, Porous Medium, Darcy Model.

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Jyoti Prakash ¹, Sanjay Kumar Gupta ¹, Renu Bala ¹, Kanu Vaid ¹

Affiliations
1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

Abstract

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.

Keywords

Triply Diffusive Convection, Oscillatory Motions, Complex Growth Rate, Principle of the Exchange of Stability.

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