International Journal of Technology

Ajaib S. Banyal ¹, Kamal Singh ²

Affiliations
1 Department of Mathematics, Sidharth Govt. College, Nadaun, Distt. Hamirpur, (HP) 177033, IN
2 Department of Mathematics, Govt. College, Indora, Distt. Kangra, (HP) 176401, IN

Abstract

Thermosolutal instability of Veronis(1965) type in a couple-stress fluid in the presence of uniform vertical magnetic field in a porous medium is considered. Following the linearized stability theory and normal mode analysis, the paper mathematically established the condition for characterizing the oscillatory motions which may be neutral or unstable, for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. It is proved analytically that all non-decaying slow motions starting from rest, in a couple-stress fluid of infinite horizontal extension and finite vertical depth in a porous medium, which is acted upon by uniform vertical magnetic field opposite to force field of gravity and a constant vertical adverse temperature gradient, are necessarily nonoscillatory, in the regime established, the result is important since the exact solutions of the problem investigated are not obtainable in closed form, for any arbitrary combination of free and rigid boundaries. A similar characterization theorem is also established for Stern (1960) type of configuration.

Keywords

Thermosolutal Convection, Couple-Stress Fluid, Magnetic Field, Rayleigh Number, Chandrasekhar Number.

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Monika Khanna ¹, Ajaib S. Banyal ²

Affiliations
1 Department of Mathematics, Singhania University, Jhunjhunu, (Raj.), IN
2 Department of Mathematics, Govt. College, Nadaun, Dist. Hamirpur, (HP) 177033, IN

Abstract

The thermal instability of a couple-stress fluid acted upon by uniform vertical rotation and heated from below in a porous medium is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical rotation in porous medium, for the case of rigid boundaries shows that the complex growth rate of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle, in the right half of a complex -plane, where is the Taylor number, is the dimensionless medium permeability of the porous medium and F is the couple-stress parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory couple-stress fluid in porous medium heated from below. Further, It is established that the existence of oscillatory motions of growing amplitude in the present configuration, depends crucially upon the magnitude of the non-dimensional number, in the sense so long as, no such motions are possible, and in particular PES is valid.

Keywords

Thermal Convection, Couple-Stress Fluid, Rotation, PES, Taylor Number.

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Kamal Singh ¹, Ajaib S. Banyal ²

Affiliations
1 Department of Maths, Singhania University, Pacheri Bari, Jhunjhunu- 333515 (Raj.), IN
2 Department of Mathematics, Govt. College, Nadaun, Dist. Hamirpur, (HP)-177033, IN

Abstract

Thermal instability of couple-stress fluid in the presence of uniform vertical rotation is considered. Following the linearized stability theory and normal mode analysis, the paper established the regime for all oscillatory and nondecaying slow motions starting from rest, in a couple-stress fluid of infinite horizontal extension and finite vertical depth in the presence of uniform vertical rotation and the necessary condition for the existence of 'overstability' and the sufficient condition for the validity of the 'exchange principle' is derived, when the bounding surfaces of infinite horizontal extension, at the top and bottom of the fluid are rigid. Further, the stationary convection at marginal state with free horizontal boundaries is analyzed numerically and graphically, showing that the couple-stress parameter and rotation has stabilizing effect on the system. However, for the constant magnitude of couple-stress parameter and rotation, the wave number has a destabilizing effect for a value less than a critical value, which varies with the magnitude of the couple-stress parameter and rotation; and for higher value than the critical value of the wave number; it has a stabilizing effect on the system.

Keywords

Thermal Convection, Couple-Stress Fluid, Rotation, PES, Taylor Number.

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Kamal Singh ¹, Ajaib S. Banyal ²

Affiliations
1 Department of Mathematics, Singhania University, Pacheri-Bari, Jhunjnu, Rajasthan, IN
2 Department of Mathematics, Govt. College, Nadaun, Dist. Hamirpur, (HP) 177033, IN

Abstract

The thermal instability of a couple-stress fluid acted upon by uniform vertical magnetic field and heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper mathematically established the condition for characterizing the oscillatory motions which may be neutral or unstable for free and perfectly conducting boundaries at the top and bottom of the fluid. It is established that all nondecaying slow motions starting from rest, in a couple-stress fluid of infinite horizontal extension and finite vertical depth, which is acted upon by uniform vertical magnetic field opposite to gravity and a constant vertical adverse temperature gradient, are necessarily nonoscillatory, in the regime

R ≤ π²(3π⁴F+1/P²)

 

where R is the thermal Rayleigh number, F is the couple-stress parameter and P₂ is the magnetic Prandtl number. The result is important since the exact solutions of the problem investigated in closed form, are not obtainable.

Keywords

Thermal Convection, Couple-Stress Fluid, PES, Magnetic Field, Chandrasekhar Number.

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