Research Journal of Science and Technology

Rajesh Kumar Chaodhary ¹, Ajaib S. Banyal ¹

Affiliations
1 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 non decaying 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 non oscillatory, in the regime

R ≤ π[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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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 non-oscillatory, 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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Pawan Kumar ¹, Ajaib S. Banyal ², Nirmal Singh ¹

Affiliations
1 Department of Mathematics, NSCBM GC Hamirpur, (HP) 177005, IN
2 Department of Mathematics, Govt. College Nadaun (Hamirpur), (HP) 177033, IN

Abstract

Thermosolutal instability of Veronis type in Rivlin-Ericksen viscoelastic 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 established that all non-decaying slow motions starting from rest, in a Rivlin-Ericksen viscoelastic fluid of infinite horizontal extension and finite vertical depth in the presence of uniform vertical magnetic field in a porous medium, are necessarily non-oscillatory, in the regime

R_s ≤ [27π⁴/4][1/E'P₃]{[1/ε+F/P₁]-[QP₂/π²]},

where R_s is the Thermosolutal Rayliegh number, Q is the Chandrasekhar number, p₂ is the magnetic Prandtl number, p₃ is the thermosolutal Prandtl number, P₁ is the medium permeability, ε is the porosity and F is the viscoelasticity parameter. The result is important since it hold for all wave numbers and for any arbitrary combination of free and rigid boundaries at the top and bottom of the fluid. A similar characterization theorem is also proved for Stern type of configuration.

Keywords

Thermosolutal Convection, Rivlin-Ericksen Fluid, Magnetic Field, PES, Rayleigh Number, Chandrasekhar Number.

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Daleep K. Sharma ¹, Ajaib S. Banyal ²

Affiliations
1 Department of Mathematics, G. C.Arki, Distt. Solan (HP), IN
2 Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP) 177033, IN

Abstract

The thermal instability of a Rivlin-Ericksen viscoelastic 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 Rivlin-Ericksen viscoelastic fluid convection with a uniform vertical rotation, for the case of rigid boundaries shows that the complex growth rate σ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside the right half of the semi-circle

σ²_r + σ²_i T_A[εP₁/P₁+εF]²,

in a σ-plane, where T_A is the Taylor number, F is the viscoelasticity parameter, ε is the porosity, P₁ is the medium permeability; which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory Rivlin-Ericksen viscoelastic fluid heated from below. The result is important since it hold for all wave numbers and for rigid boundaries of infinite horizontal extension at the top and bottom of the fluid, and the exact solutions of the problem investigated in closed form, is not obtainable.

Keywords

Thermal Convection, Rivlin-Ericksen Fluid, Rotation, PES, Rayleigh Number, Taylor Number.

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Monika Khanna ¹, Ajaib S. Banyal ²

Affiliations
1 Department of Mathematics, Govt. College Amb, Dist. Una, (HP), IN
2 Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP), IN

Abstract

The thermal instability of a couple-stress fluid acted upon by uniform vertical magnetic field and rotation heated from below 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 magnetic field and rotation, for the case of rigid boundaries shows that the complex growth rate σ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle

σ²_r + σ²_i Q²[1-T_A/{2π²(1+π²F)-1}]^-2,

in the right half of a complex σ-plane, where Q is the Chandrasekhar number, T_A is the Taylor number, 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.

Keywords

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

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