Research Journal of Engineering and Technology

Kamal Singh ¹, Ajaib S. Banyal ²

Affiliations
1 Department of Mathematics, Govt. College, Indora, Dist. Kangra, (HP), 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 magnetic field is considered. Following the linearized stability theory and normal mode analysis, the paper established the regime for all oscillatory and non-decaying slow motions starting from rest, in a couple-stress fluid of infinite horizontal extension and finite vertical depth in the presence of uniform vertical magnetic field, the necessary condition for the existence of ‘overstability’ and the sufficient condition for the validity of the ‘exchange principle’ is derived. 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. Further, the stationary convection at marginal state with free horizontal boundaries is analyzed numerically and graphically, showing that the couple-stress parameter and magnetic field has stabilizing effect on the system. However, for the constant magnitude of couple-stress parameter and magnetic field, 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 magnetic field 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, PES, Magnetic Field, Chandrasekhar Number, MSC 2000 No.:76A05, 76E06, 76E15, 76U05.

Full Text

     

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

The effect of wave number and rotation on the thermal convection in couple-stress fluid heated from below in the presence of uniform magnetic field is investigated. The paper established the regime for all non-oscillatory and nondecaying slow motions starting from rest; the necessary condition for the existence of 'overstability' and the sufficient condition for the validity of the 'exchange principle' is also derived, when both the bounding planes are rigid. Further, the stationary convection at marginal state with free horizontal boundaries is analyzed numerically and graphically, showing that the rotation has a stabilizing effect on the system. In the presence of rotation; the magnetic field and couple-stress parameter has stabilizing (or destabilizing) effect on the system depending upon conditions satisfied. In the absence of rotation, both magnetic field and couple-stress parameter has stabilizing effect. However, for the constant magnitude of any of the couple-stress parameter, magnetic field and rotation, the wave number has a destabilizing effect for a value less than the critical value, which varies with the magnitude of the couple-stress parameter, magnetic field and rotation; and for higher values than the critical value of the wave number; it has a stabilizing effect on the system.

Keywords

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

Full Text