Effect of Wave Number on the Onset of Instability in Couple-Stress Fluid and its Characterization in The Presence of Magnetic Field

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.