Instability in Couple-Stress Fluid in the Presence of Magnetic Field

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.