A semi-circle theorem in couple-stress fluid in the presence of rotation

The thermal instability of a couple-stress fluid acted upon by uniform vertical rotation and 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 rotation, for the case of rigid boundaries shows that the complex growth rate sigma of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle vertical bar sigma vertical bar(2) = T-A-(pi F-8(2)), in the right half of a complex a-plane, where 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 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 T-A/(pi F-8(2)), in the sense so long as 0 < TAP/(pi F-8(2)) <= 1, no such motions are possible, and in particular PES is valid.