Bounds for the Complex Growth Rate in Rivlin-Ericksen Viscoelastic Fluid in the Presence of Rotation in a Porous Medium

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.