Efforts to construct a general theoretical basis containing the essential features of Tollmien's counter example to the sufficiency of Rayleigh's theorem on point of inflexion have resulted in the determination of a pair of upper bounds of the rate of growth of arbitrary unstable disturbances; whereas, the necessary condition of the existence of these upper bounds have provided access to a sufficient condition of stability in its simplest form in the equilibrium of homogeneous incompressible inviscid parallel shear flows that are not known as yet and go beyond the works of Rayleigh [1], Tollmien [2], Friedrichs [3], Fjortoft [4], Hoiland [5], Howard [6, 7], Hickernell [8], and Banerjee et al. [9]. An alternative proof of the result that a wide class of such flows could be made stable by bringing the boundaries sufficiently close, although the flow has a point of inflexion inside the domain of flow with the Fjortoft's criterion satisfied, which is derived by Drazin and Howard [10] from variational formulation of the problem follows as an outcome of the expressions of these upper bounds. The counter example has played the role of a forerunner for much of the development that followed in its wake after 1935, and the present succession of papers is especially undertaken to investigate the trail left behind by the counter example and, it is hoped, to arrive at a

Resultsof the previous paper with this title (M. B. Banerjee et al., Stud. Appl. Math. 103:43-50) are extended to the case of neutrally stable perturbations.

The thermal instability of a Rivlin-Ericksen viscoelastic fluid, acted upon by uniform vertical rotation and heated from below, is investigated. Following linearized stability theory and normal mode analysis, the mathematical analysis of the governing equations of Rivlin-Ericksen viscoelastic fluid convection with a uniform vertical rotation is performed. It is shown that for the cases of rigid boundaries the complex growth rate  of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semicircle , in the right-hand half of a complex -plane with the center at the origin. This 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. Furthermore, the conditions necessary for the existence of oscillatory motions of growing amplitude in the present configuration and the sufficient condition for the validity of the Principle of Exchange of Stabilities are established.

No abstract

No abstract