In this paper, Lagrange’s equations along with the Ritz method are used to obtain the equation of motion for a flexible, slender cylinder subjected to axial flow. The cylinder is supported only by a translational and a rotational spring at the upstream end, and at the free end, it is terminated by a tapering end-piece. The equation of motion is solved numerically for a system in which the translational spring is infinitely stiff, thus acting as a pin, while the stiffness of the rotational spring is generally non-zero. The dynamics of such a system with the rotational spring of an average stiffness is described briefly. Moreover, the effects of the length of the cylinder and the shape of the end-piece on the critical flow velocities and the modal shapes of the unstable modes are investigated.
This paper presents an analysis of the blending characteristics of axial flow high-speed impellers under a turbulent regime of flow of an agitated low viscosity liquid. The conductivity method is used to determine the time course of blending (homogenisation) of miscible liquids in a pilot plant fully baffled mixing vessel, and a torquemeter is used for measuring the impeller power input in the same system. Four-blade and six-blade pitched blade impellers and three high efficiency axial flow impellers are tested for the given degree of homogeneity (98%). The experimental results and also the results of the authors' previous study, in accordance with the theoretical approach described in the literature, show that there is a universal relationship between the impeller power number and the dimensionless blending time, taking into consideration the impeller-to-vessel diameter ratio, independent of the geometry of the axial flow impeller but dependent on the degree of homogeneity. This relationship is found to be valid on a pilot plant scale under a turbulent flow regime of an agitated liquid.