For application to liquid-filled shell problems, the natural frequencies and decay rates of oscillations of liquids during spin-up in filled rotating cylinders are calculated. In this first part only the fully spun-up flow, i.e., solid-body rotation, is considered. Nevertheless, this part describes in detail the method of solution for the general case of spin-up, and presents results which check with experimental and previous theoretical data closely enough to confirm the reliability of the computational procedure. Data are shown which illustrate the variation of the eigenfrequencies of a disturbance mode with Reynolds number and aspect ratio of the cylinder. This work treats the viscous perturbation equations for flow of a rotating fluid. An eigenvalue problem results, defined by a sixth-order system which is integrated using the orthonormalization technique. (Author).

The unsteady motion of a fluid which fills a cylindrical container in a spinning projectile is considered. The spin is imparted impulsively to the cylinder and spin-up of the fluid is the basic flow which is perturbed to study the waves in the rotating fluid. The core flow is perturbed, not the Ekman layer flow. This is called the spin-up eigenvalue problem which is solved with modal solutions. Viscous perturbations are needed because of the boundary layer and critical layer. A numerical method of solution is given and several results shown. There are two times at which projectile instability might occur. The results of the theory and method are validated by comparison with experimental results.

An understanding of the instability of a spinning liquid-filled projectile requires a knowledge of the wave system in the rotating fluid. The axisymmetric wave system in a cylinder is studied by a numerical simulation; the particular case of perturbed solid body rotation is treated. Finite-difference solutions to the Navier-Stokes equations provide data from which wave frequencies and damping can be extracted using Fourier transform and digital filter techniques. The frequency and damping are compared with the values computed from a linearized eigenvalue analysis. It is found that the latter can be used with confidence for Reynolds number as low as 1,000.