Numerical study of viscous flows inside partially filled spinning and coning cylinders

This paper is concerned with the solution of the 3-D-Navier-Stokes equations describing the steady motion of a viscous fluid inside a partially filled spinning and coning cylinder. The cylinder contains either a single fluid of volume less than that of the cylinder or a central rod and a single fluid of combined volume (volume of the rod plus volume of the fluid) equal to that of the cylinder. The cylinder rotates about its axis at the spin rate omega and rotates about an axis that passes through its center of mass at the coning rate Omega. In practical applications, as in the analysis and design of liquid-filled projectiles, the parameter epsilon = tau sin theta, where tau = Omega/omega and theta is the angle between spin axis and coning axis, is small. As a result, linearization of the Navier-Stokes equations with this parameter is possible. Here, the full and linearized Navier-Stokes equations are solved by a spectral collocation method to investigate the nonlinear effects on the moments caused by the motion of the fluid inside the cylinder. In this regard, it has been Sound that nonlinear effects are negligible for tau approximate to 0.1, which is of practical interest to the design of liquid-filled projectiles, and the solution of the linearized Navier-Stokes equations is adequate for such a case. However as tau increases, nonlinear effects increase, and become significant as epsilon surpasses about 0.1. In such a case, the nonlinear problem must be solved. Complete details on how to solve such a problem is presented.