Chebyshev Expansion Of The Flow In A Spinning And Coning Cylinder

Abstract

This paper is concerned with the calculation of the moments exerted by a viscous fluid on the walls of a cylinder that is spinning about its axis and coning about an axis that passes through its center of mass. For small coning angles and/or coning frequencies, these moments are estimated by solving the linearized Navier-Stokes equations. Solving the linearized Navier-Stokes equations is computationally expensive. Fortunately, when using the control volume approach to calculate these moments, these moments depend essentially on the axial velocity, and the linearized equations describing the deviation of the fluid motion from solid body rotation can be reduced to a single sixth-order partial differential equation governing the axial velocity. This single equation is solved by expanding the axial velocity in a triple series made of Fourier functions in the azimuthal direction and Chebyshev polynomials in the radial and axial directions. For linear analysis, only the fundamental component in the azimuthal direction is needed for the evaluation of moments and the triple series is reduced to a double Chebyshev expansion in the radial and axial directions thereby reducing the three-dimensional problem into a two-dimensional one. The results obtained by Chebyshev expansion show good agreement with those obtained by using eigenfunction expansion.