Flow induced by a rotating cone: velocity field and stability properties

The flow induced by a cone which rotates around its symmetry axis is a canonical case of interest both from a fundamental and an applied viewpoints. Despite its simplicity, it has received little attention if compared to the case of a rotating disk. The boundary-layer equations in the case of a rotating cone allow for a self-similar solution (see Wu, Appl. sci. Res. 8, 1959) analogous to the von Kármán one for a rotating disk. This solution implies that the wall-normal velocity does not vanish far from the wall, at difference with what expected for a cone of finite size. Moreover, the cone apex induces a large-scale motion in comparison to the boundary layer thickness, which is absent for the rotating disk. In this work we extend the analysis by Wu (1959) to account for viscous corrections and large-scale motion due to the apex, arriving to an original correction which is also self-similar. We furthermore explore the influence of the proposed correction on the estimated stability properties of the flow using a weakly-divergent approach based on the assumption of a slow evolution in the streamwise direction. In addition to a standard approach, which is first-order and in which elliptic terms are excluded, we also propose and apply here a second-order extension.