Stability analysis of solid particle motion in rotational flows

A two-dimensional model of a rotational flow field is used to performthe stability analysis of solid particle motion.It results that the stagnation points are equilibrium points for the motion ofparticles and the stability analysis allows to estimate their rolein the general features of particle motion and to identify different regimesof motion.Furthermore, the effects of Basset history force on the motion ofparticles lighter than the fluid ({\it bubbles}) are evaluated by meansof a comparison with the analytical results found in the case of Stokes drag.Specifically, in the case of bubbles, the vortex centres are stable(attractive) points, so the motion is dominated by the stability propertiesof these points. A typical convergence time scale towards the vortexcentre is defined and studied as a function of the Stokes number $St$ andthe density ratio $\gamma$. The convergence time scale shows a minimum(nearly, in the range $0.1 < St < 1$), in the case either with or withoutthe Basset term.In the considered range of parameters, the Basset force modifies theconvergence time scalewithout altering the qualitative features of the particle trajectory. Inparticular, a systematic shift of the minimum convergence time scale towardthe inviscid region is noted.For particles denser than the fluid, there are no stable points.In this case, the stability analysis is extended to the vortex vertices.It results that the qualitative features of motion depend on thestability of both the centres and the vertices of the vortices.In particular, the different regimes of motion (diffusive or ballistic)are related to the stability properties of the vortex vertices. The criterionfound in this way is in agreement with the results of previousauthors (see, {\it e.g.}, Wang {\it et al.} \cite{wm92}).