The equation of motion of a small spherical particle in an isolated Rankine vortex is analyzed using an asymptotic scheme valid for the limiting case of small Stokes number St. The effects of particle inertia and added mass, gravity, the acceleration of the fluid, viscous drag, and the Basset history force are taken into consideration. For the case of an isolated Rankine vortex, the analysis shows that in the region where the fluid velocity is large enough, the viscous drag constrains the particle to move with a velocity equal to that of the fluid plus a perturbation of order St. This perturbative term incorporates the effects of gravity, the density difference between the particle and the fluid, and the local acceleration of the fluid. In the region where the fluid velocity is small, the particle moves with a velocity equal to the sum of the fluid velocity and the rising/settling velocity of the particle in still fluid, the effects of particle inertia and fluid acceleration appearing as small perturbations. Throughout the whole region of the flow, the effect of the Basset force always appears at higher order than the other forces acting on the particle and may, consequently, be neglected. The analysis also shows that a particle with a mass density greater than that of the fluid always escapes from the central region of the vortex, but a buoyant particle may be trapped by the equilibrium point located there.
The unsteady-state energy conservation equation for a small spherical particle in LII modeling
