On the effect of the Boussinesq–Basset force on the radial migration of a Stokes particle in a vortex

Abstract

The trajectory of an isolated solid particle dropped in the core of a vertical vortex is investigated theoretically and experimentally, in order to analyze the effect of the history force on the radial migration of the inclusion. Both the Stokes number (based on the particle radius and the fluid angular velocity) and the particle Reynolds number are small. The particle is heavier than the fluid, and is therefore expelled from the center of the vortex. An experimental device using spherical particles injected in a rotating cylindrical tank filled with silicone oil has been built. Experimental trajectories are compared to analytical solutions of the motion equations, which are obtained by making use of classical Laplace transforms. The analytical expression of the history force and the ejection rate are carried out. This force does not vanish, but increases exponentially and has to be taken into account for efficient predictions. In particular, calculations without history force overestimate particle ejection. The relative difference between the ejection rate with and without history force scales like the square root of the Stokes number, so that differences of the order of 10% are visible as soon as the Stokes number is of the order of 0.01. Also, agreement between experimental and theoretical trajectories is observed only if the acceleration term in the history integral involves the time derivative of the fluid velocity following the particle, rather than the acceleration of fluid points at the particle location, even for small particle Reynolds numbers. Finally, analytical calculations show that the particle ejection rate is more sensitive to the Boussinesq–Basset force than to Saffman’s lift.