Experimental evidence of algebraic relaxation time in particle settling under gravity in Stokes flow

Particles settling under gravity through a fluid are common in geophysics and the environment, e.g., plankton in the ocean and droplets in clouds. The Maxey-Riley equation models the dynamics of such small inertial particles when spherical and in the unsteady Stokes regime. It is standard practice to neglect the Basset-Boussinesq history force in this equation and deduce an exponentially fast relaxation of the particle to its terminal velocity. However, a few theoretical studies, including Basset (1910), cautioned that the history force could result in qualitatively different behavior. In fact, the particle relaxes to its terminal velocity algebraically slowly, asymptotically at the rate t^{-1/2}, i.e., with an algebraic relaxation rate of -0.5, not exponentially fast. We conducted experiments to validate the theoretical predictions. By employing high-speed imaging, we analyzed the trajectories of stainless-steel spherical particles settling in Silicone oil in a controlled Stokes regime. Furthermore, we explored the wake structures surrounding the particles using flow visualization. The experiments show an algebraic relaxation rate, verifying the theory. To the best of our knowledge, these experiments provide the first confirmation of the theoretical relaxation rate.