Georgia Institute of Technology College of Sciences

Recent experimental and numerical observations have shown the significance of the Basset--Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. I discuss the governing equations of motion for the inertial particles, i.e. the Maxey-Riley equation, including a fractional order derivative in time. Then I show that the observed algebraic decay is a universal property of the Maxey--Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is O(\epsilon)-close to the fluid velocity, where 0<\epsilon<<1 is proportional to the square of the ratio of the particle radius to the fluid characteristic length-scale. These results follows from a sharp analytic upper bound that we derive for the particle velocity.