Accurate solution method for the Maxey-Riley equation, and the effects of Basset history

Abstract

The Maxey-Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey-Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional heat equation, and solved using the Unified Transform Method. We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as stretched-exponential at short time while slowing it down to a power-law relaxation, $\sim t^{-3/2}$, at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity ($\sim t^{-1/2}$), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects. Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover our results indicate that the Basset history can affect dynamics significantly.