The Maxey-Riley equation as a boundary condition to the 1-D heat equation

The Maxey-Riley equation describes the motion of a small spherical particle in an ambient flow field. It has a non-local contribution in the form of the Basset history integral due to the time-dependent motion of the particle. A major hurdle in studying the collective behaviour of particles has been the accurate and efficient computation of this integral. Previously, it has been either neglected or approximated by terms that are hard to rigorously justify. We show that the Maxey-Riley equation in its entirety can be mapped as a modified-Robin boundary condition to the 1-D heat equation. Exploiting this reformulation we obtain exact solutions for the particle velocity in any homogeneous time-dependent flow. We find that for short times, the particle relaxes faster than the exponential decay due to Stokes drag and for large times it relaxes as $t^{-3/2}$ in a still environment. We provide a numerical method with spectral accuracy for general flow fields at a fixed memory cost, without approximating the history integral, unlike traditional approaches.