Inertial forces in the Maxey–Riley equation in nonuniform flows

The Maxey–Riley equation describes the motion of a spherical particle suspended in a spatially nonuniform, time-dependent flow, and finds applications in a wide range of flow situations. We reexamine the hydrodynamics underlying the Maxey–Riley equation to find additional inertial forces associated with second gradients of the background flow velocity, not accounted for in the original framework. These forces amplify inertial Faxén terms threefold, while also contributing advective terms that are quadratic in fluid velocity and may exceed Faxén forces in some flows. We present a more comprehensive form of the Maxey–Riley equation that includes these contributions, and discuss its implications for particle dynamics in flows with curvature.