ABSTRACT The motion of inertial particles in a fluid is modeled by the Maxey–Riley–Gatignol equation (MaRGE). The MaRGE contains an integral term that arises due to the viscous diffusion of vorticity in the fluid around the particle. Because it makes MaRGE difficult to solve numerically, the integral term is often neglected or approximated, despite its demonstrated importance for obtaining realistic trajectories. There are some studies that propose algorithms to solve the full MaRGE numerically for two dimensional flows fields. For simple flows like a vortex, analytical solutions exist that can serve as test cases to verify implementations. However, in most practical applications, fluids will be three dimensional. This article extends a multi‐step algorithm proposed by Daitche to the three‐dimensional case. Based on an approach by Candelier et al., it derives an analytical solution for a particle moving in a three‐dimensional vortex while being subject to gravity. Numerical examples compare empirical and theoretical convergence orders and demonstrate order reduction in particular for particles with non‐zero relative initial velocity.
Rathi et al. (Tue,) studied this question.