In this work, the authors advance the numerical modeling of particle–fluid interaction by extending a multi-step algorithm to solve the full Maxey–Riley–Gatignol equation (MaRGE) in three-dimensional flow fields. Their approach captures the demanding integral term that arises from the viscous diffusion of vorticity around inertial particles, a term that is often neglected or approximated because it makes the equation difficult to solve, even though it is essential for obtaining realistic particle trajectories.
Building on an approach by Daitche for the two-dimensional case and an analytical solution by Candelier and colleagues, the study derives an analytical solution for a particle moving in a three-dimensional vortex while subject to gravity, which serves as a test case to verify the implementation. Numerical examples compare empirical and theoretical convergence orders and reveal an order reduction, in particular for particles with non-zero initial relative velocity. This development paves the way for more accurate simulations of particle dynamics in engineering and environmental applications.
The study was carried out at the Institute of Mathematics at Hamburg University of Technology within the Collaborative Research Centre CRC 1615 (SMART Reactors).
Vamika Rathi, Daniel Ruprecht (2026). Numerical modeling of inertial particles in three-dimensional fluid flow. PAMM 26 (2), e70158.
