On the coupling between an ideal fluid and immersed particles

•Interpolation methods for velocity fields are principal connections.•The horizontal Lagrange–Poincaré equations yield particle methods for fluids.•Higher-order interpolation methods are derived by higher-order isotropy subgroups.•Higher-order isotropy groups yield finite-dimensional circulation theorems.

In this paper, we present finite-dimensional particle-based models for fluids which respect a number of geometric properties of the Euler equations of motion. Specifically, we use Lagrange–Poincaré reduction to understand the coupling between a fluid and a set of Lagrangian particles that are supposed to simulate it. We substitute the use of principal connections in Cendra et al. (2001)  [13] with vector field valued interpolations from particle velocity data. The consequence of writing evolution equations in terms of interpolation is two-fold. First, it provides estimates on the error incurred when interpolation is used to derive the evolution of the system. Second, this form of the equations of motion can inspire a family of particle and hybrid particle–spectral methods, where the error analysis is “built in”. We also discuss the influence of other parameters attached to the particles, such as shape, orientation, or higher-order deformations, and how they can help us achieve a particle-centric version of Kelvin’s circulation theorem.