A Hamiltonian formulation for the motion of an active spheroidal particle suspended in laminar straight duct flow

TL;DR

This paper develops a Hamiltonian framework for analyzing the motion of active spheroidal particles in laminar duct flow, extending previous models and enabling efficient computation of particle trajectories with new insights into their dynamical behavior.

Contribution

It introduces a Hamiltonian formulation for active spheroidal particles in arbitrary laminar flows, generalizing prior spherical models and providing explicit constants of motion.

Findings

Hamiltonian formulation simplifies particle orbit calculations

Explicit constants of motion derived for arbitrary flow fields

Model adaptable to oblate spheroidal particles

Abstract

We analyse a generalisation of Z\"{o}ttl and Stark's model of active spherical particles [Phys. Rev. Lett. 108, 218104 (2012)] and prolate spheroidal particles [Eur. Phys. J. E 36(1), 4 (2013)] suspended in cylindrical Poiseuille flow, to particle dynamics in an arbitrary unidirectional steady laminar flow through a straight duct geometry. Our primary contribution is to describe a Hamiltonian formulation of these systems and provide explicit forms of the constants of motions in terms of the arbitrary fluid velocity field. The Hamiltonian formulation provides a convenient and robust approach to the computation of particle orbits whilst also providing new insights into the dynamics, specifically the way in which orbits are trapped within basins defined by a potential well. In addition to considering spherical and prolate spheroidal particles, we also illustrate that the model can be adapted to oblate spheroidal particles.