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

We analyze a generalization of Zöttl and Stark's model of active spherical particles [Phys. Rev. Lett. 108, 218104 (2012)0031-900710.1103/PhysRevLett.108.218104] and prolate spheroidal particles [Eur. Phys. J. E 36, 4 (2013)1292-894110.1140/epje/i2013-13004-5] 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 while 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.