Flipping and scooping of curved 2D rigid fibers in simple shear: the Jeffery equations

The dynamical system (or ``Jeffery equations'') governing the orbits of a curved rigid two-dimensional fiber in simple shear is derived in analytical form. The study is motivated by the need to understand the dynamics of isolated non-axisymmetric rod-like particles in simple flows for use in suspension modeling. Solutions of the dynamical system are shown to display the ``flipping'' and ``scooping'' recently observed in computational studies of three-dimensional fibers using linked rigid rod and bead-shell models [Wang {\em et al.}, {\em Phys. Fluids}, {\bf 24}, (2012)]. Indeed the equations we derive are expected to be the same ones governing curved 3D slender fibers executing motions in the plane of shear.