  We carry out a numerical study on the dynamics of a single non-Brownian
flexible fiber in two-dimensional Couette flows at finite Reynolds numbers. We
employ the bead-spring model of flexible fibers to extend the fluid particle
dynamics (FPD) method that is originally developed for rigid particles in
viscous liquids. We implement the extended FPD method using a
multiple-relaxation-time (MRT) scheme of the lattice Boltzmann method (LBM).
The numerical scheme is validated firstly by a series of benchmark simulations
that involve liquid-solid coupling. The method is then used to study the
dynamics of flexible fibers in Couette flows. We only consider the highly
symmetric case where the fibers are placed on the symmetry center of Couette
flows and we focus on the effects of the fiber stiffness, the confinement
strength, and the finite Reynolds number (from 1 to 10). A diagram of the fiber
shape is obtained. For fibers under weak confinement and a small Reynolds
number, three distinct tumbling orbits have been identified. (1) Jeffery orbits
of rigid fibers. The fibers behave like rigid rods and tumble periodically
without any visible deformation. (2) S-turn orbits of slightly flexible fibers.
The fiber is bent to an S-shape and is straightened again when it orients to an
angle of around 45 degrees relative to the positive x direction. (3) S-coiled
orbits of fairly flexible fibers. The fiber is folded to an S-shape and tumbles
periodically and steadily without being straightened anymore during its
rotation. Moreover, the fiber tumbling is found to be hindered by increasing
either the Reynolds number or the confinement strength, or both.
