In this paper, we present a numerical method designed to simulate the
challenging problem of the dynamics of slender fibers immersed in an incompressible
fluid. Specifically, we consider microscopic, rigid fibers, that
sediment due to gravity. Such fibers make up the micro-structure of many
suspensions for which the macroscopic dynamics are not well understood.
Our numerical algorithm is based on a non-local slender body approximation
that yields a system of coupled integral equations, relating the forces
exerted on the fibers to their velocities, which takes into account the hydrodynamic
interactions of the fluid and the fibers. The system is closed by
imposing the constraints of rigid body motions.
The fact that the fibers are straight have been further exploited in the
design of the numerical method, expanding the force on Legendre polynomials
to take advantage of the specific mathematical structure of a finite-part
integral operator, as well as introducing analytical quadrature in a manner
possible only for straight fibers.
We have carefully treated issues of accuracy, and present convergence
results for all numerical parameters before we finally discuss the results from
simulations including a larger number of fibers.