  We present and analyze a series of benchmark tests regarding the application
of the immersed boundary (IB) method to viscoelastic flows through and around
non-trivial, stationary geometries. The IB method is widely used for the
simulation of biological fluid dynamics and other modeling scenarios where a
structure is immersed in a fluid. Although the IB method has been most commonly
used to model systems with viscous incompressible fluids, it also can be
applied to visoelastic fluids, and has enabled the study of a wide variety of
dynamical problems including the settling of vesicles and the swimming of
elastic filaments in fluids modeled by the Oldroyd-B constuitive equation.
However, to date, relatively little work has explored the accuracy or
convergence properties of the numerical scheme. Herein, we present benchmarking
results for an IB solver applied to viscoelastic flows in and around
non-trivial geometries using the idealized Oldroyd-B and more realistic,
polymer-entanglement-based Rolie-Poly constitutive equations. We use
two-dimensional numerical test cases along with results from rheology
experiments to benchmark the IB method and compare it to more complex finite
element and finite volume viscoelastic flow solvers. Additionally, we analyze
different choices of regularized delta function and relative Lagrangian grid
spacings which allow us to identify and recommend the key choices of these
numerical parameters depending on the present flow regime.
