A deterministic-particle-based scheme for micro-macro viscoelastic flows

In this article, we introduce a new method for discretizing micro-macro models of dilute polymeric fluids based on deterministic particles. Our approach integrates a finite element discretization for the macroscopic fluid dynamic equation with a deterministic variational particle scheme for the microscopic Fokker-Planck equation. To address challenges arising from micro-macro coupling, we employ a discrete energetic variational approach to derive a coarse-grained micro-macro model with a particle approximation first and then develop a particle-FEM discretization for the coarse-grained model. The accuracy of our method is evaluated for a Hookean dumbbell model in a Couette flow by comparing the computed velocity field with existing analytical solutions. We also use our method to study nonlinear FENE dumbbell models in different scenarios, such as extension flow, pure shear flow, and lid-driven cavity flow. Numerical examples demonstrate that the proposed deterministic particle approach can accurately capture the various key rheological phenomena in the original FENE model, including hysteresis and $\delta$-function-like spike behavior in extension flows, velocity overshoot phenomenon in pure shear flow, symmetries breaking, vortex center shifting and vortices weakening in the lid-driven cavity flow, with a small number of particles.