A semi-Lagrangian micro-macro method for viscoelastic flow calculations

We present in this paper a semi-Lagrangian algorithm to calculate the viscoelastic flow in which a dilute polymer solution is modeled by the FENE dumbbell kinetic model. In this algorithm the material derivative operator of the Navier-Stokes equations (the macroscopic flow equations) is discretized in time by a semi-Lagrangian formulation of the second order backward difference formula (BDF2). This discretization leads to solving each time step a linear generalized Stokes problem. For the stochastic differential equations of the microscopic scale model, we use the second order predictor-corrector scheme proposed in [22] applied along the forward trajectories of the center of mass of the dumbbells. Important features of the algorithm are (1) the new semi-Lagrangian projection scheme; (2) the scheme to move and locate both the mesh-points and the dumbbells; and (3) the calculation and space discretization of the polymer stress. The algorithm has been tested on the 2d 10:1 contraction benchmark problem and has proved to be accurate and stable, being able to deal with flows at high Weissenberg (Wi) numbers: specifically, by adjusting the size of the time step we obtain solutions at Wi = 444. (C) 2009 Elsevier B.V. All rights reserved.