The adaptive Lagrangian particle method for macroscopic and micro–macro computations of time-dependent viscoelastic flows

We propose a new numerical technique, referred to as the Adaptive Lagrangian Particle Method (ALPM), for computing time-dependent viscoelastic flows using either a differential constitutive equation (macroscopic approach) or a kinetic theory model (micro–macro approach). In ALPM, the Eulerian finite element solution of the conservation equations is decoupled from the Lagrangian computation of the extra-stress at a number of discrete particles convected by the flow. In the macroscopic approach, the extra-stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories. In the micro–macro approach, the extra-stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model. At each time step, ALPM automatically enforces that all elements of the mesh have a number of Lagrangian particles ranging within a user-specified interval. Results are given for the start-up flow between highly eccentric rotating cylinders, using the FENE and FENE-P dumbbell models for dilute polymer solutions.