A multiscale SPH method for simulating transient viscoelastic flows using bead-spring chain model

In this article, a multiscale smoothed particle hydrodynamics (SPH) method is proposed to simulate transient viscoelastic flows by using a bead-spring chain description of polymer molecule. This methodology couples macroscopic conservation equations for mass and momentum with a stochastic differential equation for bead-spring chain dynamics, which can be solved by a three-step semi-implicit algorithm to obtain the polymeric stress. Accordingly, a closed-form constitutive equation is not required. The multiscale SPH method is firstly verified by the plane Couette flow with Hookean and finitely extendable non-linear elastic (FENE) bead-spring chain models with the number of beads M = 2, which correspond to Hookean and FENE dumbbell models, respectively. Results for the velocity and shear stress are in excellent agreement with the available numerical and analytical solutions in literature. Then, the numerical method is extended to simulate the plane Couette flow and Poiseuille flow with FENE bead-spring chain model with M  > 2. In particular, some molecular information of bead-spring chain such as the molecular stretch, the orientation angle and the mean configuration thickness are displayed. The influence of the number of beads M on the evolutions of the molecular information is also analyzed in detail. It is demonstrated that the multiscale SPH method proposed here is a promising computational tool for the multiscale simulations of transient viscoelastic flows.