Instationary Eulerian viscoelastic flow simulations using time separable Rivlin-Sawyers constitutive equations

A time dependent method for solving integral constitutive equations of the Rivlin±Sawyers type is introduced. The deformation history is represented by a finite number of deformation fields. Using these fields the stress integral is approximated as a finite sum. When the flow evolves the deformation fields are convected and deformed. The approach presented in this paper is the first Eulerian method that can handle integral equations in a time dependent way. The method is validated by using the upper-convected Maxwell (UCM) benchmark of a sphere moving in a tube. We show that the method converges with mesh and time step refinement and that the results are accurate, comparable to the results obtained with the differential equivalent of the UCM model. To demonstrate that complicated linear spectra are easily incor! porated, results of a Rouse model simulation of 100 modes are presented. We also compare results on a falling sphere problem to the results obtained by a Lagrangian method as reported by Rasmussen and Hassager [H.K. Rasmussen, O. Hassager, On the sedimentation velocity of spheres in a polymeric liquid, Chem. Eng. Sci. 51 (1996) 1431±1440]. The model being employed is the PSM model, for which no differential equivalent exists