We present a new finite element scheme for direct simulation of inertialess particle suspensions in simple shear flows of Oldroyd-B fluids. The sliding bi-periodic frame concept of Lees & Edwards [J. Phys. C 5 (1972) 1921] has been combined with the DEVSS/DG finite element scheme, by introducing constraint equations along the domain boundary. The force-free, torque-free rigid body motion of a particle is described by the rigid-ring problem and implemented by Lagrangian multipliers only on the particle boundary, which allows general treatments for boundary-crossing particles. In our formulation, the bulk stress is obtained by simple boundary integrals of Lagrangian multipliers along the domain and particles. Concentrating on 2-D circular disk particles, we discuss the bulk rheology of suspensions as well as the micro-structural developments through the numerical examples of single-, two- and many-particle problems, which represent a large number of such systems in simple shear flow. We report the steady bulk viscosity and the first normal stress coefficient, from very dilute to highly concentrated systems. The results show shear-thickening behavior for both properties and the common experimental observation of the scaling of the first normal stress to the shear stress has been reproduced. Unlike Newtonian systems, two particles in an Oldroyd-B fluid result in kissing-tumbling-tumbling phenomena: they keep rotating around each other, when they are closely located. Many-particle problems reveal the occurrence of strong elongational flows between separating particles.
Hwang, W. R., Hulsen, M. A., & Meijer, H. E. H. (2004). Direct simulations of particle suspensions in a viscoelastic fluid in sliding bi-periodic frames. Journal of Non-Newtonian Fluid Mechanics, 121(1), 15-33. https://doi.org/10.1016/j.jnnfm.2004.03.008