Direct numerical simulations of hard particle suspensions in planar elongational flow

We present a new direct simulation technique of inertialess particle suspensions in planar elongational flow of a Newtonian fluid. The extensional bi-periodic domain concept is introduced such that a single cell problem with a small number of particles may represent a large number of repeated structures of such a cell in planar elongational flow. For implicit treatment of the hydrodynamic interaction between particles and fluid, we employ a finite-element/fictitious-domain method similar to the distributed Lagrangian multipliers (DLM) method together with a rigid-ring description of the particle. The extensional bi-periodic frame is incorporated by constraint equations with Lagrangian multipliers and is implemented by the mortar element method. In our formulation, the bulk stress is evaluated by simple boundary integrals. Concentrating on 2D circular disk particles, we present numerical examples of single-particle, two-particle and 100-particle problems in the extensional bi-periodic frame. We discuss effects of solid fraction and particle configuration on the elongational viscosity of the suspension, in comparison with simple shear flow. We found that, at zero strain, the relative elongational viscosity is almost the same as the relative shear viscosity in simple shear for moderately concentrated suspensions. There is a small increase in elongational viscosity for large strains, which is related to an anisotropic distribution of the particles.