A Lagrangian finite element method for simulation of a suspension under planar extensional flow

A numerical simulation of a suspension of two-dimensional solid particles in a Newtonian fluid under planar extensional flow is presented. The method uses a finite element solution of the flow with a unit cell within the self-replicating lattice for planar extensional flow identified by Kraynik and Reinelt [A.M. Kraynik, D.A. Reinelt, Extensional motions of spatially periodic lattices, Int. J. Multiphase Flow 18 (1992) 1045]. This is implemented using a quotient space representation that maps all points space onto points within the unit cell. This mapping is preserved by using fully Lagrangian grid movement, with grid quality preserved by a combination of Delaunay reconnection and grid adaptivity. The no-slip boundary conditions on the particles are enforced weakly via a traction force acting as a Lagrange multiplier. The method allows simulations of suspensions under planar extensional flow to be conducted to large strains in a truly periodic cell. The method is illustrated for both isotropic and anisotropic two-dimensional particles and can be easily extended to viscoelastic fluids and to non-rigid particles.