Numerical simulation of immiscible incompressible viscous, viscoelastic and elastic multiphase flows

A unified numerical framework is presented for the modelling of multiphasic viscoelastic and elastic flows. The rheologies considered range from incompressible Newtonian or Oldroyd-B viscoelastic fluids to Neo-Hookean elastic solids. The model is formulated in Eulerian coordinates. The unknowns are the volume fraction of each phase (liquid, viscoelastic or solid), the velocity, pressure and the stress in each phase. A time splitting strategy is applied in order to decouple the advection operators and the diffusion operators. The numerical approximation in space consists of a two-grid method. The advection equations are solved with a method of characteristics on a structured grid of small cells and the diffusion step uses an unstructured coarser finite element mesh. An implicit time scheme is suggested for the time discretisation of the diffusion step. Estimates for the time and space discretisation of a simplified model are presented, which proves unconditional stability. Several numerical experiments are presented, first for the simulation of one phase flows with free surfaces. The implicit time scheme is shown to be more efficient than the explicit one. Then, the model for the deformation of an elastic material is validated for several test cases. Finally, Signorini boundary conditions are implemented and presented for the simulation of the bouncing of an elastic ball. The multiphase model is validated through different test cases. Collisions between Neo-Hookean elastic solids are explored. Simulations of multiple viscoelastic flows are presented, for instance an immersed viscoelastic droplet and a Newtonian fluid in a constricted cavity. The fall of an immersed Neo-Hookean elastic solid into a Newtonian or a viscoelastic fluid is also presented. Finally, the one phase model is extended to compressible flows. The method of characteristics is updated in order to solve the advection equations, when the velocity is not divergence-free. A numerical scheme is proposed and a numerical experiment is presented.