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.