Interaction of multiphase fluids and solidstheory, algorithms and applications

Resumo

The work presented in this thesis is devoted to the study and numerical simulation of Fluid-Structure Interaction (FSI) problems involving complex uids. The nonlinear and time dependent nature of FSI problems makes the analytical solution very difficult or even impossible to obtain, requiring the use of experimental analysis and/or numerical simulations. This fact has prompted the development of a great variety of numerical models for the interaction of uids and solid structures. However, most of the efforts have been focused on classical uids governed by the Navier-Stokes equations, which cannot capture the physical mechanisms behind complex uids. Here, we try to fill this gap by proposing several models for the interplay of solids and multi-phase or multi-component ows. The proposed models are then applied to particular problems that spark interest in fields, such as engineering, microfabrication and chemistry. In this work, the behavior of the structure is described by the nonlinear equations of elastodynamics and treated as an hyperelastic solid. Two different constitutive theories are employed, a Neo-Hookean model with dilatational penalty and a Saint Venant-Kirchhoff model. The description of complex uids is based on the diffuse-interface or phase-field method. In particular, two approaches are adopted. The first one is based on the Navier- Stokes-Korteweg equations, which describe compressible uids that are composed by two phases of the same component that may undergo phase transformation, such as water vapor and liquid water. We use this model to study the in uence of surface active agents in droplet coalescence and show that droplet motion may be driven by strain gradients -tensotaxis- of the underlying substrate. We also show several problems of phase-changedriven implosion, in which a thin structure collapses due to the condensation of a uid. The second approach is based on the Cahn-Hilliard model, which we couple with the incompressible Navier-Stokes equations. We adopt an stabilization based on the residualbased variational multiscale formulation. This results in a model that describes twocomponent immiscible ows with surface tension. The potential of this model is illustrated by solving several elastocapillary problems in two and three dimensions including capillary origami, the static wetting of soft substrates and the deformation of micropillars As FSI technique, we adopt a moving mesh or boundary-fitted approach with matching discretization at the uid-structure interface. This choice permits to strongly impose the kinematic compatibility conditions and results in more accurate solutions at the uid-solid interface. In particular, we use the Lagrangian description to derive the semi-discrete form of the solid equations and the Arbitrary Lagrangian-Eulerian (ALE) description for the uid domain. This means that the uid mesh needs to be updated to accommodate the motion of the structure. For this purpose, we solve an additional linear elasticity problem subject to displacement boundary conditions coming from the motion of the solid. For the spatial discretization of the solid and uid domains, we adopt Isogeometric Analysis (IGA) based on Non-Uniform Rational B-Splines (NURBS), a generalization of the finite-element method that posseses higher-order global continuity and allows for a more precise geometric representation of complex objects. Regarding the time integration, we use a generalized-[alfa] scheme. The nonlinear system of equations is solved using a Newton-Raphson iteration procedure, which leads to a two-stage predictor-multicorrector algorithm. The resulting linear system is solved using a preconditioned GMRES method. A quasi-direct monolithic formulation is adopted for the solution of the FSI problem, that is, the fluid and solid equations are solved in a coupled fashion, while the mesh motion is solved separately using as input, data from the fluid-solid solve.