A regularized phase field model for solid-fluid dynamics description

Usually, Lagrangian or Eulerian-Lagrangian descriptions of motion are adopted to model interaction of fluids and deformable solid bodies. However, in some cases both approaches lead to serious difficulties. An example is a system with large number of solid bodies; another one is the case where topology of the phases can change. In these situations, an Eulerian description is much more convenient. The presented work is devoted to the development of a phase field mathematical model for description of dynamics of multiphase multicomponent system (mixture) with phases having whether liquid or solid rheology. All the phases and interphase boundaries (interfaces) are directly resolved. Main balance laws of the proposed model are formulated using the Eulerian description. The model is of phase field type: the interphase boundary is diffuse and is described by a thin layer of finite thickness. Mass densities of mixture components are used as order parameters. To describe a stress-strain behavior of the solid phase, we assume that the Helmholtz free energy depends on deformation gradient tensor which is defined as a solution of the corresponding evolution equation. Constitutive relations are derived by means of the well-known Coleman-Noll procedure and the second law of thermodynamics. A distinctive feature of the considered model is its preliminary regularization based on the quasi-hydrodynamic technique, which allows one to improve numerical stability properties when an explicit discretization is applied. A new family of quasi-hydrodynamic closures is obtained.