A unified HTC multiphase model of continuum mechanics

In this paper, we present a unified nonequilibrium model of continuum mechanics for compressible multiphase flows. The model, which is formulated within the framework of Hyperbolic Thermodynamically Compatible (HTC) equations, can describe an arbitrary number of phases that can be heat conducting inviscid and viscous fluids, as well as elastoplastic solids. The phases are allowed to have different velocities, pressures, temperatures, and shear stresses, while the material interfaces are treated as diffuse interfaces with the volume fraction playing the role of the interface field. To relate our model to other multiphase approaches, we reformulate the novel HTC governing equations in terms of the phase state parameters and put them in the form of Baer-Nunziato-type models. It is the Baer-Nunziato form of the HTC equations which is then solved numerically using a robust second-order path-conservative MUSCL-Hancock finite volume method on Cartesian meshes. Due to the fact that the obtained governing equations are very challenging we restrict our numerical examples to a simplified version of the model for three-phase mixtures. To address the stiffness properties of the relaxation source terms present in the model, the implemented scheme incorporates a semi-analytical time integration method specifically designed for the non-linear stiff source terms governing the strain relaxation. The validation process involves a wide range of benchmarks and several applications to compressible multiphase problems. Notably, results are presented for multiphase flows in several relaxation limit cases of the model, including inviscid and viscous Newtonian fluids, as well as non-linear hyperelastic and elastoplastic solids. In all cases, the numerical results demonstrate good agreement with established models, including the Euler or Navier-Stokes equations for fluids and the classical hypo-elastic model with plasticity for solids. Importantly, however, this approach achieves these results within a unified multiphase framework of continuum mechanics.