Diffuse interface modeling of two-phase flows based on averaging: mass and momentum equations

A diffuse interface model is derived for the direct simulation of two-phase flows with surface tension, phase-change, and density and viscosity differences between the phases. The derivation starts from the balance equations for a sharp interface and uses an ensemble averaging procedure on an atomic scale to obtain a diffuse interface version of the equations. As opposed to thermodynamically derived models, the two phases are assumed to coexist inside the diffuse interface with different properties, velocities, and pressures. Separate conservation equations are solved for each phase. The phase interactions are modeled explicitly through the inclusion of interfacial forces in the momentum equations for each phase. Based on a superposition of microscopic (atomic-scale) and macroscopic interface morphologies, an expression for the interfacial momentum source due to surface tension is introduced that is equivalent to the capillary stress term encountered in thermodynamically derived models. Also, a constitutive relation for the average viscous stresses of each phase inside the diffuse interface is presented. The model is tested for simple one-dimensional flows tangential and normal to a diffuse interface, and the results are compared to those obtained from a thermodynamically derived model.