An alternative phase-field interfacial tension force representation for binary fluid systems

The Navier-Stokes/Cahn-Hilliard (NSCH) system of equations has been extensively used for investigating the dynamics of two-phase flows of Newtonian fluids. However, the accurate calculation of interfacial tension via NSCH has been perceptibly doubted, and thus, a successive solution of NSCH equations is rarely not accompanied by mesh adaptation techniques and complex numerical schemes. In this work, it is demonstrated that the cause of such a miscalculation of the interfacial tension is inherent when following the conventional way of coupling the Navier-Stokes with the Cahn-Hilliard equation in their dimensionless form, where the capillary number is defined by assuming that the fluid/fluid interface is flat and at equilibrium. Hence, an alternative NSCH model was developed for the more accurate computation of interfacial tension that does not rely on any such a priori assumptions, and it uses a more abstract coupling by accounting for the distribution of the binary system's energy on the interfacial region. This model was implemented on two different cases: (i) an investigation of the effects of inertia and capillarity on the deformation of liquid drops in simple shear flow and (ii) a study of an interfacial instability due to viscosity stratification. To solve the set of governing equations, implicit time integration schemes based on finite differences were further developed and implemented. The results regarding the topological evolution of the fluid/fluid interface from both cases were additionally cross-validated with other methods from the literature as well as with the conventional NSCH model. The comparison suggests that our NSCH model indeed remedies the standard NSCH model, without the need of mesh adaptation or any complex numerical scheme, by more accurately computing the interfacial tension for binary systems consisting of incompressible, immiscible, and Newtonian fluids.