On the pitfalls of diffuse interface methods in problems involving non-material interfaces

In this paper, we discuss the implications of applying traditional diffuse-interface techniques to problems involving mass flux across the interface such as the phase change front. In a simplified setting of stationary radial flow and linear viscous fluid, we confirm by analytical tools in the framework of Colombeau algebra that the numerical solutions to such problems approximate in fact modified physical problems that involve additional surface tension-like stress localized in the interfacial zone. The arising dynamical surface tension depends on the viscosity and density profiles within the interface. Expanding the setting to models of power-law fluids, we show that the dynamic surface tension vanishes in the limit of interfacial width going to zero for shear-thinning fluids. In contrast, for the shear-thickening case, the diffuse interface numerical solutions to the considered class of problems cannot be assigned any straightforward physical meaning, as the dynamic surface tension becomes unbounded with decreasing interfacial width and the traction jump in the limiting case cannot be even represented by any classical distribution. Consequently, our findings raise questions regarding the broad applicability of diffuse interface techniques in scenarios involving non-material interfaces, underscoring the necessity for further investigation.