An unstructured geometrical un-split VOF method for viscoelastic two-phase flows

We present an unstructured geometrical Volume-of-Fluid (VOF) method for handling two-phase flows with a viscoelastic liquid phase. The viscoelastic behavior is modeled via generic rate-type constitutive equations. A one-field formulation is employed, which results from conditional volume averaging of the local instantaneous bulk equations and interface jump conditions. The method builds on the 'plicRDF-isoAdvector' geometrical VOF solver that is extended and combined with the modular framework 'DeboRheo' for viscoelastic CFD. A piecewise-linear geometrical interface reconstruction technique on general unstructured meshes is employed for discretizing the viscoelastic stresses across the fluid interface in a numerically consistent and highly accurate way. Because of the numerical challenges posed by the high Weissenberg number problem, we employ an appropriate stabilization approach to the constitutive equation of the viscoelastic phase to increase the robustness of the method at higher fluid elasticity. DeboRheo facilitates a flexible combination of different rheological models with appropriate stabilization methods to address the high Weissenberg number problem. We discuss the theoretical formulation and implementation of the proposed method and demonstrate its effectiveness using numerical examples of viscoelastic flows. The results highlight the method's ability to accurately capture the behavior of viscoelastic fluids in various applications. The proposed method holds promise for furthering our understanding and predictive capabilities of viscoelastic flows in various industrial and natural processes.