Phase-field based Navier-Stokes models are well established for the description of immiscible multi-phase flow e.g. since they allow to conveniently handle topological changes like jet breakup. These methods employ a locally diffuse transition between macroscopically immiscible fluids cf. e.g. [1]. Thus, modelling of the constitutive law for the fluid mixture is required. The accuracy of simulation results depends on these mixture models. In the present talk, the modelling of viscous stress in the diffuse interface region is addressed. Commonly employed schemes like the arithmetic [1] or harmonic [2] mean of phase-inherent viscosities are investigated regarding their accuracy and it is shown, that both schemes have shortcomings in particular flow situations. In the limiting case of the velocity gradient being purely normal to the interface, the harmonic mean is superior to the arithmetic one, while for purely tangential velocity gradients the arithmetic mean yields higher accuracy. Additionally, a novel jump condition approach for viscous stress is introduced which overcomes the shortcomings of the classical schemes leading to a higher accuracy for finite interface thickness [3]. Thereby, the jump approach yields a split of the velocity gradient where the viscous stress contribution of tangential components are calculated via an arithmetic mean, while the contribution of normal components is evaluated with the harmonic mean. Therefore, in the limiting cases of a vanishing normal or tangential part, the jump condition approach coincides with the more favourable classical scheme for the respective flow situation.

References
[1] Z. Guo, F. Yu, P. Lin, S. Wise, and J. Lowengrub. A diffuse domain method for two-phase
flows with large density ratio in complex geometries. Journal of Fluid Mechanics, 907:A38, 2021.
[2] Y. Sun and C. Beckermann. Diffuse interface modeling of two-phase flows based on averaging:
mass and momentum equations. Physica D: Nonlinear Phenomena, 198(3-4):281–308, 2004.
[3] M. Reder, A. Prahs, D. Schneider, and B. Nestler. Viscous stress approximations in diffuse
interface methods for two-phase flow based on mechanical jump conditions. 2023. In submission.
Preprint available at SSRN: http://dx.doi.org/10.2139/ssrn.4523250.