Small-scale kinematics of two-phase flows: Identifying relaxation processes in separated- A nd disperse-phase flow models

We propose here to investigate the impact of small-scale effects on the bulk evolution of a two-phase flow system. More precisely, we choose to examine the sole influence of a small-scale (with respect to the bulk velocity) off-equilibrium velocity on the system. In order to narrow our analysis and avoid complex well-posedness issues, we choose to examine a simple barotropic 5-equation two-phase flow model that accounts for an equilibrium common bulk velocity and a small-scale off-equilibrium velocity. A full derivation of the model is presented: It is based on a variational principle which allows us to insert the two-scale kinematics into the model by considering two different kinetic energies. Additional entropy dissipation requirements allow us to add dissipative structures to the model. This system is neutral with respect to the topology of the flow structure and is equipped with parameters that can be connected to relaxation processes. When considering instantaneous relaxations, we obtain two limit systems of the literature that are used for the simulation of separated-phase flows. In this sense we obtain a hierarchy of models. We show that the parent 5-equation model is also compatible with the description of a bubbly fluid that allows small-scale vibrations for the disperse phase. This identification is verified and discussed through comparisons with experimental measurements of sound dispersion (Silberman, J. Acoust. Soc. Am., vol. 29, 1957, pp. 925-933; Leroy et al., J. Acoust. Soc. Am., vol. 123, 2008, pp. 1931-1940) and with the dispersion relations of a reference model for bubbly flows by Cheng et al. (Trans. ASME J. Heat Transfer, vol. 107, 1985, pp. 402-408). The present work is a first contribution to a larger effort that aims at unifying models that can describe both separated and disperse two-phase flows, coupling small-scale modelling with large-scale resolution.