Vibrational convection in a heterogeneous binary mixture. Part I. Time-averaged equations

Abstract

High-frequency vibrations of a container filled with a fluid generate the pulsation flows that however are barely visible with the naked eye, and induce the slow but large-amplitude averaged flows that are important for various practical applications. In this work we derive a theoretical model that gives the averaged description of the influence of uniform high-frequency vibrations on an isothermal mixture of two slowly miscible liquids. The miscible multiphase system is described within the framework of the phase-field approach. The full Cahn-Hillard-Navier-Stokes equations are split into the separate systems for the quasi-acoustic, pulsating and averaged flow fields, eliminating the need for the resolution of the short time-scale pulsation motion and thus making the analysis of the long-term evolution much more efficient. The resultant averaged model includes the effects of concentration diffusion and barodiffusion, the dynamic interfacial stresses, and the generation of the hydrodynamic flows by non-homogeneities of the concentration field (when they are combined with the effects of gravity and vibrations). The resultant model for the vibrational convection in a heterogeneous mixture of two fluids separated by diffusive boundaries could be used for the description of processes of mixing/de-mixing, solidification/melting, polymerisation, etc. in the presence of vibrations.

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ORCID for Anatoliy Vorobev: orcid.org/0000-0002-6458-9390

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