Vibrational convection in a heterogeneous binary mixture. Part 1. Time-averaged equations

High-frequency vibrations of a container filled with a fluid generate 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 give...

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Bibliographic Details
Published in:Journal of fluid mechanics 2019-07, Vol.870, p.543-562
Main Authors: Vorobev, Anatoliy, Lyubimova, Tatyana
Format: Article
Language:English
Subjects:
Binary mixtures
Boundary conditions
Computational fluid dynamics
Convection
Diffusion effects
Evolution
Fluid mechanics
Fluids
Gravity
Gravity effects
Hydrodynamics
Interfaces
Interfacial stresses
JFM Papers
Liquids
Mathematical models
Miscibility
Navier-Stokes equations
Polymerization
Pulsation
Solidification
Velocity
Vibrations
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Summary:High-frequency vibrations of a container filled with a fluid generate 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.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2019.282