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

Published online by Cambridge University Press:  14 May 2019

Anatoliy Vorobev Open the ORCID record for Anatoliy Vorobev [Opens in a new window]  and
Tatyana Lyubimova Open the ORCID record for Tatyana Lyubimova [Opens in a new window]
Anatoliy Vorobev*
Affiliation:
Faculty of Engineering and Physical Sciences, University of Southampton, Southampton SO17 1BJ, UK
Tatyana Lyubimova
Affiliation:
Institute of Continuous Media Mechanics Ural Branch RAS, Perm, 614013, Russia Perm State University, Perm, 614990, Russia
*
†Email address for correspondence: A.Vorobev@soton.ac.uk
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Abstract

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.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 543 - 562
Copyright
© 2019 Cambridge University Press 

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