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

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.

The Zoo of Modes of Convection in Liquids Vibrated along the Direction of the Temperature Gradient

“…In addition to the acceleration amplitude, the imposed ∆T, and the aforementioned angle Θ, an additional degree of freedom influencing the dynamics is represented by the 'frequency' of vibrations. Nevertheless, only a limited number of studies are available in the literature where this variant of buoyancy flow was considered (Monti et al [8]; Alexander [9]; Alexander et al [10,11]; Feonychev and Dolgikh [12]; Gershuni and Lyubimov [13]; Monti et al [14]; Naumann [15]; Mialdun et al [16]; Melnikov et al [17]; Lyubimova et al [18]; Bouarab et al [19]; Shevtsova et al [20,21]; Maryshev et al [22]; Vorobev and Lyubimova [23]; Lappa, [24][25][26][27][28]; Lappa and Burel [29]). Still fewer articles have been devoted to the case where vibrations and temperature gradient are parallel.…”

Symmetry breaking phenomena in thermovibrationally driven particle accumulation structures

“…When a fluid is shaken in the presence of an imposed temperature difference, the emerging flow is known as thermovibrational convection (Gershuni and Lyubimov 44 ; Mialdun et al, 45 ; Lyubimova 46 ; Bourab et al, 47 ; Shevtsova et al, 48,49 , Maryshev et al, 50 ; Lappa 51 ; Vorobev and Lyubimova 52 ).…”

Kelvin-Helmholtz and Holmboe instabilities of a diffusive interface between miscible phases

“…The full equations are however too hard for direct numerical simulations, as these equations include the full continuity equation (that is called the effect of quasi-compressibility and that is explained by the dependence of the mixture density on concentration). It was later shown that the slower convective and diffusive evolution of a mixture can be determined on the basis of the fully incompressible equations, that represent the Boussinesq approximation of the full Cahn-Hilliard-Navier-Stokes equations [29,30]. The Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations are numerically solved in the current work to describe the evolution of a heterogeneous binary mixture.…”