Linear stability of an interface between two miscible liquids

Abstract

The evolution of small disturbances to a horizontal interface separating two miscible liquids is examined. Initially, the liquid-liquid binary system is assumed to be out of its thermodynamic equilibrium, and hence, the system immediately starts moving towards its equilibrium state. The aim is to investigate whether the slow interface smearing is overrun by faster and more complex mixing. The study is based on the numerical solution of the linear stability of the phase boundary subjected to normal thermo- and hydro-dynamics perturbations.

It was found that the classical results for the stability of immiscible interfaces can be re-obtained through the phase-field approach provided that the wavelength, that defines the perturbation, is greater than the thickness of the interface. Additionally the system becomes thermodynamically unstable if the value of the interface thickness is greater than the value of the interface thickness that is defined at thermodynamic equilibrium, so the gravity waves may become unstable. It was also found that the interfacial mass transfer plays the role of additional dissipation, reducing the growth rate of the Rayleigh-Taylor instability and increasing the dissipation of the gravity waves, and the mutual action of diffusive and viscous effects completely suppresses the modes with shorter wavelengths.

The flow imposed along the interface adds the mechanisms of the Kelvin-Helmholtz and Holmboe instabilities. It was found that if the heavier liquid lies above the lighter liquid, then the interface is unconditionally unstable. Viscosity, diffusivity and capillarity reduce the growth of perturbations. In the opposite case of the heavier liquid underlying the lighter one, the interface can be stable. The stability boundaries are primarily defined by the strength of the density contrast and the intensity of the imposed flow. Thinner interfaces are usually more unstable (characterised by larger zones of instability), however, the thermodynamic instability identified for interfaces with thicknesses greater than the thickness of a thermodynamically equilibrium phase boundary makes such interfaces unconditionally unstable. Surprisingly, the diffusivity and capillarity effects were found to alter the behaviour of the interface in miscible liquids. However, the viscosity effect retains its stabilising role for both immiscible and miscible liquids.

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Identifiers

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

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