Linear stability analysis of a horizontal phase boundary separating two miscible liquids

Electric-permittivity-based instability of two dielectric miscible liquids under DC field

This paper considers the mixing of two dielectric miscible viscous liquids with different electric permittivities bounded by solid walls in an external electric field normal to the interface of the liquids. The mutual diffusion of these two liquids leads to the formation of an unsteady self-similar 1D diffusion layer. This layer is found to be unstable to the perturbations of the interface. A special sophisticated mathematical approach in self-similar variables is developed to estimate its stability. The results of a linear stability theory are verified by direct numerical simulations of the full nonlinear problem. A mixing efficiency based on the separation amplitude and an optimal electric field strength to achieve the fastest mixing are proposed in the present study.

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

The stability of a shear flow imposed along a diffusive interface that separates two miscible liquids (a heavier liquid lies underneath) is studied using direct numerical simulations. The phase-field approach is employed for description of a thermo- and hydrodynamic evolution of a heterogeneous binary mixture. The approach takes into account the dynamic interfacial stresses at a miscible interface and uses the extended Fick's law for setting the diffusion transport (the diffusion flux is proportional to the gradient of chemical potential). The shear flow is unstable to two kinds of instabilities: (1) the Kelvin-Helmholtz instability, with an immovable vortex formed in the middle of an interface (in the vertical direction) and (2) the Holmboe instability, with traveling waves along the interfacial boundary. The development of the Holmboe instability results in a stronger enhancement of molecular mixing between the mixture components. Earlier, the boundaries of these instabilities were determined using the linear stability analysis and employing the concept of a "frozen interface." In the current work, through the solution of full equations, we obtain the stability boundaries for several sets of governing parameters, showing a greater variety of the possible shapes of the stability diagrams. The Kelvin-Helmholtz instability always occurs at lower gravity effects (lower density contrasts), while the Holmboe instability occurs when gravity is stronger. We show that for some parameters these two instabilities are separated by a zone where the shear flow is stable, and this zone disappears for the other sets of parameters.

Hydrodynamic instabilities in miscible fluids

Hydrodynamic instabilities in miscible fluids are ubiquitous, from natural phenomena up to geological scales, to industrial and technological applications, where they represent the only way to control and promote mixing at low Reynolds numbers, well below the transition from laminar to turbulent flow. As for immiscible fluids, the onset of hydrodynamic instabilities in miscible fluids is directly related to the physics of their interfaces. The focus of this review is therefore on the general mechanisms driving the growth of disturbances at the boundary between miscible fluids, under a variety of forcing conditions. In the absence of a regularizing mechanism, these disturbances would grow indefinitely. For immiscible fluids, interfacial tension provides such a regularizing mechanism, because of the energy cost associated to the creation of new interface by a growing disturbance. For miscible fluids, however, the very existence of interfacial stresses that mimic an effective surface tension is debated. Other mechanisms, however, may also be relevant, such as viscous dissipation. We shall review the stabilizing mechanisms that control the most common hydrodynamic instabilities, highlighting those cases for which the lack of an effective interfacial tension poses deep conceptual problems in the mathematical formulation of a linear stability analysis. Finally, we provide a short overview on the ongoing research on the effective, out of equilibrium interfacial tension between miscible fluids.

Phase-field modelling of gravity-capillary waves on a miscible interface

Using the approach of direct numerical simulations we investigate the gravity-capillary waves induced on a horizontal interface between two slowly miscible liquids. It is assumed that the liquids are just brought into contact, and thus the interface is slowly smeared by the action of interfacial diffusion. It is also assumed that the initial shape of the interface is distorted by harmonic perturbations, which results in the development of the gravity-capillary surface waves. The evolution of the binary mixture is modelled on the basis of the phase-field method. Our results show that in the limiting case of negligible diffusion the classical dispersion relations for immiscible interfaces can be reproduced. Although, for the waves with shorter wavelengths such an agreement is more difficult to obtain. The interfacial diffusion brings an additional dissipation to the fluid system, strongly damping the development of the shorter waves. We also show that the mixing (or the transition of a binary system to the state of thermodynamic equilibrium) is intensified by the presence of the surface waves, and this effect is more important when the liquids are slowly miscible, i.e. when the interfacial diffusion is weak.

On the phase-field modelling of a miscible liquid/liquid boundary

Mixing of miscible liquids is essential for numerous processes in industry and nature. Mixing, i.e. interpenetration of molecules through the liquid/liquid boundary, occurs via interfacial diffusion. Mixing can also involve externally or internally driven hydrodynamic flows, and can lead to deformation or disintegration of the liquid/liquid boundary. At the moment, the mixing dynamics remains poorly understood. The classical Fick's law, generally accepted for description of the diffusion process, does not explain the experimental observations, in particular, the recent experiments with dissolution of a liquid solute by a liquid solvent within a horizontal capillary (Stevar and Vorobev, 2012). We present the results of the numerical study aimed at development of an advanced model for the dissolution dynamics of liquid/liquid binary mixtures. The model is based on the phase-field (Cahn-Hilliard) approach that is used as a physics-based model for the thermo- and hydrodynamic evolution of binary mixtures. Within this approach, the diffusion flux is defined through the gradient of chemical potential, and, in particular, includes the effect of barodiffusion. The dynamic interfacial stresses at the miscible interface are also taken into account. The simulations showed that such an approach can accurately reproduce the shape of the solute/solvent boundary, and some aspects of the diffusion dynamics. Nevertheless, all experimentally-observed features of the diffusion motion of the solute/solvent boundary, were not reproduced.

Linear stability of a horizontal phase boundary subjected to shear motion

We investigate the stability of slowly smearing phase boundary that appears at the contact of two miscible liquids. A hydrodynamic flow is imposed along the boundary. The aim is to find out whether the slow diffusive smearing of a boundary can be overrun by faster mixing. The phase-field approach is used to model the evolution of the binary mixture. The linear stability in respect to 2D perturbations is studied. If the heavier liquid lies above the lighter liquid, the interface is unconditionally unstable due to the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The imposed flow accelerates the growth of the long-wave modes and suppresses the growth of the short-wave perturbations. Viscosity, diffusivity and capillarity reduce the growth of perturbations. If the heavier liquid underlies the lighter one, the interface can be stable. The stability boundaries are defined by the strength of gravity (density contrast) and the intensity of the imposed flow. Thinner interfaces are usually characterised by larger zones of instability. The thermodynamic instability, identified for the thicker interfaces with the thicknesses greater than the thickness of a thermodynamically equilibrium phase boundary, makes such interfaces unconditionally unstable. The zones of instability are enlarged by diffusive and capillary terms. Viscosity plays its stabilising role.