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.


Introduction
The evolution of hydrodynamic perturbations on the surface of a slowly smearing boundary is studied in order to comprehend the mechanisms of the mixing dynamics of liquid/liquid and gaseous/liquid substances, which is a pre-requirement for numerous chemical reactions, and thus needed for the majority of processes in chemical engineering. Slow miscibility frequently limits the reaction rates and the amounts of the reactants that partake in the chemical process, normally forcing an extensive supply of reactants, and leading to the problems of separation of the products from non-reacted substances. The mixing rate strongly depends on the contact area of the phases, and the surface waves may be expected to intensify the mixing through enlargement of the contact area.
In the current work, the evolution of a disturbed boundary between two miscible liquids is modelled on the bases of the phase-field approach. We consider the initial moments after the contact of two liquids, thus when a binary mixture is in a state different from the state of thermodynamical equilibrium, and when the process of interfacial diffusion occurs across the boundary. Through the direct numerical simulations we aim (i) to understand the effects of the interfacial diffusion on the dispersion relations of the gravity-capillary waves, and (ii) the effect of the surface waves on the intensity of the mixing process of two miscible liquids.

Governing equations. Phase-field approach
The continuum approach for description of a heterogeneous mixture of two miscible liquids was first proposed by van der Waals [1] and Korteweg [2]. Within the phasefield approach the interface separating two liquids is represented as a transitional boundary of finite thickness. The concentration field C, that is defined as the mass fraction of one component in the mixture, is used to trace the evolution of the interfacial boundary.
The specific free energy function for the miscible multiphase system is defined by a function of concentration and gradient of concentration [3], [4], In this expression, the second term takes into account the dynamic surface tension effects at a miscible interface. The capillary constant is assumed to be sufficiently small so to be able to neglect this term everywhere except at the places of large concentration gradients, i.e. where the interface is located.
The double-well potential is frequently used for the classical part of the free energy function, f 0 , especially, when the primary interest is tracing the evolution of the immiscible interface [5]. We however want to consider the interface between two miscible liquids. In [7], it was pro- posed to determine f 0 on the basis of the the Landau formula, This expression was originally proposed for description of the states of the binary mixtures near the thermodynamic critical point. In particular, it should well represent the equilibrium states of the binary mixture with the so-called upper critical temperature (consolute point), when the mixture is homogeneous in equilibrium above the critical temperature and may be heterogeneous below the critical temperature. For convenience, we will shift the reference point for the concentration field as C → (C − C cr ).
The parameters a and b are treated as two phenomenological coefficients with the values taken so that to provide better matching with the experimental data. We will however assume that b remains always positive, while a can be either positive or negative defining whether the system is single-and two-phase in equilibrium.
The free energy function (1) can be used to re-derive the Navier-Stokes equations for the mixture, that was done by Lowengrub and Truskinovsky [6]. The so-obtained full Cahn-Hilliard-Navier-Stokes equations are however very complex for numerical treatment because of quasi-compressibility (dependence of the mixture density on concentration) that forces to use the full continuity equation even for description of two incompressible liquids. The Boussinesq approximation of the full equations, derived in [7], is used in the current work to define the evolution of the gravitycapillary waves on a surface of two miscible liquids.
The governing equations reflect the conservation of momentum, species, and mass ∂u ∂t Here, the standard notations are used for the variables.
These equations are applied to the whole multiphase system, including the interface. One sees that the Navier- the Reynolds number, the Peclet number, the capillary number, In these formulae L * is the typical size, ρ 1 is used as the density scale, µ * is the unit of the chemical potential, η 1 is the viscosity scale, α is the mobility constant, and φ = ρ2−ρ1 ρ1 is the density contrast, with ρ 2 and ρ 1 being the densities of the pure components of the binary mixture.
The parameter A = a/b is used to define the thermodynamic state of the mixture.
It is necessary to note that the non-dimensional parameters are different from the classical parameters, that would be defined on the basis of the standard set of phenomenological parameters such as the surface tension and diffusion coefficients. We use the common names for these parameters, as they appear in front of the corresponding terms in the governing equations and generally have the meanings similar to those associated with the classical non-dimensional numbers. Ca is called here the capillary number, however, in some other works (e.g. [6]), the same parameter is called the Cahn number.
It is also necessary to note that the governing equations (3) were derived for two liquids with different vis-cosities. The difference in viscosities was however assumed to be small, and only one single Reynolds number is used to characterize the binary mixture. For the case when the viscosity difference is large, the governing equations need to be reconsidered, bringing several new terms to the equations. It has not been done in the present work.
The surface tension coefficient associated with the phase boundary can be determined as where x is the coordinate across the interface. For a heterogeneous binary mixture in the state of thermodynamic equilibrium and with no gravity effect with a flat interface, the thickness of the interface can be evaluated as δ 0 = −Ca/A, and the surface tension coefficient as On macroscopic scales, the interface between two liquids is infinitely sharp, and the limit of δ → 0 is of the primary interest for the phase-field approach. As shown in [8], the limit of the sharp interface is not easy to obtain, as both the interface thickness and the surface tension coefficients depend on Ca, and both values would decrease if Ca tends to zero. Hence, a single decrease in the values of the capillary number would make the interface thinner, but a thinner interface would be also endowed with smaller surface tension.

Problem statement and numerical approach
It is assumed that two pure liquids of different densities are brought into contact, so that the heavier liquid is placed at the bottom (i.e. Gr < 0). The initial shape of the interface is given by the following concentration profile, The pure liquids are defined by the concentration levels, For the numerical solution, the equations were re-written into the streamfunction (ψ)-vorticity(Ω) formulation, defined by the following relations, The resultant equations are solved using the finitedifference method on a uniform mesh. The explicit first order in time and second-order in space discretization scheme was used.

Results
The typical snapshots of the fields of concentration and velocity are depicted in Figures 1a,b. In these figures, one sees the sharp interfaces that separate two phases and the flow vortices that penetrate through the interfaces. Figure   1c shows These formulas are valid for the weakly viscous case. Here, σ = Ca/(3δ) is the surface tension coefficient (11) calculated for the initial state (12). The finite thickness of the interface introduced within the phase-field approach is an artificial parameter, as the thickness of a real interface is just several molecular layers, i.e. zero for the macroscopic theory. Figure 3 shows the behavior of the binary mixture for different values of the parameter δ. The results demonstrate that in the limit of longer wavelengths, the dependence on the interface thickness is rather weak, and the data obtained for three different values of δ almost coincide. For the shorter waves the difference in the data is getting more significant. It is also interesting, that the gradual decrease of the interface thickness makes the difference of the calculated frequency and decay rate even more different from the theoretical curve (15).
The classical dispersion relation (15) says that the capillary effects are weaker for longer waves, and these effects get much stronger for shorter waves. For longer waves, the limiting behavior of a sharp interface can be obtained by a gradually decrease of the interface thickness. This however is not working for shorter waves, when the capillarity effects are essential.
The coefficient of surface tension is given as the ratio of the capillary number over the interface thickness.
If the interface thickness is changed at the same capillary number, then at least the initial value of the surface ten-

Conclusions
In the current work we aimed to assess whether the phasefield approach could accurately reproduce the dynamics of an interface between two miscible liquids that are brought into contact. We assume that at the time of the contact, the binary mixture is thermodynamically non-equilibrium, and equilibration is primarily driven by the interfacial diffusion, a very slow process. Traditionally, it is assumed that the surface waves increase the contact area of two liquids, and hence should enhance the mixing, or the transition of the binary mixture to the state of thermodynamic equilibrium. Thus, the second aim of the work was to as-sess the capability of phase field approach to capture the dynamics of the waves propagating on a slowly miscible interface, and, finally, to understand whether the mixing dynamics can be controlled by the surface waves.
The results of our study show that the overall dynamics of the miscible boundary could be correctly reproduced with the help of phase-field boundary. The boundary remains sharp, and by variation of the values of the Peclet and Reynolds number, the typical time scales for the diffusive and viscous processes can be separately set. At very high Peclet numbers, the interfacial diffusion is negligible, and at sufficiently high Reynolds numbers the gravitycapillary waves could be induced at the boundary. For the waves with longer wavelengths, we managed to obtain a good agreement with the classical relationships.
The shorter waves on a miscible interface, are characterized by much stronger decay rates, due to additional diffusive damping. We also showed that the surface waves do also affect the diffusion process, especially at higher Peclet numbers, when diffusion is weaker, clearly intensifying the smearing of the interfacial boundary, and thus increasing the rate of transition to the state of thermodynamic equilibrium.

Acknowledgements
The financial support of the Royal Society (Ref. All data supporting this study is openly available from the University of Southampton repository at http://doi.org/10.5258/SOTON/D0257.