Diffuse interface modelling of soluble surfactants in two-phase flow

Abstract

Abstract. Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are derived from balance equations and an energy inequality so that thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they are related to sharp interface models. Both cases of dynamic as well as instantaneous adsorption are covered. Flexibility with respect to the choice of bulk and surface free energies allows us to realise various isotherms and relations of state between surface tension and surfactant. Some numerical simulations display the effectiveness of the presented approach.Key words. Two-phase flow, surfactant, phase field model, adsorption isotherm subject classifications. 35R35, 35R01, 76T99, 76D45, 35C20, 35Q351. Introduction Surface active agents (surfactants) reduce the surface tension of fluid interfaces and, via surface tension gradients, can lead to tangential forces resulting in the Marangoni effect. Biological systems take advantage of their impact on fluids with interfaces, but surfactants are also important for industrial applications such as processes of emulsification or mixing. While often much experience and knowledge is available on how surfactants influence the rheology of multi-phase fluids, the goal is to understand how exactly the presence of a surfactant influences coalescence and segregation of droplets.Surfactants can be soluble in at least one of the fluid phases and the exchange of surfactants between the bulk phases and the fluid interfaces is governed by the process of adsorption and desorption. Ward and Tordai [55] derived a time-dependent relation for the surfactant density at the interface and the surfactant density at the adjacent bulk phase (known as the sub-layer or sub-surface). To compute the interfacial density, a closure relation between the two quantities has been proposed in the form of several different equilibrium isotherms [18,33,32], where the underlying assumption is that the interface is in equilibrium with the sub-layer at all times. This corresponds to the case of diffusion-limited adsorption studied in Diamant and Andelman [16], where the process of adsorption to the interface is fast compared to the kinetics in the bulk phases. However, instantaneous adsorption is not valid in the context of ionic surfactant systems [16] or when the diffusion is not limited to a thin layer [12,13,14]. Therefore, we would like to be able to account for non-instantaneous adsorption in our models.Two-phase flow with surfactant is classically modelled with moving hypersurfaces describing the interfaces separating the two fluids. We will derive the following sharp interface model for a domain Ω containing two fluids of different mass densities. We denote by Ω(1) (t), Ω (2) (t) the domains of the fluids which are separated by an interface

Stable finite element approximations of two-phase flow with soluble surfactant

“…This equation relates the bulk fluxes to differences of the chemical potentials [26]. Here G ± ∈ C(R ≥0 ) ∩ C 2 (R >0 ) are bulk free energy densities that satisfy…”

“…In particular, we consider the system with the novel free boundary condition in [26] that allows for a stability bound. For a particular instance of this model, where the bulk surfactant concentration is assumed to be continuous across the interface, we are able to prove a stability bound for our semidiscrete finite element approximation.…”

“…The level set method, which describes the interface as the level set of a function, was considered in [43]. Numerical computations based on diffuse interface models have been presented in [35,42,24,26]. The immersed boundary method has been used in [34,15].…”

“…In [13] energy methods and semigroup theory were used to study the stability of equilibria in the soluble case. In [26] a diffuse interface model was introduced to describe two-phase flow with soluble surfactants for which an energy inequality can be shown. Moreover, by using matched asymptotic expansions the authors of [26] could show that a novel sharp interface model can be recovered, which also satisfies an energy law.…”

“…In [26] a diffuse interface model was introduced to describe two-phase flow with soluble surfactants for which an energy inequality can be shown. Moreover, by using matched asymptotic expansions the authors of [26] could show that a novel sharp interface model can be recovered, which also satisfies an energy law. We refer to Section 2 for the precise details of this sharp interface model, which has already been outlined above.…”