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Title: Diffuse interface models of soluble surfactants in two-phase fluid flows
Author: Lam, Kei Fong
ISNI:       0000 0004 5356 0108
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2014
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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. 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. One can compute the interfacial surfactant density from the bulk surfactant density by assuming that the interface is in equilibrium with the adjacent bulk phase and imposing a closure relation (known as adsorption isotherm) between the two quantities. The assumption (known as instantaneous adsorption) is valid when the process of adsorption to the interface is fast compared to the kinetics in the bulk phases. However, it is not valid in the context of ionic surfactant systems, or when the diffusion is not limited to a thin layer. In this thesis, we derive two types of mathematical models for two-phase flow with a soluble surfactant that can account for both instantaneous and non-instantaneous adsorption. The first type is a sharp interface model, in which the interface is modelled by moving hypersurfaces. While the second type is a phase field model, in which the interface is a region of small, nonzero thickness where there is some microscopic mixing of the two fluids. Both types of models are shown to satisfy energy inequalities which guarantee thermodynamical consistency. Via a formal asymptotic analysis, we show the phase field models are related to sharp interface models in the limit that the interfacial width tends to zero. 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. We present some numerical simulations to support the asymptotic analysis and display the effectiveness of the our approach. As a first step towards an analysis of our models, we consider sharp interface and phase field models for soluble surfactants in a static situation. The surfactant equations become a linear elliptic coupled bulk-surface partial differential equation, and our main result is the rigorous convergence of the weak solution of the phase field models to the weak solution of the sharp interface models.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EP/H023364/1)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics