Dunbar, Oliver (2017) A diffuse interface model of surfactants in multi-phase flow. PhD thesis, University of Warwick.

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Abstract

We investigate a free boundary problem arising in fluid dynamics, by modelling multiple incompressible fluids over subdomains with different material quantities, and in the presence of surface tension reducing chemicals known as surfactants. We construct a free energy for this system, and we require it obey the second law of thermodynamics, leading to the formulation of an energy minimisation problem (the sharp problem). This problem is degenerate, so we regularise it by constructing a new energy of Ginzburg-Landau type, parametrised by a (small) constant ε > 0 and when ε → 0 the sharp problem is recovered in the sense of Γ-convergence. This multi-phase energy is formed from a multiwell potential and gradient term, and the minimisers are known as phase field variables. The phase field variables approximate characteristic functions of the subdomains, and the model is rewritten as functions of them. Beneficially, the energy analysis can be repeated as before to obtain a diffuse interface model.

We construct and perform numerical analysis of a novel discretisation scheme for a Cahn-Hilliard Navier-Stokes system. Here we create a fractional-theta coupling scheme which is importantly proved to be of second order in time. The key property of this scheme is that it uses weighted operator splitting to separate the different nonlinearities that appear in a Cahn-Hilliard Navier-Stokes system. That is, the Cahn-Hilliard multiwell potential, the incompressibility condition and the convection. We discuss stability and the extension to surfactants. We implement the novel scheme in DUNE (Distributed Unified Numerics Environment), a finite element package and use simulation to run tests to validate the stability and consistency of the schemes, convergence of the diffuse interface model with respect to its parametrisation, and flexibility for the code development.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Fluid dynamics -- Mathematical models, Surface active agents, Multiphase flow -- Mathematical models, Boundary value problems, Surface chemistry, Liquid-liquid interfaces -- Mathematical models
Official Date:	September 2017
Dates:	Date Event September 2017 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Stinner, Björn ; Dedner, Andreas
Sponsors:	Engineering and Physical Sciences Research Council
Format of File:	pdf
Extent:	xii, 159 leaves : illustrations, charts
Language:	eng

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