Modeling the Curvature of a Ferrofluid Interface Using a Height Function Method

The behavior of an interface embedded in a fluid is central to a wide range of biological, chemical, environmental and physical problems and engineering processes. Modeling the evolution of a fluid interface is thus a critical and important problem. In many instances, including two-phase (e.g. liquid-gas) flows, the interface is an internal boundary within a PDE model. A model of the interface properties and its evolution is then typically performed by numerical computation, within the framework of the PDE solution method, such as finite differences (FD). Volume of Fluid (VOF) is a simple FD based method which exhibits excellent volume conservation and topological properties [13]. Because of the FD framework, however, differential properties such as the normal and curvature of the interface are poorly computed. Recently, the use of a local integral property, or “height function,” has been shown to allow more accurate curvature computation within VOF for some interface configurations. The height function (HF) method is a technique for estimating interface normals and curvatures from well-resolved volume fraction data that shows second-order convergence with grid refinement. One drawback is that HF’s best results occur when the interface curvature is weak and its orientation aligns with one of the underlying computational grid directions. This thesis will use a height function method to approximate the geometrical properties of a fluid interface and analyze the errors that result for arbitrary curvatures and orientations with respect to a wide range of grids. Additionally, this thesis examines the impact of the height function method in two-dimensional simulations of ferrofluids in an imposed uniform magnetic field, which causes these bubbles or droplets to become roughly elliptical in shape