Phase field methods for flows with elastic membranes

An open area of research in computational fluid dynamics is the study of multi-fluid flows, phenomena which arise in many real-world fluid mechanics applications. The study of these flows necessitates an understanding of dynamic and deforming interfaces. Numerically, accurately resolving and tracking the complex motion of the interfaces in both time and space presents many numerical and algorithmic difficulties. Incorporating mechanical effects at the interface, such as surface tension and elasticity, into a model for the fluid motion presents additional challenges.; Existing numerical techniques for representing an interface are generally characterized as either front-tracking or front-capturing. Front-tracking methods have the advantage that the interface is sharply resolved; however, numerically, large deformations result in degenerate elements in the finite element mesh, necessitating frequent re-discretization of the domain. The computational geometry required to maintain a quality propagating mesh as it deforms is particularly difficult, especially in three dimensions. Front-capturing methods are an attractive alternative for interfaces and boundaries undergoing large deformations and topological changes. However, front-capturing techniques do not exactly resolve the location of the interface; thus, interfacial forces are smeared across a finite region near the interface.; This work proposes a phase field model, a front-capturing Eulerian method, representing multi-fluid flow. In addition, this work introduces an Eulerian formulation for an elastic membrane embedded in the fluid motion. Elasticity is an inherently Lagrangian quantity; the elastic response requires knowledge of the history of the elastic deformation. Prior attempts to incorporate interface mechanics have generally either (1) been limited to incorporating surface tension, an inherently Eulerian quantity, into front-capturing methods or (2) for elasticity, required a Lagrangian description of the interface. This work demonstrates that front-capturing Eulerian methods can be used to model Lagrangian quantities.; This thesis introduces the phase field method for representing multi-fluid flow. A sharp interface model of the elastic membrane stresses is derived and, using the phase field methods, integrated into the Eulerian model for fluid flow. Examples demonstrating the effect of the elastic membrane on the deformation and flow of a binary fluid are presented.