Micro-foams with the Arbitrary Lagrangian Eulerian Method

Title of dissertation: CAPTURING MICRO-EMULSIONS AND MICRO-FOAMS WITH THE ARBITRARY LAGRANGIAN EULERIAN METHOD Andrew Brandon, Doctor of Philosophy, 2015 Dissertation directed by: Professor Konstantina Trivisa Department of Mathematics The goal of this work is to develop a 2D, heterogenous model for the purpose of studying liquid drainage from micro-foams and micro-emulsions. Liquid drainage within foams and emulsions is an important phenomenon because it affects several intrinsic properties of foams and emulsions, including their viscoelastic and thermophysical properties. While previous attempts have explained some aspects of drainage with a homogenous treatment, there are fundamental questions regarding drainage that remain unanswered. In our heterogenous model, we treat the bubbles and droplets of micro-foams and micro-emulsions as rigid particles because their small radii result in large surface tension forces, which cause them to remain spherical for a significant portion of time. To track the sharp, rigid interfaces of the droplets and bubbles, we build our heterogenous model upon the Arbitrary Lagrangian Eulerian (ALE) method. The ALE method is a moving grid method that can be expensive because the finite element matrices must be reassembled each time the grid is moved and grid deformation can lead to low quality elements. We reduce the cost of running ALE simulations by employing a second order accurate, semi-implicit-explicit time integration scheme designed for low Reynolds number flows and by utilizing a novel function for controlling the deformation of the mesh. With this heterogenous model, we first investigate coarsening induced drainage within an idealized micro-foam. Our heterogenous model shows that the foam’s liquid channel thicknesses must be allowed to vary with time in the bubble’s coarsening equations. In addition, our model also shows that bubble position is as important as bubble size when it comes to the coarsening process. We conclude this work with a series of studies designed to determine the algorithm’s ability to capture the bubbles and droplets of micro-foams and micro-emulsions as a collection of free bodies. The results of this testing indicate that our algorithm is capable of capturing a microemulsion as a collection of free body droplets. However, the testing also indicates that our algorithm is not capable of capturing micro-foams as a collection of free body bubbles at this time. CAPTURING MICRO-EMULSIONS AND MICRO-FOAMS WITH THE ARBITRARY LAGRANGIAN EULERIAN METHOD