The effect of surfactants on expanding free surfaces

This paper develops a systematic theory for the flow observed in the so-called ``overflowing cylinder'' experiment. The basic phenomenon to be explained is the order of magnitude increase in the surface velocity of a slowly overflowing beaker of water that is caused by the addition of a small amount of soluble surfactant. We perform analyses of (i) an inviscid bulk flow in which diffusion is negligible, (ii) a hydrodynamic boundary layer in which viscous effects become important, (iii) a diffusive boundary layer where diffusion is significant, and by matching these together arrive at a coupled problem for the liquid velocity and surfactant concentration. Our model predicts a relation between surface velocity and surface concentration which is in good agreement with experiment. However a degeneracy in the boundary conditions leaves one free parameter which must be taken from experimental data. We suggest an investigation that may resolve this indeterminacy

Mathematical analysis of some diffusion problems associated to the modeling of surfactant compounds at the air-water interface

Surfactants are chemical compounds with a particular structure, that is the responsible of their behavior in a water solution. When a new surface is formed in a surfactant solution, surfactant molecules tend to migrate from the bulk of the solution to the surface, consequently varying its surface tension. The dynamic surface tension is a very important property since it plays a major role in several biochemical, biological and industrial processes. All the huge applications of the dynamic surface tension make it a subject of study for a long period of time. There are several publications in the chemical literature dealing with the numerical solutions of the models that describe the surfactant behavior. However, none of them deal with their mathematical analysis. In this thesis, we introduce the mathematical treatment of those models, that consist of the diffusion equation coupled with an adsorption model

Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems

We show global well-posedness and exponential stability of equilibria for a general class of nonlinear dissipative bulk-interface systems. They correspond to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes nonlinear slow and fast diffusion in the bulk and nonlinear coupled diffusion on the interface. Additional driving mechanisms can be included and non-smooth geometries and coefficients are admissible, to some extent. An important application are volume-surface reaction-diffusion systems with nonlinear coupled diffusion.Comment: 21 page

New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries

The field of mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a flourishing area of research. Many such systems arise from mathematical models in material science, fluid dynamics and biology, for example phase separation in alloys, epitaxial growth, dynamics of multiphase fluids, evolution of cell membranes and in industrial processes such as crystal growth. The governing equations for the dynamics of the interfaces in many of these applications involve surface tension expressed in terms of the mean curvature and a driving force. Here the forcing terms depend on variables that are solutions of additional partial differential equations which hold either on the interface itself or in the surrounding bulk regions. Often in applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimization problems with partial differential equation constraints including free boundaries. Because of the maturity of the field of computational free boundary problems it is now timely to consider such control problems