Analysis and numerical computations of free boundaries in fluid dynamics: surfactant solubility and elastic fibers

This proposal is for the investigation of fundamental free boundary problems in fluid mechanics that are motivated by different applications but share similar mathematical features. The first class of problems concerns the effect that the solubility of a surfactant in the bulk flow of immiscible fluids can have on the deformation and breakup of liquid drops, bubbles, and jets. A significant difficulty is introduced by the slow diffusion (or large Peclet number) of surfactant in the bulk flow, which causes a large gradient in concentration of surfactant to occur across a thin layer adjacent to the interface between immiscible fluids. Resolving the surfactant gradient near the interface must be accomplished with great accuracy to determine the interface's dynamics but presents a significant challenge for traditional numerical methods. In this project, the slenderness of the layer will be used to develop fast and accurate 'hybrid' numerical methods that incorporate a separate analytical reduction of the layer's dynamics into numerical solution of the interfacial free boundary problem. The second class of problems concerns the deformation of a slender elastic fiber or filament when it is influenced by capillary forces at a fluid interface. This project will develop a robust and efficient numerical method that uses one-dimensional integral equations from slender body theory to describe the fiber's dynamics and takes into account the interaction between fluid, filament, and free surface.

Surfactants, such as alcohols, sulfates, and detergents, are widely used to control the dynamics of emulsification or blending of fluid systems in applications in the chemical and pharmaceutical industries. At a fundamental level, the emulsification process occurs via the behavior of single fluid drops and jets as they break up by bursting or tip-streaming in a straining flow. This project seeks to determine details of the mode of breakup that are important in practice but have so far been difficult to capture accurately by traditional numerical methods. Similarly, sticking and stiction of slender elastic filaments and cantilevers is important in the proper operation of specific micro-electromechanical devices, and filament-fluid interfacial forces are being investigated as a means to assemble filaments into larger structures, such as nanotube rings and biological filamented structures. This project seeks to elucidate fundamental effects that occur in such processes by incorporating analytical techniques for the resolution of small-scale structures into fast and accurate, robust numerical methods.