Analytical methods for multi-phase, multi-physics systems

Abstract

This thesis proposes novel analytical approaches to a range of boundary value problems, typically modelling flow past liquid-infused and superhydrophobic surfaces.

It finds exact solutions to fully-coupled, two-phase, free boundary flows: the first of their kind. These solutions are then used to quantify a mode of failure for liquid-infused surfaces of considerable contemporary interest, avoiding common simplifying assumptions. These tools are used to demonstrate the improved drag reduction offered by a currently theoretical class of liquid-infused surface: those with `re-entrant grooves'.

New solutions to the heat transfer of flow through superhydrophobic pipes are also found. These are used to quantify convective heat transfer of such flow by deriving a formula for the Nusselt number. A family of superhydrophobic pipes that cause a marked improvement to the Nusselt number are discovered: a departure from present consensus that superhydrophobicity always decreases the Nusselt number of flow past a surface.

A significant motivating physical configuration of this thesis is pressure-driven flow through a channel patterned with long, thin, lubricant-filled longitudinal grooves. Asymptotic approaches used by Game et al., Kirk, and Crowdy are incorporated to partially solve the full 3-D flow, reducing the problem to one in the 2-D plane. A novel complex variable transform method, generalising that of Fokas & Kapaev and more recently Crowdy to fully coupled two-phase problems, is then developed to solve this. Further extensions of this theory are also explored.

Lastly, conformal mapping and the theory of analytic functions are used to study the flow of pressure-driven fluid through a superhydrophobic, cylindrical pipe. Philip found the solution to the flow in this configuration, and this is used to solve the thermal advection-diffusion problem. The exact solutions found are then used to derive a closed form for the Nusselt number, prompting the aforementioned surprising results.