Surfactant Dynamics in Confined Geometries

Abstract

Surfactants reduce the tension of liquid-liquid or liquid-gas interfaces, and they create fluid flows via the Marangoni effect. The ubiquity of surfactants in the industrial and biological environment often causes unexpected Marangoni flows and stresses, and they act like hidden variables, sometimes affecting applications in profound ways. In this thesis we theoretically investigate three model problems concerning the surfactant-induced Marangoni effect and how confinement of the surface affects such flows. First, we tackle the problem of surfactant spreading on the surface of a viscous liquid contained in a two-dimensional rectangular cavity. We reformulate the governing equations as an eigenvalue problem, and we find that surfactant concentrations can be decomposed into a sum of normal modes. The flow is singular at the corners of the domain near pinned contact lines, around which an asymptotic solution shows that the pressure has a logarithmic singularity, and the tangential surface stress oscillates with exponentially decreasing strength as we approach the corner. We show how diffusion can regularise these singularities, and create boundary layers so small that direct numerical simulation would almost inevitably miss them. Next, we model a previously published experiment, where surfactant added to red dye spreads to `solve' a fluid-filled maze with minimum penetration into lateral branches. We hypothesise that this is caused by endogenous surfactant present within the fluid, and we build a model to test this theory. Using a lubrication theory approximation to Stokes flow, we solve the governing equations on a graph representing the geometry of the maze, implementing a so-called mimetic finite-difference scheme, which numerically conserves mass of surfactant to machine precision. The model qualitatively reproduces the dominant dynamic behaviours of the experiment. Lastly, we tackle the problem of material particle tracing on the surface of a thin liquid film undergoing surfactant-driven flow. We solve the dynamic problem in two ways demonstrating corroboration between results. First, we solve for concentration evolution and particle dynamics on separate finite difference grids; secondly, we combine the governing equations for concentration and particle motion into a single Lagrangian vector equation. The results show how confinement creates drift and flow reversals. We demonstrate a link between surfactant dynamics and the optimal transport theory, allowing us to find an approximation of equilibrium locations of material surface particles, given only their initial locations, by solving a Monge-Amp\`ere equation. We show how the edges of deposits of exogenous surfactant added to a small endogenous concentration deform into near-polygonal structures, with self-similar corner structures.

Date of Award	31 Dec 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Oliver Jensen (Supervisor) & Julien Landel (Supervisor)