Special function solutions of complex differential equations and their application to Marangoni flows
File(s)
Author(s)
Curran, Anna Elizabeth
Type
Thesis
Abstract
This thesis is concerned with the mathematical study of steady two-dimensional Marangoni flow problems in both planar and radial geometries. By exploiting the recently discovered connection between the forced complex Burgers equation and Marangoni flows in a viscous fluid region, first discovered by D. G. Crowdy, it shall be shown that three physically distinct problems can each be reduced to a single forced complex Burgers equation, satisfied by a complex function that is analytic in the fluid domain. The contribution of this thesis is to demonstrate that, in all three cases, steady equilibrium solutions to these equations are given by special function solutions of certain well-known ordinary differential equations. Specifically, the parabolic cylinder and doubly-confluent Heun equations are considered.
The first physical problem considers a localised concentration of insoluble surfactant occupying an infinite flat interface between a deep viscous fluid region experiencing a linear extensional flow and a constant pressure region. This scenario is then generalised to a two-phase analogue, where the surfactant is now soluble to one of the fluids. In both cases, solutions are found in terms of logarithmic derivatives of the parabolic cylinder function. The solutions and their singularities are analysed in the limit of vanishing surface diffusion using asymptotic techniques and Liouville-Green analysis. These results are then discussed in the context of the Stokes phenomenon for nonlinear ODEs.
The third and final problem considered in this thesis is that of a bubble laden with insoluble surfactant translating steadily through a viscous fluid region that is experiencing a linear temperature gradient. Equilibrium solutions are found in terms of a logarithmic derivative of a solution of the doubly-confluent Heun equation, which is determined by a monodromy condition imposed on the equilibrium solution. A semi-explicit expression for the bubble speed is derived.
The first physical problem considers a localised concentration of insoluble surfactant occupying an infinite flat interface between a deep viscous fluid region experiencing a linear extensional flow and a constant pressure region. This scenario is then generalised to a two-phase analogue, where the surfactant is now soluble to one of the fluids. In both cases, solutions are found in terms of logarithmic derivatives of the parabolic cylinder function. The solutions and their singularities are analysed in the limit of vanishing surface diffusion using asymptotic techniques and Liouville-Green analysis. These results are then discussed in the context of the Stokes phenomenon for nonlinear ODEs.
The third and final problem considered in this thesis is that of a bubble laden with insoluble surfactant translating steadily through a viscous fluid region that is experiencing a linear temperature gradient. Equilibrium solutions are found in terms of a logarithmic derivative of a solution of the doubly-confluent Heun equation, which is determined by a monodromy condition imposed on the equilibrium solution. A semi-explicit expression for the bubble speed is derived.
Version
Open Access
Date Issued
2025-09-26
Date Awarded
01/02/2026
Advisor
Crowdy, Darren
Papageorgiou, Demetrios
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)