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A theoretical study of the effect of surface diffusion on a stagnant cap of surfactant
File(s)
Author(s)
Curran, Anna
Crowdy, Darren
Papageorgiou, Demetrios
Type
Journal Article
Abstract
A steady equilibrium solution is presented showing how a stagnant cap of insoluble surfactant is affected by surface diffusion in the limit of low Reynolds and capillary numbers. Exploiting a recently discovered connection between the complex Burgers equation and Marangoni flows in a half-plane of viscous fluid with surfactants, it is shown how, in the steady case, a linearizable Riccati-type ordinary differential equation (ODE) governs the system. This allows the equilibrium solution, including surface diffusion effects, to be expressed in terms of parabolic cylinder functions. The stagnant cap equilibrium is recovered using a Liouville–Green approximation, in the limit of vanishing surface diffusion, through a modification of large-parameter expansions of the parabolic cylinder function. Moreover, connection formulae for such functions can be used to show how simple poles of the solution in the non-physical upper half-plane coalesce, as surface diffusion lessens, to select a choice of branch cut joining square root branch points at the two edges of a limiting stagnant cap. This singularity structure, which is expected to be generic for such Marangoni systems, is discussed in the context of the Stokes phenomenon for solutions of nonlinear ODEs.
Date Issued
2025-04-30
Date Acceptance
2025-01-30
Citation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2025, 481 (2312)
URI
URL
DOI
ISSN
1364-5021
Publisher
The Royal Society
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
481
Issue
2312
Copyright Statement
© 2025 The Authors. Published by the Royal Society under the terms of theCreative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author andsource are credited.
License URL
Identifier
10.1098/rspa.2024.0778
Subjects
applied mathematics
complex analysis Marangoni flows
surfactant
asymptotic analysis
Publication Status
Published
Article Number
20240778
Date Publish Online
2025-04-30