On the consistency between fluid surface tension and membrane geometric stiffness: How thick is the surface membrane?

Abstract

Although liquid surfaces exhibiting surface tension are often qualitatively compared to membranes in solid mechanics, there is a lack of rigorous theoretical correspondence between them. This study addresses this gap by providing a mathematical equivalence between the Young-Laplace equation, which governs the behaviour of fluid interfaces, and Kirchhoff plate theory, which describes the behaviour of homogeneous thin membranes under moderate rotations. We show that the configuration of a fluid interface is identical to that of a pre-stretched thin membrane with negligible bending stiffness when the surface-tension coefficient equals the membrane's pre-tension per unit length. This equivalence has both theoretical and practical significance. The study also explores the maximum equivalent membrane thickness and unusable boundary width, offering guidance on selecting appropriate thin membranes to replicate fluid interface behaviour in practice. These findings provide a foundation for future research and practical applications, particularly when there is a need to apply experimental methods from solid mechanics to fluid mechanics or vice versa.

Publication Title

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

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