# Quasi-static motion of a liquid droplet on a deformable substrate

Wang, Chung-Hao (2023) Quasi-static motion of a liquid droplet on a deformable substrate. Doctoral thesis, University of East Anglia.

## Abstract

We investigate the dynamics of a three-dimensional droplet sitting on a time-dependent solid substrate. The study focuses on droplet motion caused by slow deformations of the substrate, where inertial effects are negligible, and the shape of the droplet at each time instant is governed by the balance of the gravity and capillary forces. Four models of the contact line dynamics are considered in the present study.

A quasi-static approximation is employed to analyse this dynamic problem. Our approach to solving the quasi-static problem involves using asymptotic methods by considering a small perturbation to the elevation of the solid substrate. Initially, the solid substrate is flat and horizontal, and the droplet is axisymmetric and at rest. The first-order correction of the droplet shape, induced by slow motions of the substrate, depends on the specific contact line model applied.

In the first model, the contact line remains fixed during the dynamic process. Apart from determining the droplet shape, the current local contact angle needs to be calculated. By contrast, in the second model, the contact line is free to move, and the current local contact angle along the moving contact line is assumed to be equal to the equilibrium contact angle. The first-order corrections of both the position of the contact line and the droplet shape are determined simultaneously.

The third model combines the previous two models. Depending on the current local contact angle, the different parts of the contact line have different characteristics, termed pinned and unpinned regions. Lastly, the fourth model of contact line motion is known as the Cox-Voinov model, in which the contact line is always moving in the direction of its normal.

Item Type: Thesis (Doctoral) Faculty of Science > School of Mathematics Chris White 26 Jun 2024 14:19 26 Jun 2024 14:19 https://ueaeprints.uea.ac.uk/id/eprint/95694