# Wetting States of Two-Dimensional Drops under Gravity

## TL;DR

This paper develops an analytical model for 2D droplet shapes under gravity, accounting for contact line pinning and mobility, and provides insights into wetting states on flat and inclined surfaces.

## Contribution

It introduces a new analytical framework for 2D droplet profiles under gravity, considering contact line behavior and extending to inclined surfaces.

## Key findings

- Analytical expressions for droplet profiles on flat surfaces.
- Contact line pinning determines droplet shape independently of Young contact angle.
- Droplet wetting states on inclined surfaces depend on free energy minimization.

## Abstract

An analytical model is proposed for the Young-Laplace equation of two-dimensional (2D) drops under gravity. Inspired by the pioneering work of Landau & Lifshitz (1987), we derive analytical expressions of the profile of drops on flat surfaces, for arbitrary contact angles and drop volume. We then extend our theory for drops on inclined surfaces and reveal that the contact line plays a key role on the wetting state of the drops: (1) when the contact line is completely pinning, the advancing and receding contact angles and the shape of the drop can be uniquely determined by the predefined droplet volume, sliding angle and contact area, which does not rely on the Young contact angle; (2) when the drop has a movable contact line, it would achieve a wetting state with a minimum free energy resulting from the competition between the surface tension and gravity. Our theory is in excellent agreement with numerical results.