# Dynamics of Taylor Rising

## TL;DR

This paper investigates the dynamics of liquid rising in a tilted corner, deriving a PDE model that predicts a universal $t^{1/3}$ scaling law for the liquid height, influenced by the tilting angle, with applications in biomimetic surface design.

## Contribution

The study derives a PDE model for liquid rise in tilted corners and demonstrates the universal $t^{1/3}$ scaling law, extending previous vertical corner results to tilted geometries.

## Key findings

- Liquid height follows a $t^{1/3}$ scaling law for large times.
- The tilting angle affects the coefficient in the scaling law.
- The Onsager principle accurately predicts the coefficient.

## Abstract

We study the dynamics of liquid climbing in a narrow and tilting corner, inspired by recent work on liquid transportation on the peristome surface of Nepenthes alata. Considering the balance of gravity, interfacial tension and viscous force, we derive a partial differential equation for the meniscus profile, and numerically study the behavior of the solution for various tilting angle $\beta$. We show that the liquid height $h(t)$ at time $t$ satisfy the same scaling law found for vertical corner, i.e., $h(t) \propto t^{1/3}$ for large $t$, but the coefficient depends on the tilting angle $\beta$. The coefficient can be calculated approximately by Onsager principle, and the result agrees well with that obtained by numerical calculation. Our model can be applied for a weakly curved corner and may provide guidance to the design of biomimetic surfaces for liquid transportation.