Analytical Analysis of a Dripping Faucet Model based on the Perturbation Method

Ziyang Xu

doi:10.54691/28yasx98

Authors

Ziyang Xu

DOI:

https://doi.org/10.54691/28yasx98

Keywords:

Interfacial mechanics, perturbation method, dripping faucet, stability analysis, bond number.

Abstract

The dripping faucet model is a classical system in fluid mechanics and nonlinear dynamics, and has application value in engineering problems such as inkjet printing and droplet manipulation. Previous studies mostly relied on numerical integration or variational methods to solve the Young–Laplace equation, which leads to high computational cost and makes it difficult to obtain analytical expressions for the hanging drop profile and critical volume. This paper proposes an analytical approximation method based on the perturbation method, used to describe the morphology and stability of a hanging drop at the end of a vertically downward circular thin-walled faucet. Under the condition of small Bond number, a perturbation expansion is applied to the Young–Laplace equation to derive analytical expressions for the hanging drop profile and volume, and a unique critical solution criterion is given based on the convexity of the profile function. In the experiment, a controlled liquid supply system and high-speed photography were used to measure the critical volume of hanging drops under six different faucet inner diameters. The results show that the theoretical predictions are consistent with the experimental results within an error range of 2–12%, and the critical volume increases with faucet radius and Bond number, while being independent of the intrinsic contact angle of the liquid.

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