A singularity-free analytic solution of rise dynamics of a liquid in a vertical cylindrical capillary

TL;DR

This paper presents a novel, convergent analytic solution for the rise dynamics of a liquid in a vertical cylindrical capillary using the homotopy analysis method, valid across all physical parameters and capable of predicting oscillatory or monotonic behavior.

Contribution

The authors develop the first known convergent analytic solution for capillary rise in a cylindrical tube applicable over all physical parameters.

Findings

Solution agrees well with numerical simulations.

Valid across entire physical parameter range.

Can predict oscillatory versus monotonic rise.

Abstract

Capillary driven flow is a famous problem in fluid dynamics which dates back to Leonardo da Vinci. In this paper, we apply an analytic approximation method for highly nonlinear problem, namely the homotopy analysis method (HAM), to a model of the meniscus movement in a uniform vertical circular tube. Convergent explicit series solution is successfully obtained. Our results agree well with the numerical results given by the symbolic computing software Mathematica using six-order Runge-Kutta methods. More importantly, our analytic solution is valid in the whole region of physical parameters, and therefore can predict whether the path of liquid is monotonic or oscillatory. This kind of solution, to the best knowledge of the authors, has never been reported in the past, which might greatly deepen our understandings about capillarity.