  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.
