On the rise height in a capillary tube of general cross section

Abstract

In a recent paper on the circular capillary tube we proved the asymptotic correctness of a formal expansion of the rise height given by Laplace in 1806. Here we extend this result to tubes of more general cross section.

We prove the existence of an asymptotic expansion of the rise height with respect to the (small) Bond number under the main assumption that the zero gravity solution exists. The proof is completely based on the comparison principle of Concus and Finn. As examples, the annulus and the regular n-gon will be considered. In the case of the annulus the expansion is uniform with respect to the boundary contact angle which is a consequence of the special nonlinearity of the problem.