On the Laplace formula for the capillary tube

Abstract

The asymptotic correctness is shown of a formal expansion given by Laplace in 1806 of the rise height of a fluid in a circular capillary tube. The proof is completely based on the comparison principle of Concus and Finn. The first two non-constant terms in the expansion are calculated. It is of special interest that the expansion is uniform with respect to the boundary contact angle although the governing quasilinear elliptic equation becomes singular on the boundary if the contact angle tends to zero.