On the theory of the capillary tube

Abstract

A recent paper by Richards and Coombs discusses in some detail the determination of surface tension by the rise of the liquid in capillary tubes, and reflects mildly upon the inadequate assistance afforded by mathematics. It is true that no complete analytical solution of the problem can be obtained, even when the tube is accurately cylindrical. We may have recourse to graphical constructions, or to numerical calculations by the method of Runge, who took an example from this very problem. But for experimental purposes all that is really needed is a sufficiently approximate treatment of the two extreme cases of a narrow and of a wide tube. The former question was successfully attacked by Poisson, whose final formula [(18) below] would meet all ordinary requirements. Unfortunately doubts have been thrown upon the correctness of Poisson's results, especially by Mathieu, who rejects them altogether in the only case of much importance, i. e. when the liquid wets the walls of the tube-a matter which will be further considered later on. Mathieu also reproaches Poisson's investigation as implying two different values of h , of which the second is really only an improvement upon the first, arising from a further approximation. It must be admitted, however, that the problem is a delicate one, and that Poisson's explanation at a critical point leaves something to be desired. In the investigation which follows I hope to have succeeded in carrying the approximation a stage beyond that reached by Poisson. In the theory of narrow tubes the lower level from which the height of the meniscus is reckoned is the free plane level. In experiment the lower level is usually that of the liquid in a wide tube connected below with the narrow one, and the question arises how wide this tube needs to be in order that the inner part of the meniscus may be nearly enough plane.