The meniscus on a needle – a lesson in matching

Abstract

We reconsider the second approximation for the height of the meniscus on a slender vertical cylinder, which has been calculated by James using the method of matched asymptotic expansions. Fraenkel has warned that the asymptotic matching principle may fail in certain cases, and we confirm that failure here. This meniscus problem is used to discuss the kind of failures that may occur if Fraenkel's restricted form of the matching principle is not used. A less restricted matching principle is also suggested in this paper. With this matching principle, we show that the last known term of order Re³ In Re in the drag on a sphere in low-Reynolds-number flow is correct, even though it was found in a way that violates Fraenkel's warning.