The problem of obtaining the first and second derivatives of the profile of a pendant droplet is formulated as an integral equation of the first kind. This equation is solved by Tikhonov regularization in which the method of general cross validation is used to guide the selection of the regularization parameter. These derivatives are converted into mean curvature as a function of droplet height. Surface tension is then obtained by regression computation between the mean curvature and two possible algebraic expressions suggested by the Laplace-Young equation. This way of obtaining surface tension is demonstrated by applying it to a number of published droplet profiles. Some of the problems encountered are discussed and solutions suggested.
Yeow, Y. L., Mcgowan, K. L., Ong, S. I., & Leong, Y-K. (2005). Applying Tikhonov regularization to process pendant droplet tensiometry data. Langmuir, 21(24), 11241-11250. https://doi.org/10.1021/la051650p