Three-dimensional analogues of the Blasius-Chaplygin formulas
Abstract
The classical Blasius--Chaplygin formula provides an elegant method for calculating the lift force on a two-dimensional body in steady, irrotational flow. The key ingredient is the definition of a complex-valued potential function \begin{equation*} f(z) = \varphi(x, y) + \mathbf{i}\psi(x, y), \end{equation*} which can then be integrated using Cauchy's theorem along any closed contour surrounding the body. In this paper, we propose a three-dimensional extension of the classical Blasius--Chaplygin formula using quaternionic analysis. After presenting the basics of quaternionic analysis, we discuss how \emph{monogenic functions} -- the quaternionic analog of classical holomorphic functions -- can be used to describe problems in fluid dynamics. Finally, we present the Blasius--Chaplygin formula in quaternionic form.
Keywords
Cite
@article{arxiv.2508.03382,
title = {Three-dimensional analogues of the Blasius-Chaplygin formulas},
author = {Dmitrii Legatiuk and Heikki Orelma},
journal= {arXiv preprint arXiv:2508.03382},
year = {2025}
}