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arXiv · 2508.03382

Three-dimensional analogues of the Blasius-Chaplygin formulas

Dmitrii Legatiuk·Heikki Orelma
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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.

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Dmitrii Legatiuk, Heikki Orelma. 2025-08-05. Three-dimensional analogues of the Blasius-Chaplygin formulas. https://arxiv.org/abs/2508.03382

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