Three-dimensional analogues of the Blasius-Chaplygin formulas

TL;DR

This paper extends the classical Blasius--Chaplygin formula to three dimensions using quaternionic analysis, enabling calculation of lift forces in 3D steady, irrotational flows.

Contribution

It introduces a quaternionic framework for the Blasius--Chaplygin formula, generalizing it from 2D to 3D fluid dynamics problems.

Findings

Develops quaternionic analysis foundation for fluid dynamics

Derives 3D Blasius--Chaplygin formula in quaternionic form

Provides a new mathematical tool for 3D lift force calculations

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.