Three-dimensional analogues of the Blasius-Chaplygin formulas

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) + $\mathbf{i}$ψ(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.