Quaternion wavefunction theory bridges quantum formalism and classical fluid dynamics: a zero-parameter derivation of sphere drag

TL;DR

This paper introduces a quaternion wavefunction approach linking quantum mechanics and classical fluid dynamics, deriving the sphere drag coefficient from first principles with high accuracy using geometric constraints.

Contribution

It presents a novel quaternion wavefunction formulation that reduces Euler equations to a Schrödinger-type equation, providing a geometric solution to classical fluid problems and deriving the sphere drag coefficient without empirical parameters.

Findings

Derived the sphere drag coefficient as 4/9 with 0.04% accuracy

Established quaternion holomorphicity as a selection principle for physical solutions

Unified quantum and classical fluid mechanics through geometric quaternion structures

Abstract

We present a quaternion wavefunction formulation that reduces the incompressible Euler equations to a single nonlinear Schr\"odinger-type equation with a holomorphic constraint, revealing hidden geometric structure connecting quantum and classical fluid mechanics. The velocity field emerges from a complex quaternion wavefunction $\Psi \in \mathbb{C} \otimes \mathbb{H}$ satisfying a constrained Gross-Pitaevskii equation, with incompressibility enforced through quaternion analyticity conditions that generalize the Cauchy-Riemann equations to three dimensions. This geometric structure provides a selection principle for physically realized Euler solutions, resolving D'Alembert's 270-year-old paradox through geometry rather than phenomenology. The key insight is that incompressibility corresponds to quaternion holomorphicity, known as the Cauchy-Riemann-Fueter conditions, which selects physical solutions from among the infinitely many weak solutions established by De~Lellis and Sz\'ekelyhidi. Application to steady flow past a sphere yields the Newton regime drag coefficient $C_{D,\infty} = 4/9 \approx 0.44$ as a \textbf{zero-parameter prediction} from quaternion orthogonality constraints, achieving 0.04\% agreement with experiment. This represents the first derivation of this fundamental fluid mechanics constant from first principles. The mechanism parallels how the Kutta condition determines airfoil circulation: quaternion orthogonality constraints break fore-aft pressure symmetry, producing finite drag within inviscid theory.