Quaternion wavefunction formulation of incompressible inviscid fluid dynamics

TL;DR

This work recasts the incompressible Euler equations as a single constrained quaternion wavefunction evolution, transforming four coupled PDEs into a complex quaternion Gross-Pitaevskii-type equation with a holomorphic incompressibility constraint. By encoding velocity in a unit quaternion and enforcing incompressibility via a quaternionic Cauchy-Riemann-like condition, the authors derive conservation laws from a natural Lagrangian framework and predict fluid behavior across regimes, including a topological derivation of the Newton drag plateau and a wake-topology-based onset of vortex shedding. The key contributions are the explicit quaternion construction of the flow field, the constrained evolution equation, and the robust validation against sphere-drag experiments over five orders of magnitude in Reynolds number, along with a quantitative topological explanation for critical shedding. The approach promises computational advantages through local constraint enforcement and FFT-compatible evolution, and suggests wide-ranging extensions to viscous, turbulent, and compressible flows, potentially transforming analytical and numerical fluid dynamics.

Abstract

We present a quaternion wavefunction formulation that reduces the incompressible Euler equations to a single nonlinear Schrödinger-type equation. The velocity field emerges from a complex quaternion wavefunction $Ψ\in \mathbb{C} \otimes \mathbb{H}$ satisfying a constrained Gross-Pitaevskii equation, with incompressibility enforced through a holomorphic constraint on quaternion space. This formulation preserves all conservation laws through a natural Lagrangian structure and reduces the system from four coupled nonlinear equations (three velocity components plus pressure) to one quaternion field equation with an algebraic constraint. We demonstrate the utility of this approach by deriving analytical solutions for three-dimensional flow past a sphere, obtaining the Newton regime drag coefficient $C_{D,\infty} = 0.44$ from pure quaternion topology (achieving $C_D = 0.488$ at Reynolds number $\text{Re} = 1000$, within $3.8\%$ of experimental measurements) and establishing topological consistency with the experimentally observed onset of vortex shedding at $\text{Re}_c = 270$ through quaternion circulation quantization. The formulation provides a new mathematical framework for inviscid fluid dynamics and suggests efficient numerical algorithms exploiting quaternion structure.

Figures (1)

• Figure 1: Comprehensive validation of quaternion formulation for sphere drag prediction. Panel (a) shows drag coefficient versus Reynolds number on log-log scale, comparing quaternion predictions (red curve) with experimental data (black points with error bars) across Re = 0.1 to 5000. The vertical orange dashed line marks the critical Reynolds number at Re$_c = 270$ predicted from quaternion topological consistency. Regime boundaries (gray dotted lines) separate Stokes (Re $< 1$), transition (1 $<$ Re $< 100$), and Newton (Re $> 1000$) regions. Panel (b) displays relative prediction error, color-coded by formula source: blue for Stokes-Oseen theory, purple for empirical transition regime, and green for Newton theory. The green band indicates $\pm 10\%$ accuracy. Panel (c) presents direct comparison between predicted and experimental drag coefficients, with theoretical predictions (Stokes and Newton regimes) shown as blue and green points, and empirical regime as purple. The dashed line indicates perfect agreement. Key Reynolds numbers (Re = 1, 100, 270, 1000) are annotated. Panel (d) illustrates flow regime classification across Reynolds number, with the critical topological transition at Re$_c = 270$ marked by vertical dashed line. Panel (e) summarizes validation statistics: mean absolute error (MAE) = 5.8%, root mean square error (RMSE) = 6.7%, correlation coefficient $R^2 = 0.9996$, with 16 of 17 test cases achieving $< 10\%$ error. The quaternion formulation correctly recovers the Stokes regime, predicts the Newton plateau, and establishes topological consistency with vortex shedding onset.