Quaternion wavefunction theory bridges quantum formalism and classical fluid dynamics: a zero-parameter derivation of sphere drag
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912867990634496 |
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| author | Chishtie, Farrukh A. |
| author_facet | Chishtie, Farrukh A. |
| contents | We present a quaternion wavefunction formulation that reduces the incompressible Euler equations to a single nonlinear Schrödinger-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 $Ψ\in \mathbb{C} \otimes \mathbb{H}$ 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 Székelyhidi. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07469 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quaternion wavefunction theory bridges quantum formalism and classical fluid dynamics: a zero-parameter derivation of sphere drag Chishtie, Farrukh A. Fluid Dynamics We present a quaternion wavefunction formulation that reduces the incompressible Euler equations to a single nonlinear Schrödinger-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 $Ψ\in \mathbb{C} \otimes \mathbb{H}$ 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 Székelyhidi. 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. |
| title | Quaternion wavefunction theory bridges quantum formalism and classical fluid dynamics: a zero-parameter derivation of sphere drag |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2511.07469 |