Schrödinger–Navier–Stokes–Quantum-π: A Unified Model and Hybrid Numerical Method for Quantum Fluids with π-Phase Structure - Barack Ndenga (Publisher, 2025)
I present a unified theoretical and numerical framework that couples quantum wave dynamics (Schrödinger) with classical viscous flow (Navier–Stokes) through an emergent quantum-π field (π_q) that encodes phase-topology and coherence. The model reproduces Schrödinger dynamics in the conservative limit, Navier–Stokes turbulence in the dissipative limit, and novel intermediate regimes where quantum coherence and fluid turbulence coexist and interact. I derive the governing equations by (i) applying a Madelung decomposition to a complex field ψ, (ii) introducing a controlled viscous regularization and non-linear coupling terms, and (iii) coupling a dynamically evolving π_q scalar (or tensor) field that modulates local coherence, effective mass, and information flux. I then present a robust hybrid numerical method (QHFVM — Quantum Hydrodynamic Finite-Volume Method) combining split-step spectral propagation for dispersive quantum terms and conservative finite-volume solvers for advective, viscous and pressure dynamics. I validate the approach on a set of canonical problems (quantum vortex shedding in a viscous background, π-phase revival in confined geometries, and turbulence spectra with quantum corrections), show numerical convergence and conservation properties, and outline applications spanning quantum fluids, nanoscale bio-fluids, materials, and hybrid QEC architectures. Flash explanations highlight intuition and immediate experimental tests.
This work introduces a unified Schrödinger–Navier–Stokes–Quantum-π (SNS-π) model that couples quantum wave dynamics, viscous fluid flow, and a dynamic phase-topology field π_q. I derive the governing equations, present a hybrid numerical method (QHFVM) combining split-step spectral and finite-volume solvers, validate on canonical test cases, and describe applications in quantum fluids, nanomaterials, biophysics, and quantum error correction. Keywords: Schrödinger–Navier–Stokes, quantum hydrodynamics, quantum turbulence, π quantum, hybrid numerical method, quantum fluids, nano-biofluid, quantum-classical coupling, quantum error correction, turbulent spectra.