Schrödinger–Navier–Stokes–π Unified Computational Framework : A Unified Theoretical and Numerical Architecture for Quantum-Coherent Fluid Dynamics Across Physical and Biological Scales

The coexistence of quantum coherence, nonlinear fluid dynamics, and biomolecular π-fields within biological and nanoscale systems requires a unified mathematical description. Classical hydrodynamics alone fails to capture coherence propagation, while the linear Schrödinger equation cannot model dissipation, viscosity, or turbulence. Here, I introduce the Schrödinger–Navier–Stokes–π Unified Computational Framework, a novel equation set combining: the quantum phase field of the Schrödinger equation, the viscous and nonlinear transport of Navier–Stokes dynamics, the curvature-driven quantum π-potential, and a hybrid Hamiltonian–dissipative evolution capturing both coherence and irreversible flow. This framework predicts quantum-fluid transitions, π-induced tunneling acceleration, coherence-enhanced mixing, nanoscale turbulence, and dynamic switching between classical and quantum transport regimes. It provides a general model applicable to biology, nanotechnology, photonics, and quantum materials. Keywords : Quantum Hydrodynamics Navier–Stokes Schrödinger Equation Unified Computational Framework π-field Dynamics Quantum Fluid Mechanics Hybrid Quantum-Classical Transport Coherence-Based Models Quantum Biology Nanoscale Transport Bio-Quantum Modeling Quantum Potential Quantum Fluid Engineering Biophysical Modeling Nonlinear Dynamics Computational Physics High-Order Numerical Methods Dissipative Quantum Systems Biomolecular Channels Quantum-Inspired Computing