Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

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

Ndenga, Barack

Published Dec 5, 2025• 0 citations
Publisher

Abstract

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

DOI
10.60763/africarxiv/10342

Related papers

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

Ndenga, Barack

This article introduces a unified model combining the Schrödinger equation, Navier–Stokes dynamics, and π-field coherence into a single computational framework. The approach describes systems where quantum coherence, viscosity, dissipation, and structural curvature coexist at the nanoscale. The framework predicts hybrid transport regimes, coherence-modulated flow, and π-induced quantum potentials. High-order numerical simulations (4th-order finite differences + RK4) demonstrate transitions between quantum-dominant, hybrid, and classical behaviours. To my knowledge, this is the first formulation unifying Schrödinger dynamics, viscous fluid flow, and π-field curvature into one model.

•Dec 5, 2025•0 citations

Schrödinger–Navier–Stokes–Quantum-π: A Unified Model and Hybrid Numerical Method for Quantum Fluids with π-Phase Structure

Ndenga, Barack

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.

•Nov 30, 2025•0 citations

Schrödinger–Navier–Stokes–Quantum-π: A Unified Model and Hybrid Numerical Method for Quantum Fluids with π-Phase Structure

Ndenga, Barack

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.

•Nov 30, 2025•0 citations

Quantum computing of fluid dynamics using the hydrodynamic Schrödinger equation

Zhaoyuan Meng, Yue Yang

Simulating fluid dynamics on a quantum computer is intrinsically difficult due to the nonlinear and non-Hamiltonian nature of the Navier-Stokes equation (NSE). We propose a framework for quantum computing of fluid dynamics based on the hydrodynamic Schr\"odinger equation (HSE), which can be promising in simulating three-dimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible or incompressible flows with finite vorticity and dissipation. Since the HSE is expressed as a unitary operator on a two-component wave function, it is more suitable than the NSE for quantum computing. The flow governed by the HSE can resemble a turbulent flow consisting of tangled vortex tubes with the five-thirds scaling of energy spectrum. We develop a prediction-correction quantum algorithm to solve the HSE. This algorithm is implemented for simple flows on the quantum simulator Qiskit with partial exponential speedup.

arXiv:2302.09741•Physical Review Research•Sep 12, 2023•76 citations

Quantum-Fluid Interpretation of Enzymatic Tunnels and Energy Transport

Ndenga, Barack

This work introduces a novel theoretical framework in which enzymatic tunnels are reinterpreted as quantum-fluid coherence channels rather than simple geometric pathways. Guided by π-field hydrodynamics, these tunnels support coherent, directed energy transport at the nanoscale, providing a unified explanation for ultrafast proton movement, long-range communication between protein sites, low-dissipation energy flow, and the remarkable sensitivity of catalytic efficiency to single-point mutations. In this article, I develop a quantum-hydrodynamic model incorporating coherence density, phase-field dynamics, and the π-induced quantum potential 𝑄𝜋. The resulting framework predicts: ballistic-like transport within enzymatic tunnels, π-gradient–driven directionality, coherence-supported reduction of effective energy barriers, mutation-induced π-defects disrupting transport, and the emergence of nanoscale quantum-fluid vortices. To the best of my knowledge, this is the first scientific work to formally characterize enzymatic tunnels as quantum-fluid conduits governed by π-field dynamics. This establishes a new research direction at the intersection of quantum biology, enzymology, and nanophysical modeling, which I refer to as Quantum Fluid Enzymology (QFE). The deposition includes conceptual figures, theoretical equations, and a computational π-fluid simulation approach for tunnel transport

•Dec 4, 2025•0 citations

Schrödinger-Navier-Stokes Equation for the Quantum Simulation of Navier-Stokes Flows

Luca Cappelli, Sauro Succi, Monica Lacatus +2 more

The search for quantum-like wave formulations of the Navier-Stokes (Schrödinger-Navier-Stokes, SNS for short) equations describing classical dissipative fluids has met with increasing attention in the recent years, due to the large portfolio of potential applications in science and engineering. A SNS formulation of classical fluids was first presented in a largely un-noticed paper by Dietrich and Vautherin back in 1985(Journal de Physique). In this paper, we revisit this specific SNS approach and assess its viability for quantum implementations based on Carleman embedding/linearization techniques. Specifically, we i) Clarify in full mathematical detail why the SNS dissipator presents a steep challenge for quantum computers and propose a way out strategy based on the Hamilton-Jacobi (HJ) formulation of fluid dynamics; ii) Develop a corresponding quantum algorithm using a new technique based on a tensor-network representation of Carleman embedding of the HJ equations (CHJ) which permits substantial memory savings; iii) Emulate the CHJ quantum algorithm on a classical computer and analyse its convergence and accuracy for the specific case of Kolmogorov-like flows at moderate Reynolds numbers. To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity.

quant-pharXiv:2604.11113•Quantum Science and Technology•Apr 14, 2026•1 citation
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
This article introduces a unified model combining the Schrödinger equation, Navier–Stokes dynamics, and π-field coherence into a single computational framework. The approach describes systems where quantum coherence, viscosity, dissipation, and structural curvature coexist at the nanoscale. The framework predicts hybrid transport regimes, coherence-modulated flow, and π-induced quantum potentials. High-order numerical simulations (4th-order finite differences + RK4) demonstrate transitions between quantum-dominant, hybrid, and classical behaviours. To my knowledge, this is the first formulation unifying Schrödinger dynamics, viscous fluid flow, and π-field curvature into one model.
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.
Simulating fluid dynamics on a quantum computer is intrinsically difficult due to the nonlinear and non-Hamiltonian nature of the Navier-Stokes equation (NSE). We propose a framework for quantum computing of fluid dynamics based on the hydrodynamic Schr\"odinger equation (HSE), which can be promising in simulating three-dimensional turbulent flows in various engineering applications. The HSE is derived by generalizing the Madelung transform to compressible or incompressible flows with finite vorticity and dissipation. Since the HSE is expressed as a unitary operator on a two-component wave function, it is more suitable than the NSE for quantum computing. The flow governed by the HSE can resemble a turbulent flow consisting of tangled vortex tubes with the five-thirds scaling of energy spectrum. We develop a prediction-correction quantum algorithm to solve the HSE. This algorithm is implemented for simple flows on the quantum simulator Qiskit with partial exponential speedup.
This work introduces a novel theoretical framework in which enzymatic tunnels are reinterpreted as quantum-fluid coherence channels rather than simple geometric pathways. Guided by π-field hydrodynamics, these tunnels support coherent, directed energy transport at the nanoscale, providing a unified explanation for ultrafast proton movement, long-range communication between protein sites, low-dissipation energy flow, and the remarkable sensitivity of catalytic efficiency to single-point mutations. In this article, I develop a quantum-hydrodynamic model incorporating coherence density, phase-field dynamics, and the π-induced quantum potential 𝑄𝜋. The resulting framework predicts: ballistic-like transport within enzymatic tunnels, π-gradient–driven directionality, coherence-supported reduction of effective energy barriers, mutation-induced π-defects disrupting transport, and the emergence of nanoscale quantum-fluid vortices. To the best of my knowledge, this is the first scientific work to formally characterize enzymatic tunnels as quantum-fluid conduits governed by π-field dynamics. This establishes a new research direction at the intersection of quantum biology, enzymology, and nanophysical modeling, which I refer to as Quantum Fluid Enzymology (QFE). The deposition includes conceptual figures, theoretical equations, and a computational π-fluid simulation approach for tunnel transport
The search for quantum-like wave formulations of the Navier-Stokes (Schrödinger-Navier-Stokes, SNS for short) equations describing classical dissipative fluids has met with increasing attention in the recent years, due to the large portfolio of potential applications in science and engineering. A SNS formulation of classical fluids was first presented in a largely un-noticed paper by Dietrich and Vautherin back in 1985(Journal de Physique). In this paper, we revisit this specific SNS approach and assess its viability for quantum implementations based on Carleman embedding/linearization techniques. Specifically, we i) Clarify in full mathematical detail why the SNS dissipator presents a steep challenge for quantum computers and propose a way out strategy based on the Hamilton-Jacobi (HJ) formulation of fluid dynamics; ii) Develop a corresponding quantum algorithm using a new technique based on a tensor-network representation of Carleman embedding of the HJ equations (CHJ) which permits substantial memory savings; iii) Emulate the CHJ quantum algorithm on a classical computer and analyse its convergence and accuracy for the specific case of Kolmogorov-like flows at moderate Reynolds numbers. To the best of our knowledge, this is the first quantum algorithm based on a quantum-like wave formulation of the genuine Navier-Stokes equations, including pressure, dissipation and vorticity.