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Quantum hydrodynamics of the Bingham fluid and the CNT enhanced fluid flow

Rafaela Alvarenga, Felipe S. Escórcio, Alan Barros de Oliveira, Hermano Endlich Schneider Velten

Published Dec 16, 2025• 0 citations
Publisher

Abstract

Abstract The superflow of water at nanoscale remains an open problem. Classical hydro-dynamics predicts flow rates that are 2 to 5 orders of magnitude lower than thoseobserved experimentally. Therefore, it is natural to question the validity of suchmodels while exploring alternative approaches. In this work, we investigate thebehavior of viscoplastic fluids, particularly Bingham fluids, from the perspectiveof de Broglie–Bohm quantum mechanics, incorporating the effects of a quantumpotential (quantum hydrodynamics). We assume that the physics behind theseunexplained experimental results might be associated to introducing an effectiveanti-dissipative term induced by the quantum potential. The study combinesa theoretical approach with its numerical solutions, and comparative analysis,employing the Navier–Stokes equations and the Bingham model. Our results indi-cate that the inclusion of the quantum potential is directly associated with theincreased flow velocity, reinforcing that quantum effects may play a significantrole in the dynamics of fluids confined within nanotubes. Moreover, well-behavedsolutions for the system’s wavefunction were identified even after the inclusionthe of effective anti-dissipative term. This evidence provides a possible theoret-ical explanation for the superflux phenomenon, reinforcing the hypothesis thatquantum effects may underlie the discrepancies relative to classical hydrodynamic predictions.

DOI
10.21203/rs.3.rs-8222487/v1

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This paper is intended to provide the reader with a review of the authors’ latest results dealing with the modeling of quantum dissipation/diffusion effects at the level of Schrodinger systems, in connection with the corresponding phase space and fluid formulations of such kind of phenomena, especially in what concerns the role of the Fokker–Planck mechanism in the description of open quantum systems and the macroscopic dynamics associated with some viscous hydrodynamic models of Euler and Navier–Stokes type.

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Abstract The superflow of water at nanoscale remains an open problem. Classical hydro-dynamics predicts flow rates that are 2 to 5 orders of magnitude lower than thoseobserved experimentally. Therefore, it is natural to question the validity of suchmodels while exploring alternative approaches. In this work, we investigate thebehavior of viscoplastic fluids, particularly Bingham fluids, from the perspectiveof de Broglie–Bohm quantum mechanics, incorporating the effects of a quantumpotential (quantum hydrodynamics). We assume that the physics behind theseunexplained experimental results might be associated to introducing an effectiveanti-dissipative term induced by the quantum potential. The study combinesa theoretical approach with its numerical solutions, and comparative analysis,employing the Navier–Stokes equations and the Bingham model. Our results indi-cate that the inclusion of the quantum potential is directly associated with theincreased flow velocity, reinforcing that quantum effects may play a significantrole in the dynamics of fluids confined within nanotubes. Moreover, well-behavedsolutions for the system’s wavefunction were identified even after the inclusionthe of effective anti-dissipative term. This evidence provides a possible theoret-ical explanation for the superflux phenomenon, reinforcing the hypothesis thatquantum effects may underlie the discrepancies relative to classical hydrodynamic predictions.
Abstract The superflow of water at nanoscale remains an open problem. Classical hydrodynamics predicts flow rates that are 2 to 5 orders of magnitude lower than those observed experimentally. Therefore, it is natural to question the validity of such models while exploring alternative approaches. In this work, we investigate the behavior of viscoplastic fluids, particularly Bingham fluids, from the perspective of de Broglie–Bohm quantum mechanics, incorporating the effects of a quantum potential (quantum hydrodynamics). We phenomenologically assume that the physics behind these unexplained experimental results might be associated to an effective quantum potential which properties that can be associated to an effective anti-dissipative hydrodynamical term in the classical approach. We combine a theoretical approach with its numerical solutions to understand the interplay between the Navier–Stokes-Bingham hydrodynamical system and the quantum hydrodynamics. Our results indicate that the quantum potential can be directly associated with the increased flow velocity, reinforcing that quantum effects may play a significant role in the dynamics of fluids confined within nanotubes. Moreover, well-behaved solutions for the system’s wavefunction were identified even in the exotic case of anti-dissipative fluid flow. This evidence provides a possible theoretical explanation for the superflux phenomenon, reinforcing the hypothesis that quantum effects may underlie the discrepancies relative to classical hydrodynamic predictions.
We present a mapping between a Schrödinger equation with a shifted nonlinear potential and the Navier–Stokes equation. Following a generalization of the Madelung transformations, we show that the inclusion of the Bohm quantum potential plus the laplacian of the phase field in the nonlinear term leads to continuity and momentum equations for a dissipative incompressible Navier–Stokes fluid. An alternative solution, built using a complex quantum diffusion, is also discussed. The present models may capture dissipative effects in quantum fluids, such as Bose–Einstein condensates, as well as facilitate the formulation of quantum algorithms for classical dissipative fluids.
The aim of this paper is to derive the quantum hydrodynamic system associated with the most general class of nonlinear Schrödinger equations accounting for Fokker--Planck type diffusion of the probability density, called of Doebner--Goldin. This 'Doebner--Goldin hydrodynamic system' is shown to be reduced in most cases to a simpler one of quantum Euler type by means of the introduction of a nonlinear gauge transformation that changes the fluid mean velocity into a new effective velocity corrected by an osmotic contribution. Finally, we also discuss some particular situations of especial interest and compare the structure of the resulting fluid systems with that of the viscous quantum hydrodynamic and the quantum Navier--Stokes equations stemming from maximization of the quantum entropy for Wigner--BGK models.
Abstract By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where the effective gauge potential results from the introduction of weak contact interactions in an ultracold dilute Bose gas of optically-addressed two-level atoms. In the Cauchy equation of mechanical momentum transport of the superfluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the stress tensor features a canonical flow pressure term given by the inner-product of the gauge potential and the canonical current density. By numerical simulation, we illustrate an interesting effect of the nonlinear gauge potential on the groundstate wavefunction of a superfluid in the presence of a foreign impurity. We find that the groundstate adopts a non-trivial local phase, which is antisymmetric under reversal of the gauge potential. The phase profile leads to a canonical-flow or phase-flow dipole about the impurity, resulting in a skirting mechanical flow. As a result, the pressure becomes asymmetric about the object and the condensate undergoes a deformation. Graphical abstract
Recent experiments on walking droplets suggest an underlying connection between fluid dynamics and the quantum mechanics. Such experiments may be used to support the de Broglie-Bohm pilot-wave theory. In this paper we show that many quantum-like features of hydrodynamic excitations can be explained in the framework of the parametrically driven nonlinear Schrödinger equation (DNLSE). It is shown that the nonlinear Schrödinger equation (NLSE) describes the pseudoquantum features of a given pseudoparticle in the Sagdeev pseudopotential just as the ordinary linear Schrödinger equation (LSE) describes the quantum nature of real particles. The NLSE is shown to reduce to nonlinear Hamilton-Jacobi equation (NHJE) for the phase function of the complex wavefunction. The origin of the wave-particle dual-character in both LSE and NLSE is explained and basic similarities between the two models is highlighted. It is shown that many more pseudoquantum effects (yet to be explored) such as pseudoentanglement, self-interference effect and quantization of pseudoparticle energy are inherent features of classical fluids. Current investigation suggests that emergence of the quantum nature of fluids is independent from dynamics of the bouncing droplets or their interactions with the surface waves instantaneously created by them. In contrast to previous suggestions, it is remarked that intrinsically nonlinear collective excitations of a fluid can drive any particle even a pseudoparticle to behave quantum mechanically. These collective interactions in hydrodynamic model confirm the important nonlocal character (which is a key aspect of the de Broglie-Bohm pilot-wave theory) of all the pseudoquantum phenomena. The recent hydrodynamic quantum analog experiments and extensive new theoretical models may hopefully enable physicists to unlock the decade long hidden mystery of the quantum weirdness.
This paper is intended to provide the reader with a review of the authors’ latest results dealing with the modeling of quantum dissipation/diffusion effects at the level of Schrodinger systems, in connection with the corresponding phase space and fluid formulations of such kind of phenomena, especially in what concerns the role of the Fokker–Planck mechanism in the description of open quantum systems and the macroscopic dynamics associated with some viscous hydrodynamic models of Euler and Navier–Stokes type.