Hydrodynamic Analogy Research Articles

Introduction to focus issue on hydrodynamic quantum analogs.

Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269-292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is "path memory" [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech. 675, 433-463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting on the droplet depends explicitly on its past. The second is the resonance condition between droplet and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient is chaos, which in several systems is characterized by unpredictable switching between unstable periodic orbits. This focus issue is devoted to recent studies of and relating to pilot-wave hydrodynamics, a field that attempts to answer the following simple but provocative question: Might deterministic chaotic pilot-wave dynamics underlie quantum statistics?

Faraday-Talbot effect: Alternating phase and circular arrays.

A hydrodynamic analog to the optical Talbot effect may be realized on the surface of a vertically shaken fluid bath when a periodic array of pillars protrudes from the fluid surface. When the pillar spacing is twice or one and a half times the Faraday wavelength, we observe repeated images of the pillars projected in front of the array. Sloshing inter-pillar ridges act as sources of Faraday waves, giving rise to self-images. Here, we explore the emergence of Faraday-Talbot patterns when the sloshing ridges between pillars have alternating phases. We present a simple model of linear wave superposition and use it to calculate the expected self-image locations, comparing them to experimental observations. We explore how alternating phase sources affect the Faraday-Talbot patterns for linear and circular arrays of pillars, where curvature allows for magnification and demagnification of the self-imaging pattern. The use of an underlying wavefield is a subject of current interest in hydrodynamic quantum analog experiments, as it may provide a means to trap walking droplets.

Black Hole Quasibound States from a Draining Bathtub Vortex Flow.

Quasinormal modes are a set of damped resonances that describe how an excited open system is driven back to equilibrium. In gravitational physics these modes characterize the ringdown of a perturbed black hole, e.g., following a binary black hole merger. A careful analysis of the ringdown spectrum reveals the properties of the black hole, such as its angular momentum and mass. In more complex gravitational systems, the spectrum might depend on more parameters and hence allows us to search for new physics. We present a hydrodynamic analog of a rotating black hole that illustrates how the presence of extra structure affects the quasinormal mode spectrum. The analogy is obtained by considering wave scattering on a draining bathtub vortex flow. We show that due to vorticity of the background flow, the resulting field theory corresponds to a scalar field on an effective curved spacetime which acquires a local mass in the vortex core. The obtained quasinormal mode spectrum exhibits long-lived trapped modes, commonly known as quasibound states. Our findings can be tested in future experiments building upon recent successful implementations of analog rotating black holes.

Free Streamline Hydrodynamic Analogy for a Linear Elastic Antiplane Slot Problem with Perfectly Plastic Ligaments at Its Ends

It is shown that a two-dimensional analogy exists between an inviscid incompressible fluid flow around two finite-length flat plates (Riabouchinsky problem) and that of a straight slot cut in a linear elastic plate having rounded ends that are on the verge of yielding plastically. For the plates, a uniform flow at infinity is assumed in a direction perpendicular to the side-by-side parallel lines that constitute the two-dimensional geometry of the plates. Correspondingly, the slot has a boundary condition at infinity of a uniform shear traction in the antiplane direction. In the fluid flow problem, two free streamlines form at both ends of the flat plates, while in the slotted plate problem, the rounded ends of the traction-free slot surfaces constitute ligaments of perfectly plastic material. The specific shape of the rounded ends of the slot problem can be inferred by analogy with the fluid flow problem.

Quantized superfluid vortex dynamics on cylindrical surfaces and planar annuli

Superfluid vortex dynamics on an infinite cylinder differs significantly from that on a plane. The requirement that a condensate wave function be single valued upon once encircling the cylinder means that such a single vortex cannot remain stationary. Instead, it acquires one of a series of quantized translational velocities around the circumference, the simplest being $\pm \hbar/(2MR)$, with $M$ the mass of the superfluid particles and $R$ the radius of the cylinder. A generalization to a finite cylinder automatically includes these quantum-mechanical effects through the pairing of the single vortex and its image in either the top or bottom end of the surface. The dynamics of a single vortex on this surface provides a hydrodynamic analog of Laughlin pumping. The interaction energy for two vortices on an infinite cylinder is proportional to the classical stream function $\chi({\bf r}_{12})$, and it crosses over from logarithmic to linear when the intervortex separation ${\bf r}_{12}$ becomes larger than the cylinder radius. An Appendix summarizes the connection to an earlier study of Ho and Huang for one or more vortices on an infinite cylinder. A second Appendix reviews the topologically equivalent planar annulus, where such quantized vortex motion has no offset, but Laughlin pumping may be more accessible to experimental observation.

Hydrodynamic analog of particle trapping with the Talbot effect

A hydrodynamic analog of the Talbot or self-imaging effect is observed on the surface of a vertically shaken fluid bath when a periodic row of pillars protrudes from the surface. The self-images are used to trap bouncing and walking droplets.

Mathematical Models and Calculation of the Coefficients of Heat and Mass Transfer in the Packings of Mechanical-Draft Towers

Semiempirical expressions for calculating the average coefficients of heat and mass transfer in the blocks of film-type sprayers are considered. The equations of the Chilton–Colburn hydrodynamic analogy, Prandtl model, generalizations of the hydrodynamic analogy, as well as dimensionless expressions and experimental data of various authors have been used. It is shown that the best agreement with experiment is provided by equations obtained with the aid of the hydrodynamic analogy and Prandtl model.

External and internal problems of modeling the heat and mass transfer coefficients at particles motion in liquids

An approach to the determination of the heat and mass transfer coefficients from dispersed particles by the development of the hydrodynamic analogy is considered. The equations for computing the heat and mass transfer coefficients in continuous phase at a laminar regime of the flow around solid particles as well as the mass transfer coefficients in droplets are obtained. Comparisons with the experimental data of different authors are presented.

Modeling heterogeneous traffic flow: A pragmatic approach

Modeling dynamics of heterogeneous traffic flow is central to the control and operations of today’s increasingly complex transportation systems. We develop a macroscopic heterogeneous traffic flow model. This model considers interplay of multiple vehicle classes, each of which is assumed to possess homogeneous car-following behavior and vehicle attributes. We propose the concepts of road capacity split and perceived equivalent density for each class to model both lateral and longitudinal cross-class interactions across neighboring cells. Rather than leveraging hydrodynamic analogies, it establishes pragmatic cross-class interaction rules aspired by capacity allocation and approximate inter-cell fluxes. This model generalizes the classical Cell Transmission Model (CTM) to three types of traffic regimes in general, i.e. free flow, semi-congestion, and full congestion regimes. This model replicates prominent empirical characteristics exhibited by mixed vehicular flow, including formation and spatio-temporal propagation of shockwaves, vehicle overtaking, as well as oscillatory waves. Those features are validated against numerical experiments and the NGSIM I-80 data. Realistic class-specific travel times can be computed from this model efficiently, which demonstrates the feasibility of applying this multi-class model to large-scale real-world networks.

A thermodynamic cycle for the solar cell

A solar cell is a heat engine, but textbook treatments are not wholly satisfactory from a thermodynamic standpoint, since they present solar cells as directly converting the energy of light into electricity, and the current in the circuit as maintained by an electrostatic potential. We propose a thermodynamic cycle in which the gas of electrons in the p phase serves as the working substance. The interface between the p and n phases acts as a self-oscillating piston that modulates the absorption of heat from the photons so that it may perform a net positive work during a complete cycle of its motion, in accordance with the laws of thermodynamics. We draw a simple hydrodynamical analogy between this model and the “putt-putt” engine of toy boats, in which the interface between the water’s liquid and gas phases serves as the piston. We point out some testable consequences of this model.

Shock Waves in Dispersive Hydrodynamics with Nonconvex Dispersion

Dissipationless hydrodynamics regularized by dispersion describe a number of physical media including water waves, nonlinear optics, and Bose-Einstein condensates. As in the classical theory of hyperbolic equations where a non-convex flux leads to non-classical solution structures, a non-convex linear dispersion relation provides an intriguing dispersive hydrodynamic analogue. Here, the fifth order Korteweg-de Vries (KdV) equation, also known as the Kawahara equation, a classical model for shallow water waves, is shown to be a universal model of Eulerian hydrodynamics with higher order dispersive effects. Utilizing asymptotic methods and numerical computations, this work classifies the long-time behavior of solutions for step-like initial data. For convex dispersion, the result is a dispersive shock wave (DSW), qualitatively and quantitatively bearing close resemblance to the KdV DSW. For non-convex dispersion, three distinct dynamic regimes are observed. For small jumps, a perturbed KdV DSW with positive polarity and orientation is generated, accompanied by small amplitude radiation from an embedded solitary wave leading edge, termed a radiating DSW or RDSW. For moderate jumps, a crossover regime is observed with waves propagating forward and backward from the sharp transition region. For jumps exceeding a critical threshold, a new type of DSW is observed we term a translating DSW or TDSW. The TDSW consists of a traveling wave that connects a partial, non-monotonic, negative solitary wave at the trailing edge to an interior nonlinear periodic wave. Its speed, a generalized Rankine-Hugoniot jump condition, is determined by the far-field structure of the traveling wave. The TDSW is resolved at the leading edge by a harmonic wavepacket moving with the linear group velocity. The non-classical TDSW exhibits features common to both dissipative and dispersive shock waves.

Hydrodynamic Analogy to Special Relativity

By studying of a slender body moving in a fluid wave-medium, e.g., in air or in shallow water, it was found that the hydrodynamic momentum mass and the total energy of the fluid field can be expressed in forms of and E=mc2, where v is the body moving speed, c is the wave speed and is the hydrodynamic mass at the zero speed. Thus a hydrodynamic analogy to the relativistic particle motion in vacuum can be traced. The velocity dependence of mass and the mass-energy equivalence are universal for any wave medium, which should not be regarded as a consequence of relative Lorentz time-space, but one of the existence of wave in the medium. Its further inference leads to an even more significant physical picture. If the mass particle moves in an unbounded space at a supercritical speed, i.e. , waves are generated and radiated from it, like the Mach waves by the supersonic plane, and the particle itself experiences a resistance as reaction from the wave radiation. By an extension of this analogy, it can be interred from a hydrodynamic superconductive phenomenon that particles or waves can move possibly at a superluminal speed without experiencing any resistance through a tunnel (a bounded space) under certain conditions. Therefore the speed of light is not the limit of our physical world and superluminal phenomena are possible.

The model of heat and mass transfer in rough and irrigated ducts

To determine the coefficients of the heat and mass transfer in the ducts with rough and irrigated walls the development of the classical hydrodynamic analogy of the momentum, mass, and energy transfer has been continued. The conservation properties of the skin-friction laws with respect to various disturbances are used for this purpose, and the “effective velocity” is found the value of which enables us to correct the hydrodynamic analogy. The examples of computations of the Nusselt, Sherwood, and Stanton numbers are shown for rough ducts, cooler, and film flow in the counter-flow. A comparison with the results of other researchers is given.

English

The work analyzes the compatibility between the classical freedom, the local relativistic causality and the non-local behavior of quantum mechanics in the frame of the stochastic approach of the quantum hydrodynamic analogy (SQHA). The work describes the role of the quantum potential in generating the quantum non-local dynamics in a fluctuating environment. The analysis shows that it is possible to maintain the concept of classical freedom between far away weakly bounded systems (moderate non-locality) as well as to make compatible the uncertainty principle with the relativistic postulate of invariance of light speed. The work shows that the paradox of instantaneous quantum non local behavior at infinite distances of the standard formalism is an artifact due to the non-relativistic non-stochastic ambit of such theory where the light speed is infinite and the non-local interaction owns an infinite range of action. The work envisages that the SQHA can possibly lead to a fully theoretically self-standing quantum mechanics where the wave function collapse, during a measurement process, can be described by the theory itself without empirical postulates. Under this light the paper discusses the need of searching for (both local and non-local) hidden variables quantum mechanics as well as the need of superluminal communications in quantum experiments. The analysis shows that all these hypotheses are attempts of interpreting the outputs of quantum measurements that cannot be fully explained by the semi-empirical formalism of quantum mechanics, based on the statistical postulates of the measuring process as well as the existence of a classical observer. A two photon experiment is discussed to the light of the SQHA approach. Key words: Quantum non-locality, superluminal transmission of quantum information, classical freedom, local relativistic causality, Einstein, Podolsky, and Rosen (EPR) paradox, macroscopic quantum decoherence, Bell’s inequalities, quantum hydrodynamic analogy.

Implications of a deeper level explanation of the deBroglie–Bohm version of quantum mechanics

Elements of a "deeper level" explanation of the deBroglie-Bohm (dBB) version of quantum mechanics are presented. Our explanation is based on an analogy of quantum wave-particle duality with bouncing droplets in an oscillating medium, the latter being identified as the vacuum's zero-point field. A hydrodynamic analogy of a similar type has recently come under criticism by Richardson et al., because despite striking similarities at a phenomenological level the governing equations related to the force on the particle are evidently different for the hydrodynamic and the quantum descriptions, respectively. However, said differences are not relevant if a radically different use of said analogy is being made, thereby essentially referring to emergent processes in our model. If the latter are taken into account, one can show that the forces on the particles are identical in both the dBB and our model. In particular, this identity results from an exact matching of our emergent velocity field with the Bohmian "guiding equation". One thus arrives at an explanation involving a deeper, i.e. subquantum, level of the dBB version of quantum mechanics. We show in particular how the classically-local approach of the usual hydrodynamical modeling can be overcome and how, as a consequence, the configuration-space version of dBB theory for $N$ particles can be completely substituted by a "superclassical" emergent dynamics of $N$ particles in real 3-dimensional space.

Hydrodynamic‐Analogy‐Based Modeling Approach for Distillative Separation of Organic Systems with Elevated Viscosity

AbstractA modeling approach based on hydrodynamic analogies is applied to predict the influence of elevated viscosity on the separation efficiency of distillation processes in structured packed columns. An advantage of this approach compared to traditional models based on the film theory is that separation columns can be described without empirical mass transfer coefficient correlations. The separation efficiency was evaluated for two systems with distinct viscosities at different gas and liquid loads and various operating pressures. Simulated concentration profiles and height of a theoretical plate (HETP) values were compared with the data obtained from own experiments. The developed model is capable of correctly predicting the differences in the HETP values of the two evaluated systems.

The Sponge Resistor Model — A Hydrodynamic Analog to Illustrate Ohm's Law, the Resistor Equation R=ρℓ/A, and Resistors in Series and Parallel

Physics students encountering electric circuits for the first time often ask why adding more resistors to a circuit sometimes increases and sometimes decreases the resulting total resistance. It appears that these students have an inadequate understanding of current flow and resistance. Students who do not adopt a model of current, voltage, and resistance necessarily resort to memorizing formulas for calculating, e.g., the resistance of a resistor network. For these students, certain properties of electric circuits may remain mysterious or puzzling.

Hydrodynamic analogy approach for modelling reactive absorption

Abstract Design of separation units for reactive gas–liquid systems is usually based on conventional modelling methods employing the stage concept, either equilibrium or rate-based. The parameters of such models (e.g., mass transfer coefficients) are determined experimentally, and, most often, significant experimental effort is necessary when new processes are explored or new types of column internals are developed. In the context of a novel approach based on hydrodynamic analogies, rigorous partial differential transport equations are applied for modelling of non-reactive and reactive separation processes in columns filled with structured packings. The approach has already been verified for distillation and reactive desorption processes. In this work, its application to reactive absorption – also known as chemical absorption – is examined. Different types of reactions appearing in a wide range of absorption systems are considered. To account for absorption processes operated at high gas and liquid loads, an adjustment of the turbulent gas flow description is performed. Therefore, different test systems are used, namely physical absorption of ammonia into water, chemical absorption of sulphur dioxide and chemical absorption of carbon dioxide into sodium hydroxide solutions. The absorption of ammonia and sulphur dioxide took place under turbulent gas flow conditions, while in the absorption of carbon dioxide, the gas flow was laminar. Based on the carbon dioxide system, different types of corrugated sheet packings were compared. The hydrodynamic analogy model was successfully validated using available experimental data.

Are micro and macro arrows of time connected each other?

The connection between the micro and macro-arrow of time is discussed in the frame of the stochastic quantum hydrodynamic analogy (SQHA). The presence of fluctuations that in the case on non-linear interactions leads to the breaking of the quantum mechanics on large scale and then to the macroscopic irreversibility with a defined arrow of time, gives also rise to the time reversal breaking in the micro-scale quantum evolution (micro-arrow of time). The quantum irreversibility with time reversal asymmetry is briefly discussed.

Hydrodynamic analogies between the equations of classical hydrodynamics and electrodynamics in electrochemistry

It is shown that the use of equations of hydrodynamics of an incompressible and compressible fluid gives similar results for a number of experimental data from the field of classical electrodynamics used in electrochemistry. The analogue of electric current in wires is a stream that creates around itself a flow of a fluid. The analogue of electric field is the acceleration of a flow, whereas the analogue of magnetic induction is the frequency of a rotational motion of the fluid. Ampere’s law in hydrodynamics describes the interaction of flows with real bodies in terms of the Zhukovsky equation. The power laws in the fluid are similar, with some distinctions, to Maxwell equations. The expansion of the equations of conservation of momentum and mass in a series in perturbations leads to wave equations also similar to the Maxwell equations for the propagation of electromagnetic waves.