Classical Fluid Research Articles

Hydrodynamic and ballistic AC transport in two-dimensional Fermi liquids

Electron transport in clean 2D systems with weak electron-phonon (e-ph) coupling can transition from an Ohmic to a ballistic or a hydrodynamic regime. The ballistic regime occurs when electron-electron (e-e) scattering is weak whereas the hydrodynamic regime arises when this scattering is strong. Despite this difference, we find that vortices and a negative nonlocal resistance believed to be quintessentially hydrodynamic are equally characteristic of the ballistic regime. These non-Ohmic regimes cannot be distinguished in DC transport without changing experimental conditions. Further, as our kinetic calculations show, the hydrodynamic regime in DC transport is highly fragile and is wiped out by even sparse disorder and e-ph scattering. We show that microwave-frequency AC sources by contrast readily excite hydrodynamic modes with current vortices that are robust to disorder and e-ph scattering. Indeed, current reversals in the non-Ohmic regimes occur via repeated vortex generation and mergers through reconnections, as in classical 2D fluids. Crucially, AC sources give rise to strong correlations across the entire device that unambiguously distinguish all regimes. These correlations in the form of nonlocal current-voltage and voltage-voltage phases directly check for the presence of a nonlocal current-voltage relation signifying the onset of non-Ohmic behavior as well as also for the dominance of bulk interactions, needed to confirm the presence of a hydrodynamic regime. We use these probes to demarcate all regimes in an experimentally realizable graphene device and find that the ballistic regime has a much larger extent in parameter space than the hydrodynamic regime.

Two-fluid plasma model for radial Langmuir probes as a converging nozzle with sonic choked flow, and sonic passage to supersonic flow

Using the Lambert function, Guittienne et al. [Phys. Plasmas 25, 093519 (2018)] derived two-fluid solutions for radial Langmuir probes in collisionless and isothermal plasma. In this Brief Communication, we point out the close analogy with classical compressible fluid dynamics, where the simultaneous flows of the ion and electron fluids experience opposite electrostatic body forces in the inward radial flow of the plasma, which behaves as a converging nozzle. Hence, the assumed boundary condition of sonic flow of the repelled species at the probe is explained as choked flow. The sonic passage from subsonic to supersonic flow of the attracted species at the sonic radius is also interpreted using classical fluid dynamics. Moreover, the Lambert function can provide a general solution for one-dimensional, isothermal compressible fluids, with several applications.

Effects of quantum correction on the density profile, adsorption, and phase behavior of confined hydrogen and deuterium fluids in small systems: A DFT study

The density functional theory based on the effective Wigner-Kirkwood potential is employed to investigate the effects of quantum correction on the structure, adsorption, and phase behavior of hydrogen and deuterium fluids confined in nano-spherical pores. Results show that quantum correction in the classical method for hydrogen and deuterium fluids influences all their thermodynamic properties when confined in nano-pores. Moreover, they indicate that the repulsion between quantum-mechanical molecules and intermolecular energy effects play important roles in the structure and thermodynamic properties of confined fluids before they come into contact at low temperatures or densities. The wall pressure of the fluid at the wall of the pore increases but the height of the profile in the center of the pore decreases with increased quantum effect, or the de Boer's quantumness parameter Λ. At low temperatures and low density limits, the adsorption and structure of the fluid in the pore are determined by the intermolecular energy effect; this will be reversed for high density limits in which the entropy effect is the dominant factor. It is shown in this study that, at low bulk density limits, the rising trend of adsorption takes a more steep slope in the classical fluid than in the deuterium one and that it is steeper in the deuterium fluid than it is in the hydrogen one; this behavior is reversed for high bulk density limits. Finally, it is shown that the critical temperature decreases with increasing Λ. Also, the locus of the phase transition is shown to shift toward higher values of bulk density as a result of increasing values of quantum effect, Λ.

Theory of thermophoresis of a spherical particle embedded in a micropolar fluid

Analytical investigations are considered for the related problems of thermophoresis, photophoresis and thermophoresis with chemical reactions of a spherical aerosol particle embedded in a gaseous micropolar medium. The Knudsen number is assumed to be in the slip-flow regime. Viscous and gyrational dissipations are ignored in the energy equation. At the surface of the particle, a thermal stress slip and the creep slip are considered in the analysis of motion. Expressions for thermophoretic and photophoretic velocities and forces are obtained as functions of the micropolarity parameter characterizing the micropolar medium and the thermal properties of the medium and particle. Results of the effect of the chemical reaction within the particle on its thermophoresis velocity are presented for various distributions of the composition-dependent factor or heat generation parameter of the chemical reaction as well as the relative thermal and surface properties of the aerosol spherical particle. The corresponding limiting cases of the classical viscous fluids are obtained.

Equation of State of Detonation Products Based on Exponential‐6 Potential Model and Analytical Representation of the Excess Helmholtz Free Energy

AbstractOne of the most important tasks associated with theoretical prediction of performance of explosives by thermochemical calculation is accurate description of the equation of state (EOS) of detonation products. Widely used empirical equations of state, such is Becker‐Kistiakowsky‐Wilson (BKW EOS), have limited accuracy when it comes to energetic materials with unusual elemental composition. Replacing empirical EOS with more sophisticated and theoretically based EOS, the accuracy of thermochemical calculations can be increase. We present an equation of state of detonation products based on the Buckingham α‐exponential‐6 equation (Exp‐6) and an analytical representation of the excess thermodynamic functions for classical fluid mixture. The excess Helmholtz free energy, as a function of reduced density, temperature, and stiffness parameter, is calculated applying KLRR perturbation statistical mechanical theory, and so obtained data are approximated by multinomial coefficients. The interpolation domain is defined so to describe accurately the range of pressures, densities, and temperatures relevant for detonation studies. The equation is incorporated in EXPLO5 thermochemical code. Good agreement with Monte‐Carlo calculations is obtained, as well as between calculated and experimental detonation parameters for a series of explosives.

Prediction of Broadband Noise from Symmetric and Cambered Airfoils

Aerodynamic sound generation and self-noise mechanisms from lifting surfaces such as airfoil involve the fields of classical acoustics and fluid mechanics. In this paper, trailing edge noise production is evaluated using empirical model for NACA 0012 and NACA 6320 airfoils. The sound pressure levels from trailing edge surface are calculated for different flow configurations. The growth of boundary layer thickness and displacement thickness, for different chord lengths and Mach numbers with varying angles of attack, is illustrated for NACA 0012. The sound pressure levels were computed numerically between 00 to 60 angles of attack and at constant chord length of 1.2m using Brookes Pope Marcolini method. The results showed a change of ~2-5dB in peak amplitude for mid frequencies region of spectrum. The effects of varying chord length and Mach number on sound pressure levels are illustrated for both airfoils. The relative velocity field for airfoils was computed using the boundary element method. The combined effect of thickness and camber on sound power level is demonstrated at a 40 angle of attack and for a Mach number of 0.191. Validation of sound pressure levels is done based on the results obtained for NACA0012 for similar flow conditions.

Three dimensional Darcy-Forchheimer radiated flow of single and multiwall carbon nanotubes over a rotating stretchable disk with convective heat generation and absorption

Several important characteristics of CNTs (carbon nanotubes) like excellent electrical conductivities, extraordinary thermal conductivities, good mechanical and chemical stability and extremely light mass make CNTs a model material which is utilized in various electro-chemical mechanisms. Keeping the said significance of carbon nanotubes in notice our main aim in this analysis is to establish the import relation of Darcy Forchheimer in three dimensional (3D) flows of classical fluid (water) based CNTs in presence of slip velocity due to rotating stretchable disk. Analysis of nanoparticles like single and multi-walled (SWCNTs, MWCNTs) carbon nanotubes are developed and equated. Present investigation further comprises non-linear thermal radiation, heat generation/absorption as well as convective thermal boundary condition aspects. The pertinent non-linear ordinary differential equations after utilizing the appropriate variables are calculated. HAM (Homotopy Analysis Method) technique is utilized for the analysis of velocities and thermal distribution. Various graphs have been plotted in order to examine that by what means the velocities and thermal field are altered by different model quantities. The main features of flow like the CFReα, CGReα (skin friction), Nu(Reα)−1/2 (heat transfer rate) have been intended numerically and graphically.

Confinement-induced modulation of elastic properties of nano-confined fluids in slit pore**Project supported by the National Natural Science Foundation of China (Grant No. 21503077), the Fundamental Research Fund for the Central Universities of China (Grant No. 2016MS156), and the Research Project from Hebei Education Department, China (Grant No. QN2018119).

Based on statistical mechanics for classical fluids, the general expressions for the elastic moduli of inhomogeneous fluid are derived and expressed as functionals of density functions. Within the framework of classical density functional theory, the bulk modulus of confined argon in slit pore is calculated under different conditions. The effects of vapor pressure, temperature, and pore width on modulus are calculated and investigated. Obvious confinement-induced effect has been observed in the confined argon. In addition, the solvation pressure dependence of the bulk modulus is also investigated, and the results suggest that the Tait–Murnaghan equation is still valid for the confined fluids.

Dynamical similarity and instabilities in high-Stokes-number oscillatory flows of superfluid helium

We present a unified analysis of the drag forces acting on oscillating bodies submerged in superfluid helium such as a vibrating wire resonator, tuning forks, a double-paddle oscillator, and a torsionally oscillating disk. We find that for high-Stokes-number oscillatory flows, the drag force originating from the normal component of superfluid helium exhibits a clearly defined universal scaling. Following classical fluid dynamics, we derive the universal scaling law and define relevant dimensionless parameters such as the Donnelly number. We verify this scaling experimentally using all of our oscillators in superfluid $^{4}\mathrm{He}$ and validate the results by direct comparison with classical fluids. We use this approach to illustrate the transition from laminar to turbulent drag regime in superfluid oscillatory flows and compare the critical velocities associated to the production of quantized vortices in the superfluid component with the critical velocities for the classical instabilities occurring in the normal component. We show that depending on the temperature and geometry of the flow, either type of instability may occur first and we demonstrate their crossover due to the temperature dependence of the viscosity of the normal fluid. Our results have direct bearing on present investigations of superfluids using nanomechanical devices [Bradley et al., Sci. Rep. 7, 4876 (2017)].

Density and time scaling effects on the velocity autocorrelation function of quantum and classical dense fluid para-hydrogen.

We report the results of a ring polymer molecular dynamics study of the Kubo velocity autocorrelation function of a quantum fluid as para-hydrogen aimed at the comparison with its classical counterpart. Quite different density conditions were considered for both the classical and quantum cases, in order to compare the two systems before and after the dynamical crossover typically undergone by the velocity autocorrelation function (VAF) of fluids at densities around the triple point, where the shape of the function changes from a monotonic to an oscillatory behavior with a negative minimum. A detailed study of the phase diagram of classical para-hydrogen was necessary for a reasonable choice of the classical states to be taken into consideration, in the spirit of the classical principle of corresponding states. The shape of the quantum and classical VAF was thoroughly analyzed, exhibiting at all studied densities clear differences that might be taken as evidence of quantum effects. We show that these differences are substantially reduced by applying a state-dependent time scaling with respect to a reference time identified with the inverse of the collision rate. An even better coincidence in shape is found by comparing the two systems at slightly non-corresponding reduced densities, suggesting that the quantum system behaves almost like the classical one, but at systematically less dense reduced states of the latter. We also find an unexpected and quite interesting density trend of the collision rate of both classical and quantum para-hydrogen, which accounts for the effectiveness of the scaling throughout the explored density range. The mean kinetic energy and the diffusion coefficients are also discussed in some detail.

Experimental investigation of the flow characteristics within a vortex tube with different configurations

The energy separation in a vortex tube is a combined result of different factors and its explanation remains debatable. As a classical fluid mechanics phenomenon, understanding of the complex helical flow mechanism within a vortex tube is a necessary foundation. The small scale of an industrial vortex tube and the extremely complex flow conditions are the two main challenges in obtaining the internal flow properties. This paper reports the results of an experimental investigation on the flow behaviour within a confined cylindrical system having different configurations corresponding to the actual flow field in a vortex tube at different conditions. Transparent devices were used to enable flow visualisation and Particle Image Velocimetry (PIV) measurement. The results of the flow visualisation and PIV experiments show that a precessing vortex core is significant only in a specific range of swirling strength. A good agreement between the observed flow characteristics and previously published results was observed.

Dropping slender-body theory into the mud

The equations describing classical viscous fluid flow are notoriously challenging to solve, even approximately, when the flow is host to one or many immersed bodies. When an immersed body is slender, the smallness of its aspect ratio can sometimes be used as a basis for a ‘slender-body theory’ describing its interaction with the surrounding environment. If the fluid is complex, however, such theories are generally invalid and efforts to understand the dynamics of immersed bodies are almost entirely numerical in nature. In a valiant effort, Hewitt & Balmforth (J. Fluid Mech., vol. 856, 2018, pp. 870–897) have unearthed a theory to describe the motion of slender bodies in a viscoplastic fluid, ‘fluids’ such as mud or toothpaste which can be coaxed to flow, but only with a sufficiently large amount of forcing. Mathematical theories for some tremendously complicated physical systems may now be within reach.

Decay of the melt stream during dispersion in granulation devices

The aim of the article is a theoretical description and experimental study of the melt jet expiration process from a perforated shell of a vibrating granulator. Mathematical modeling of hydrodynamic flows was carried out based on the points of classical fluid and gas mechanics and technical hydromechanics. Reliability of the obtained experimental results is based on the application of time-tested in practice methods. Hydrodynamic properties of the liquid jet outflow were obtained. The presented mathematical model allows calculation of the radial component of the jet outflow velocity, as well as determination of the influences of physical and chemical properties of the liquid and the outflow hole diameter on the jet length and flow velocity along the axis to its disintegration into separated drops. The developed mathematical model extended with the theoretical description of the melt dispersion process from rotating perforated shells allowed us to improve design of the granulator to stabilize hydrodynamic parameters of the melt movement. The nitrogen fertilizers melt disperser was investigated regarding industrial-scale production and operating parameters of the process of jet decay into drops, drop size and monodispersity level were optimized.

Distinguished Limits of the Navier Slip Model for Moving Contact Lines in Stokes Flow

The no-slip boundary condition in classical fluid mechanics is violated at a moving contact line, and it leads to an infinite rate of energy dissipation when combined with hydrodynamic equations. To overcome this difficulty, the Navier slip condition associated with a small parameter $\lambda$, named the slip length, has been proposed as an alternative boundary condition. In a recent work of Ren, Trinh, and E [J. Fluid Mech., 772 (2015), pp. 107--126], the distinguished limits of a spreading droplet when $\lambda$ tends to zero were studied using a thin film equation with the Navier slip condition. In this paper, we extend this analysis to the more general situation where the flow is modeled by the Stokes equation. In particular, we consider two distinguished limits as the slip length $\lambda$ tends to zero: one where time is held constant $t=O(1)$, and the other where time goes to infinity at the rate $t=O(|\ln\lambda|)$. It is found that when time is held constant, the contact line dynamics converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. On the other hand, when time goes to infinity, significant contact line displacement occurs and the contact line slippage becomes a leading-order singular effect. In this latter case, we recover the earlier analysis, e.g., by Cox [J. Fluid Mech., 168 (1986), pp. 169--194], after rescaling time.

Analysis of dynamic characteristics of two-component granular mixture segregation in thin shear cell

Particle separation is important in industrial production. The granular powder exhibits the property that is like both fluid property and solid property, which makes it difficult to establish a mathematical model to reveal the particle motion mechanism. The fluid property of the granular powder can be partly explained by the classical fluid theory, but the solid property cannot be covered. Theories combining the fluid and solid properties are also used to explain the particle separation phenomenon. However, they are not in consensus about the granular theory to explain the particle separation mechanism. Friction dissipation, which represents the particle pairwise damp interactions, greatly influences the particle separation process. In order to understand the particle separation mechanism and the effect of friction coefficient on the particle motion, a three-dimensional discrete element model is used to simulate the separation of three-dimensional spherical binary particles in a cylindrical groove （shear cell）in this paper. Initially, the large particle is placed at the bottom and the other small particles pile into the groove. The shear flow of the particles is established by rotating the bottom plate of the groove. The large particles gradually jump to the top of the groove under the shearing action. The effect of particle friction coefficient on the separation is studied. Focusing on the characteristics of kinematics and dynamics in the jumping process of large particle, the influence of the friction coefficient on the trajectory, velocity and acceleration of particle are quantitatively analyzed. The conclusions are obtained as follows. 1) The process of large particle jumping can be divided into three stages: relaxation stage (the large particle stays at the bottom of the groove), the take-off stage (the large particle rises up), and the equilibrium stage(the large particle moves to the top and stays there). 2) The relaxation time decreases with friction coefficient increasing. 3) The equilibrium height of particles increases with friction coefficient increasing. 4) The amplitude of the force pulsation of the large particle increases with friction coefficient increasing. For the behavior analysis of the ascending motion of the large particle, we propose a neighborhood analysis method and define a floating factor. The ratio of the number of small particles arranged in the upper to that in the lower adjacent space of the large particle is defined as the buoyancy factor. It is found that the buoyancy factor drops sharply at the jumping point of the large particle, forming the opportunity for the large particle jumping. It is revealed that the take-off of the large particle is the result of both the high-frequency characteristics of force fluctuation and the steep drop of buoyancy factor. The rising motion of the big particle is determined by the force and the surrounding space.

Classical harmonic model for the behavior of pure fluids at the critical point.

We present a simple model to explain the limiting behavior of many thermodynamic properties of pure classical fluids as one approaches the critical point. The model consists of four main aspects: (i) a focus on the thermodynamic properties as described by Fluctuation Solution Theory (FST); (ii) the removal of all intramolecular energy contributions from the FST expressions; (iii) particle fluctuations (and not the energy fluctuations) dominate the thermodynamics close to the critical point; and (iv) the equipartition theorem applies to the resulting low frequency collective modes of the fluid at the critical point. Using this approach, we predict and compare the thermodynamic results for 121 pure fluids as provided by accurate equations of state. The model explains why some ratios of diverging thermodynamic properties remain finite at the critical point, and reliably predicts the value of these ratios for equations of state that mimic both mean field and Ising system types, suggesting universal behavior.

Nonequilibrium free-energy calculations of fluids using LAMMPS

We present a guide to compute the absolute free energies of classical fluids using nonequilibrium free-energy techniques within the LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) code. The main approach is based on the construction of a thermodynamic path connecting the fluid of interest to either atomic or molecular variants of the Uhlenbeck-Ford (UF) model as reference systems. We describe these reference systems in detail, discuss their implementation in the LAMMPS package and make available source code, scripts as well as auxiliary files. As an illustration we detail a number of distinct applications, involving systems characterized by fundamentally different interactions. In addition to two different atomic models (mW water and the MEAM-2NN CuZr liquid binary alloy), we consider three molecular models for water, two of them rigid (TIP4P and SPC/E) and one flexible (q-SPC/Fw). For the molecular models we develop UF-based reference systems for which the free energies are given by a sum of two contributions: an intermolecular part described by the known UF free energy and an intramolecular contribution that can be determined analytically. The tools described in this paper provide a platform on which fluid-phase free energies can be easily and efficiently computed using the LAMMPS code. In addition to being useful for the development of new models for liquid phases, the tools may also find applications in the construction of community databases containing thermodynamic properties of existing models.

A modified discrete-vortex method algorithm with shedding criterion for aerodynamic coefficients prediction at high angle of attack

• New algorithm of the LDVM with a gain of a factor 2 in computing time. • Transient pitching motion: the leading edge vortex formation is two-dimensional. • Merging of vortices: decrease of the computing cost for long time simulations. • Evidence of the development of the lift coefficient between 30° and 90 ∘ . Low-order methods require less computing power than classical computational fluid dynamics and can be implemented on a laptop computer, which is needed for engineering tasks. Discrete vortex methods are such low order methods that can describe the unsteady separated flow around an airfoil . After a presentation of the leading edge suction parameter discrete vortex method, a modified algorithm is proposed, in order to reduce the computing cost, and compared with the previous one. Several reference unsteady airfoil motions are discussed in terms of gain in the computation time with comparisons between the previous scheme and the present one. The accuracy of the new method is demonstrated through aerodynamic coefficients . The application of the present discrete vortex method to a transient pitching motion of an airfoil is also presented, in order to understand the leading edge vortex formation, and its implication in terms of lift and drag coefficients. The method is not limited to unsteady or transient motions but can also simulate the flow around a constant angle of attack airfoil. In that case, an original method of fast summation of the vortices located far away from the airfoil, allows a linear dependence of the computation time versus the number of vortices shed, which is a great improvement over the quadratic dependence observed in the classical discrete vortex methods. The development of the aerodynamic coefficients with angle of attack, from values ranging between −10° and 90°, is obtained for a purely two-dimensional flow. In particular, the shape of the lift coefficient of the airfoil in the fully detached flow region is established. Comparisons with relevant experimental or computational fluid dynamics data are discussed in order to grasp the influence of upstream turbulence level and three-dimensional effects in the measured data in the fully detached flow region.

Visualization of a Coflow Jet in Superfluid Helium

We present preliminary results of the visualization of a submerged coflow jet of liquid helium produced by a fountain pump. The jet propagating inside the bulk superfluid is visualized with particle tracking velocimetry using hydrogen particles. We compare the characteristics of the coflow jet with those measured in classical fluids such as helium gas or water. In contrast to the classical experiments, a temperature-dependent angle of the jet is observed, suggesting that the flow may not be described quasi-classically, despite the strong coupling between normal and superfluid components by mutual friction. We report on the statistics of the velocities inferred from the particle trajectories recorded by a high-speed camera at 1.68 and 1.95 K, for jet velocities ranging from 47 to 4500 mm/s.

Bridging the gap between correlation entropy functionals in the mean spherical and the hypernetted chain approximations: a field theoretic description

The correlation entropy as a functional of radial distribution function (or the total correlation function ) in classical fluids has been obtained from the second Legendre transform of the grand potential. We focus on the correlation entropy difference between the two typical functionals in the mean spherical approximation (MSA) and the hypernetted chain (HNC) approximation. While the entropy functional difference between these approximations is of a simple form, the diagrammatic approaches in the liquid state theory are quite different from each other. Here we clarified the gap between the MSA and HNC functionals by developing a field theoretic description of the correlation functional theory that combines the variational principle of lower bound free energy, the conventional saddle-point approximation of a reference system to be optimized based on the variational principle, and the hybrid treatment of the saddle-point approximation and the fugacity expansion for modifying the primary optimization. Our formulation demonstrates that the MSA functional is reproduced by the first maximization of the variational functional in the saddle-point approximation, and that the HNC functional is obtained from the improved maximization of the virial term due to the fugacity expansion around the MSA functional. The virial term leads to the modification of a reference system interacting via the direct correlation function, thereby creating the correlation entropy difference.