Low Velocity Limit Research Articles

Phase transitions in materials with thermal memory

A model for thermally induced phase transitions in materials with thermal memory is explored. From a general starting point of the thermodynamics of materials with memory, field equations with an order parameter, temperature and temperature gradient are derived. The heat flux is given by an integral over the history of the temperature gradient. Asymptotic analysis yields that to leading order, there is a relation connecting the discontinuity in the temperature field with the normal velocity of the transition zone. The temperature discontinuity is negligible for low transition zone velocity. The latent heat must depend on velocity and becomes negligible compared with the specific heat as the normal velocity approaches the speed of propagation of thermal disturbances. There is also a condition which is different from the Stefan condition connecting discontinuities in the temperature gradient with velocity, but reduces to this form, under certain approximations, in the low transition zone velocity limit. The circumstances where the transition zone velocity may exceed the speed of thermal disturbances are discussed.

Statistical features of traffic flow on urban freeways

Urban freeways play an important role in urban traffic networks. Compared with highway traffic, urban freeway traffic has different characteristics, such as denser on- and off-ramps, more complex road conditions, and lower velocity limits. Until now, however, there has been no comprehensive analysis of urban traffic flow. In this paper, through an analysis of the density dependence of velocity distribution, we investigate the fundamental velocity–density relationships of urban freeways, compare them with those of highway traffic, and explain them using existing traffic flow theories.

Threshold effect in the energy-loss straggling of protons channeled in Au⟨100⟩ at very low velocities

The energy-loss straggling of protons channeled in gold crystals in the ⟨100⟩ direction has been experimentally and theoretically studied at the low-velocity limit. The experimental results show a threshold effect in the straggling values, which is similar to that observed earlier in the energy loss. This effect is described by a theoretical model which explains the velocity dependence by considering the individual features of the contributions of free electrons in the conduction band and the nearly free $(5{d}^{10})$ electrons which generate a threshold effect at low energies due to their binding energy. The model predicts a deviation of proportionality with ion velocity of the energy-loss spread and yields a good description of the obtained experimental data.

Particles dispersed in a dilute gas: Limits of validity of the Langevin equation

The problem of the limits of validity of the Langevin equation is considered in detail in the case of (microscopic) test-particles in very dilute gases. It is shown that, in this case, the current Langevin equation follows from the Newton’s law in an exact way only in the Maxwell test-particle–gas-particle interaction model, and in an approximate way only in the Rayleigh-gas limit and in the low-velocity limit, while in any other interaction model, or limit, only a Langevin-like equation with speed-dependent friction coefficient and speed-dependent fluctuating force can be written. Such a circumstance, although probably limited to the particular physical situation considered in this paper, suggests that, in general, some preliminary, specific check of the validity of the Langevin equation should be performed before using the said equation to interpret laboratory experiments.

Computer simulation of model cohesive powders: Influence of assembling procedure and contact laws on low consolidation states

Molecular dynamics simulations are used to investigate the structure and mechanical properties of a simple two-dimensional model of a cohesive granular material. Intergranular forces involve elasticity, Coulomb friction, and a short-range attraction akin to the van der Waals force in powders. The effects of rolling resistance (RR) at intergranular contacts are also studied. The microstructure of the cohesive packing under low pressure is shown to depend sensitively on the assembling procedure which is applied to the initially isolated particles of a granular gas. While a direct compression produces a final equilibrated configuration with a similar density to that of cohesionless systems, the formation of large aggregates prior to the application of an external pressure results in much looser stable packings. A crucial state variable is the ratio P;{*}=PaF_{0} of applied pressure P , acting on grains of diameter a , to maximum tensile contact force F0 . At low P;{*} the force-carrying structure and force distribution are sensitive to the level of velocity fluctuations in the early stages of cluster aggregation. The coordination number of packings with RR approaches 2 in the limit of low initial velocities or large rolling friction. In general the force network is composed of hyperstatic clusters, typically comprising four to a few tens of grains, in which forces reach values of the order of F0 , joined by barely rigid arms, where contact forces are very small. Under growing P;{*} , it quickly rearranges into force chainlike patterns that are more familiar in dense systems. Density correlations are interpreted in terms of a fractal structure, up to a characteristic correlation length xi of the order of ten particle diameters for the studied solid fractions. The fractal dimension in systems with RR coincides, within measurement uncertainties, with the ballistic aggregation result, in spite of a possibly different connectivity, but is apparently higher without RR. Possible effects of micromechanical and assembling process parameters on mechanical strength of packings are evoked.

Hetero-/homogeneous combustion and stability maps in methane-fueled catalytic microreactors

The hetero-/homogeneous steady combustion and the stability limits of methane-fueled catalytic microreactors (Pt-coated) have been investigated numerically in a 1-mm-gap channel at pressures of 1 and 5 bar. Computations were carried out with a full-elliptic two-dimensional model for the gas- and solid-phases that included elementary heterogeneous and homogeneous chemical reaction schemes, heat conduction in the solid wall, surface radiation heat transfer, and external heat losses. Gas-phase chemistry extended the low-velocity stability limits due to the establishment of strong flames and to an even greater degree the high-velocity blowout limits due to the heat release originating primarily from the incomplete homogeneous oxidation of methane. When considering the same mass throughput, the stable combustion envelope at 5 bar was substantially wider than its 1 bar counterpart due to the increased reactivity of both catalytic and gaseous pathways at elevated pressures. Stable combustion could be sustained with solid thermal conductivities at least as low as 0.1 W/mK, while the stability limits reached their larger extent between 20 and 50 W/mK, a range that covers many practical metallic compounds. The stability limits of catalytic microreactors were wider than those reported for non-catalytic systems. Surface radiation heat transfer greatly impacted the microreactor energy balance and combustion stability. At conditions well-below the stability limits, surface radiation provided an efficient heat loss mechanism that moderated the surface temperatures, whereas close to the limits it could stabilize combustion by transferring heat from the hotter rear of the channel to the colder front. Investigation of smaller confinements has shown that gas-phase combustion could be sustained in catalytic microreactors with gaps as low as 0.3 mm.

Modulation of dust acoustic waves with a quantum correction

The one-dimensional quantum hydrodynamic model is considered in the limit of low phase velocity (compared to the Fermi thermal velocities) as well as the limit of low frequency (compared to the particle-neutral collision frequency) to study the amplitude modulation of dust-acoustic waves in a three-species quantum dusty plasma. Using the standard reductive perturbation technique, a nonlinear Schrödinger equation containing the quantum effects is derived. The quantum mechanical effects containing the quantum diffraction and quantum statistics on the modulational instability are studied both analytically and numerically. It is found that quantum effects are to suppress the instability.

Role of gas phase in the deposition dynamics of fine particles in trickle-bed reactors

Experiments were conducted to study the role of gas velocity in the capture of fine particles from non-aqueous suspensions circulated in co-current down-flow trickle flow reactors. The rate of filtration and pressure drop in the trickle bed were investigated using surfactant-stabilized kaolin-containing kerosene suspensions. It was determined that the filter coefficient was sensitive to liquid holdup and specific deposit. The initial collection efficiencies were compared with predictions based on existing theories. Agreement was generally not good with the exception for the limit of low superficial gas velocity. A general correlation establishing the relationship between the filtration rate and the liquid holdup in trickle beds was proposed to reconcile the experimental data with existing filtration theories.

Determination of an equivalent pore size from acoustic flow measurements

The hydraulic radius, rh, is defined as the ratio of a channel’s cross-sectional area to its perimeter. This parameter is important for specification of the performance of a porous medium that can be used as a regenerator in a Stirling engine or refrigerator. It is easy to calculate rh for pores of regular geometry, but difficult in more complex media. Two techniques which use oscillating flow to determine this parameter will be presented and compared. One technique extracts rh by finding the low velocity limit of the standard expression for viscous pressure drop in the Poiseuille flow regime. The other involves a plot of the nondimensional viscous flow resistance, Δpvis/Δxωρu, versus the reciprocal of the viscous penetration depth, 1/δν, in the laminar flow regime. When rh<δν, the flow behavior is frequency independent and the dynamics is characterized by rh only. When rh>δν, the flow resistance is frequency dependent and the dynamics is characterized by both rh and δν. It is possible to identify an effective hydraulic radius by equating it to the value of δν where that transition occurs. [Work supported by ONR.]

Canonical quantization of Galilean covariant field theories

The Galilean-invariant field theories are quantized by using the canonical method and the five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. This method is motivated by the fact that the extended Galilei group in 3 + 1 dimensions is a subgroup of the inhomogeneous Lorentz group in 4 + 1 dimensions. First, we consider complex scalar fields, where the Schrödinger field follows from a reduction of the Klein–Gordon equation in the extended space. The underlying discrete symmetries are discussed, and we calculate the scattering cross-sections for the Coulomb interaction and for the self-interacting term λΦ 4. Then, we turn to the Dirac equation, which, upon dimensional reduction, leads to the Lévy-Leblond equations. Like its relativistic analogue, the model allows for the existence of antiparticles. Scattering amplitudes and cross-sections are calculated for the Coulomb interaction, the electron–electron and the electron–positron scattering. These examples show that the so-called ‘non-relativistic’ approximations, obtained in low-velocity limits, must be treated with great care to be Galilei-invariant. The non-relativistic Proca field is discussed briefly.

PRESENT STATUS OF THE STUDY OF THE ANOMALOUS ACCELERATION OF THE PIONEER 10/11 SPACECRAFT

Results from an almost twenty years study of radiometric data from Pioneer 10/11, Galileo and Ulysses spacecraft indicate an anomalous time depending blueshift1, 2 which bares hardly prosaic explanations. Local manifestation of the expansion of the universe or new force terms are not favored either by the observational data. So, we explore the possibility that the reported anomaly, referred to as the "Pioneer anomaly", does not result from a real change in velocity. It turns out that the main Pioneer anomaly may be looked at as a new validation of general relativity (GR) in the weak field and low velocity limit on account of the "machian" behavior of quintessence like dark energy.

Time-dependent density-functional theory for the stopping power of an interacting electron gas for slow ions

Based on the time-dependent density-functional theory, we have derived a rigorous formula for the stopping power of an {\it interacting} electron gas for ions in the limit of low projectile velocities. If dynamical correlation between electrons is not taken into account, this formula recovers the corresponding stopping power of {\it noninteracting} electrons in an effective Kohn-Sham potential. The correlation effect, specifically the excitonic one in electron-hole pair excitations, however, is found to considerably enhance the stopping power for intermediately charged ions, bringing our theory into good agreement with experiment.

Nonlinear screening and stopping power in two-dimensional electron gases

We have used density-functional theory to study the nonlinear screening properties of a two-dimensional (2D) electron gas. In particular, we consider the screening of an external static point charge of magnitude $Z$ as a function of the distance of the charge from the plane of the gas. The self-consistent screening potentials are then used to determine the 2D stopping power in the low-velocity limit based on the momentum transfer cross section. Calculations as a function of $Z$ establish the limits of validity of linear and quadratic response theory calculations, and show that nonlinear screening theory already provides significant corrections in the case of protons. In contrast to the 3D situation, we find that the nonlinearly screened potential supports a bound state even in the high-density limit. This behavior is elucidated with the derivation of a high-density screening theorem which proves that the screening charge can be calculated perturbatively in the high-density limit for arbitrary dimensions. However, the theorem has particularly interesting implications in 2D where, contrary to expectations, we find that perturbation theory remains valid even when the perturbing potential supports bound states.

On the low-velocity limit of the Bohr stopping formula

The low-velocity limit of the classical Bohr stopping model is investigated by applying non-perturbative methods. For the repulsive Coulomb interaction between a heavy projectile and a harmonically bound electron, the stopping cross-section S+(v) is found to scale as v5/3 in the limit of v ↦0, where v is the projectile velocity. This scaling is obtained by establishing a corresponding scaling law for the energy transfer T+(v,b) in a single collision with an impact parameter b, namely, that T+/v1/3 is a function of a scaled variable b/v2/3. For the opposite case of the Coulomb attraction, direct numerical calculations reveal that the energy transfer T-(v,b) exhibits sharp resonances along the b axes when v becomes sufficiently small. The latter results in a characteristic non-regular behaviour of S-(v) near maximum. Suitable fitting formulae are proposed for the corresponding stopping numbers L±(v).

Hybrid continuum-atomistic simulation of singular corner flow

A hybrid numerical method is used to study cavity flow driven by a moving wall. Continuum equations with no-slip boundary conditions predict singular stresses at the corners between moving and static walls. Molecular dynamics simulations are used to resolve these singular regions, and the flow field in the remainder of the cavity is obtained from the Navier-Stokes (NS) equations. This hybrid solution agrees well with fully atomistic simulations on small systems, and allows calculations to be accelerated by orders of magnitude in larger systems. Fully continuum and hybrid solutions for the stress and velocity also agree over most of the cavity. Both yield a shear stress that scales as the inverse of the distance from the corner over almost two orders of magnitude. However, in the hybrid solution, this divergence is cut off within a distance S from the corners. In the limit of low wall velocities U, S is a few molecular diameters and corresponds to the length over which slip occurs. By comparing the hybrid solution to NS solutions, we show that the slip cannot be quantitatively described by the Navier slip condition. At higher U, non-Newtonian behavior near the corner causes S to rise linearly with U.

Wrapped D-branes as BPS monopoles: The moduli space perspective

We study the four dimensional effective action of a system of D6-branes wrapped on the $K3$ manifold times a torus, allowing the volume of the internal manifolds to remain dynamical. An unwrapped brane is at best a Dirac monopole of the dual RR sector field to which it couples. After wrapping, a brane is expected to behave as a BPS monopole, where the Higgs vacuum expectation value is set by the size of the $K3.$ We determine the moduli space of an arbitrary number of these wrapped branes by introducing a time dependent perturbation of the static solution and expanding the supergravity equations of motion to determine the dynamics of this perturbation, in the low velocity limit. The result is the hyper-K\"ahler generalization of the Euclidean Taub-NUT metric presented by Gibbons and Manton. We note that our results also pertain to the behavior of bound states of Kaluza-Klein monopoles and wrapped NS5-branes in the ${T}^{4}$ compactified heterotic string.

Eulerian Simulation Strategy for Scaling up a Bubble Column Slurry Reactor for Fischer−Tropsch Synthesis

CFD simulations were carried out in the Eulerian framework using both two-dimensional (2D) axisymmetric and transient three-dimensional (3D) strategies to describe the influence of column diameter on the hydrodynamics and dispersion characteristics of the bubble column slurry reactor for Fischer−Tropsch (FT) synthesis. Interactions between the bubbles and the slurry were taken into account by means of a momentum-exchange, or drag, coefficient; this coefficient was estimated from experimental data on the bubble swarm velocity in the limit of low superficial gas velocity U. The turbulence in the slurry phase was described using the k−ε model. For an FT slurry reactor operating at U = 0.15 m/s, simulations were also carried out for columns with diameters of 0.38, 1, 2, 4, 6, and 10 m using the 3D simulation strategy to determine gas hold-up; the liquid circulation velocity; and the axial dispersion coefficient of the liquid phase, Dax,L. The results demonstrate the strong increase of liquid circulation and Dax,L with increasing column diameter. We conclude that 3D Eulerian simulations can provide a powerful tool for hydrodynamic scale-up of bubble columns, obviating the need for large-scale experiments.

Energy deposition of heavy ions in the regime of strong beam-plasma correlations.

The energy loss of highly charged ions in dense plasmas is investigated. The applied model includes strong beam-plasma correlation via a quantum T-matrix treatment of the cross sections. Dynamic screening effects are modeled by using a Debye-like potential with a velocity dependent screening length that guarantees the known low and high beam velocity limits. It is shown that this phenomenological model is in good agreement with simulation data up to very high beam-plasma coupling. An analysis of the stopping process shows considerably longer ranges and a less localized energy deposition if strong coupling is treated properly.

Massive transformation and absolute stability

Under carefully chosen conditions, solidification theory may be applied to solid-state transformations, and this has been done here for composition-invariant diffusion transformations. The predictions of the modeling are compared with isovelocity experiments in two iron systems, Fe-7.29 wt pct Cr and Fe-3.1 wt pct Ni. The ferrite to austenite phase transformation is used to demonstrate that stabilization of a planar transformation front at absolute stability is the natural lower velocity limit for a composition-invariant (massive) transformation. The results of the model, which includes nonequilibrium effects, clearly show that steady-state plane-front growth leading to composition invariance can be obtained at various temperatures depending on the growth velocity. In the lower velocity range, at the limit of absolute stability (of the order of 10 µm/s in the systems studied), the transformation interface moves under conditions of local equilibrium, and the temperature corresponds to the lower solvus temperature. At higher velocity (of the order of the interface diffusion rate, which in these systems is of the order of cm/s), the transformation is predicted to proceed at temperatures close to T0. At even higher rates, atom attachment kinetic undercooling will decrease the transformation temperature with respect to T0. In some cases, this temperature might even drop below the lower solvus.

Mechanical Entropy and Its Implications

It is shown that the classical laws of thermodynamics require that mechanical systems must exhibit energy that becomes unavailable to do useful work. In thermodynamics, this type of energy is called entropy. It is further shown that these laws require two metrical manifolds, equations of motion, field equations, and Weyl's quantum principles. Weyl's quantum principle requires quantization of the electrostatic potential of a particle and that this potential be non-singular. The interactions of particles through these non-singular electrostatic potentials are analyzed in the low velocity limit and in the relativistic limit. It is shown that writing the two particle interactions for unlike particles allows an examination in two limiting cases: large and small separations. These limits are shown to have the limiting motions of: all motions are ABOUT the center of mass or all motion is OF the center of mass. The first limit leads to the standard Dirac equation. The second limit is shown to have equations of which the electroweak theory is a subset. An extension of the gauge principle into a five-dimensional manifold, then restricting the generality of the five-dimensional manifold by using the conservation principle, shows that the four-dimensional hypersurface that is embedded within the 5-D manifold is required to obey Einstein's field equations. The 5-D gravitational quantum equations of the solar system are presented.