A Simple Method to Mix Granular Materials

The effects of subgrid scale (SGS) motions on the dispersion of heavy particles raise a challenge to the large-eddy method of simulation (LES). As a necessary first step, we propose the use of a stochastic differential equation (SDE) to represent the SGS contributions to the relative dispersions of heavy particles in LES of isotropic turbulence. The main difficulty is in closing the SGS-SDE model whilst accounting for the effects of particle inertia, filter width and gravity. The physics of the interaction between heavy particles and SGS turbulence is explored using the filtered direct numerical simulation method. It is found in the present work that (i) the ratio of the SGS Lagrangian and Eulerian timescales is different from that of the full-scale Lagrangian and Eulerian timescales. The ratios are also dependent on filter widths. (ii) In the absence of gravity, the SGS timescale seen by heavy particles non-monotonically changes with particle Stokes number and has a maximum at particle Stokes number (St = τp/δTE) near 0.5. (iii) In the presence of gravity, a similarity law exists between the SGS Lagrangian correlation function seen by a heavy particle within a time-delay τ and the SGS spatial correlation function with the displacement 〈w〉τ, where 〈w〉 is the average settling velocity of a heavy particle. The joint effects of particle inertia and gravity are accounted for using the elliptic model for pair correlation of SGS velocity seen by heavy particles. The SGS timescale seen by heavy particles is extracted from the elliptic model and used to close the SGS-SDE model. The validations of the model against direct numerical simulation show that the SGS-SDE model can improve the performance of LES on relative dispersions especially when their initial separations are in the inertial subrange. Furthermore, we assess the performance of the SGS-SDE model by comparing the results with the approximate deconvolution method. The results show that the SGS-SDE model is more suitable for particles with small Stokes numbers, StK < 2. The model developed here provides a basis for the development of a more advanced SGS model for particles in non-homogeneous and anisotropic turbulent flows in pipes or channels.

The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the twospecies PASEP. In this model, there are two species of particles, one “heavy” and one “light”, on a one-dimensional finite lattice with open boundaries. Both particles can hop into adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can switch places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our first result is a combinatorial interpretation for the stationary distribution at $q=0$ in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and hence directly explains the connection between the two. Finally, we extend our formula for the stationary distribution to the $q=1$ case, using certain two-species alternative tableaux. Le but de ce document est de fournir une expression combinatoire décrivant les probabilités de l’état d’équilibre de PASEP à deux espèces. Dans ce modèle, il existe deux espèces de particules, une “lourde” et une “légère”, disposées sur un réseau fini unidimensionnel. Les deux particules peuvent sauter dans les trous adjacents à droite et à gauche, avec des probabilités proportionnelles à 1 et $q$. Par ailleurs, lorsque les particules lourdes et légères sont à côté l’une de l’autre, elles peuvent changer de place, comme si la particule légère était un trou. En outre, la particule lourde peut sauter dans et hors de la frontière du réseau. Notre premier résultat est une interprétation combinatoire de la distribution stationnaire dans le cas $q=0$, en termes de certains tableaux “multi-Catalan”. Nous proposons une formule explicite déterminantale pour les probabilités stationnaires, ainsi que plusieurs résultats énumératifs généraux pour ce cas. Nous décrivons aussi un processus de Markov sur ces tableaux, qui se projette sur le PASEP à deux espèces, et qui fournit donc directement une connexion entre les deux. Enfin, nous exprimons notre formule pour la distribution stationnaire dans le cas $q=1$, en utilisant certains tableaux alternatifs de deux espèces.

We investigate the steady-state solutions of granular Couette flows with a bidisperse particle mixture. The Couette geometry allows us to study segregation in a simple nonuniform flow. Two continuum models using kinetic theory are examined, and their results are compared with particle dynamic (PD) simulations. A notable difference between the two models is their treatment of energy partition (equipartition versus nonequipartition). We consider mixtures of particles with different sizes, masses, and densities. For Couette flows, we find that energy equipartition breaks down with an increase in the system inelasticity and the mass ratio. The effect of the size ratio on nonequipartition of granular energy is very small if the two particle species have the same mass. Two forms of segregation are studied in the present work: total solids segregation in the system and solids species segregation. Total solids segregation is related to the distribution of the granular energy across the walls. Solids species segregation is due to a competition of three diffusion forces: the thermal diffusion force, the ordinary diffusion force, and the pressure diffusion force. For equal density but different size particles, we observe quantitative differences between the two continuum models, where one model predicts a much greater degree of both total solids and solids species segregation than the other model. For equal size but different mass particles, we observe qualitative differences between the models where one model and PD simulations predict a segregation transition based on the particle mass, which is not seen in the other model for the conditions we have examined.

Gravity separation of bidisperse suspensions: Light and heavy particle species

Purpose The purpose of this study is to obtain a general solution to the field equations of thermoelastic solid with voids and micro-temperatures under the gravitational field in the context of the three theories, namely, coupled theory (CT), Lord and Shulman theory and Green and Lindsay theory. Design/methodology/approach The normal mode analysis is used to obtain the exact expressions for the considered variables. Comparisons are made with the results obtained in the three theories with and without gravity. Some particular cases are also deduced from the present investigation. Findings The effect of the gravity on the displacement, the micro-temperature vector, the temperature distribution, the normal stress, the changes in the volume fraction field and the heat flux moments have been depicted graphically. Research limitations/implications Some particular cases are also deduced from the present investigation. Originality/value The results of the physical quantities have been illustrated graphically by a comparison between three different theories in the presence and absence of gravity.

For an interceptor that follows a Keplerian trajectory, we have obtained a closed‐form linear expression for the miss distance in terms of the perturbations of the booster cut‐off conditions, where the miss distance reflects the predicted miss at the Point of Closest Approach (PCA) between the interceptor and the target. We use this analysis result to develop a new guidance law which, in the absence of gravity, ensures (1) that the magnitude of the predicted PCA miss decays exponentially, and (2) that the magnitude of the relative velocity is constant. The same guidance law has been applied to interceptors flying in a gravity field. In the presence of random navigation errors in the new guidance law, the numerically simulated results show that increasing the guidance law gain increases the rms of the predicted PCA miss, which results in a degradation of the interception performance. A trade‐off in gain magnitude is required to prevent this degradation.

Combining activation energy asymptotics, suitable scalings and numerical methods, we study how flame-balls move under the action of the free convection that they themselves generate in the presence of a weak, uniform gravity field. Attention is focused on steady configurations (in a suitable reference frame), on an isolated flame-ball of size comparable to what is obtained in the absence of gravity, and on deficient reactants that are characterized by a low Lewis number. For the sake of simplicity, we consider an adiabatic combustion process, in the sense that the radiative exchanges are neglected. This work provides one with:(a) a description of the free-convection field around the flame-ball, along with an asymptotic estimate of the drift velocity;(b) a relationship between the flame-ball radius, strength of gravity and physico-chemical properties of the reactive premixture;(c) extinction conditions, caused by the net effect of heat extraction from the flame-ball to its surroundings by the free-convection field.Hints on generalizations currently under consideration are also given.

Three-dimensional Voronoï analysis is used to quantify the clustering of inertial particles in homogeneous isotropic turbulence using data sets from numerics in the point particle limit and one experimental data set. We study the clustering behaviour at different density ratios, particle response times (i.e. Stokes numbers $\mathit{St}$) and two Taylor–Reynolds numbers (${\mathit{Re}}_{\lambda } = 75$ and 180). The probability density functions (p.d.f.s) of the Voronoï cell volumes of light and heavy particles show different behaviour from that of randomly distributed particles, i.e. fluid tracers, implying that clustering is present. The standard deviation of the p.d.f. normalized by that of randomly distributed particles is used to quantify the clustering. The clustering for both light and heavy particles is stronger for higher ${\mathit{Re}}_{\lambda } $. Light particles show maximum clustering for $\mathit{St}$ around 1–2 for both Taylor–Reynolds numbers. The experimental data set shows reasonable agreement with the numerical results. The results are consistent with previous investigations employing other approaches to quantify the clustering. We also present the joint p.d.f.s of enstrophy and Voronoï volumes and their Lagrangian autocorrelations. The small Voronoï volumes of light particles correspond to regions of higher enstrophy than those of heavy particles, indicating that light particles cluster in higher vorticity regions. The Lagrangian temporal autocorrelation function of Voronoï volumes shows that the clustering of light particles lasts much longer than that of heavy or neutrally buoyant particles. Due to inertial effects arising from the density contrast with the surrounding liquid, light and heavy particles remain clustered for much longer times than the flow structures which cause the clustering.

While traditional thermodynamic equilibrium requires uniform temperature across coexisting phases, underdamped active matter systems can sustain nonequilibrium hot-cold coexistence through motility-induced phase separation. We investigate particle demixing and emergent temperature gradients in binary mixtures of inertial active and passive particles. Remarkably, within specific parameter ranges of intermediate particle inertia and self-propulsion strength, the system simultaneously achieves pronounced particle demixing and sustains significant hot-cold coexistence. Activity differences drive rapid species separation, which is further enhanced over time by persistent rotational diffusion. The synergy between inertia and activity significantly amplifies temperature differences both between particle species and across coexisting gas-liquid phases. These temperature disparities originate from inertia-enabled energy storage, collision-mediated energy transfer, propulsion-driven acceleration, and weakened liquid-phase cohesion. Unlike equilibrium systems, active-passive mixtures circumvent thermal homogenization by maintaining kinetic temperature gradients through continuous energy injection and dissipation. These findings elucidate fundamental principles of nonequilibrium self-organization in hybrid systems, with implications for bio-inspired materials, microbial ecology, and energy transport in active composites.

Early in the operation of the SLD CCD vertex detector (VXD3) at the SLC, radiation damage to the CCDs was observed. It is well known that low-energy light particles (electrons and photons) are a few orders of magnitude less effective than heavy particles (neutrons or heavy charged particles) in the generation of radiation damage effects in silicon. The SLD environment was known to be dominated by electrons and photons with a small fraction of neutrons. The estimated radiation damage by these particles cannot account for the observed damage. Therefore, this damage is puzzling. A CCD-based detector is a leading option for vertex detection at the future linear collider (LC). A full understanding of background models in LCs and the associated damage is needed. Earlier results on neutron damage to an SLD CCD were reported at the 1999 IEEE NSS, and these new results complement our old results. In addition to tests on controlled exposures of individual CCDs, we have studied the nature of the traps produced in the SLD vertex detector to assess their origin-whether heavy or light particles.

Early in the operation of the SLD CCD vertex detector (VXD3) at the SLC, radiation damage to the CCDs was observed. It is well known that low energy light particles (electrons and photons) are a few orders of magnitude less effective than heavy particles (neutrons or heavy charged particles) in the generation of radiation damage effects in silicon. The SLD environment was known to be dominated by electrons and photons, and a large fluence of neutrons was not expected. Therefore, this damage is puzzling. A CCD based detector is a leading option for vertex detection at the future linear collider. A full understanding of background models in linear colliders; and the associated damage is needed. Earlier results on neutron damage to an SLD CCD were reported at the 1999 IEEE NSS, and these new results complement our old results. In addition to tests on controlled exposures of individual CCDs, we have studied the nature of the traps produced in the SLD vertex detector to assess their origin heavy or light particles?.

The relativistically invariant equation of radiative transfer with arbitrary velocity, flows temperature and density gradients, and gravitational fields can be expanded in projected, symmetric, trace‐free tensors. In the optically‐thick limit, the monopole moment of the radiation field is Planckian plus a small term that depends on the proper velocity dotted with the temperature gradient (if any) divided by the absorption (but not the scattering) cross‐section. The dipole moment contains a term proportional to the temperature gradient and another term proportional to the proper acceleration, both divided by the sum of the absorption cross‐section and the scattering cross‐section. The quadrupole term is proportional to the rate of strain (if any), divided by the sum of the absorption cross‐section and nine‐tenths the scattering cross‐section. All higher‐order moments vanish identically. Thus the cosmic microwave background should not show higher‐order multipole moments, in agreement with COBE observations....

Double-diffusive Hadley–Prats flow in a porous medium subject to gravitational variation

Gravitational effect and initial stress on generalized magneto-thermo-microstretch elastic solid for different theories

In accordance with the special theory of relativity all forms of energy, including heat, have inertia and hence in accordance with the equivalence principle also have weight. The purpose of the present article is to investigate the thermodynamic implications of the idea that heat has weight. In particular an investigation is made to see if a temperature gradient is a necessary accompaniment of thermal equilibrium in a gravitational field, in order to prevent the flow of heat from regions of higher to those of lower gravitational potential.A preliminary non-rigorous treatment of this problem is first given by attempting to modify the classical thermodynamics only to the extent of associating with each intrinsic quantity of energy an additional amount of potential gravitational energy. In this way an expression is obtained for increase in equilibrium temperature with decrease in gravitational potential which, however, could in any case only be correct as a first approximation in a weak gravitational field. A discussion of the uncertainties and lack of rigor of this preliminary treatment is then given and the necessity pointed out for a rigorous treatment based on the principles of general relativity.A rigorous relativistic treatment is then undertaken using the extension of thermodynamics to general relativity previously presented by the author. The system to be treated is taken as a static spherical distribution of perfect fluid which has come to gravitational and thermodynamic equilibrium. The principles of relativistic mechanics are first applied to such a system in order to obtain results needed in the later work. And it is then shown that these mechanical principles themselves are sufficient to determine the temperature distribution as a function of potential in the simple case of black-body radiation. The principles of relativistic thermodynamics are then applied to this same case of pure black-body radiation and the same expression for temperature as a function of potential obtained by the thermodynamic as by the mechanical treatment. This may be regarded as giving some measure of check on the validity of the proposed relativistic thermodynamics.Following this, a thermodynamic treatment is given for the temperature distribution in the more general case of matter and radiation and a result found which harmonizes with that for radiation alone. A treatment is then given to the distribution of a perfect monatomic gas in a gravitational field both on the assumption that the total number of atoms must remain constant and on the assumption of the ready interconvertibility of matter and radiation. In the latter case the same dependence of concentration on temperature is obtained as was found by Stern and by the author for the case of flat space-time.Using a system of coordinates such that the line element for the sphere of fluid takes the form$d{s}^{2}=\ensuremath{-}{e}^{u}(d{r}^{2}+{r}^{2}d{\ensuremath{\theta}}^{2}+{r}^{2}{sin}^{2}\ensuremath{\theta}d{\ensuremath{\varphi}}^{2})+{e}^{\ensuremath{\nu}}d{t}^{2}$ the general result for the relation between gravitational potential and equilibrium temperature ${T}_{0}$ as measured by a local observer in proper coordinates can be given by the equation $\frac{d \mathrm{ln}{T}_{0}}{\mathrm{dr}}=\ensuremath{-}\frac{1}{2}\frac{d\ensuremath{\nu}}{\mathrm{dr}}$This equation reduces in the case of a weak field to that obtained by the preliminary non-rigorous treatment, and gives a very small change of temperature with position in fields of ordinary intensity. The result, however, is one of great theoretical interest, since constant temperature throughout any system which has come to thermal equilibrium has hitherto been regarded as an inescapable thermodynamic conclusion. It is also not out of the question that the effect might sometime be of experimental or observational importance.