Hard-sphere Collision Research Articles

Erratum: Extended hard-sphere model and collisions of cohesive particles [Phys. Rev. E84, 031303 (2011)

Erratum: Extended hard-sphere model and collisions of cohesive particles [Phys. Rev. E<b>84</b>, 031303 (2011)

Investigation of convective heat transfer through constant wall heat flux micro/nano channels using DSMC

In this research, convective heat transfer of the argon gas flow through a micro/nano channel with uniform heat flux wall boundary condition is investigated using the direct simulation Monte Carlo (DSMC) method. Both of the hot wall (qw>0) and the cold wall (qw<0) cases are considered. Implementation of wall heat flux in the DSMC method is performed using the recently developed “iterative” technique. Our investigation considers heat transfer behavior in both of slip and transition flow regimes. We investigate the influence of rarefaction, i.e., Knudsen number, and viscous dissipation, i.e., Brinkman number, on the Nusselt number behavior. We use the generalized hard sphere (GHS) collision model to consider accurate variation of the heat conductivity with the temperature. The DSMC solutions for the Nusselt number are compared with different analytical expressions reported in the literature with suitable accuracy through the slip regime. We observe that the dependency of the Nusselt number on the Knudsen number decreases in nanochannels as Knudsen number increases into the transition regime, i.e., Nusselt number approaches nearly a constant value as Knudsen number goes beyond 1. Additionally, it is shown that the Nusselt number is a weak function of the Brinkman number.

Internal energy relaxation in shock wave structure

The Wang Chang-Uhlenbeck (WCU) equation is numerically integrated to characterize the internal structure of Mach 3 and Mach 5 shock waves in a gas with excitation in the internal energy states for the treatment of inelastic collisions. Elastic collisions are modeled with the hard sphere collision model and the transition rates for the inelastic collisions modified appropriately using probabilities based on relative velocities of the colliding particles. The collision integral is evaluated by the conservative discrete ordinate method [F. Tcheremissine, “Solution of the Boltzmann kinetic equation for high-speed flows,” Comput. Math. Math. Phys. 46, 315–329 (2006); F. Cheremisin, “Solution of the Wang Chang-Uhlenbeck equation,” Dokl. Phys. 47, 487–490 (2002)] developed for the Boltzmann equation. For the treatment of the diatomic molecules, the internal energy modes in the Boltzmann equation are described quantum mechanically given by the WCU equation. As a first step in the treatment of the inelastic collisions by the WCU equation, a two- and three-quantum system is considered to study the effect of the varying of (1) the inelastic cross section and (2) the energy gap between the quantum energy states. An alternative method, the direct simulation Monte Carlo method, is used for the Mach 3 shock wave to ensure the consistency of implementation in the two methods and there is an excellent agreement between the two methods. The results from the WCU implementation showed consistent trends for the Mach 3 and Mach5 standing shock waves simulations. Inelastic contributions change the downstream equilibrium state and allow the flow to transition to the equilibrium state further upstream.

Spectral gap for the Kac model with hard sphere collisions

We prove the analog of the Kac conjecture for hard sphere collisions, giving a computable, close estimate on the spectral gap that is independent of the number of particles. Previous work has focused on the case in which the collision rates are independent of the particle velocities, the case of so-called Maxwellian molecules. The new methods introduced here allow us to deal with collision rates that are not bounded from below. We also obtain information on the structure of the gap eigenfunction.

GLOBAL DRAG-INDUCED INSTABILITIES IN PROTOPLANETARY DISKS

We use the Fokker-Planck equation and model the dispersive dynamics of solid particles in annular protoplanetary disks whose gas component is more massive than the particle phase. We model particle--gas interactions as hard sphere collisions, determine the functional form of diffusion coefficients, and show the existence of two global unstable modes in the particle phase. These modes have spiral patterns with the azimuthal wavenumber $m=1$ and rotate slowly. We show that in ring-shaped disks, the phase space density of solid particles increases linearly in time towards an accumulation point near the location of pressure maximum, while instabilities grow exponentially. Therefore, planetesimals and planetary cores can be efficiently produced near the peaks of unstable density waves. In this mechanism, particles migrating towards the accumulation point will not participate in the formation of planets, and should eventually form a debris ring like the main asteroid belt or classical Kuiper belt objects. We present the implications of global instabilities to the formation of ice giants and terrestrial planets in the solar system.

Time resolved laser induced fluorescence on argon intermediate pressure microwave discharges: Measuring the depopulation rates of the 4p and 5p excited levels as induced by electron and atom collisions

The reaction kinetics in the excitation space of Ar is explored by means of Laser Induced Fluorescence (LIF) experiments using the combination of high rep-rate YAG–Dye laser systems with a well defined and easily controllable surfatron induced plasma setup. The high rep-rate favors the photon statistics while the low energy per pulse avoids intrusive plasma laser interactions. An analysis shows that, despite the low energy per pulse, saturation can still be achieved even when the geometrical overlap and spectral overlap are optimal. Out of the various studies that can be performed with this setup we confine the current paper to the study of the direct responses to the laser pump action of three 4p and one 5p levels of the Ar system. By changing the plasma in a controlled way one gets for these levels the rates of electron and atom quenching and therewith the total destruction rates of electron and atom collisions. Comparison with literature shows that the classical hard sphere collision rate derived for hydrogen gives a good description for the observed electron quenching (e-quenching) in Ar whereas for heavy particle quenching (a-quenching) this agreement was only found for the 5p level. An important parameter in the study of electron excitation kinetics is the location of the boundary in the atomic system for which the number of electron collisions per radiative life time equals unity. It is observed that for the Ar system this boundary is positioned lower than what is expected on grounds of H-like formulas.

Lattice Boltzmann based discrete simulation for gas–solid fluidization

Discrete particle simulation, a combined approach of computational fluid dynamics and discrete methods such as DEM (discrete element method), DSMC (direct simulation Monte Carlo), SPH (smoothed particle hydrodynamics), PIC (particle-in-cell), etc., is becoming a practical tool for exploring lab-scale gas–solid systems owing to the fast development of parallel computation. However, gas–solid coupling and the corresponding fluid flow solver remain immature. In this work, we propose a modified lattice Boltzmann approach to consider the effect of both the local solid volume fraction and the local relative velocity between particles and fluid, which is different from the traditional volume-averaged Navier–Stokes equations. A time-driven hard sphere algorithm is combined to simulate the motion of individual particles, in which particles interact with each other via hard-sphere collisions, the collision detection and motion of particles are performed at constant time intervals. The EMMS (energy minimization multi-scale) drag is coupled with the lattice Boltzmann based discrete particle simulation to improve the accuracy. Two typical fluidization processes, namely, a single bubble injection at incipient fluidization and particle clustering in a fast fluidized bed riser, are simulated with this approach, with the results showing a good agreement with published correlations and experimental data. The capability of the approach to capture more detailed and intrinsic characteristics of particle–fluid systems is demonstrated. The method can also be used straightforward with other solid phase solvers.

Classical aspects of the reduced collision cross section for elastic scattering in ion mobility spectrometry

A classical theory for the collision cross section associated with elastic scattering in ion mobility spectrometry is developed. The collision cross section is found to be equal to a hard-sphere collision cross section evaluated at the turning radius plus a potential scattering term inversely proportional to the effective temperature. The results of the theory are compared to current theories by fitting reduced collision cross sections for both small-ion and large-ion mobility data using a molecular modeled hard-sphere collision cross section as reference. The turning point radius is found to be 0.7 of the hard-sphere collision radius and the interaction potential evaluated at the turning point varies inversely as the square of the turning radius. Taken in combination, classical theory provides an incomplete description of the scattering event.

Heat conduction during collisions of cohesive and viscoelastic particles

This paper investigates heat transfer between particles during particle–particle and particle–wall collisions in fluid–solid flows. A model is described, which is based on the work by Sun and Chen (1988) [27], who described heat conduction between elastic, non-cohesive particles. The model of Sun and Chen is, in this work, extended to investigate how cohesion and viscoelasticity, elements that are always present in real systems, of the colliding particles influence the process of heat transfer. This was done by adopting three different soft-sphere models. At the end of the paper dimensional analysis is applied and, by fitting relations between the resulting dimensionless groups to numerical simulations using the models developed, “empirical” relations for the heat transfer during particle–particle and particle–wall collisions are derived. These empirical relations are suitable for use with hard-sphere collision models.

On Boundedness of Higher Velocity Moments for the Linear Boltzmann Equation with Diffuse Boundary Conditions

This article considers the time and space-dependent linear Boltzmann equation for elastic or inelastic (granular) collisions. First, in the angular cut-off case or with hard sphere collisions, mild L 1-solutions are constructed as limits of iterate functions. Then, in the case of hard potentials together with diffuse boundary conditions, global boundedness in time of higher velocity moments is proved, using our old collision velocity estimates together with a Jensen inequality.

Fan affinity laws from a collision model

The performance of a fan is usually estimated using hydrodynamical considerations. The calculations are long and involved and the results are expressed in terms of three affinity laws. In this paper we use kinetic theory to attack this problem. A hard sphere collision model is used, and subsequently a correction to account for the flow behaviour of air is incorporated. Our calculations prove the affinity laws and provide numerical estimates of the air delivery, thrust and drag on a rotating fan.

A generalized Fokker–Planck equation for anomalous diffusion in velocity space

A more general quasi-Fokker–Planck equation is derived to describe particle kinetics in situations when the usual Fokker–Planck equation is not applicable. The equation is valid for an arbitrary value of the transferred in a collision act momentum and for the arbitrary mass ratio of the interacting particles. The only assumption is smallness of the typical velocity of the particles, undergoing diffusion. The developed approach is another tool to study anomalous diffusion, avoiding conventional in such problems fractional differentiation. In this Letter anomalous diffusion in velocity space is considered for hard-sphere, Coulomb and dusty plasma collision models.

Coulomb-influenced collisions in aerosols and dusty plasmas

In aerosol and dusty plasma systems, the behavior of suspended particles (grains) is often strongly influenced by collisions occurring between ions and particles, as well as between particles themselves. In determining the collision kernel or collision rate coefficient for such charged entities, complications arise in that the collision process can be completely described neither by continuum transport mechanics nor by free molecular (ballistic) mechanics; that is, collisions are transition regime processes. Further, both the thermal energy and the potential energy between colliding entities can strongly influence the collision rate and must be considered. Flux-matching theory, originally developed by Fuchs, is frequently applied for calculation of collision rate coefficients under these circumstances. However, recent work suggests that crucial assumptions in flux-matching theory are not appropriate to describe transition regime collisions in the presence of potential interactions. Here, we combine dimensional analysis and mean first passage time calculations to infer the collision kernel between dilute charged entities suspended in a light background gas at thermal equilibrium. The motion of colliding entities is described by a Langevin equation, and Coulombic interactions are considered. It is found that the dimensionless collision kernel for these conditions, H, is a function of the diffusive Knudsen number, Kn(D) (in contrast to the traditional Knudsen number), and the potential energy to thermal energy ratio, Ψ(E). For small and large Kn(D), it is found that the dimensionless collision kernels inferred from mean first passage time calculations collapse to the appropriate continuum and free molecular limiting forms, respectively. Further, for repulsive collisions (Ψ(E) negative) or attractive collisions with Ψ(E)<0.5, calculated results are in excellent agreement with flux-matching theory predictions, and the dimensionless collision kernel can be determined conveniently via use of the H(Kn(D)) relationship found for hard-sphere collisions with modified definitions of H and Kn(D) to account for potential energy. However, for Ψ(E)>0.5, it is found that flux-matching theory predictions substantially underestimate the collision kernel. We find that the collision process in this regime is governed by the minimum of Kn(D) and Kn(Ψ) (Kn(Ψ) = 3Kn(D)/2Ψ(E)), and based on calculations, propose a function H(Kn(D), Kn(Ψ)) for collision kernel evaluation. The situations for which Ψ(E)>0.5 apply to singly charged nanoparticles and multiply charged submicrometer and supermicrometer particles, and are thus prevalent in both aerosol and dusty plasma environments.

Inter-particle collision in particle-laden isotropic turbulence

Direct numerical simulations of heavy particles in isotropic homogeneous turbulence are performed to study inter-particle collisions in a wide range of Stokes numbers. Dynamics of the particles are described by Stokes drag including particle-particle interactions via hard-sphere collisions while fluid turbulence is solved using a pseudo-spectral method. The statistics related to rotation and dissipation rates as well as collision frequency are investigated. A particular emphasis is placed on the statistics of the fluid experienced by heavy particles, which provide essential information on the collision process. A plausible explanation of the collision mechanism for various Stokes numbers is provided.

Simulations of granular bed erosion due to laminar shear flow near the critical Shields number

Direct numerical simulations of granular beds consisting of uniformly sized spherical particles being eroded by a shear flow of Newtonian liquid have been performed. The lattice-Boltzmann method has been used for resolving the flow of the interstitial liquid. Fluid and solid dynamics are fully coupled with the particles having finite size and undergoing hard-sphere collisions. Only laminar flow has been considered with particle-based Reynolds numbers in the range 0.02 to 0.6. The parameter range of the simulations covers the transition between static and mobilized beds. The transition occurs for 0.10&amp;lt;θ&amp;lt;0.15 with θ the Shields number. The transition is insensitive of the Reynolds number and the solid-over-liquid density ratio. Incipient bed motion has been interpreted in terms of the probability density functions of the hydrodynamic forces acting on the spheres in the upper layer of the bed.

Extended hard-sphere model and collisions of cohesive particles

In two earlier papers the present authors modified a standard hard-sphere particle-wall and particle-particle collision model to account for the presence of adhesive or cohesive interaction between the colliding particles: the problem is of importance for modeling particle-fluid flow using the Lagrangian approach. This technique, which involves a direct numerical simulation of such flows, is gaining increasing popularity for simulating, e.g., dust transport, flows of nanofluids and grains in planetary rings. The main objective of the previous papers was to formally extend the impulse-based hard-sphere model, while suggestions for quantifications of the adhesive or cohesive interaction were made. This present paper gives an improved quantification of the adhesive and cohesive interactions for use in the extended hard-sphere model for cases where the surfaces of the colliding bodies are "dry," e.g., there is no liquid-bridge formation between the colliding bodies. This quantification is based on the Johnson-Kendall-Roberts (JKR) analysis of collision dynamics but includes, in addition, dissipative forces using a soft-sphere modeling technique. In this way the cohesive impulse, required for the hard-sphere model, is calculated together with other parameters, namely the collision duration and the restitution coefficient. Finally a dimensional analysis technique is applied to fit an analytical expression to the results for the cohesive impulse that can be used in the extended hard-sphere model. At the end of the paper we show some simulation results in order to illustrate the model.

Energy limitation for models to simulate the buffer gas cooling

Abstract Detailed simulations of buffer gas cooling, which is widely applied at the rare-isotope facilities, have been investigated by using three frequently used models: viscous damping force (VDF) model, hard sphere collision (HSC) model, and realistic interaction potential (RIP) model. By comparing the stopping ranges from the simulated data with those from SRIM code, we found that the VDF and HSC models are better in low energy range, but the RIP model is better in relatively high energy range. In helium buffer gas, which is always used in cooler and bunchers, the VDF model should be used when the ion’s energy is less than ∼5 eV/u, and the RIP model less than ∼80 eV/u. Above this energy, all three models cannot be used.

A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

AbstractA moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles. Different orders of accuracy in terms of the moments of the velocity distribution function are considered, accounting for moments up to seventh order. Quadrature-based closures for four different models for inelastic collision-the Bhatnagar-Gross-Krook, ES-BGK, the Maxwell model for hard-sphere collisions, and the full Boltzmann hard-sphere collision integral-are derived and compared. The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls. Results obtained from the kinetic models are compared with the predictions of molecular dynamics (MD) simulations of a nearly equivalent system withfinite-sizeparticles. The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed. Results for constitutive quantities such as the stress tensor and the heat flux are provided, and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions.

The dynamics of evaporation from a liquid surface

We explore the collisional energy transfer dynamics of benzene molecules spontaneously evaporating from an in vacuo water–ethanol liquid beam. We find that rotations are cooled significantly more than the lowest-energy vibrational modes, while the rotational energy distributions are Boltzmann. Within experimental uncertainty, the rotational temperatures of vibrationally-excited evaporating molecules are the same as the ground state. Collision-induced gas phase energy transfer measurements reveal that benzene undergoes fast rotational relaxation, from which we deduce that the rotational temperature measured in the evaporation experiments (200–230 K) is an indication of the translational energy of the evaporate. Conversely, vibrational relaxation of the high frequency mode, ν 6, is very inefficient, suggesting that the ν 6 temperature (260–270 K) is an indication of the liquid surface temperature. Modelling of the relaxation dynamics by both ‘temperature gap’ and ‘Master Equation’ approaches indicates that the equivalent of 150–260 hard-sphere collisions occur during the transition from liquid to vacuum.

Thin Metal Film Deposition on Microparticles Trapped in RF Magnetron Discharge Plasma

AbstractWe present the technique for metal film deposition on spherical microparticles in plasma trap. An RF magnetron discharge was applied both to sputter Ag or Cu electrodes and to confine the microparticles in the immediate proximity to the sputtered area. We estimated the stopping power of Ar buffer gas on the sputtered atoms in van der Waals collisions which provide a main thermalization channel with cross‐section of the order of magnitude larger than for the hard sphere collision model. During our deposition experiments the microparticles were trapped close to the border zone of ballistic bombardment by the sputtered atoms. The deposition rate was of the order of 10−1 nm · s−1, the coatings have continuous surface with uniform thickness. magnified image