Dissipative Collisions Research Articles

Inelastic wedge billiards

Billiards are simple systems used to investigate Hamiltonian dynamics in physics. When real billiards are examined experimentally, the energy dissipated in each collision must be replaced by an external stimulus to maintain the dynamics. We focus on a specific system of a driven billiard using a wedge shaped boundary to examine nonlinear and chaotic behavior. Mathematical models such as the logistic map are simple low dimensional systems that exhibit nonlinear and chaotic behavior as a single parameter is varied. This logistic map can then be used to identify a very specific mathematical parameter known as the Lyapunov exponent, which helps in identifying chaos more clearly. In the current experiment, the dynamics of a particle free to move near a horizontally shaken vertical boundary will be examined for the presence of chaos. The goal of the research is to extract a Lyapunov exponent between any two trajectories in the system. In addition, the manner in which the dynamics evolve freely through dissipative collisions provides a testbed for measurements of the velocity dependent coefficients of restitution for the billiard. A better description of hard sphere coefficients of restitution would be beneficial to a host of experiments and numerical simulations in granular physics.

ON THE ISOTROPIZATION OF SOLAR WIND PROTONS

ABSTRACT Protons observed in the solar wind are characterized by temperature anisotropies whose upper and lower bounds can be partially explained by marginal instability conditions associated with various plasma instabilities. However, an outstanding problem is that the majority of data distributed in space are located away from the boundaries and occupy a broad region with the peak near isotropic condition. The present paper employs macroscopic-kinetic theory that includes adiabatic effects arising from various non-monotonic inhomogeneities of magnetic field and density, the influence from proton-cyclotron and parallel firehose instabilities, and collisional dissipation in order to explore the consequence of each effect. It is found that spatial inhomogeneities are the leading cause of the scattering of the data points away from the marginal stability boundaries in space, thus providing a potential explanation for observations.

Phase diagram for inertial granular flows.

Flows of hard granular materials depend strongly on the interparticle friction coefficient μ_{p} and on the inertial number I, which characterizes proximity to the jamming transition where flow stops. Guided by numerical simulations, we derive the phase diagram of dense inertial flow of spherical particles, finding three regimes for 10^{-4}≲I≲10^{-1}: frictionless, frictional sliding, and rolling. These are distinguished by the dominant means of energy dissipation, changing from collisional to sliding friction, and back to collisional, as μ_{p} increases from zero at constant I. The three regimes differ in their kinetics and rheology; in particular, the velocity fluctuations and the stress ratio both display nonmonotonic behavior with μ_{p}, corresponding to transitions between the three regimes of flow. We rationalize the phase boundaries between these regimes, show that energy balance yields scaling relations between microscopic properties in each of them, and derive the strain scale at which particles lose memory of their velocity. For the frictional sliding regime most relevant experimentally, we find for I≥10^{-2.5} that the growth of the macroscopic friction μ(I) with I is induced by an increase of collisional dissipation. This implies in that range that μ(I)-μ(0)∼I^{1-2b}, where b≈0.2 is an exponent that characterizes both the dimensionless velocity fluctuations L∼I^{-b} and the density of sliding contacts χ∼I^{b}.

Nonlinear coherent structures of Alfvén wave in a collisional plasma

The Alfvén wave dynamics is investigated in the framework of two-fluid approach in a compressible collisional magnetized plasma. In the finite amplitude limit, the dynamics of the nonlinear Alfvén wave is found to be governed by a modified Korteweg-de Vries Burgers equation (mKdVB). In this mKdVB equation, the electron inertia is found to act as a source of dispersion, and the electron-ion collision serves as a dissipation. The collisional dissipation is eventually responsible for the Burgers term in mKdVB equation. In the long wavelength limit, this weakly nonlinear Alfvén wave is shown to be governed by a damped nonlinear Schrödinger equation. Furthermore, these nonlinear equations are analyzed by means of analytical calculation and numerical simulation to elucidate the various aspects of the phase-space dynamics of the nonlinear wave. Results reveal that nonlinear Alfvén wave exhibits the dissipation mediated shock, envelope, and breather like structures. Numerical simulations also predict the formation of dissipative Alfvénic rogue wave, giant breathers, and rogue wave holes. These results are discussed in the context of the space plasma.

Modification of kinetic theory of granular flow for frictional spheres, Part I: Two-fluid model derivation and numerical implementation

We derive a kinetic theory of granular flow (KTGF) for frictional spheres in dense systems, including rotation, sliding and sticking collisions. We use the Chapman–Enskog solution procedure of successive approximations, where the single-particle velocity distribution is assumed to be nearly Maxwellian both translationally and rotationally, as assumed by McCoy et al. (1966). An expression for the first-order particle velocity distribution function is derived, which includes the effects of particle rotation and friction. Using a simple moment method, balance equations for mass, momentum and energy are derived with closure equations for viscosities, and thermal conductivities and collisional energy dissipation rates of angular and translational kinetic energy. Because the internal angular momentum changes are coupled to the flow field, the stress tensor contains anti-symmetric components which are associated with a rotational viscosity. In the resulting closure equations, the rheological properties of the particles are explicitly described in terms of the friction coefficient. The model has been incorporated into our in-house two-fluid model (TFM) code for the modeling of dense gas–solid fluidized beds. For verification, a comparison of the present model in the limit of zero friction with the original (frictionless) KTGF model is carried out. Simulation results of both models agree well. In the next part, simulation results obtained with the new model will be compared with experimental data and discrete particle model (DPM) simulations.

Granular dynamics of cohesive powders in a rotating drum as revealed by speckle visibility spectroscopy and synchronous measurement of forces due to avalanching

We have used speckle visibility spectroscopy (SVS) and synchronized force measurements to compare the granular dynamics of two cohesive lactose powders, with Sauter mean diameters of ~29 and ~151μm, in a rotating drum. A load cell (LC) was used to measure forces on the drum mounting frame and enable monitoring of bulk powder motion; SVS is a dynamic light scattering technique particularly suited for studying dynamics in dense, non-ergodic granular systems. Our results reveal that surface slumping and intermittent collisional dynamics in the bulk of the bed are correlated, especially for the fine more cohesive particles (Geldart group C/A boundary), but not as much for the less cohesive larger particles (Geldart group A/B boundary). The specific dissipation energy of the particles in the drum is similar for both powders, and increases linearly with increasing drum speed. However, the dependencies of the load cell and SVS signals on rotation speed have opposing trends for these two powders, indicating different dissipation mechanisms for the different Geldart Groups; collisional dissipation is more important for the Geldart C/A powder, while for the Geldart A/B powder avalanche dissipation is dominant.

Hypocoercivity for a Linearized Multispecies Boltzmann System

A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cutoff condition. Two proofs are given: a non-constructive one, based on the decomposition of the collision operator into a compact and a coercive part, and a constructive one, which exploits the “cross effects” coming from collisions between different species and which yields explicit constants. Furthermore, the essential spectra of the linearized collision operator and the linearized Boltzmann operator are calculated. Based on the spectral-gap estimate, the exponential convergence towards global equilibrium with explicit rate is shown for solutions to the linearized multispecies Boltzmann system on the torus. The convergence is achieved by the interplay between the dissipative collision operator and the conservative transport operator and is proved by using the hypocoercivity method of Mouhot and Neumann.

Spontaneous magnetic reconnection

The present review concerns the relevance of collisionless reconnection in the astrophysical context. Emphasis is put on recent developments in theory obtained from collisionless numerical simulations in two and three dimensions. It is stressed that magnetic reconnection is a universal process of particular importance under collisionless conditions, when both collisional and anomalous dissipation are irrelevant. While collisional (resistive) reconnection is a slow, diffusive process, collisionless reconnection is spontaneous. On any astrophysical time scale, it is explosive. It sets on when electric current widths become comparable to the leptonic inertial length in the so-called lepton (electron/positron) “diffusion region”, where leptons de-magnetise. Here, the magnetic field contacts its oppositely directed partner and annihilates. Spontaneous reconnection breaks the original magnetic symmetry, violently releases the stored free energy of the electric current, and causes plasma heating and particle acceleration. Ultimately, the released energy is provided by mechanical motion of either the two colliding magnetised plasmas that generate the current sheet or the internal turbulence cascading down to lepton-scale current filaments. Spontaneous reconnection in such extended current sheets that separate two colliding plasmas results in the generation of many reconnection sites (tearing modes) distributed over the current surface, each consisting of lepton exhausts and jets which are separated by plasmoids. Volume-filling factors of reconnection sites are estimated to be as large as $${<}10^{-5}$$ per current sheet. Lepton currents inside exhausts may be strong enough to excite Buneman and, for large thermal pressure anisotropy, also Weibel instabilities. They bifurcate and break off into many small-scale current filaments and magnetic flux ropes exhibiting turbulent magnetic power spectra of very flat power-law shape $$W_b\propto k^{-\alpha }$$ in wavenumber k with power becoming as low as $$\alpha \approx 2$$ . Spontaneous reconnection generates small-scale turbulence. Imposed external turbulence tends to temporarily increase the reconnection rate. Reconnecting ultra-relativistic current sheets decay into large numbers of magnetic flux ropes composed of chains of plasmoids and lepton exhausts. They form highly structured current surfaces, “current carpets”. By including synchrotron radiation losses, one favours tearing-mode reconnection over the drift-kink deformation of the current sheet. Lepton acceleration occurs in the reconnection-electric field in multiple encounters with the exhausts and plasmoids. This is a Fermi-like process. It results in power-law tails on the lepton energy distribution. This effect becomes pronounced in ultra-relativistic reconnection where it yields extremely hard lepton power-law energy spectra approaching $$F(\gamma )\propto \gamma ^{-1}$$ , with $$\gamma $$ the lepton energy. The synchrotron radiation limit becomes substantially exceeded. Relativistic reconnection is a probable generator of current and magnetic turbulence, and a mechanism that produces high-energy radiation. It is also identified as the ultimate dissipation mechanism of the mechanical energy in collisionless magnetohydrodynamic turbulent cascades via lepton-inertial-scale turbulent current filaments. In this case, the volume-filling factor is large. Magnetic turbulence causes strong plasma heating of the entire turbulent volume and violent acceleration via spontaneous lepton-scale reconnection. This may lead to high-energy particle populations filling the whole volume. In this case, it causes non-thermal radiation spectra that span the entire interval from radio waves to gamma rays.

Fluid vs. kinetic magnetic reconnection with strong guide fields

The fast rates of magnetic reconnection found in both nature and experiments are important to understand theoretically. Recently, it was demonstrated that two-fluid magnetic reconnection remains fast in the strong guide field regime, regardless of the presence of fast-dispersive waves. This conclusion is in agreement with recent results from kinetic simulations, and is in contradiction to the findings in an earlier two-fluid study, where it was suggested that fast-dispersive waves are necessary for fast reconnection. In this paper, we give a more detailed derivation of the analytic model presented in a recent letter and present additional simulation results to support the conclusions that the magnetic reconnection rate in this regime is independent of both collisional dissipation and system-size. In particular, we present a detailed comparison between fluid and kinetic simulations, finding good agreement in both the reconnection rate and overall length of the current layer. Finally, we revisit the earlier two-fluid study, which arrived at different conclusions, and suggest an alternative interpretation for the numerical results presented therein.

Stability of freely cooling granular mixtures at moderate densities

The formation of velocity vortices and density clusters is an intriguing phenomenon of freely cooling granular flows. In this work, the critical length scale Lc for the onset of instability is determined via stability analysis of the linearized Navier–Stokes hydrodynamic equations of d-dimensional granular binary mixtures at moderate densities. In contrast to previous attempts, the analysis is not restricted to nearly elastic systems since it takes into account the nonlinear dependence of the transport coefficients and the cooling rate on the collisional dissipation. As expected from previous results obtained in the very dilute regime, linear stability shows d−1transversal (shear) modes and a longitudinal (“heat”) mode to be unstable with respect to long enough wavelength excitations. The theoretical predictions also show that the origin of the instability is driven by the transversal component of the velocity field that becomes unstable when the system length L > Lc. An explicit expression of Lc is obtained in terms of the masses and diameters of the mixture, the composition, the volume fraction and the coefficients of restitution. Previous results derived in the limit of both mechanically equivalent particles and low-density mixtures are consistently recovered. Finally, a comparison with previous theoretical works which neglect the influence of dissipation on the transport coefficients shows quantitative discrepancies for strong dissipation.

On the avalanche generation of runaway electrons during tokamak disruptions

A simple zero dimensional model for a tokamak disruption is developed to evaluate the avalanche multiplication of a runaway primary seed during the current quench phase of a fast disruptive event. Analytical expressions for the plateau runaway current, the energy of the runaway beam, and the runaway energy distribution function are obtained allowing the identification of the parameters dominating the formation of the runaway current during disruptions. The effect of the electromagnetic coupling to the vessel and the penetration of the external magnetic energy during the disruption current quench as well as of the collisional dissipation of the runaway current at high densities are investigated. Current profile shape effects during the formation of the runaway beam are also addressed by means of an upgraded one-dimensional model.

Dynamical approach to weakly dissipative granular collisions.

Granular systems present surprisingly complicated dynamics. In particular, nonlinear interactions and energy dissipation play important roles in these dynamics. Usually (but admittedly not always), constant coefficients of restitution are introduced phenomenologically to account for energy dissipation when grains collide. The collisions are assumed to be instantaneous and to conserve momentum. Here, we introduce the dissipation through a viscous (velocity-dependent) term in the equations of motion for two colliding grains. Using a first-order approximation, we solve the equations of motion in the low viscosity regime. This approach allows us to calculate the collision time, the final velocity of each grain, and a coefficient of restitution that depends on the relative velocity of the grains. We compare our analytic results with those obtained by numerical integration of the equations of motion and with exact ones obtained by other methods for some geometries.

Multiscale Nature of the Dissipation Range in Gyrokinetic Simulations of Alfvénic Turbulence.

Nonlinear energy transfer and dissipation in Alfvén wave turbulence are analyzed in the first gyrokinetic simulation spanning all scales from the tail of the MHD range to the electron gyroradius scale. For typical solar wind parameters at 1 AU, about 30% of the nonlinear energy transfer close to the electron gyroradius scale is mediated by modes in the tail of the MHD cascade. Collisional dissipation occurs across the entire kinetic range k(⊥)ρ(I)≳1. Both mechanisms thus act on multiple coupled scales, which have to be retained for a comprehensive picture of the dissipation range in Alfvénic turbulence.

Dynamics of a bouncing ball

We describe an experiment dedicated to the study of the trajectories of a ball bouncing on a vibrating plate. Using an experimental device of our own design, it is possible to impose arbitrary trajectories on the plate and we show that the entire trajectory of the ball can be reconstructed solely from measurement of the times the ball hits the plate. In this paper, we make use of our apparatus to introduce the notion of dissipative collisions and to propose three different ways to measure the associated restitution coefficient. We then report on correlations in the chaotic regime and theoretically discuss the complex patterns that are exhibited in the case of a sinusoidal vibration. Lastly, we show that the use of an aperiodic driving vibration makes it possible to minimize part of these correlations.

Subjective valuation of cushioning in a human drop landing task as quantified by trade-offs in mechanical work

Humans can perform motor tasks in a variety of ways, yet often favor a particular strategy. Some factors governing the preferred strategy may be objective and quantifiable, (e.g. metabolic energy or mechanical work) while others may be more subjective and less measurable, (e.g. discomfort, pain, or mental effort). Subjectivity can make it challenging to explain or predict preferred movement strategies. We propose that subjective factors might nevertheless be characterized indirectly by their trade-offs against more objective measures such as work. Here we investigated whether subjective costs that influence human movement during drop landings could be indirectly assessed by quantifying mechanical work performed. When landing on rigid ground, humans typically absorb much of the collision actively by bending their knees, perhaps to avoid the discomfort of stiff-legged landings. We measured how work performed by healthy adults (N=8) changed as a function of surface cushioning for drop landings (fixed at about 0.4m) onto varying amounts of foam. Landing on more foam dissipated more energy passively in the surface, thus reducing the net dissipation required of subjects, due to relatively fixed landing energy. However, subjects actually performed even less work in the dissipative collision, as well as in the subsequent active, positive work to return to upright stance (approximately linear decrease of about 1.52J per 1cm of foam thickness). As foam thickness increased, there was also a corresponding reduction in center-of-mass vertical displacement after initial impact by up to 43%. Humans appear to subjectively value cushioning, revealed by the extra work they perform landing without it. Cushioning is thus worth more than the energy it dissipates, in an amount that indicates the subjective discomfort of stiff landings.

Pseudo-turbulent gas-phase velocity fluctuations in homogeneous gas–solid flow: fixed particle assemblies and freely evolving suspensions

Gas-phase velocity fluctuations due to mean slip velocity between the gas and solid phases are quantified using particle-resolved direct numerical simulation. These fluctuations are termed pseudo-turbulent because they arise from the interaction of particles with the mean slip even in ‘laminar’ gas–solid flows. The contribution of turbulent and pseudo-turbulent fluctuations to the level of gas-phase velocity fluctuations is quantified in initially ‘laminar’ and turbulent flow past fixed random particle assemblies of monodisperse spheres. The pseudo-turbulent kinetic energy $k^{(f)}$ in steady flow is then characterized as a function of solid volume fraction ${\it\phi}$ and the Reynolds number based on the mean slip velocity $\mathit{Re}_{m}$. Anisotropy in the Reynolds stress is quantified by decomposing it into isotropic and deviatoric parts, and its dependence on ${\it\phi}$ and $Re_{m}$ is explained. An algebraic stress model is proposed that captures the dependence of the Reynolds stress on ${\it\phi}$ and $Re_{m}$. Gas-phase velocity fluctuations in freely evolving suspensions undergoing elastic and inelastic particle collisions are also quantified. The flow corresponds to homogeneous gas–solid systems, with high solid-to-gas density ratio and particle diameter greater than dissipative length scales. It is found that for the parameter values considered here, the level of pseudo-turbulence differs by only 15 % from the values for equivalent fixed beds. The principle of conservation of interphase turbulent kinetic energy transfer is validated by quantifying the interphase transfer terms in the evolution equations of kinetic energy for the gas-phase and solid-phase fluctuating velocity. It is found that the collisional dissipation is negligible compared with the viscous dissipation for the cases considered in this study where the freely evolving suspensions attain a steady state starting from an initial condition where the particles are at rest.

Mechanical and energetic consequences of reduced ankle plantar-flexion in human walking.

The human ankle produces a large burst of 'push-off' mechanical power late in the stance phase of walking, reduction of which leads to considerably poorer energy economy. It is, however, uncertain whether the energetic penalty results from poorer efficiency when the other leg joints substitute for the ankle's push-off work, or from a higher overall demand for work due to some fundamental feature of push-off. Here, we show that greater metabolic energy expenditure is indeed explained by a greater demand for work. This is predicted by a simple model of walking on pendulum-like legs, because proper push-off reduces collision losses from the leading leg. We tested this by experimentally restricting ankle push-off bilaterally in healthy adults (N=8) walking on a treadmill at 1.4 m s(-1), using ankle-foot orthoses with steel cables limiting motion. These produced up to ∼50% reduction in ankle push-off power and work, resulting in up to ∼50% greater net metabolic power expenditure to walk at the same speed. For each 1 J reduction in ankle work, we observed 0.6 J more dissipative collision work by the other leg, 1.3 J more positive work from the leg joints overall, and 3.94 J more metabolic energy expended. Loss of ankle push-off required more positive work elsewhere to maintain walking speed; this additional work was performed by the knee, apparently at reasonably high efficiency. Ankle push-off may contribute to walking economy by reducing dissipative collision losses and thus overall work demand.

The fluctuation energy balance in non-suspended fluid-mediated particle transport

Here, we compare two extreme regimes of non-suspended fluid-mediated particle transport, transport in light and heavy fluids (“saltation” and “bedload,” respectively), regarding their particle fluctuation energy balance. From direct numerical simulations, we surprisingly find that the ratio between collisional and fluid drag dissipation of fluctuation energy is significantly larger in saltation than in bedload, even though the contribution of interparticle collisions to transport of momentum and energy is much smaller in saltation due to the low concentration of particles in the transport layer. We conclude that the much higher frequency of high-energy particle-bed impacts (“splash”) in saltation is the cause for this counter-intuitive behavior. Moreover, from a comparison of these simulations to particle tracking velocimetry measurements which we performed in a wind tunnel under steady transport of fine and coarse sand, we find that turbulent fluctuations of the flow produce particle fluctuation energy at an unexpectedly high rate in saltation even under conditions for which the effects of turbulence are usually believed to be small.

Different singularities in the functions of extended kinetic theory at the origin of the yield stress in granular flows

We use previous results from discrete element simulations of simple shear flows of rigid, identical spheres in the collisional regime to show that the volume fraction-dependence of the stresses is singular at the shear rigidity. Here, we identify the shear rigidity, which is a decreasing function of the interparticle friction, as the maximum volume fraction beyond which a random collisional assembly of grains cannot be sheared without developing force chains that span the entire domain. In the framework of extended kinetic theory, i.e., kinetic theory that accounts for the decreasing in the collisional dissipation due to the breaking of molecular chaos at volume fractions larger than 0.49, we also show that the volume fraction-dependence of the correlation length (measure of the velocity correlation) is singular at random close packing, independent of the interparticle friction. The difference in the singularities ensures that the ratio of the shear stress to the pressure at shear rigidity is different from zero even in the case of frictionless spheres: we identify that with the yield stress ratio of granular materials, and we show that the theoretical predictions, once the different singularities are inserted into the functions of extended kinetic theory, are in excellent agreement with the results of numerical simulations.

Transport description of dissipative heavy ion collisions at Fermi energies

Peripheral collisions of heavy ions at Fermi energies are a powerful tool for producing new isotopes far from stability line. The mechanism of these reactions has features between direct and dissipative reactions, as is seen from the velocity distributions of the projectile-like fragments, which are peaked near the beam velocity and have a long tail to the smaller velocities. In this report we treat these heavy ion collisions in a Boltzmannlike transport approach, which describes the evolution of the phase space distribution function under the influence of a mean field and a two-body collision term. It is used to calculate the primary fragments after the collision phase, which are, however, still highly excited. The de-excitation of the fragments is calculated in a statistical model. We calculate isotope distributions and velocity spectra for the reaction of 40Ar as a projectile on targets of 9Be and 181Ta at E/A = 57 MeV and find reasonably good agreement with experimental data.