Thesis Presentation

Energy Dissipation, Clustering, and Velocity Distribution of a OneDimensional Freely-Cooling Granular Gas: A Computer Simulation

Julius F. Madrid Meriska Monica F. Paglinawan

OUTLINE I. Abstract II. Introduction III. Related Literature and Studies IV. Methodology V. Results and Discussion VI. Conclusions and Recommendations

Abstract This study investigated the energy decay and the cluster evolution in time of a freely-cooling one-dimensional granular gas. On the idealization of Inelastic Hard Sphere model (IHS), the computer simulated the inelastic collision of granular gases. It is supposed that the relative velocity of two colliding particles is reduced by a factor r, the coefficient of restitution, where 0 ≤ r ≤ 1. This paper concluded that energy dissipation increases as r decreases in agreement to Haff’s Law (1/t2).

Energy dissipation with

different solid fraction, s, is also examined. It was found that as s increases, early occurrence of inelastic collapse is observed. Lastly, the authors determine the non-Gaussian and non-Maxwellian behavior of the velocity distributions of a freely-cooling inelastic granular gas. This study

Chapter 1

Introduction

Granular materials

any materials which consist of grains or relatively small particles.

LIQUID SOLID GAS

Granular Gas • A set of solid, macroscopic particles or grains which are in relative motion with respect to each other. • Energy is not conserved

Dissipation of energy

• Far from equilibrium state

FREELY COOLING System No energy input that will compensate the energy loss.

DRIVEN System Energy is injected to compensate the energy loss. Therefore equation of state is possible

Behavior of Granular Gas DISSIPATIVE ENERGY

VELOCITY DISTRIBUTION INELASTIC COLLISION

CLUSTERING

INELASTIC COLLAPSE

1.1 Objectives The study will try to give insights on the behavior of a freelycooling one dimensional dissipative granular gas. Clustering and Energy Dissipation •different coefficients of restitution, r •different solid fractions The study aims to report the velocity distribution of a freely-cooling one dimensional dissipative granular gas

1.2 Significance •The study demonstrates the fundamental difference between granular gases and molecular gases. •The simulation results resolve outstanding issues concerning clustering and velocity distributions in freely cooling granular gases. •Particularly, the established velocity distribution is noteworthy because this single particle distribution function is often sufficient to characterize a system that is far from equilibrium (equation of state is not possible) •It also provides a connection between granular gases and hydrodynamic theory of the formulation of large scale structure in the universe.

1.3 Scope and Limitation The study deals with the freelycooling case of a one-dimensional granular gas. There are no other forces that interact with the ensemble of particles. The behavior observed is limited only to the effect of varying degree of inelasticity as well as to varying solid fraction. Lastly velocity distribution of a freely-cooling granular gas is obtained.

Chapter 2

RELATED LITERATURE AND STUDIES • Inelastic Hard Sphere (IHS) model [3-4, 12-19] where there are no attractive forces between particles but in each collision energy is lost . • Many one dimensional gas systems [2-3, 12-17, 20-26] were studied since 1D idealization makes it possible the easy implementation. • The dynamics of granular gas is completely determined by pure mechanical collisions of its particles.

RELATED LITERATURE AND STUDIES • Inelastic Hard Sphere (IHS) model [3-4, 12-19] where there are no attractive forces between particles but in each collision energy is lost . • Many one dimensional gas systems [2-3, 12-17, 20-26] were studied since 1D idealization makes it possible the easy implementation. • The dynamics of granular gas is completely determined by pure mechanical collisions of its particles.

• In the case of inelastic collapse, there is no attractive force between particles to cause particle collisions. • Remarkably, ‘‘inelastic collapse’’ also persists in higher dimensions, where it produces dense chainlike clusters, as shown in Fig. 2.1. FIG. 2.1 A two-dimensional simulation of hard disks colliding inelastically in a container with periodic boundary conditions. The line of particles that are solid circles consists of those that have undergone ‘‘inelastic collapse.’’ Adapted from McNamara and Young (1994) [24].

• The study of dissipative granular gas permits to consider the problem of inelastic collisions in its generality. This rate of decay in hard-sphere models is usually referred as the Haff’s Law [2, 4, 5,11,17,24] and is stated as the energy is decaying proportional to t-2. • Since this paper limits only the non-equilibrium system of particles equation of state is not possible, instead velocity distribution of the particles is exploited to describe the system. Many studies reported a non-Gaussian velocity distribution for a granular gas [8, 10, 13, 15, 26, 29].

CHAPTER 3: METHODOLOGY 3.1 Model 3.2 Simulation Details 3.3 Algorithm

• 3.1 Model The model considers N point particles of unit mass, m = 1, confined in a line of length L = 1. Periodic boundary condition was implemented, so that the particles lie on a circle of unit circumference.

Figure 3.1.1. Picture of the one-dimensional system of particles. Periodic boundary conditions are used, and the particle’s diameter depends on the input parameters of the simulation. Adapted from A Study of Granular Gas using a One-Dimensional Model:Computer Simulation by K. Gargar [14]  

When two particles i and j collide in this one-dimensional system, the final (primed) velocities are given in terms of the initial (unprimed) velocities by

and Eqs. 1

Assumptions/Idealizations: • Inelastic hard sphere contact duration is implicitly zero assures the conservation of mass in each collision

• Freely-Cooling System Energy is continuously dissipated through each collision and the system is freely evolved until the system cools down and eventually the collision of each particle stop

• Constant Coefficient of Restitution assumption of a constant coefficient of restitution r = const. is very helpful when performing calculations since it simplifies the mathematics significantly. situations arise when the assumption of a constant coefficient of restitution is well justified, provided there is a narrow velocity distribution.

• Identical Size and Mass to simplify calculations

• 3.2 Simulation Details The algorithm used is an event driven algorithm. -The collisions are the events which have to be treated by the algorithm. The algorithm processes the events one after the other. After a collision the positions and velocities of the two involved particles are updated, the state of all other particles remains unchanged.

- The researcher utilized a high programming language QBasic (Beginner's All-purpose Symbolic Instruction Code, it is formerly known simply as Basic) to do the simulation. QBasic was chosen because of its high applicability to numerical simulations and the syntax used in writing the algorithm is easily understood.

• 3.3 Algorithm The algorithm developed for this study is divided into five phases. These are (a) initialization, (b) calculation of minimum time, (c) evolution of the system, (d) minimum time update, and (e) finalization.

Figure 3.3.1 Flow chart of the simulation algorithm.

• 3.3.1 Initialization Input parameters are use to initialize the system. The system will run accordingly to the value of inputs supplied to the algorithm. These input parameters are: a. number of collision b. number of particles c. coefficient of restitution d. solid fraction

• Also in the initialization phase, the initial velocity and position of each particle are randomly and uniformly distributed respectively.

• 3.3.2 Calculation of minimum time Event driven algorithm processes the events (collision) particle by particle. The time of collision is calculated for every adjacent pair of particles in the system. The least time obtained from the calculations was gathered and recorded. Collision of particles will stop when the least time of collision is already recorded.

• 3.3.3 Evolution of the system The program calculates the total kinetic energy of the system as well as the mean free path.

• 3.3.4 Minimum time update The time of collision obtained in phase two of the program was updated every collision. Updates of collision time for every particle was searched and only the least time obtained from this calculation was considered.

• 3.3.5 Finalization The steps performed in phases 1-4 of the algorithm was repeated 33 times to obtain number of samples that will be treated statistically to yield reliable results

• Data gathered were processed using Microsoft Excel and SigmaPlot 10.0. Scaling distribution function is used for the obtained granular gas distribution. Scaling technique is adopted from the work of Brito and Ernst [7], and is widely used in the study of the velocity distribution of granular gas.

CHAPTER 4: RESULTS and DISCUSSION 4.1 Kinetic Energy and Coefficient of Restitution 4.2 Kinetic Energy and Solid Fraction 4.3 Inelastic Collapse and Clustering 4.4 Non-Gaussian Velocity Distribution 4.5 Agreement with Haff’s Law

• 4.1 Kinetic Energy and Coefficient of Restitution Energy dissipation with different r is shown below

The kinetic energy of the system as a function of number of collision for a freely-cooling inelastic gas with different r. Number of particles is set to 210 , number of collision is 50000 and solid fraction is set to 0.1  

• It is evident that energy dissipates when coefficients of restitution value are less than 1. When the value of r is set to 1, it is observed that there is no energy dissipation as what is expected for a perfectly elastic case. The energy decay for a granular gas collision is proportional to the degree of inelasticity. As the value of r decreases, that is approaching a perfectly inelastic limit, the energy dissipation from initial configuration to its final state intensifies.

• 4.2 Kinetic Energy and Solid Fraction Figure 4.2.1 shows the effect of solid fraction on the kinetic energy of the granular gas system.

Figure 4.2.1. The kinetic energy of the system as a function of number of collision for a freely-cooling inelastic gas with different s. Number of particles is set to 29, number of collision is 1200 and coefficient of restitution is set to 0.99.

• 4.3 Inelastic Collapse and Clustering The energy of the granular gas system, as evident on the results of the simulation, continuously decreases as number of collisions is raised. It is expected that it will assume a constant value after the particles experienced a finite amount of collisions. This behavior is called the inelastic collapse. The consequence of this phenomenon is that after a certain amount of collision, and the energy of the system is already assuming a less fluctuating value, clustering could occur.

• Figures 4.3.3 to 4.3.5 shows the evolution of cluster as the number of collision is raised. 1200, 25000 and 50000 number of collisions are observed.

Figure 4.1.3. Position velocity plot for 32 particles after 1,200 collisions. Coefficient of restitution is set to 0.3 and solid fraction is 0.1. A single cluster is evident with these conditions.

Figure 2.3.4. Position velocity plot for 32 particles after 25,000 collisions. Coefficient of restitution is set to 0.3 and solid fraction is 0.1. Number of clusters increase from the previous figure.

Figure 4.3.5. Position velocity plot for 32 particles after 50,000 collisions. Coefficient of restitution is set to 0.3 and solid fraction is 0.1. Clustering is more evident and particles tend to compress more in a single region in space.

• Saturation of clusters happens when all the particles have the same velocity and the total energy of the system already attain its steady value. This fact is evident on a freelycooling granular gas system. The system cools down until the particles tend to “crystallize” or assemble together into a one huge cluster having a constant value of kinetic energy.

• 4.4 Non-Gaussian Velocity Distribution One basic property of ordinary gases is the velocity distribution, which is a Maxwell Boltzmann or a Gaussian distribution. Granular gas systems however deviate from the Gaussian distribution that one would expect if the collisions were elastic.

• Figure 4.4.1 shows the velocity distribution of 1024 granular gas particles simulated to 50000 collisions and with r = 0.1 as the degree of inelasticity

Figure 4.4.1 Velocity distribution of the granular gas for N = 1024 and r = 0.1.

• It is shown that the velocity distribution has a fluctuating right end-tail as compared to a Maxwell Boltzmann distribution which has an asymptotic flat right-end tail. • Fluctuating tails, as what is observed on the simulated granular gas, corresponds to the clusters of particles having different velocities. This fluctuation is shown as a peak on the distribution of granular gas.

• Emphasizing the non-Gaussianity of the distribution is shown below 1.0

Relative Probability

0.8

0.6

0.4

0.2

0.0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Velocity Granular Gas Velocity Distribution Gaussian Distribution

Figure 4.4.2. Scaled velocity distribution for the simulated granular gas. Scaling function used was adapted from Brito and Ernst [3]. Gaussian fitting is inserted to the figure for the comparison of the two velocity distributions. R2 for the fit is equal to 0.9887.

• The fact that the granular gas has a non-Maxwellian and a non-Gaussian distribution has the following implication. If the velocity distribution is not Gaussian, then there are correlations between the components of the velocity distribution.

• Figure below shows the linearize probability versus the velocity of the granular gas.

Figure 4.4.3. . Linearize plot of the relative probability distribution function versus velocity for the coefficient of restitution r=0.3 and N = 1024 particles. Strong correlation was observed for extremely low velocities (v<0.2) and a fluctuating correlation for high velocity particles (v>0.35).  

• 4.5 Agreement with Haff’s Law Dissipative collisions lead a decay of kinetic energy as what is evident on part 4.1. Haff’s law states that energy dissipated are proportional to t-2.

83 82

Kinetic Energy

81 80 79 78 77 76 0.00116

0.00118

0.00120

0.00122

0.00124

0.00126

0.00128

Collision Time Kinetic energy vs Collision Time Col 6 vs Col 7

•

Figure 4.5.1. Kinetic energy as a function of collision time for r = 0.8. Number of particles is 512 and solid fraction is set to 0.1. Number of collision is set to 50000. A curve for Haff’s Law is fitted to compare the kinetic energy decay curve with the law.

•

Figure above shows the graph for the energy decay as a function of time. Haff’s law is fitted to the curve to show the comparison of the data obtained with the given law. Standard deviation of 0.0284 is obtained for the energy decay curve as fitted on the Haff’s Law curve.

CHAPTER 5: CONCLUSIONS AND RECOMMENDATIONS

• 5.1 CONCLUSIONS • 5.2 RECOMMENDATIONS

• 5.1 CONCLUSIONS b. It is concluded that granular gas deviates considerably with the ordinary molecular gases. c. Granular gas behavior depends primarily on two main parameters, namely: coefficient of restitution and solid fraction. It is observed that large values of coefficient of restitution correspond to large decay of energy when simulated at a certain number of collisions. Also, it is evident that, as the solid fraction of the granular gas system takes a large value, the granular gas system tends to collapse (inelastic collapse) faster. Thus the formation of cluster is faster on a dense system.

c. It is also concluded that the velocity distribution of a freely-cooling granular gas system is non-Maxwellian and non-Gaussian. d. The energy decay of the simulated granular gas conforms to the Haff’s Law. It is concluded that in the regime of the number of collisions observed, the energy decay still follows the t-2 rule.

• 5.2 RECOMMENDATIONS a. It is recommended that future studies on the 2-D and 3-D case shall follow. b. Study on granular gas wherein heat inputs are applied is also suggested. c. Utilization of super-fast computers is highly recommended to deal with the longtime behavior (such as 1015 number of collisions) of the granular gas.

d. Analytical and experimental studies are also suggested to further understand the physics of granular gas. e. Detailed investigation on the effect of varying coefficient of restitution on the velocity distribution of granular gas system is recommended.

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