Shock propagation in locally driven granular systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Shock propagation in locally driven granular systems
Jilmy P. Joy,
1, 2, ∗ Sudhir N. Pathak, † Dibyendu Das, ‡ and R. Rajesh
1, 2, § The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai-600113, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India Division of Physics and Applied Physics, School of Physical andMathematical Sciences, Nanyang Technological University, Singapore Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai-400076, India (Dated: September 26, 2018)We study shock propagation in a system of initially stationary hard spheres that is driven by acontinuous injection of particles at the origin. The disturbance created by the injection of energyspreads radially outward through collisions between particles. Using scaling arguments, we deter-mine the exponent characterizing the power law growth of this disturbance in all dimensions. Thescaling functions describing the various physical quantities are determined using large-scale event-driven simulations in two and three dimensions for both elastic and inelastic systems. The resultsare shown to describe well the data from two different experiments on granular systems that aresimilarly driven.
PACS numbers: 45.70.Qj, 45.70.-n, 47.57.Gc
I. INTRODUCTION
Granular materials are ubiquitous in nature. Exam-ples include geophysical flows [1], large-scale structureformation of the universe [2], sand dunes [3], craters [4],etc. The dissipative nature of the interactions amongthe constituent particles can lead to diverse physical phe-nomena such as pattern formation, clustering instability,granular piles, jamming, segregation, stratification, shearflows, surface waves, fingering instability, and fluidization(see the reviews in [5–7]). A subclass of problems thathave been of experimental and theoretical interest is theresponse of a granular system at rest to an external per-turbation that is applied either as an instantaneous im-pulse or continuously in time. This phenomenon has beenstudied in many different contexts, examples of which in-clude avalanches in sand piles as a response to the addi-tion of sand grains [8], crater formation on granular bedsdue to the impact of an external object [9, 10], growingcraters due to impinging jets on granular piles [11], shockformation in flowing granular media due to external im-pact [10], viscous fingering due to constant injection ofparticles [12–16], and formation of bastwaves in astro-physical systems [17]. The externally applied perturba-tion often results in a disturbance that grows in timeas a power law and the power-law exponents may oftenbe obtained by studying simple tractable models of suit-ably excited spherical particles where energy dissipationis only through inelastic collisions [18]. We discuss be-low the response to perturbation in the context of suchmodels.One of the most commonly studied examples is the ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] globally perturbed freely cooling granular gas, where ho-mogeneously distributed macroscopic particles with ran-dom initial velocities move ballistically and dissipate en-ergy through inelastic collisions, in the absence of any ex-ternal driving. Here the perturbation is the energy thatis initially given. In the early stage of evolution, whenthe system is spatially homogeneous, kinetic energy of thesystem E ( t ) decays with time t as t − (Haff’s law) [19] inall dimensions. At later times, due to inelastic collisions,the system becomes spatially inhomogeneous [20, 21] andenergy decreases as t − θ d , where θ d is less than 2 and de-pends on dimension d [22–31]. Haff’s law for the homoge-neous regime has been confirmed in experiments [32, 33],while θ d characterizing the inhomogeneous regime hasstill not been observed in any experiment.A different limit is the locally perturbed freely cool-ing granular gas, where initially all particles are at restand kinetic energy is imparted to a few localized par-ticles. Due to collisions, the disturbance grows radiallyoutward, with a shock front separating the moving par-ticles from the stationary ones. The elastic version ofthis problem has great similarity to the problem of shockpropagation following an intense explosion. The hydro-dynamic description of the propagation in a conservativefluid is the famous Taylor-von Neumann-Sedov (TvNS)solution [34–36]. This solution is relevant in the experi-mental studies of the production of a cylindrically sym-metric blast wave produced by ultrafast laser pulses [37].Numerical simulations of the elastic system are consis-tent with the TvNS exponents [38, 39]. In the inelasticsystem, the disturbance is concentrated in dense bandsthat move radially outward, and the relevant exponentsmay be obtained through scaling arguments based on theconservation of radial momentum [18, 39, 40]. The varia-tion of physical quantities inside the dense band may beobtained through a hydrodynamic description [41, 42].The exponents obtained thus may be used to describe [18]experiments on shock propagation in flowing glass beadsthat are perturbed by the impact of steel balls [10].In both cases discussed above, the perturbation wasan impulse. One could also consider continuous and lo-cally perturbed driven granular systems, where particlesat rest are driven by a continuous injection of energy ina small domain. This scenario has been investigated inmany recent experiments and includes pattern formationin granular material due to the injection of a gas [12, 15],grains [14], or fluid [16]. There is currently no modelthat determines the exponents for such situations. Inthis paper we study a simple model of spheres at restthat is driven at the origin by a continuous injection ofparticles from outside. From a combination of event-driven simulations and scaling arguments, we determinethe exponents governing the growth of the disturbance.The results are compared with the data from two exper-iments [12, 15] and excellent agreement is obtained.The remainder of the paper is organized as follows. InSec. II we define the model precisely and give details ofthe event-driven simulations that we performed. The ex-ponents characterizing the growth of the different physi-cal quantities in the problem are determined using scalingarguments in Sec. III. The assumptions and predictionsof the scaling argument are tested using large-scale sim-ulations in Sec. IV for both the elastic and the inelasticsystem. In Sec. V we show that the results in this paperare able to explain data from two experiments on drivengranular systems. Section VI contains a brief summaryand a discussion of results. II. MODEL
Consider a d -dimensional system of hard spheres whosemass and diameter are set to one. The particles moveballistically until they undergo momentum-conserving bi-nary collisions with other particles. If ~u and ~u are thevelocities of two particles 1 and 2 before collision, thenthe velocities after collision, ~v and ~v , are given by ~v = ~u − r n · ( ~u − ~u )]ˆ n, (1) ~v = ~u − r n · ( ~u − ~u )]ˆ n, (2)where r is the coefficient of restitution and ˆ n is the unitvector along the line joining the centers of particles 1and 2. In a collision, the tangential component of therelative velocity remains unchanged, while the magnitudeof the longitudinal component is reduced by a factor r .The collisions are elastic when r = 1, and inelastic anddissipative otherwise.Initially, all particles are at rest and uniformly dis-tributed in space. The system is driven locally by acontinuous input of energy restricted to a small regionby injecting particles at a constant rate J at the origin.The injected particles have a speed v in a randomlychosen direction until they undergo their first collision,after which the injected particles are removed from the system. Driving in this manner injects energy into thesystem, but conserves the total number of particles. Wewill refer to this model as the conserved model.We also consider a nonconserved model. This model isidentical to the conserved model described above, but theinjected particles stay in the system, thereby increasingthe total number of particles at a constant rate J . Whilethe conserved model is applicable to two-dimensionalgranular systems driven by a gas (where the gas mayescape in the third dimension), the non-conserved modelis applicable to two-dimensional granular systems drivenby granular material. We will show in Sec. III that thescaling laws at large times are identical for both models.We will therefore present numerical results only for theconserved model.We simulate systems with number density 0 .
25 (pack-ing fraction 0 . .
40 (packingfraction 0 . × and is large enoughsuch that the disturbance induced by the injection of par-ticles does not reach the boundary up to the simulationtimes considered in this paper. We set v = 1, the rateof injection of particles J is set to 1, and the injectedparticles have the same mass and diameter as the otherparticles in the system. In the simulations, the collisionsare inelastic with constant restitution coefficient r whenthe relative velocities of the particles are greater than acut off velocity δ and considered to be elastic otherwise.This procedure prevents the occurrence of the inelasticcollapse of infinite collisions within a finite time, whichis a hindrance in simulations, and is also in accordancewith the fact that the coefficient of restitution tends to1 with decreasing relative velocity between the collidingparticle [44]. The value of δ is 10 − , unless specified oth-erwise. The results are independent of δ .The numerical results in this paper are shown onlyfor the conserved model and are typically averaged over48 different realizations of the initial particle configura-tions. All lengths are measured in units of the particlediameter and time in units of initial mean collision time t = v − n − /d , where n is the number density. III. SCALING ARGUMENT
In order to develop scaling arguments to describe thepropagation of energy, it is important to first visualizehow the inelastic system evolves in comparison to theelastic system. When the energetic particles are injectedfrom the center, in both cases particles get disturbed upto a distance and the zone of disturbance propagates radi-ally outward. Figures 1 and 2 show the time evolution ofthe elastic and inelastic systems with r = 1 and r = 0 . (a) (b)(c) (d) FIG. 1. (Color online) Moving (red) and stationary (blue)particles at times (a) t = 500, (b) t = 1000, (c) t = 1500 and(d) t = 2000. Energetic particles are injected at the center.All collisions are elastic with r = 1. The data are for theconserved model. contrast, in the case of the inelastic system, particles clus-ter together and form a dense band adjacent to the frontof the disturbance, forming a vacant region around thecenter. This circular band moves outward with time andgrows by absorbing more particles. We observe the samefeatures in the simulations of the nonconserved model.We look for scaling solutions, similar to that found forthe problem with a single impact in Ref. [39]. Let R t be the typical radius of the disturbance at time t . Weassume that it is the only relevant length scale in theproblem. We assume a power-law growth for the radiusof disturbance, R t ∼ t α . The typical velocity v t is thengiven by, v t ∼ dR/dt ∼ t α − . The total number of mov-ing particles that have undergone collisions N t is given bythe volume swept out by the disturbance in the conservedmodel, and the sum of the volume swept out by the dis-turbance and the injected particles for the nonconservedmodel. The volume swept out by the disturbance scalesas R dt ∼ t αd , where d is the spatial dimension, while thenumber of injected particles scales as Jt . Therefore, inthe limit of large time, N t ∼ R dt ∼ t αd for the conservedmodel and N t ∼ R dt ∼ t max[ αd,t ] for the nonconservedmodel. We discuss the two models separately. (a) (b)(c) (d) FIG. 2. (Color online) Moving (red) and stationary (blue)particles at times (a) t = 1000, (b) t = 2000, (c) t = 4000 and(d) t = 8000. Energetic particles are injected at the center.All collisions are inelastic with r = 0 .
1. The data are for theconserved model.
A. Conserved model
The energy of the system scales as E t ∼ N t v t ∼ t α ( d +2) − . (3)The exponent α may be determined for the elastic andinelastic cases using different conservation laws. For theelastic system, energy is not dissipated during collisions.However, due to the constant driving, the total energymust increase linearly with time, i.e., E t ∼ t . Comparingit with the scaling behavior of energy E t ∼ t α ( d +2) − , weconclude α = 3 d + 2 , r = 1 . (4)This result coincides with the power-law scaling exponentobtained in the case of astrophysical blast waves [17].For the inelastic system, the total energy is no longerconserved. However, the formation of the bands, as canbe seen in Fig. 2, implies that there is no transfer of mo-mentum from a point in the band to a point diametricallyopposite to it by particles streaming across. Thus, oncethe bands form, radial momentum is conserved duringcollisions and flows radially outward [18, 39]. Due to thecontinuous driving, the radial momentum must increaselinearly with time t [45]. We confirm this in simulationsby measuring radial momentum as the sum of the radialvelocities of all the moving particles. As shown in Fig. 3, R a d i a l m o m e n t u m t d = 2d = 3 -1 R a d i a l m o m e n t u m tt d=2d=3 FIG. 3. (Color online) Radial momentum as a function of time t for two and three-dimensional inelastic systems, showing alinear increase. The inset shows the data on a log-log scale,which show an initial transient regime before the linear growthis attained. The data are for the conserved model. radial momentum increases linearly with time, at largetimes, in both two and three dimensions. There is an ini-tial transient period (see the inset of Fig. 3), where theinitial growth is not linear, reflecting the time taken toform stable dense bands. The radial momentum, in termsof the exponent α , scales as N t v t ∼ t α ( d +1) − . Compar-ing it with the linear increase in t , we obtain α = 2 d + 1 , r < . (5) B. Non-conserved model
We show that the non-conserved model has the samescaling laws as described in Eqs. (4) and (5). The energyof the system scales as E t ∼ N t v t ∼ t max[ αd, α − . (6)In the elastic case, energy is conserved and E t ∼ t . Com-paring with Eq. (6), we obtain α = 3 / ( d + 2) if αd ≥ α = 1 if αd <
1. For d ≥
1, the only solution is α = 3 / ( d + 2), as obtained for the conserved model [seeEq. (4)].For the inelastic case, the radial momentum increaseslinearly with time (see Sec. III A). The radial momentumscales as N t v t ∼ t max[ αd, α − . Comparing it with thelinear increase in t , we obtain α = 2 / ( d + 1) if αd ≥ α = 1 if αd <
1. For d ≥
1, the only solution is α = 2 / ( d + 1), as obtained for the conserved model [seeEq. (5)].We conclude that the scaling laws are identical forboth the conserved and non-conserved models. In the re-maining part of the paper, we discuss only the conservedmodel. IV. NUMERICAL RESULTS
All the numerical results presented in this section arefor the conserved model. The results for the noncon-served model are similar and omitted for the sake ofbrevity.
A. Elastic
We first show that the power-law growth of the shockradius R t , the number of moving particles N t , and thetotal energy E t , as obtained in Sec. III using scaling argu-ments, is correct, using event-driven molecular dynamicssimulations. For the elastic system, the scaling argu-ments predict R t ∼ t / , E t ∼ t , and N t ∼ t / in twodimensions and R t ∼ t / , E t ∼ t , and N t ∼ t / inthree dimensions. The results from simulations, shownin Figs. 4(a)–4(c) for R t , E t , and N t , respectively, are inexcellent agreement with the above scaling and confirmthe value of the exponent α as given by Eq. (4).The scaling argument leading to the exponent inEq. (4) assumes the existence of only one length and onevelocity scale, and leads to the correct scaling of the bulkquantities R t , N t , and E t with time. This assumptionmay be further checked by studying the scaling behaviorof local space-dependent physical quantities. We definecoarse-grained radial density distribution function ρ ( r, t )as the number of moving particles per unit volume, lo-cated within a shell of radius of r to r + dr . Similarly, theradial velocity distribution function v ( r, t ) and the radialenergy distribution function e ( r, t ) are defined as the av-erage radial velocity of particles and the average kineticenergy per unit volume, respectively, contained withinthe shell at any time t . We expect these local coarse-grained quantities to have the following scaling forms: ρ ( r, t ) ∼ f ρ ( r/t α ) ,v ( r, t ) ∼ t α − f v ( r/t α ) ,e ( r, t ) ∼ t − β f e ( r/t α ) , (7)where β = 2(1 − α ), since e scales as v .In Figs. 5(a) and 5(b), when ρ ( r, t ) for the elastic sys-tem is plotted against the scaled distance r/t α , the datafor different times collapse onto a single curve for α = 3 / α = 3 / r/t α ≈
1) and scaleddistances approximately equal to 0 .
5. However, the curveis nonzero and decreases to zero (as a power law) for smalldistances. Thus the region of disturbed particles doesnot have an empty core, unlike the case of the inelasticsystem, as we will see below. From Figs. 5(c)–5(f) weobserve that data for v ( r, t ) and e ( r, t ) also collapse ontoa single curve in both two and three dimensions whenscaled as in Eq. (7) with the same values of α . Both R t t t (a) d = 2d = 310 E t tt(b) d = 2d = 310 N t t t t (c) d = 2d = 3 FIG. 4. (Color online) Simulation results for the elastic sys-tem ( r = 1) for the temporal variation of (a) radius R t , (b)kinetic energy E t , and (c) number of moving particles N t intwo and three dimensions. The solid lines are power laws withexponents as predicted by the scaling arguments presented inthe text. The data are for the conserved model. radial velocity and density initially increase as the dis-tance from the shock front increases. This leads to morecompaction near the shock front due to faster particlespushing against the slower particles. Finally, in order tounderstand better the direction of motion of the particlesin this driven gas, we calculate the distribution functionof h cos θ ( r, t ) i , where θ is the angle made by the instanta-neous particle velocity with respect to the outward unitradial vector at its location, and the averaging is per-formed over all particles contained within the shell fromradius r to r + dr . In Figs. 5(g) and 5(h) we see thatfor small values of the scaled distance less than 0 . .
8, its value is positive and close to 1, forboth two and three dimensions. This implies that nearthe shock front the particles are mostly directed radiallyoutward, while near the center of the sphere the particlesare on average moving inward, a feature related to thefact that the particle collisions are elastic. The inward-moving particles are responsible for the transfer of radial ρ (a) d=2t=2500t=5000t=10000 0 0.2 0.4 0.6(b) d=3t=1000t=2000t=3000 0 0.1 0.2 0.3 0.4 0.5 t - α v r (c) d=2t=2500t=5000t=10000 0 0.1 0.2 0.3(d) d=3t=1000t=2000t=3000 0 0.02 0.04 0.06 0.08 e t β (e) d=2t=2500t=5000t=10000 0 0.02 0.04 0.06 0.08(f) d=3t=1000t=2000t=3000-0.4 0 0.4 0.8 0 0.2 0.4 0.6 0.8 1 〈 c o s θ 〉 r/t α (g) d=2t=2500t=5000t=10000 0 0.2 0.4 0.6 0.8 1 -0.4 0 0.4 0.8 r/t α (h) d=3t=1000t=2000t=3000 FIG. 5. (Color online) Scaled radial distribution functionsagainst scaled distances r/t α for the elastic gas: (a) ρ ( r, t ),(c) v ( r, t ), (e) e ( r, t ), and (g) h cos θ ( r, t ) i in two dimensionsand (b), (d), (f), and (h) corresponding quantities in threedimensions. Here α = 3 / ( d + 2), as in Eq. (4), and β =2(1 − α ). The data are for the conserved model. momentum across the origin and lead to the breakdownof conservation of radial momentum in a particular di-rection. B. Inelastic
Now we turn to the case more relevant to granular mat-ter, namely, of systems with particles suffering inelastic collisions. The scaling dependence on time t of variousquantities in such systems relies on the basic assump-tion of radial momentum growing linearly as a functionof time t (see Sec. III). In Fig. 2 we saw that the per-turbed particles cluster in an outward moving narrowband. For the inelastic system, the scaling argumentspredict R t ∼ t / , E t ∼ t / , and N t ∼ t / in two di-mensions and R t ∼ t / , E t ∼ t / , and N t ∼ t / inthree dimensions. The results from simulations, shownin Fig. 6(a)–6(c) for R t , E t , and N t , respectively, are inexcellent agreement with the above scaling and confirmthe value of the exponent α as given by Eq. (5).Next we study the radial distribution functions for the R t t t (a)d = 2d = 310 -1 E t t t (b)d = 2d = 310 N t t t t (c)d = 2d = 3 FIG. 6. (Color online) Simulation results for the inelasticsystem ( r = 0 .
1) for the temporal variation of (a) radius R t ,(b) kinetic energy E t , and (c) number of moving particles N t in two and three dimensions. The solid lines are powerlaws with exponents as predicted by the scaling argumentspresented in the text. The data are for the conserved model. inelastic gas and compare them with the elastic casesconsidered in Sec. IV A. The data for the different dis-tributions for different times collapse onto a single curvewhen scaled as in Eq. (7) with α as in Eq. (5) for bothtwo dimensions [see Figs. 7(a), 7(c), 7(e), and 7(g)] andthree dimensions [see Figs. 7(b), 7(d), 7(f), and 7(h)].From Figs. 7(a) and 7(b) we see that the particle den-sity is highly localized between scaled distances 0 . h cos θ ( r, t ) i approaches thevalue 1 [see Figs. 7(g) and 7(h)]. Like for the elastic case,the radial velocity increases as one moves away from theshock front, stabilizing the dense bands containing the ρ (a)d=2t=5000t=10000t=20000 0 0.4 0.8 1.2(b)d=3t=2500t=5000t=10000 0 0.2 0.4 0.6 t - α v r (c)d=2t=5000t=10000t=20000 0 0.1 0.2 0.3 0.4(d)d=3t=2500t=5000t=10000 0 0.05 0.1 0.15 0.2 0.25 e t β (e)d=2t=5000t=10000t=20000 0 0.02 0.04 0.06 0.08(f)d=3t=2500t=5000t=10000 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 〈 c o s θ 〉 r/t α (g)d=2t=5000t=10000t=20000 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 r/t α (h)d=3t=2500t=5000t=10000 FIG. 7. (Color online) Scaled radial distribution functionsagainst scaled distances r/t α for the inelastic gas: (a) ρ ( r, t ),(c) v ( r, t ), (e) e ( r, t ), and (g) h cos θ ( r, t ) i in two dimensionsand (b), (d), (f), and (h) corresponding quantities in threedimensions. Here α = 2 / ( d + 1), as in Eq. (5), and β =2(1 − α ). The data are for the conserved model. particles. V. COMPARISON WITH EXPERIMENTS
There are quite a few experiments [12–16] that studypattern formation in a layer of granular matter drivenlocally at the center through the injection of another ma-terial, gas or liquid, but not all of them study physicalquantities, which is relevant for the predictions of this pa-per. In this section we discuss two experiments that pro-vide quantitative data on driven granular particles andwe show how our scaling theory and simulations providean explanation for the radial growth law as seen in theseexperiments.The first experiment of interest is pattern formationin spherical glass beads that are distributed uniformlywithin a circular Hele-Shaw cell [12]. The beads, ini-tially at rest, were perturbed by the continuous injec-tion of pressurized nitrogen through a hole at the centerof the bottom plate of the cell. The driving was uni-form (similar to what we assume in this work). The cell -1 -1 R / R t/t tt t ∆ P=0.15 atm ∆ P=0.20 atm ∆ P=0.34 atm ∆ P=0.68 atm ∆ P=1.02 atm
FIG. 8. (Color online) Experimental data (taken fromRef. [12]) for the scaled radius R of the longest finger fromthe center, as a function of normalized time t/t . Here R = R ( t ). The data have been plotted for different gasoverpressures. The solid lines are power laws t / , t / , and t and are shown for reference. boundary was open so that any bead driven to the edgecould freely flow out of the cell. The patterns formedwere recorded with high-speed camera. When the driv-ing pressure was high enough, the continuous pertur-bation led to the formation of a time-dependent grow-ing viscous fingering pattern. Our interest is the radialgrowth law of this pattern in the early stage; at the latestage beyond some characteristic time t , a wild growthin radius due to effect of boundaries is seen, which isnot of interest in this paper. We replot the publisheddata [Fig. 1(c) in Ref. [12]] in Fig. 8 for scaled radius R/R against scaled time t/t , where R = R ( t ). Quitestrikingly, we find that the data converge close to thepower law R t ∼ t / , as shown in Fig. 8, consistent withour theoretical prediction for the two-dimensional inelas-tic system [see Eq. (5)]. However, the scaling analysisassumes that the only means of dissipation is inelastic-ity. The experiment has dissipative frictional forces too,but it is evident from the data being consistent with thepower law that possibly the frictional effect is nullifiedby the critical pressure, beyond which beads start mov-ing, and eventually inelasticity remains as the dominantmechanism of dissipation. We note that the experimen-tal paper [12] erroneously mentions a linear growth ofradius, but it is clear that the line proportional to t inFig. 8 describes the data poorly. We also note that thepower law t / in Fig. 8 is a poorer fit to the data thanthe power law t / .We look at another similar experiment with granularmaterial confined in a circular Hele-Shaw cell with cen-tral air injection [15]. When the injection pressure issufficient enough, the particles in the system move outby forming a central (roughly circular) region devoid ofparticles. Around this central region, there is a zonewhere the granular material is compacted. The patterns r a d i a l l e ng t h ( c m ) time (s)t t t P=2.58kPaP=2.20kPa FIG. 9. (Color online) Experimental data (taken fromRef. [15]) for the growth of maximum radial coordinate ofthe central zone of disturbance with time for two differentvalues of injection pressures. The solid lines are power laws t / , t / , and t and are shown for reference. formed have been recorded by using a high-speed, high-resolution CCD camera. The data obtained from thisexperiment [Fig. 13(a) in Ref. [15]] also follow the powerlaw R t ∼ t / as shown in Fig. 9, consistent with ourgrowth-law exponent [see Eq. (5) with d = 2]. We notethat the power laws t / and t in Fig. 9 are poorer fitsto the data than the power law t / . Thus, again we seethat the simple scaling law obtained the from dominanceof inelastic dissipation, and band formation, is experi-mentally relevant. VI. CONCLUSION AND DISCUSSION
We studied shock propagation in a granular systemthat is continuously driven in a localized region. Weanalyzed both the elastic and inelastic systems throughscaling arguments and extensive event-driven moleculardynamics simulations. By identifying that energy growslinearly in the elastic system and radial momentum growslinearly in the inelastic system, the exponents governingthe power-law growth of bulk quantities such as radiusof disturbance and number of moving particles were ob-tained. For the inelastic system, the linear growth of ra-dial momentum crucially depended on the formation ofdense bands enclosing an empty region, due to inelasticcollision, as seen in the simulations. There are very fewdriven granular systems where exact results can be ob-tained. The solution in this paper provides an examplewhere the exponents, presumably exact, may be deter-mined through scaling arguments.We analyzed two experiments on pattern formationthat arise due to the injection of a gas at localized pointin a two-dimensional granular medium. The experimen-tally obtained radial growth of the pattern was shown tobe consistent with the results in this paper, even thoughthe present study ignores friction that would appear tobe relevant in experiments. The experimental patternsshow the formation of bands that have fractal structure,which is not captured by our model. However, the de-tailed structure of the bands does not play a role in de-termining the growth-law exponent, as the scaling argu-ments required only conservation of radial momentum,which in turn depends only on the existence of a bandenclosing an empty region and not on its structure.We described numerical results for the model wherethe injected energetic particles were removed from thesystem after their first collision. However, we presentedscaling arguments to show that the power-law exponentsfor the nonconserved model, in which the injected en-ergetic particles remain in the system, are identical tothat of the conserved model. Simulations are also consis-tent with the predictions of scaling theory. Such modelsmay be valid for experiments where granular material isdriven through injection of other granular material.Unlike the power-law exponents, it does not appearto be possible to analytically determine the form of the scaling functions for the different local densities. For theelastic system, one might ask whether the TvNS solu-tion [34–36] that describes shock propagation followingan intense blast may be modified to the case of continuousdriving. The local conservation laws of density, energy,and momentum continue to hold for localized continuousdriving away from the source. However, we find in ourpreliminary studies that the solution develops singulari-ties at a finite distance between the origin and the shockfront. This could be because the additional assumptionof local thermal equilibrium made in the TvNS solutionmay not hold when the driving is continuous. A detailedanalysis of the elastic case is a promising area for futurestudy.
ACKNOWLEDGMENTS
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