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1、arXiv:cond-mat/0006181v1 cond-mat.soft 12 Jun 2000 Continuum description of avalanches in granular media Igor S. Aranson1and Lev S. Tsimring2 1 Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439 2 Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA
2、92093-0402 (February 6, 2008) We develop a continuum description of partially fl uidized granular fl ows. Our theory is based on the hydrodynamic equation for the fl ow coupled with the order parameter equation which describes the transition between fl owing and static components of the granular sys
3、tem. This theory captures important phenomenology recently observed in experiments with granular fl ows on rough inclined planes (Daerr and Douady, Nature (London) 399, 241 (1999): layer bistability, and transition from triangular avalanches propagating downhill at small inclination angles to balloo
4、n-shaped avalanches also propagating uphill for larger angles. PACS: 45.70.-n, 45.70.Ht, 45.70.Qj, 83.70.Fn Fundamental understanding of the dynamics of gran- ular media still poses a challenge for physicists 13 and engineers 4. The intrinsic dissipative nature of the in- teractions between the cons
5、tituent macroscopic particles sets granular matter apart from conventional gases, liq- uids, or solids. One of the most interesting phenomena pertinent to the granular systems is the transition from a static equilibrium to a granular fl ow. The most spec- tacular manifestation of such a transition o
6、ccurs during an avalanche. There has been a number of experimental studies of avalanche fl ows in large sandpiles 8,9 as well as in thin layers of grains on rough inclined surfaces 57. On the theoretical side, a signifi cant progress had been achieved by large-scale molecular dynamics simulations 10
7、,11 and by continuum theory 1215. The current continuum approach to the description of avalanche fl ows in the physics community was pioneered by Bouchaud, Cates, Ravi Prakash and Edwards (BCRE) 13 and sub- sequently developed by de Gennes, Boutreux and and Rapha el 12,14,15. In their model is the g
8、ranular sys- tem is spatially separated into two phases, static and rolling.The interaction between the phases is imple- mented through certain conversion rates.This model described certain features of thin near-surface granular fl ows including avalanches. However, due to its intrinsic assumptions,
9、 it only works when the granular material is well separated in a thin surface fl ow and an immo- bile bulk. In many practically important situations, this distinction between “liquid” and “solid” phases is more subtle and itself is controlled by the dynamics. In this Letter we propose a new continuu
10、m model for multi-phase granular matter. The underlying idea of our approach is borrowed from the Landau theory of phase transitions 16. We assume that the shear stresses in a partially fl uidized granular matter are composed of two parts: the dynamic part proportional to the shear strain, and the s
11、train-independent (or “static”) part. The rel- ative magnitude of the static shear stress is controlled by the order parameter (OP) which varies from 0 in the “liquid” phase to 1 in the “solid” phase. Unlike ordinary matter, the phase transition in granular matter is con- trolled not by the temperat
12、ure, but the dynamics stresses themselves. In particular, the Mohr-Coloumb yield fail- ure condition 4 is equivalent to a critical melting tem- perature of a solid. The OP can be related to the local entropy 17 of the granular material. OP dynamics is then coupled to the hydrodynamic equation for th
13、e gran- ular fl ow. We apply this model to study the transition to fl ow in thin granular layer on inclined planes with rough bottom. Our model captures important phenomenology observed by Pouliquen 7 and Daerr and Douady 5, in- cluding the structure of the stability diagram, triangular shape of dow
14、nhill avalanches at small inclination angles and balloon shape of uphill avalanches for larger angles. Model. The continuum description of the granular fl ow is based on the Navier-Stokes equation 0Dvi/Dt = ij xj + 0gi, j = 1,2,3.(1) where viare the components of velocity, 0= const is the density of
15、 material (we set 0= 1), g is acceleration of gravity, and D/Dt = t+ vixidenotes material deriva- tive. Since the relative density fl uctuations are small, the velocity obeys the incompressibility condition v = 0. The central conjecture of our theory is that in partially fl uidized fl ows, some of t
16、he grains are involved in plastic motion, while others maintain prolonged static contacts with their neighbors. Accordingly, we write the stress tensor as a sum of the hydrodynamic part proportional to the fl ow strain rate eij, and the strain-independent part, s ij, i.e. ij = eij+ s ij. We assume t
17、hat the diagonal elements of the tensor s iicoincide with the corresponding components of the “true” static stress tensor 0 ii for the immobile grain confi guration in the same geometry, and the shear stresses are reduced by the value of the order parameter characterizing the “phase state” of granul
18、ar matter. Thus, we write the stress tensor in the form 1 ij= ? vi xj + vj xi ? + 0 ij( + (1 )ij). (2) Here is the viscosity coeffi cient. In a static state, = 1, ij= 0 ij, vi = 0, whereas in a fully fl uidized state = 0, and the shear stresses are simply proportional to the strain rates as in ordin
19、ary fl uids. To close the system we need a set of constitutive rela- tions between static shear and normal stresses, as well as an equation for the order parameter .The issue of constitutive relations in granular materials is complex and not completely understood 4,18. It appears that in many cases,
20、 the constitutive relations are determined by the construction history 19. Recent studies indicated a fundamental role of the network of the force chains which carry forces longitudinally 20. We will assume that for any given problem, the corresponding static constitutive relations have been specifi
21、 ed. For the order parameter , we apply pure dissipative dynamics which can be derived from the “free-energy” type functional F, i.e., = F/. We adopt the stan- dard Landau form for F R dr(D|2+f(,), which includes a “local potential energy” and the diff usive spa- tial coupling. The potential energy
22、f(,) should have extrema at = 0 and = 1 corresponding to uniform solid and liquid phases. According to the Mohr-Coulomb yield criterion for non-cohesive grains 4 or its general- ization 20, the static equilibrium failure and transition to fl ow is controlled by the value of the non-dimensional ratio
23、 = max|0 mn/0nn|, where the maximum is sought over all possible orthogonal directions n and m in the bulk of the granular material.We simply use this ra- tio as a parameter in the potential energy for the OP . Without loss of generality, we write the equation for : = D2 a(1 )F(,)(3) Further, accordi
24、ng to observations we assume that the static equilibrium is unstable if 1, where tan11 is the internal friction angle for a particular granular ma- terial. Additionally, we assume that if 0, the “dy- namic” phase = 0, is unstable. Values of 0and 1 do not coincide in general. Typically there is a ran
25、ge in which both static and dynamics phases co-exist (this is related to the so-called Bagnold hysteresis 8). The sim- plest form of F(,) which satisfi es these constraints, is F(,) = +, where = (0)/(10). Setting D = 1 and a = 1 we arrive at = 2 + (1 )( ).(4) For 0 1 it is stable at small h, but los
26、es stability at a certain threshold hc 1. The most “dan- gerous”mode of instability satisfying the above boundary conditions, is acos(z/2h). The eigenvalue of this mode is (h) = 1 2/4h2, hence the neutral curve = 0 for the linear stability of the solution = 1 is given by hc= 2 1. (7) For h hc() grai
27、ns spontaneously start to roll, and a granular fl ow ensues. In addition to the trivial state = 1, for h hs() there exists a unique non-trivial stationary solution satisfying the above BC. The value of hscan be found as a minimum of the following integral as a function of 0, the value of at the surf
28、ace z = 0, hs= min Z 1 0 d q 4 2 2(+1)3 3 + 2 c(0) ,(8) 2 where c(0) = 4 0/2 2( + 1)30/3 + 20. This integral can be calculated analytically for and 1/2. It is easy to show that for large , the critical solution of Eq.(4) has a form = 1 + acos(kz) with a 1 and k = ( 1)1/2, and therefore, hs() hc(). F
29、or 1/2, the critical phase trajectory comes close to two saddle points = 0 and = 1, and an asymptotic evaluation of (8) gives hs= 2log(1/2)+const. This expression agrees with the empirical formula 0 exphs/h0 proposed in Ref. 5. Neutral stability curve hc() and the critical line hs() limiting the reg
30、ion of existence of non-trivial granular fl ow solutions, are shown in Fig.2. They divide the pa- rameter plane (,h) in three regions. At h hs(), the trivial static equilibrium = 1 is the only stationary so- lution of Eq.(4) for chosen BC. For hs() h hc(), the static regime is linearly unstable, and
31、 the only stable regime corresponds to the granular fl ow. This qualitative picture completely agrees with the recent experimental fi ndings 5,7. Moreover, for rough bottom BC (corresponding to our = 1), authors of Ref. 5 found a region of bistabil- ity in the parameter plane (h,) which has a shape
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