×

zbMATH — the first resource for mathematics

Scalar conservation laws with nonconstant coefficients with application to particle size segregation in granular flow. (English) Zbl 1209.35080
Summary: Granular materials segregate by particle size when subjected to shear, as occurs, for example, in avalanches. The evolution of a bidisperse mixture of particles can be modeled by a nonlinear first order partial differential equation, provided the shear (or velocity) is a known function of position. While avalanche-driven shear is approximately uniform in depth, boundary-driven shear typically creates a shear band with a nonlinear velocity profile.
In this paper, we measure a velocity profile from experimental data and solve initial value problems that mimic the segregation observed in the experiment, thereby verifying the value of the continuum model. To simplify the analysis, we consider only one-dimensional configurations, in which a layer of small particles is placed above a layer of large particles within an annular shear cell and is sheared for arbitrarily long times. We fit the measured velocity profile to both an exponential function of depth and a piecewise linear function which separates the shear band from the rest of the material. Each solution of the initial value problem is nonstandard, involving curved characteristics in the exponential case, and a material interface with a jump in characteristic speed in the piecewise linear case.

MSC:
35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations
74L10 Soil and rock mechanics
35L50 Initial-boundary value problems for first-order hyperbolic systems
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bachmann, F., Vovelle, J.: Existence and uniqueness of entropy solutions of scalar conservation laws with a flux function involving discontinuous coefficients. Commun. Partial Differ. Equ. 31, 371–395 (2006) · Zbl 1102.35064 · doi:10.1080/03605300500358095
[2] Chen, G.-Q., Even, N., Klingenberg, C.: Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems. J. Differ. Equ. 245, 3095–3126 (2008) · Zbl 1195.35211 · doi:10.1016/j.jde.2008.07.036
[3] Golick, L.A., Daniels, K.E.: Mixing and segregation rates in sheared granular materials. Phys. Rev. E 80, 042301 (2009) · doi:10.1103/PhysRevE.80.042301
[4] Gray, J.M.N.T., Thornton, A.R.: A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. A 461, 1447–1473 (2005) · Zbl 1145.76461 · doi:10.1098/rspa.2004.1420
[5] Hill, K.M., Fan, Y.: Isolating segregation mechanisms in a split-bottom cell. Phys. Rev. Lett. 101, 088001 (2008)
[6] Jimenez, J.: Analysis of a conservation law with space-discontinuous advection function. Monogr. Semin. Mat. Garcia Galdeano 33, 425–432 (2006) · Zbl 1124.35332
[7] Klingenberg, C., Risebro, N.H.: Convex conservation laws with discontinuous coefficients. existence, uniqueness and asymptotic behavior. Commun. Partial Differ. Equ. 20, 1959–1990 (1995) · Zbl 0836.35090 · doi:10.1080/03605309508821159
[8] Kruzkov, S.N.: First order quasilinear equations in several independent variables. Math. Sb. 10, 217–243 (1970) · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156
[9] May, L.B.H., Phillips, K.C., Daniels, K.E., Shearer, M.: Shear-driven size segregation of granular materials: modelling and experiment. Phys. Rev. E 81(1) (2010, to appear)
[10] Metcalfe, G., Shinbrot, T., Mccarthy, J.J., Ottino, J.M.: Avalanche mixing of granular solids. Nature 374, 39–41 (1995) · doi:10.1038/374039a0
[11] MiDi, G.D.R.: On dense granular flows. Eur. Phys. J. E 14, 341 (2004) · doi:10.1140/epje/i2003-10153-0
[12] Ottino, J.M., Khakhar, D.V.: Mixing and segregation of granular materials. Annu. Rev. Fluid Mech. 32, 55–91 (2000) · Zbl 0989.76087 · doi:10.1146/annurev.fluid.32.1.55
[13] Pouliquen, O., Vallance, J.W.: Segregation induced instabilities of granular fronts. Chaos 9, 621–630 (1999) · Zbl 1055.74513 · doi:10.1063/1.166435
[14] Reynolds, O.: On the dilatancy of media composed of rigid particles in contact, with experimental illustrations. Philos. Mag. 20, 469–481 (1885)
[15] Savage, S.B., Hutter, K.: The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199, 171–205 (1989) · Zbl 0659.76044 · doi:10.1017/S0022112089000340
[16] Savage, S.B., Lun, C.K.K.: Particle size segregation in inclined chute flow of dry cohesionless granular solids. J. Fluid Mech. 189, 311–335 (1988) · doi:10.1017/S002211208800103X
[17] Seguin, N., Vovelle, J.: Analysis and approximation of a scalar conservation with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13, 221–257 (2003) · Zbl 1078.35011 · doi:10.1142/S0218202503002477
[18] Wang, G., Sheng, W.: Interaction of elementary waves of scalar conservation laws with discontinuous flux function. J. Shanghai Univ. 10, 381–387 (2006) · Zbl 1131.35363 · doi:10.1007/s11741-006-0077-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.