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Mechanics of collisional motion of granular materials. Part 3. Self-similar shock wave propagation

Published online by Cambridge University Press:  26 April 2006

A. Goldshtein ,
M. SHAPIRO  and
C. Gutfinger
A. Goldshtein
Affiliation:
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
M. SHAPIRO
Affiliation:
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
C. Gutfinger
Affiliation:
Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
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Abstract

Shock wave propagation arising from steady one-dimensional motion of a piston in a granular gas composed of inelastically colliding particles is treated theoretically. A self-similar long-time solution is obtained in the strong shock wave approximation for all values of the upstream gas volumetric concentration v0. Closed form expressions for the long-time shock wave speed and the granular pressure on the piston are obtained. These quantities are shown to be independent of the particle collisional properties, provided their impacts are accompanied by kinetic energy losses. The shock wave speed of such non-conservative gases is shown to be less than that for molecular gases by a factor of about 2.

The effect of particle kinetic energy dissipation is to form a stagnant layer (solid block), on the surface of the moving piston, with density equal to the maximal packing density, vM. The thickness of this densely packed layer increases indefinitely with time. The layer is separated from the shock front by a fluidized region of agitated (chaotically moving) particles. The (long-time, constant) thickness of this layer, as well as the kinetic energy (granular temperature) distribution within it are calculated for various values of particle restitution and surface roughness coefficients. The asymptotic cases of dilute (v0 [Lt ] 1) and dense (v0 ∼ vM) granular gases are treated analytically, using the corresponding expressions for the equilibrium radial distribution functions and the pertinent equations of state. The thickness of the fluidized region is shown to be independent of the piston velocity.

The calculated results are discussed in relation to the problem of vibrofluidized granular layers, wherein shock and expansion waves were registered. The average granular kinetic energy in the fluidized region behind the shock front calculated here compared favourably with that measured and calculated (Goldshtein et al. 1995) for vibrofluidized layers of spherical granules.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 316 , 10 June 1996 , pp. 29 - 51
Copyright
© 1996 Cambridge University Press

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References

Alder, B. J. & Hoover, W. G. 1968 Numerical statistical mechanics. In: Physics of Simple Liquids, pp. 79–114. North-Holland.
Alder, B. J. & Wainwright, T. E. 1960 Studies in molecular dynamics. II. Behavior of a small number of elastic spheres. J. Chem. Phys. 33, 1439–1451.Google Scholar
Bachmann, D. 1940 Bewegungsvorgänge in schwingmühlen mit trockner mahlkörperfüllung. Verfahrenstechnik Z. VDI-Beiheft 2, 43–55.Google Scholar
Campbell, C. S. 1990 Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92.Google Scholar
Carnahan, N. F. & Starling, K. E. 1969 Equations of state of nonattracting rigid spheres. J. Chem. Phys. 51, 635–636.Google Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. 3rd edn. Cambridge University Press.
Condiff, D. W., Lu, W. K. & Dahler, J. S. 1965 Transport properties of polyatomic fluids, a dilute gas of perfectly rough spheres. J. Chem. Phys. 42, 3445–3475.Google Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Interscience.
Goldshtein, A. & Shapiro, M. 1995 Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282, 75–114.Google Scholar
Goldshtein, A., Shapiro, M. & Gutfinger, C. 1996 Mechanics of collisional motion of granular materials. Part 4. Expansion wave. J. Fluid Mech. (submitted).Google Scholar
Goldshtein, A., Shapiro, M., Moldavsky, L. & Fichman, M. 1995 Mechanics of collisional motion of granular materials: Part 2. Wave propagation through a granular layer. J. Fluid Mech. 287, 349–382.Google Scholar
Goldsmith, W. 1960 Impact: The Theory and Physical Behavior of Colliding Solids. E. Arnold.
Grad, H. 1949 On the kinetic theory of rarefied gas. Commun. Pure. Appl. Maths 2, 331–407.Google Scholar
Homsy, G. M., Jackson, R. & Grace, J. R. 1992 Report of a symposium on mechanics of fluidized beds. J. Fluid Mech. 236, 447–495.Google Scholar
Jenkins, J. T. & Richman, M. W. 1985a Grad's 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355–377.Google Scholar
Jenkins, J. T. & Richman, M. W. 1985a Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485–3494.Google Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.Google Scholar
Lan, Y. & Rosato, A. D. 1995 Macroscopic behavior of vibrating beds of smooth inelastic spheres. Phys. Fluids 7, 1818–1831.Google Scholar
Luding, S., Herrmann, H. J. & Blumen, A. 1994 Simulations of two-dimensional arrays of beads under external vibrations: Scaling behavior. Phys. Rev. E 50, 3100–3108.Google Scholar
Lun, C. K. K. 1991 Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech. 233, 539–559.Google Scholar
Lun, C. K. K. & Bent, A. A. 1994 Numerical simulations of inelastic frictional spheres in simple shear flow. J. Fluid Mech. 258, 335–353.Google Scholar
Lun, C. K. K. & Savage, S. B. 1986 The effect of an impact dependent coefficient of restitution on stresses developed by sheared granular materials. Acta Mech. 63, 15–44.Google Scholar
Lun, C. K. K. & Savage, S. B. 1987 A simple kinetic theory for granular flow of rough, inelastic, spherical particles. Trans. ASME E: J. Appl. Mech. 54, 47–53.Google Scholar
Lun, C. K. K., Savage, S. B., Jeffery, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256.Google Scholar
Matveev, S. K. 1983 Rigid-particle gas model with allowance for inelastic collisions. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 6, 12–16.Google Scholar
McCoy, B. J., Sandler, S. I. & Dahler, J. S. 1966 Transport properties of polyatomic fluids. IV. The kinetic theory of a dense gas of perfectly rough spheres. J. Chem. Phys. 45, 3485–3500.Google Scholar
Ogawa, S., Umemura, A. & Oshima, N. 1980 On the equations of fully fluidized granular materials. Z. Angew. Math. Phys. 31, 483–493.Google Scholar
Raskin, Kh. I. 1975 Application of the methods of physical kinetics to problems of vibrated granular media. Dokl. Acad. Nauk SSSR 220, 54–57.Google Scholar
Savage, S. B. 1988 Streaming motions in a bed of vibrationally fluidized dry granular material. J. Fluid Mech. 194, 457–478.Google Scholar
Theodosopulu, M. & Dahler, J. S. 1974 The kinetic theory of polyatomic liquids. II. The rough sphere, rigid ellipsoid, and square-well ellipsoid models. J. Chem. Phys. 60, 4048–4057.Google Scholar
Zel'dovich, Ya. B. & Raizer, Yu. P. 1966 Physics of shock waves and high temperature hydrodynamic phenomena. Academic