Abstract
The analysis of compressible flow of granular materials is considered. The necessary thermodynamic background has recently been developed, and this is reviewed from the viewpoint of compressible flow theory. Elementary problems can be solved rather easily following traditional methods of classical gas dynamics. However, even moderately complex problems present subtle unexpected difficulties, which are shown to be related to the inelasticity of particle-particle collisions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anderson, T.B., and R. Jackson, A Fluid Mechanical Description of Fluidized Beds, Ind. Eng. Chem. Fund., 6, 527–539 (1967).
Astarita, G., Thermodynamics. An Advanced Textbook for Chemical Engineers, Plenum, New York, 1989.
Astarita, G., and Sarti, G.C., Thermomechanics of Compressible Materials with Entropic Elasticity, in Theoretical Rheology, J.F. Hutton, J.R.A. Pearson and K. Walters Eds., Applied Science Publ., Barking, 1975.
Astarita, G., and R. Ocone, Large-Scale Statistical Thermodynamics and Wave Propagation in Granular Flow, I & EC Research, 33, 2280–2287 (1994).
Atkinson, C.M., and H.K. Kytomaa, Acoustic Wave Speed and Attenuation in Suspensions, Intl. J. Multiphase Flow, 15, 577–592 (1992).
Bagnold, R.A., Experiments on a Gravity-Free Dispersion of Large Solid Particles in a Newtonian Fluid under Shear, Proc. R.y. Soc. London, A225, 49–63 (1954).
Batchelor, G.K., A New Theory for the Instability of a Uniform Fluidized Bed, J. Fluid Mech., 193, 75–110 (1988).
Borderies, N.P. Goldreich, and S.A. Tremaine, A Granular Flow Model for Dense Planetary Rings, Icarus, 63, 406–420 (1985).
Bowen, R.M., Continuum Physics, A.C. Eringen Ed., Academic Press, vol. II, New York, 1971.
Coleman, B.D., and W. Noll, The Thermodynamics of Elastic Materials with Heat Conduction and Viscosity, Arch. Ratl. Mech. Anal., 13, 167–178 (1963).
Courant, R., and D. Hilbert, Methods of Mathematical Physics, Interscience, New York, 1952.
Da Vinci, Leonardo, Opere, Ediz. Bibl. Vatican, 1916. The work on sound waves dates back to about 1500.
Ding, J., and D. Gidaspow, A Bubbling Fluidization Model using Kinetic Theory of Granular Flow, AIChEAJ, 36, 523–538 (1990).
Drew, D., Mathematical Modeling of Two-Phase Flow, Ann. Rev. Fluid Mech., 15, 261–291 (1983).
Foscolo, P.U., and L.G. Gibilaro, Fluid Dynamic Stability of Fluidized Suspensions: the Particle Bed Model, Chem. Eng. Sci., 42, 1489–1500 (1987).
Foscolo, P.U., L.G. Gibilaro and S. Rapagna’, Infinitesimal and Finite Voidage Perturbations in the Compressible Particle Phase Description of a Fluidized Bed, AIChE Symp. Series, Fluidization and Fluid-Particle Systems, 1995.
Goldreich, P., and S.A. Tremaine, The Velocity Dispersion in Saturn’s Rings, Icarus, 34, 227–239 (1978).
Haff, P.K., Grain Flow as a Fluid Mechanical Phenomenon, J. Fluid Mech., 134, 401–430 (1983).
Hugoniot, H., Mémoire sur la Propagation du Mouvement dans les Corps et spécialemnet dan les Gaz Parfaits, J. Ecole Polyt. 58, 1–17 (1888).
Hugoniot, H., Mémoire sur la Propagation du Mouvement dans un Fluide Indefini, J. Math. Pur. Appl., 3, 477–491; 4, 153–167 (1887).
Jackson, R., The Flow of Granular Materials and Aerated Granular Material, J. Rheol., 30, 907–930 (1986).
Jean, R.H. and L.S. Fan, On the Model Equations of Gibilaro and Foscolo with Corrected Buoyancy Forces, Powder Techn., 62, 201–205 (1992).
Jenkins, J.T. and S.B. Savage, A Theory for the Rapid Flow of Identical, Smooth, Nearly Elastic Spherical Particles, J. Fluid Mech., 130, 187–202 (1983).
Joseph, D.D., Lundgren, T.S., (with an Appendix by R. Jackson and D.A. Saville), Ensemble Averaged and Mixture Theory Equations for Incompressible Fluid-Particle Suspensions, Int. J. Multiphase Flow, 16, 35–42 (1990).
Laplace, P.S., “Oeuvres”, (Imprimerie Royale, Paris 1846). The work on the speed of sound dates back to 1816.
Lax, P.D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973.
Lundgren, T.S., Slow Flow through Stationary Random Beds and Suspensions of Spheres, J. Fluid Mech., 51, 273–299 (1972).
Maddox, J., Towards Unequal Partition of Energy?,Nature, 374 (6517), 11, 1995.
Maxwell, J.C., On the Dynamical Theory of Gases, Phil. Trans. Roy. Soc. London, 157, 49–98 (1867).
Mc Tigue, D.F., and J.T. Jenkins, Channel Flow of Concentrated Suspensions, in: Advances in Micromechanics of Granular Materials, H.H. Shen, M. Statake, M. Mehrabadi, C.S. Chang, C.S. Campbell Eds., Elsevier, Amsterdam, 1992.
Morse, P.M., and H. Feshbach, Methods of Theoretical Physics, McGraw Hill, New York, 1953.
Mutsers, S.M.P., and K. Rietema, The Effect of Interparticle Forces on the Expansion of a Homogeneous Gas-Fluidized Bed, Powder Techn., 18, 239–248 (1977).
Nayfeh, A.H., Perturbation Methods, J. Wiley, New York, 1973.
Newton, I., Philosophiae Naturalis Principia Mathematica, Streeter, London 1687.
Ocone, R., and G. Astarita, A Pseudo-Thermodynamic Theory of Granular Flow Rheology, J. Rheol., 37, 727–742 (1993).
Ocone, R., and G. Astarita, On Waves of Particulate Phase Pressure in Granular Materials, J. Rheol., 38, 129–139 (1994).
Ocone, R., and G. Astarita, Compression and Rarefaction Waves in Granular Flow, Powder Techn., 82, 231–237 (1995a).
Ocone, R., and G. Astarita, Energy Partitions, Nature, 375 (6530), 254 (1995b).
Ocone, R., and G. Astarita, Grain Inertia and Fluid Viscosity Dominated Granular Flow Rheology: A Thermodynamics Analysis, Rheol. Acta, 34, 323–328 (1995c).
Ogawa, S., Multitemperature Theory of Granular Materials, in Proc. of the US-Japan Symp. on Cont. Mech. and Statist. Appr. in Mech. of Granular Materials, S.C. Cowin and M. Satake Eds., Gukujatsu Bunken Fukyukai, 1978.
Rhee, K.H., R. Axis, and N.R. Amundson, First Order Partial Differential Equations, Prentice Hall, Englewood Cliffs, 1989.
Richtmeyer, R.D., and K.W. Morton, Difference Methods for Initial Value Problems, Interscience, New York, 1967
Rosensweig, R.E., Magnetic Stabilization of the State of Uniform Fluidization, Ind. Eng. Chem. Fund., 18, 260–269 (1979).
Savage, S.B., Streaming Motions in a Bed of Vibrationally Fluidized Dry Granular Material, J. Fluid Mech., 194, 457–478 (1988).
Truesdell, C.A., Rational Thermodynamics, 2nd Ed., Springer-Verlag, Berlin, 1984.
Waterston, J.J., The Physics of Media that are Composed of Free and Perfectly Elastic Molecules in a State of Motion, Phil. Trans. Roy. Soc. London, A183, 1–79 (1893). Published posthumously, 48 years after its submission in 1845.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Astarita, T., Ocone, R., Astarita, G. (1998). Compressible Flow of Granular Materials. In: Drew, D.A., Joseph, D.D., Passman, S.L. (eds) Particulate Flows. The IMA Volumes in Mathematics and its Applications, vol 98. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-7109-0_1
Download citation
DOI: https://doi.org/10.1007/978-1-4684-7109-0_1
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-7111-3
Online ISBN: 978-1-4684-7109-0
eBook Packages: Springer Book Archive