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Grain flow as a fluid-mechanical phenomenon

Published online by Cambridge University Press:  20 April 2006

P. K. Haff
Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125


The behaviour of granular material in motion is studied from a continuum point of view. Insofar as possible, individual grains are treated as the ‘molecules’ of a granular ‘fluid’. Besides the obvious contrast in shape, size and mass, a key difference between true molecules and grains is that collisions of the latter are inevitably inelastic. This, together with the fact that the fluctuation velocity may be comparable to the flow velocity, necessitates explicit incorporation of the energy equation, in addition to the continuity and momentum equations, into the theoretical description. Simple ‘microscopic’ kinetic models are invoked for deriving expressions for the ‘coefficients’ of viscosity, thermal diffusivity and energy absorption due to collisions. The ‘coefficients’ are not constants, but are functions of the local state of the medium, and therefore depend on the local ‘temperature’ and density. In general the resulting equations are nonlinear and coupled. However, in the limit s [Lt ] d, where s is the mean separation between neighbouring grain surfaces and d is a grain diameter, the above equations become linear and can be solved analytically. An important dependent variable, in this formulation, in addition to the flow velocity u, is the mean random fluctuation (‘thermal’) velocity $\overline{v}$ of an individual grain. With a sufficient flux of energy supplied to the system through the boundaries of the container, $\overline{v}$ can remain non-zero even in the absence of flow. The existence of a non-uniform $\overline{v}$ is the means by which energy can be ‘conducted’ from one part of the system to another. Because grain collisions are inelastic, there is a natural (damping) lengthscale, governed by the value of d, which strongly influences the functional dependence of $\overline{v}$ on position. Several illustrative examples of static (u = 0) systems are solved. As an example of grain flow, various Couette-type problems are solved analytically. The pressure, shear stress, and ‘thermal’ velocity function $\overline{v}$ are all determined by the relative plate velocity U (and the boundary conditions). If $\overline{v}$ is set equal to zero at both plates, the pressure and stress are both proportional to U2, i.e. the fluid is non-Newtonian. However, if sufficient energy is supplied externally through the walls ($\overline{v} \ne 0$ there), then the forces become proportional to the first power of U. Some examples of Couette flow are given which emphasize the large effect on the grain system properties of even a tiny amount of inelasticity in grain–grain collisions. From these calculations it is suggested that, for the case of Couette flow, the flow of sand is supersonic over most of the region between the confining plates.

Research Article
© 1983 Cambridge University Press

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Ackermann, N. L. & Shen, H. T. 1978 Flow of granular material as a two-component system. In Cowin & Satake (1978), pp. 258265.
Ackermann, N. L. & Shen, H. 1982 Stresses in rapidly sheared fluid-solid mixtures. J. Engng Mech. Div. ASCE 108 (EM1), 95113.
Bagnold, R. A. 1954 Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A225, 4963.Google Scholar
Bagnold, R. A. 1956 The flow of cohesionless grains in fluids Proc. R. Soc. Lond. A249, 235297.Google Scholar
Campbell, C. S. & Brennen, C. E. 1982 Computer simulations of chute flows of granular materials. In Deformation and Failure of Granular Materials (ed. P. A. Vermeer & H. J. Luger), pp. 515521. A. A. Bulkenna, Rotterdam.
Carlos, C. R. & Richardson, J. F. 1968 Solids movement in liquid fluidised beds - I. Particle velocity distribution Chem. Engng Sci. 23, 813824.Google Scholar
Cowin, S. C. 1978 Microstructural continuum models for granular materials. In Cowin & Satake (1978), pp. 162170.
Cowin, S. C. & M. SATAKE (eds.) 1978 Proc. US-Japan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials. Gakujutsu Bunken Fukyukai, Tokyo, Japan.
Hirschfelder, J. O., Curtiss, D. F. & Bird, R. F. 1964 Molecular Theory of Gases and Liquids. Wiley.
Jenkins, J. T. & Cowin, S. C. 1979 Theories for flowing granular materials. In Mechanics Applied to the Transport of Bulk Materials (ed. S. Cowin). ASME AMD-31, pp. 7989.
Jenkins, J. T. & Savage, S. B. 1981 The mean stress resulting from interparticle collisions in a rapid granular shear flow. In Continuum Models of Discrete Systems 4 (ed. O. Brulin & R. K. T. Hsieh), pp. 365371. North-Holland.
Mctigue, D. F. 1978 A model for stresses in shear flow of granular material. In Cowin & Satake (1978), pp. 266271.
Marble, F. E. 1964 Mechanism of particle collision in the one-dimensional dynamics of gas-particle mixtures Phys. Fluids 7, 12701282.Google Scholar
Ogawa, S. 1978 Multitemperature theory of granular materials. In Cowin & Satake (1978), pp. 208217.
Ogawa, S., Umemura, A. & Oshima, N. 1980 On the equations of fully fluidized granular materials Z. angew. Math. Phys. 31, 483493.Google Scholar
Oshima, N. 1978 Continuum model of fluidized granular media. In Cowin & Satake (1978), pp. 189207.
Savage, S. B. 1979 Gravity flow of cohesionless granular materials in chutes and channels J. Fluid Mech. 92, 5396.Google Scholar
Savage, S. B. & Jeffrey, J. D. 1981 The stress tensor in a granular flow at high shear rates J. Fluid Mech. 110, 255272.Google Scholar
Shen, H. & Ackermann, N. L. 1982 Constitutive relationships for fluid-solid mixtures. J. Engng Mech. Div. ASCE 108 (EM5). 748763.
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