Granular materials are the most industrially important materials after water. In nature, they are found as landslides and avalanches. Their constitutive response is not understood, and their description depends on the manner in which they move. This course develops continuum descriptions of rapidly flowing granular materials, through the application of kinetic theory of gases, which is modified to take into account the finite size of the grains, and the dissipative nature of their collisions.


First course at the graduate level in engineering mathematics and continuum mechanics

Course Contents

Introduction to granular materials; Review of classical thermodynamics; Kinetic theory of perfect gases: Fundamental equations of fluid mechanics, Viscosity, Thermal conductivity; Kinetic description of flow of grains: smooth inelastic grains, rough inelastic grains; Examples; Effect of air on granular flow.

Topics with suggested lectures in parenthesis

I. Introduction to granular materials: Illustrative examples. (1)

II. Thermodynamics: First and second laws; Entropy; Pressure; Temperature. (3)

III. Elementary kinetic theory: Equation of state of a perfect gas; Kinetic definitions of pressure and temperature; Maxwellian distribution; van der Waals equation; Mean free path; Viscosity; Thermal conductivity. (5)

IV. Exact kinetic theory: Maxwell—Boltzmann collision equation; Molecular chaos; H—Theorem; Maxwellian distribution; Fundamental equations of fluid mechanics; Integration of the collision equation; Viscosity; Thermal conductivity. (10)

V. Elementary description of flowing granular materials. (2)

VI. Computational modeling: Introduction to DE modeling. (1)

VII. Rapid granular flow: Smooth inelastic particles. (8)

VIII. Applications: Plane Couette flow, Inclined chutes. (2)

IX. Rapid granular flow: Rough inelastic particles. (4)

X. Application to mixing and segregation: Brazil-nut effect, Rotating drum and Avalanches. (3)

XI. Effect of air: Porous beds, Barchan dunes and Hourglasses. (3)

Textbooks, alternate sources and further readings:

  1. Rao, K., and P. R. Nott 2008. An Introduction to Granular Flow. Cambridge U. Press.

  2. Sommerfeld, A. 1956. Thermodynamics and Statistical Mechanics. Associate Press.

  3. Jeans, J. H. 2009. An Introduction to the Kinetic Theory of Gases. Cambridge U. Press.

  4. Chapman, S. and T. G. Cowling 1995. The Mathematical Theory of Non-Uniform Gases. Cambridge U. Press.

  5. Ferziger, J. H. and H. G. Kaper 1972. Mathematical Theory of Transport Processes in Gases. NorthHolland.

Prepared by

Ishan Sharma