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Boundary conditions at a rigid wall for rough granular gases

Published online by Cambridge University Press:  18 April 2011

PRABHU R. NOTT
PRABHU R. NOTT*
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India
*
†Email address for correspondence: [email protected]
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Abstract

We derive boundary conditions at a rigid wall for a granular material comprising rough, inelastic particles. Our analysis is confined to the rapid flow, or granular gas, regime in which grains interact by impulsive collisions. We use the Chapman–Enskog expansion in the kinetic theory of dense gases, extended for inelastic and rough particles, to determine the relevant fluxes to the wall. As in previous studies, we assume that the particles are spheres, and that the wall is corrugated by hemispheres rigidly attached to it. Collisions between the particles and the wall hemispheres are characterized by coefficients of restitution and roughness. We derive boundary conditions for the two limiting cases of nearly smooth and nearly perfectly rough spheres, as a hydrodynamic description of granular gases comprising rough spheres is appropriate only in these limits. The results are illustrated by applying the equations of motion and boundary conditions to the problem of plane Couette flow.

JFM classification

Granular media: Granular media Rarefied Gas Flow: Kinetic theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 678 , 10 July 2011 , pp. 179 - 202
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A. 1998 Hydrodynamics for granular flow at low density. Phys. Rev. E 58, 4638–4653.CrossRefGoogle Scholar
Bryan, 1894 Brit. Assoc. Reports p. 83.Google Scholar
Burnett, D. 1935 The distribution of velocities in a slightly non-uniform gas. Proc. Lond. Math. Soc. 39, 385–430.CrossRefGoogle Scholar
Cercignani, C. 1988 The Boltzmann Equation and Its Applications. Springer.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1964 The Mathematical Theory of Nonuniform Gases, 2nd edn. Cambridge University Press.Google Scholar
Garzo, V. & Dufty, J. W. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911.CrossRefGoogle ScholarPubMed
Garzo, V., Dufty, J. W. & Hrenya, C. M. 2007 Enskog theory for polydisperse granular mixtures. I. Navier–Stokes order transport. Phys. Rev. E 76, 031303.CrossRefGoogle ScholarPubMed
Goldhirsch, I., Noskowicz, S. H. & Bar-Lev, O. 2005 a Hydrodynamics of nearly smooth granular gases. J. Phys. Chem. 109, 21449–21470.CrossRefGoogle ScholarPubMed
Goldhirsch, I., Noskowicz, S. H. & Bar-Lev, O. 2005 b Nearly smooth granular gases. Phys. Rev. Lett. 95, 068002–1–4.CrossRefGoogle ScholarPubMed
Goldsmith, W. 1960 Impact. Edward Arnold.Google Scholar
Herbst, O., Huthmann, M. & Zippelius, A. 2000 Dynamics of inelastically colliding spheres with Coulomb friction: relaxation of translational and rotational energy. Granular Matter 2, 211–219.CrossRefGoogle Scholar
Hui, K., Haff, P. K., Ungar, J. E. & Jackson, R. 1984 Boundary conditions for high-shear grain flows. J. Fluid Mech. 145, 223–233.CrossRefGoogle Scholar
Jenkins, J. T. & Richman, M. W. 1985 Grad's 13-moment system for a dense gas of inelastic particles. Arch. Rat. Mech. Anal. 87, 355–377.CrossRefGoogle Scholar
Jenkins, J. T. & Richman, M. W. 1986 Boundary conditions for plane flows of smooth, nearly elastic, circular disks. J. Fluid Mech. 171, 53–69.CrossRefGoogle Scholar
Jenkins, J. T. & Richman, M. W. 1988 Plane simple shear of smooth inelastic circular disks: the anisotropy of the second moment in the dilute and dense limits. J. Fluid Mech. 192, 313–328.CrossRefGoogle Scholar
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202.CrossRefGoogle Scholar
Johnson, K. L. 1987 Contact Mechanics. Cambridge University Press.Google Scholar
Johnson, P. C., Nott, P. & Jackson, R. 1990 Frictional-collisional equations of motion for particulate flows and their application to chutes. J. Fluid Mech. 210, 501–535.CrossRefGoogle Scholar
Kumaran, V. 2004 Constitutive relations and linear stability of a sheared granular flow. J. Fluid Mech. 506, 1–43.CrossRefGoogle Scholar
Lun, C. K. K. 1991 Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech. 223, 539–559.CrossRefGoogle Scholar
Lun, C. K. K., Savage, S. B., Jeffrey, 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.CrossRefGoogle Scholar
Maw, N., Barber, J. R. & Fawcett, J. N. 1976 The oblique impact of elastic spheres. Wear 38, 101–114.CrossRefGoogle Scholar
Maw, N., Barber, J. R. & Fawcett, J. N. 1981 The role of tangential elastic compliance in oblique impact. Trans. ASME 103, 74–80.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, 3485–3512.CrossRefGoogle Scholar
Mohan, L. S., Nott, P. R. & Rao, K. K. 1999 A frictional Cosserat model for the flow of granular materials through a vertical channel. Acta Mech. 138, 75–96.CrossRefGoogle Scholar
Mohan, L. S., Nott, P. R. & Rao, K. K. 2002 A frictional Cosserat model for the slow shearing of granular materials. J. Fluid Mech. 457, 377–409.CrossRefGoogle Scholar
Montanero, J. M., Garzo, V., Santos, A. & Brey, J. J. 1999 Kinetic theory of simple granular shear flows of smooth hard spheres. J. Fluid Mech. 389, 391–411.CrossRefGoogle Scholar
Noll, W. 1955 On the continuity of the solid and fluid states. Arch. Rat. Mech. Anal. 4, 3–81.Google Scholar
Pidduck, F. B. 1922 The kinetic theory of a special type of rigid molecule. Proc. R. Soc. A 101, 101–112.Google Scholar
Rao, K. K. & Nott, P. R. 2008 An Introduction to Granular Flow. Cambridge University Press.CrossRefGoogle Scholar
Richman, M. W. 1988 Boundary conditions based upon a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227–240.CrossRefGoogle Scholar
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41–74.CrossRefGoogle Scholar
Sela, N., Goldhirsch, I. & Noskowicz, S. H. 1996 Kinetic theoretical study of a simply sheared two dimensional granular gas to Burnett order. Phys. Fluids 8, 2337–2353.CrossRefGoogle Scholar
Sharipov, F. M. & Kremer, G. M. 1995 On the frame dependence of constitutive equations. I. Heat transfer through a rarefied gas between two rotating cylinders. Continuum Mech. Thermodyn. 7, 57–71.CrossRefGoogle Scholar
Stronge, W. J. 2000 Impact Mechanics. Cambridge University Press.CrossRefGoogle Scholar
Walton, O. R. 1993 Numerical simulation of inelastic frictional particle–particle interactions. In Particle Two-Phase Flow (ed. Roco, M. C.). Butterworth–Heinemann.Google Scholar