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On the role of the Knudsen layer in rapid granular flows

  • J. E. GALVIN (a1) (a2), C. M. HRENYA (a1) and R. D. WILDMAN (a3)


A combination of molecular dynamics simulations, theoretical predictions and previous experiments are used in a two-part study to determine the role of the Knudsen layer in rapid granular flows. First, a robust criterion for the identification of the thickness of the Knudsen layer is established: a rapid deterioration in Navier–Stokes order prediction of the heat flux is found to occur in the Knudsen layer. For (experimental) systems in which heat flux measurements are not easily obtained, a rule-of-thumb for estimating the Knudsen layer thickness follows, namely that such effects are evident within 2.5 (local) mean free paths of a given boundary. Secondly, comparisons of simulation and experimental data with Navier–Stokes order theory are used to provide a measure as to when Knudsen-layer effects become non-negligible. Specifically, predictions that do not account for the presence of a Knudsen layer appear reliable for Knudsen layers collectively composing up to 20% of the domain, whereas deterioration of such predictions becomes apparent when the domain is fully comprised of the Knudsen layer.


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Arnarson, B. O. & Jenkins, J. T. 2004 Binary mixtures of inelastic spheres: simplified constitutive theory. Phys. Fluids 16, 4543.
Behringer, R. P., van Doorn, E., Hartley, R. R. & Pak, H. K. 2002 Making a rough place ‘plane’: why heaping of vertically shaken sand must stop at low pressure. Gran. Matter 4, 9.
Bird, G. A. 1994 Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon.
Brey, J. J., Ruiz-Montero, M. J. & Moreno, F. 2001 Hydrodynamics of an open vibrated system. Phys. Rev. E 63, 061305.
Carnahan, N. F. & Starling, K. E. 1969 Equation of state of non-attracting rigid spheres. J. Chem. Phys. 51, 635.
Cercignani, C. 1987 The Boltzmann Equation and its Applications. Springer.
Cercignani, C., Illner, R. & Pulvirenti, M. 1994 The Mathematical Theory of Dilute Gases. Springer.
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.
Ciccotti, G. & Tenebaum, A. 1980 Canonical ensemble and nonequilibrium states by molecular dynamics. J. Stat. Phys. 23, 767.
Clause, P. J. & Mareschal, M. 1988 Heat-transfer in a gas between parallel plates – moment method and molecular-dynamics. Phys. Rev. A 38, 4241.
Dahl, S. R. & Hrenya, C. M. 2004 Size segregation in rapid, granular flows with continuous size distributions. Phys. Fluids 16, 1.
Ferziger, J. H. & Kaper, H. G. 1972 Mathematical Theory of Transport Processes in Gases. Elsevier.
Forterre, Y. & Pouliquen, O. 2001 Longitudinal vortices in granular flows. Phys. Rev. Lett. 86, 5886.
Galvin, J. E., Dahl, S. R. & Hrenya, C. M. 2005 On the role of non-equipartition in the dynamics of rapidly flowing granular mixtures. J. Fluid Mech. 528, 207.
Garzó, V. & Dufty, J. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895.
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267.
Goldhirsch, I., Noskowicz, S. H. & Bar-Lev, O. 2004 Theory of granular gases: some recent results and some open problems. J. Phys.: Condens. Matter 17, 2591.
Herbst, O., Müller, P. & Zippelius, A. 2005 Local heat flux and energy loss in a two-dimensional vibrated granular gas. Phys. Rev. E 72, 141303.
Hopkins, M. & Louge, M. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47.
Hrenya, C. M., Galvin, J. E. & Wildman, R. D. 2007 Evidence of higher-order effects in thermally-driven, granular flows. J. Fluid Mech. (Submitted).
Jenkins, J. T. 1998 Kinetic theory for nearly elastic spheres. In Physics of Dry Granular Media (ed. Hermann, H. J., Hovi, J. P. & Luding, S.. Kluwer.
Jin, S. & Slemrod, M. 2001 Regularization of the Burnett equations for rapid granular flows via relaxation. Physica D 150, 207.
Kadanoff, L. P. 1999 Built upon sand: theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71, 435.
Kierzenka, J. & Shampine, L. F. 2001 A BVP solver based on residual control and the Matlab pse. ACM Trans. Math. Software 27, 299.
Kumaran, V. 1997 Velocity distribution function for a dilute granular material in shear flow. J. Fluid Mech. 340, 319.
Kumaran, V. 2005 Kinetic model for sheared granular flows in the high Knudsen number limit. Phys. Rev. Lett. 95, 108001.
Mackowski, D., Papadopoulos, D. H. & Rosner, D. E. 1999 Comparison of Burnett and DSMC predictions of pressure distributions and normal stress in one-dimensional, strongly nonisothermal gases. Phys. Fluids 11, 2108.
Mareschal, M., Kestemont, E., Baras, F., Clementi, E. & Nicolis, G. 1987 Nonequilibrium states by molecular-dynamics – transport-coefficients in constrained fluids. Phys. Rev. A 35, 3883.
Martin, T. W., Huntley, J. M. & Wildman, R. D. 2006 Hydrodynamic model for a vibrofluidized granular bed. J. Fluid Mech. 535, 325.
Montanero, J. M., Alaoui, M., Santos, A. & Garzó, V. 1994 Monte Carlo simulation of the Boltzmann equation for steady Fourier flow. Phys. Rev. E 49, 367.
Pan, L. S., Xu, D., Lou, J. & Yao, Q. 2006 A generalized heat conduction model in rarefied gas. Europhys. Lett. 73, 846.
Pasini, J. M. & Jenkins, J. T. 2005 Aeolian transport with collisional suspension. Phil. Trans. Royal Soc. A – Math. Phys. Engng Sci. 363, 1625.
Poschel, T. & Schwager, T. 2005 Computational Granular Dynamics. Springer.
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1992 Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press.
Rericha, E. C., Bizon, C., Shattuck, M. D. & Swinney, H. L. 2002 Shocks in supersonic sand. Phys. Rev. Lett. 88, 014302.
Rosner, D. E. & Papadopoulos, D. H. 1996 Jump, slip and creep boundary conditions at nonequilibrium gas/solid interfaces. Ind. Engng Chem. Res. 35, 3210.
Santos, A. & Garzó, V. 1995 In Rarefied Gas Dynamics 19 (ed. Harvey, J. & Lord, G.). Oxford University Press.
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41.
Shattuck, M. D., Bizon, C., Swift, J. B. & Swinney, H. L. 1999 Computational test of kinetic theory of granular media. PhysicaA 274, 158.
Sone, Y. 2002 Kinetic Theory and Fluid Dynamics. Birkhause.
Soto, R., Mareschal, M. & Risso, D. 1999 Departure from Fourier's law for fluidized granular media. Phys. Rev. Lett. 83, 5003.
Viswanathan, H., Wildman, R. D., Huntley, J. M. & Martin, T. W. 2006 Comparison of kinetic theory predictions with experimental results for a vibrated three-dimensional granular bed. Phys. Fluids 18, 113302.
Wassgren, C. R., Cordova, J. A., Zenit, R. & Karion, A. 2003 Dilute granular flow around an immersed cylinder. Phys. Fluids 15, 3318.
Wildman, R. D., Jenkins, J. T., Krouskop, P. E. & Talbot, J. 2006 A comparison of the predictions of a simple kinetic theory with experimental and numerical results for a vibrated granular bed consisting of nearly elastic particles of two sizes. Phys. Fluids 18, 073301.
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On the role of the Knudsen layer in rapid granular flows

  • J. E. GALVIN (a1) (a2), C. M. HRENYA (a1) and R. D. WILDMAN (a3)


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