Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T03:01:03.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady base states for Navier–Stokes granular hydrodynamics with boundary heating and shear

Published online by Cambridge University Press:  25 September 2009

FRANCISCO VEGA REYES  and
JEFFREY S. URBACH
FRANCISCO VEGA REYES*
Affiliation:
Departamento de Física, Universidad de Extremadura, 06071 Badajoz, Spain
JEFFREY S. URBACH
Affiliation:
Department of Physics, Georgetown University, Washington, DC 20057, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the Navier–Stokes steady states for a low density monodisperse hard sphere granular gas (i.e a hard sphere ideal monatomic gas with inelastic inter-particle collisions). We present a classification of the uniform steady states that can arise from shear and temperature (or energy input) applied at the boundaries (parallel walls). We consider both symmetric and asymmetric boundary conditions and find steady states not previously reported, including sheared states with linear temperature profiles. We provide explicit expressions for the hydrodynamic profiles for all these steady states. Our results are validated by the numerical solution of the Boltzmann kinetic equation for the granular gas obtained by the direct simulation Monte Carlo method, and by molecular dynamics simulations. We discuss the physical origin of the new steady states and derive conditions for the validity of Navier–Stokes hydrodynamics.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 636 , 10 October 2009 , pp. 279 - 293
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Agarwal, R., Yun, K.-Y. & Balakrishnan, R. 2001 Beyond Navier–Stokes: Burnett equations for flows in the continuum-transition regime. Phys. Fluids 13, 3061.CrossRefGoogle Scholar
Alam, M. 2006 Streamwise structures and density patterns in rapid granular Couette flow: a linear stability analysis. J. Fluid Mech. 553, 1–32.CrossRefGoogle Scholar
Alam, M. & Nott, P. R. 1998 Stability of plane Couette flow of a granular material. J. Fluid Mech. 377, 99–136.CrossRefGoogle Scholar
Barrat, A. & Trizac, E. 2002 a Lack of energy equipartition in homogeneous heated binary granular mixtures. Granular Matter 4, 57–63.CrossRefGoogle Scholar
Barrat, A. & Trizac, E. 2002 b Molecular dynamics simulations of vibrated granular gases. Phys. Rev. E 66, 051303.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Brey, J. J. & Cubero, D. 2001 Hydrodynamic transport coefficients of granular gases. In Granular Gases (eds. Pöschel, T. and Luding, S.), pp. 59–78. Springer-Verlag.CrossRefGoogle Scholar
Brey, J. J., Cubero, D., Moreno, F & Ruiz-Montero, M. J. 2001 Fourier state of a fluidized granular gas. Europhys. Lett. 53, 432–437.CrossRefGoogle Scholar
Brey, J. J., Dufty, J. W., Kim, C. S. & Santos, A. 1998 Hydrodynamics for granular flow at low density. Phys. Rev. E 58, 4638.CrossRefGoogle Scholar
Campbell, C. S. 1989 The stress tensor for simple shear flows of a granular material. J. Fluid Mech. 58, 449–473.CrossRefGoogle Scholar
Chapman, C. & Cowling, T. G. 1970 The Mathematical Theory of Non-Uniform Gases, 3rd ed. Cambridge University Press.Google Scholar
Ernst, M. H. 1973 The modified Enskog equation. Physica 68, 437–456.Google Scholar
Feitosa, K. & Menon, N. 2002 Breakdown of energy equipartition in a 2D binary vibrated granular gas. Phys. Rev. Lett. 88, 198301.CrossRefGoogle Scholar
Galvin, J. E., Hrenya, C. M. & Wildman, R. D. 2007 On the role of the Knudsen layer in rapid granular flows. J. Fluid Mech. 585, 73.CrossRefGoogle Scholar
Garzó, V. 2006 Transport coefficients for an inelastic gas around uniform shear flow: linear stability analysis. Phys. Rev. E 73, 021304.CrossRefGoogle ScholarPubMed
Garzó, V. & Dufty, J. 1999 a Homogeneous cooling state for a granular mixture. Phys. Rev. E 60, 5706–5713.CrossRefGoogle ScholarPubMed
Garzó, V. & Dufty, J. W. 1999 b Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911.CrossRefGoogle ScholarPubMed
Garzó, V., Santos, A. & Montanero, J. M. 2007 Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas. Physica A 376, 94–107.CrossRefGoogle Scholar
Garzó, V. & Vega Reyes, F. 2009 Mass transport of impurities in a moderately dense granular gas. Phys. Rev. E 79, 041303.CrossRefGoogle Scholar
Garzó, V., Vega Reyes, F. & Montanero, J. M. 2009 Modified Sonine approximation for granular binary mixtures. J. Fluid Mech. 623, 387–411.CrossRefGoogle Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293.CrossRefGoogle Scholar
Grad, H. 1963 Asymptotic theory of the Boltzmann equation. Phys. Fluids 6, 147–181.CrossRefGoogle Scholar
Hilbert, D. 1912 Begründung der kinetischen Gastheorie. Math. Ann. 72, 562–577.CrossRefGoogle Scholar
Hrenya, C. M., Galvin, J. E. & Wildman, R. D. 2008 Evidence of higher-order effects in thermally driven granular flows. J. Fluid Mech. 598, 429–450.CrossRefGoogle Scholar
Jenkins, J. T. & Richman, M. W. 1985 Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485–3494.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
Khain, E., Meerson, B. & Sasorov, P. V. 2008 Knudsen temperature jump and the Navier–Stokes hydrodynamics of granular gases driven by thermal walls. Phys. Rev. E 78, 041303.CrossRefGoogle ScholarPubMed
Lun, C. K. K. 1996 Granular dynamics of inelastic spheres in Couette flow. Phys. Fluids 8, 2868–2883.CrossRefGoogle Scholar
Lutsko, J. F. 2006 Chapman–Enskog expansion about nonequilibrium states with application to the sheared granular fluid. Phys. Rev. E 73, 021302.CrossRefGoogle Scholar
Montanero, J. M., Garzó, 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
Nott, P. R., Alam, M., Agarwal, K., Jackson, R. & Sundaresan, S. 1999 The effect of boundaries on the plane Couette flow of granular materials: a bifurcation analysis. J. Fluid Mech. 397, 203–229.CrossRefGoogle Scholar
Rapaport, D. C. 2004 The Mathematical Theory of Non-Uniform Gases, 2nd ed. Cambridge University Press.Google Scholar
Santos, A., Brey, J. J. & Garzó, V. 1986 Kinetic model for steady heat flow. Phys. Rev. A 34, 5047–5050.CrossRefGoogle ScholarPubMed
Santos, A., Garzó, V. & Dufty, J. W. 2004 Inherent rheology of a granular fluid in uniform shear flow. Phys. Rev. E 69, 061303.CrossRefGoogle ScholarPubMed
Vega Reyes, F., Santos, A. & Garzó, V. Couette granular flow with zero heat flow gradient (“LTu” class), in preparation.Google 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
Tij, M., Tahiri, E. E., Montanero, J. M., Garzó, V., Santos, A. & Dufty, J. W. 2001 Nonlinear Couette flow in a low density granular gas. J. Stat. Phys. 103, 1035–1068.CrossRefGoogle Scholar
Vega Reyes, F., Garzó, V. & Santos, A. 2007 Granular mixtures modelled as elastic hard spheres subject to a drag force. Phys. Rev. E 75, 061306.CrossRefGoogle ScholarPubMed
Vega Reyes, F., Garzó, V. & Santos, A. 2008 Impurity in a granular gas under nonlinear Couette flow. J. Stat. Mech. P09003, 1–30.Google Scholar
Vega Reyes, F. & Urbach, J. S. 2008 Effect of inelasticity on the phase transitions of a thin vibrated granular layer. Phys. Rev. E 78, 051301.CrossRefGoogle Scholar
Wang, C.-W., Jackson, R. & Sundaresan, S. 1996 Stability of bounded rapid shear flows of a granular material. J. Fluid Mech. 308, 31–62.CrossRefGoogle Scholar
Wildman, R. D. & Parker, D. J. 2002 Coexistence of two granular temperatures in binary vibrofluidized beds. Phys. Rev. Lett. 88, 064301.CrossRefGoogle ScholarPubMed
Yang, X., Huan, C., Candela, D., Mair, R. W. & Walsworth, R. L. 2002 Measurements of grain motion in a dense, three-dimensional granular fluid. Phys. Rev. Lett. 69, 044301.CrossRefGoogle Scholar