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Dynamic properties in a collisional model of a confined quasi-two-dimensional granular fluid driven by a stochastic bath with friction

David González Méndez*

Rubén Gómez González†

Vicente Garzó‡,§

  • Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, E-06071 Badajoz, Spain

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Phys. Rev. E 114, 025419 – Published 21 August, 2026

DOI: https://doi.org/10.1103/rwfk-w3vv

Abstract

This paper investigates the dynamic properties of a confined quasi-two-dimensional granular fluid at moderate densities, modeled within the framework of the Enskog kinetic equation. The confinement is not treated explicitly at the geometrical level; instead, it is incorporated in an effective way through the so-called Δ model (a collisional model that includes energy injection through modified collision rules). The referred model is extended in this paper to account for the influence of an interstitial gas via a viscous drag force and a stochastic Langevin-like term. By applying the Chapman-Enskog method, the Navier-Stokes transport coefficients and the cooling rate are derived analytically considering the leading terms in a Sonine polynomial expansion. The study focuses on steady-state conditions and examines how the combined effects of inelastic collisions and external driving influence transport properties such as the viscosity and the thermal conductivity. Theoretical predictions for the steady temperature and the kurtosis are validated against direct simulation Monte Carlo results, showing excellent agreement. The findings reveal that the external driving significantly alters the transport coefficients compared to dry (no gas phase) granular systems, challenging previous assumptions that neglected these effects. Additionally, a linear stability analysis demonstrates that the homogeneous steady state is stable across the explored parameter space.

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References (79)

  1. J. S. Olafsen and J. S. Urbach, Clustering, order, and collapse in a driven granular monolayer, Phys. Rev. Lett. 81, 4369 (1998).
  2. W. Losert, D. G. W. Cooper, and J. P. Gollub, Propagating front in an excited granular layer, Phys. Rev. E 59, 5855 (1999).
  3. A. Prevost, P. Melby, D. A. Egolf, and J. S. Urbach, Nonequilibrium two-phase coexistence in a confined granular layer, Phys. Rev. E 70, 050301(R) (2004).
  4. G. Castillo, N. Mujica, and R. Soto, Fluctuations and criticality of a granular solid-liquid-like phase transition, Phys. Rev. Lett. 109, 095701 (2012).
  5. G. Castillo, N. Mujica, and R. Soto, Universality and criticality of a second-order granular solid-liquid-like phase transition, Phys. Rev. E 91, 012141 (2015).
  6. M. Guzmán and R. Soto, Critical phenomena in quasi-two-dimensional vibrated granular systems, Phys. Rev. E 97, 012907 (2018).
  7. G. Castillo, N. Mujica, N. Sepúlveda, J. C. Sobarzo, M. Guzmán, and R. Soto, Hyperuniform states generated by a critical friction field, Phys. Rev. E 100, 032902 (2019).
  8. M. Mayo, J. J. Brey, M. I. García Soria, and P. Maynar, Kinetic theory of a confined quasi-one-dimensional gas of hard disks, Physica A 597, 127237 (2022).
  9. P. Maynar, M. I. García Soria, and J. J. Brey, Dynamics of an inelastic tagged particle under strong confinement, Phys. Fluids 34, 123321 (2022).
  10. M. Mayo, J. C. Petit, M. I. García Soria, and P. Maynar, Confined granular gases under the influence of vibrating walls, J. Stat. Mech. (2023) 123208.
  11. R. Brito, D. Risso, and R. Soto, Hydrodynamic modes in a confined granular fluid, Phys. Rev. E 87, 022209 (2013).
  12. R. Brito, R. Soto, and V. Garzó, Dynamic properties in a collisional model for confined granular fluid: A review, Entropy 28, 454 (2026).
  13. J. J. Brey, M. I. García de Soria, P. Maynar, and V. Buzón, Homogeneous steady state of a confined granular gas, Phys. Rev. E 88, 062205 (2013).
  14. J. J. Brey, P. Maynar, M. I. García de Soria, and V. Buzón, Homogeneous hydrodynamics of a collisional model of confined granular gases, Phys. Rev. E 89, 052209 (2014).
  15. R. Soto, D. Risso, and R. Brito, Shear viscosity of a model for confined granular media, Phys. Rev. E 90, 062204 (2014).
  16. J. J. Brey, V. Buzón, P. Maynar, and M. I. García de Soria, Hydrodynamics for a model of a confined quasi-two-dimensional granular gas, Phys. Rev. E 91, 052201 (2015).
  17. J. J. Brey, V. Buzón, M. I. García de Soria, and P. Maynar, Stability analysis of the homogeneous hydrodynamics of a model for a confined granular gas, Phys. Rev. E 93, 062907 (2016).
  18. V. Garzó, R. Brito, and R. Soto, Enskog kinetic theory for a model of a confined quasi-two-dimensional granular fluid, Phys. Rev. E 98, 052904 (2018).
  19. V. Garzó, R. Brito, and R. Soto, Enskog kinetic theory for a model of a confined quasi-two-dimensional granular fluid, Phys. Rev. E 102, 059901(E) (2020).
  20. R. Brito, R. Soto, and V. Garzó, Energy nonequipartition in a collisional model of a confined quasi-two-dimensional granular mixture, Phys. Rev. E 102, 052904 (2020).
  21. V. Garzó, R. Brito, and R. Soto, Navier-Stokes transport coefficients for a model of a confined quasi-two dimensional granular binary mixture, Phys. Fluids 33, 023310 (2021).
  22. V. Garzó, R. Brito, and R. Soto, Applications of the kinetic theory for a model of a confined quasi-two dimensional granular binary mixture: Stability analysis and thermal diffusion segregation, Phys. Fluids 36, 033326 (2024).
  23. R. Gómez González, V. Garzó, R. Brito, and R. Soto, Diffusion of impurities in a moderately dense confined granular gas, Phys. Fluids 36, 123387 (2024).
  24. D. González Méndez and V. Garzó, Transport properties in a model of confined granular mixtures at moderate densities, Phys. Fluids 38, 033343 (2026).
  25. M. Joyce, J. Morand, and P. Viot, Attractor nonequilibrium stationary states in perturbed long-range interacting systems, Phys. Rev. E 93, 052129 (2016).
  26. R. Maire, A. Plati, M. Stockinger, E. Trizac, F. Smallenburg, and G. Foffi, Interplay between an absorbing phase transition and synchronization in a driven granular system, Phys. Rev. Lett. 132, 238202 (2024).
  27. R. Maire, A. Plati, F. Smallenburg, and G. Foffi, Dynamical and structural properties of an absorbing phase transition: A case study from granular systems, J. Stat. Mech. (2025) 123206.
  28. A. Plati, R. Maire, E. Fayen, F. Boulogne, F. Restagno, F. Smallenburg, and G. Foffi, Quasi-crystalline order in vibrating granular matter, Nat. Phys. 20, 465 (2024).
  29. R. Maire and A. Plati, Enhancing (quasi-)long-range order in a two-dimensional driven crystal, J. Chem. Phys. 161, 054902 (2024).
  30. R. Maire, A. Plati, F. Smallenburg, and G. Foffi, Non-equilibrium coexistence between a fluid and a hotter or colder crystal of granular hard disks, J. Chem. Phys. 162, 124901 (2025).
  31. R. Maire and L. Chaix, Hyperuniformity and conservation laws in non-equilibrium systems, J. Chem. Phys. 163, 214507 (2025).
  32. R. Gómez González and V. Garzó, Transport coefficients for granular suspensions at moderate densities, J. Stat. Mech. (2019) 093204.
  33. R. Gómez González and V. Garzó, Kinetic theory of granular particles immersed in a molecular gas, J. Fluid Mech. 943, A9 (2022).
  34. M. Gómez González, R. García Chamorro, and V. Garzó, Rheology of granular particles immersed in a molecular gas under uniform shear flow, Phys. Rev. E 109, 064901 (2024).
  35. R. Gómez González and V. Garzó, Diffusion of intruders in a granular gas thermostatted by a bath of elastic hard spheres, J. Fluid Mech. 1022, A36 (2025).
  36. D. L. Koch and R. J. Hill, Inertial effects in suspensions and porous-media flows, Annu. Rev. Fluid Mech. 33, 619 (2001).
  37. V. Garzó, S. Tenneti, S. Subramaniam, and C. M. Hrenya, Enskog kinetic theory for monodisperse gas-solid flows, J. Fluid Mech. 712, 129 (2012).
  38. H-K Tsao and D. L. Koch, Simple shear flows of dilute gas-solid suspensions, J. Fluid Mech. 296, 211 (1995).
  39. A. S. Sangani, G. Mo, H.-K. Tsao, and D. L. Koch, Simple shear flows of dense gas-solid suspensions at finite Stokes numbers, J. Fluid Mech. 313, 309 (1996).
  40. D. L. Koch and A. S. Sangani, Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed: Kinetic theory and numerical simulations, J. Fluid Mech. 400, 229 (1999).
  41. J. J. Wylie, Q. Zhang, Y. Li, and X. Hengyi, Driven inelastic-particle systems with drag, Phys. Rev. E 79, 031301 (2009).
  42. C. Heussinger, Shear thickening in granular suspensions: Interparticle friction and dynamically correlated clusters, Phys. Rev. E 88, 050201(R) (2013).
  43. T. Wang, M. Grob, A. Zippelius, and M. Sperl, Active microrhelogy of driven granular particles, Phys. Rev. E 89, 042209 (2014).
  44. S. Saha and M. Alam, Revisiting ignited-quenched transition and the non-Newtonian rheology of a sheared dilute gas-solid suspension, J. Fluid Mech. 833, 206 (2017).
  45. H. Hayakawa, S. Takada, and V. Garzó, Kinetic theory of shear thickening for a moderately dense gas-solid suspension: From discontinuous thickening to continuous thickening, Phys. Rev. E 96, 042903 (2017).
  46. S. Saha and M. Alam, Burnett-order constitutive relations, second moment anisotropy and co-existing states in sheared dense gas-solid suspensions, J. Fluid Mech. 887, A9 (2020).
  47. S. Takada, H. Hayakawa, A. Santos, and V. Garzó, Enskog kinetic theory of rheology for a moderately dense inertial suspension, Phys. Rev. E 102, 022907 (2020).
  48. R. Gómez González, N. Khalil, and V. Garzó, Enskog kinetic theory for multicomponent granular suspensions, Phys. Rev. E 101, 012904 (2020).
  49. A. M. Puertas and R. Gómez González, Tracer diffusion in granular suspensions: Testing the Enskog kinetic theory with direct-simulation Monte Carlo and molecular dynamics, Phys. Rev. E 113, 035414 (2026).
  50. S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases (Cambridge University Press, Cambridge, UK, 1970).
  51. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows (Clarendon, Oxford, UK, 1994).
  52. J. F. Lutsko, Rheology of dense polydisperse granular fluids under shear, Phys. Rev. E 70, 061101 (2004).
  53. M. A. Van der Hoef, R. Beetstra, and J. A. M. Kuipers, Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: Results for the permeability and drag force, J. Fluid Mech. 528, 233 (2005).
  54. R. Beetstra, M. A. Van der Hoef, and J. A. M. Kuipers, Erratum: Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres [AIChE J. 53, 489 (2007)], AIChE J. 53, 3020 (2007).
  55. X. Yin and S. Sundaresan, Fluid-particle drag in low-Reynolds-number polydisperse gas-solid suspensions, AIChE J. 55, 1352 (2009).
  56. N. Brilliantov and T. Pöschel, Kinetic Theory of Granular Gases (Oxford University Press, Oxford, UK, 2004).
  57. V. Garzó, Granular Gaseous Flows (Springer Nature, Cham, 2019).
  58. D. L. Koch, Kinetic theory for a monodisperse gas-solid suspension, Phys. Fluids A 2, 1711 (1990).
  59. J. J. Wylie, D. L. Koch, and J. C. Ladd, Rheology of suspensions with high particle inertia and moderate fluid inertia, J. Fluid Mech. 480, 95 (2003).
  60. T. P. C. van Noije and M. H. Ernst, Velocity distributions in homogeneous granular fluids: The free and heated case, Granular Matter 1, 57 (1998).
  61. G. A. Bird, Direct simulation and the Boltzmann equation, Phys. Fluids 13, 2676 (1970).
  62. J. M. Montanero and A. Santos, Simulation of the Enskog equation à la Bird, Phys. Fluids 9, 2057 (1997).
  63. N. Khalil and V. Garzó, Homogeneous states in driven granular mixtures: Enskog kinetic theory versus molecular dynamics simulations, J. Chem. Phys. 140, 164901 (2014).
  64. S. Torquato, Nearest-neighbor statistics for packings of hard spheres and disks, Phys. Rev. E 51, 3170 (1995).
  65. N. V. Brilliantov and T. Pöschel, Breakdown of the Sonine expansion for the velocity distribution of granular gases, Europhys. Lett. 74, 424 (2006).
  66. M. Huthmann, J. A. G. Orza, and R. Brito, Dynamics of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastic particles, Granular Matter 2, 189 (2000).
  67. V. Garzó, R. Brito, and R. Soto, Enskog kinetic theory for a model of a confined quasi-two-dimensional granular fluid, arXiv:1803.03429v7.
  68. While the present study was being conducted, some typos were found in the expressions reported in Ref. [18]. The corrected expressions are given in Ref. [67] and are the ones employed here.
  69. J. C. Pérez-Fuentes and V. Garzó, Influence of a drag force on linear transport in low-density gases. Stability analysis, Physica A 410, 428 (2014).
  70. J. J. Brey, J. W. Dufty, C. S. Kim, and A. Santos, Hydrodynamics for granular flow at low density, Phys. Rev. E 58, 4638 (1998).
  71. S. R. de Groot, W. A. van Leeuwen, and C. G. van Weert, Relativistic Kinetic Theory: Principles and Applications (North-Holland, Amsterdam, 1980).
  72. C. Cercignani and G. Kremer, The Relativistic Boltzmann Equation: Theory and Applications (Birkhäusser, Berlin, 2002).
  73. V. Garzó, Instabilities in a free granular fluid described by the Enskog equation, Phys. Rev. E 72, 021106 (2005).
  74. V. Garzó, J. M. Montanero, and J. W. Dufty, Mass and heat fluxes for a binary granular mixture at low density, Phys. Fluids 18, 083305 (2006).
  75. N. S. Nise, Control Systems Engineering (Wiley, New York, NY, 2020).
  76. V. Garzó and J. M. Montanero, Diffusion of impurities in a granular gas, Phys. Rev. E 69, 021301 (2004).
  77. V. Garzó and F. Vega Reyes, Mass transport of impurities in a moderately dense granular gas, Phys. Rev. E 79, 041303 (2009).
  78. V. Garzó, J. A. Murray, and F. Vega Reyes, Diffusion transport coefficients for granular binary mixtures at low density: Thermal diffusion segregation, Phys. Fluids 25, 043302 (2013).
  79. R. Gómez González, E. Abad, S. Bravo Yuste, and V. Garzó, Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation, Phys. Rev. E 108, 024903 (2023).

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Phys. Rev. E 114, 025419– Published 21 August, 2026
  • Received 6 May 2026
  • Accepted 16 July 2026
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