## Fast spectral solution of the generalized Enskog equation for dense gases.(English)Zbl 1349.76778

Summary: We propose a fast spectral method for solving the generalized Enskog equation for dense gases. For elastic collisions, the method solves the Enskog collision operator with a computational cost of $$O(M^{d - 1} N^d \log N)$$, where $$d$$ is the dimension of the velocity space, and $$M^{d - 1}$$ and $$N^d$$ are the number of solid angle and velocity space discretizations, respectively. For inelastic collisions, the cost is $$N$$ times higher. The accuracy of this fast spectral method is assessed by comparing our numerical results with analytical solutions of the spatially-homogeneous relaxation of heated granular gases. We also compare our results for force-driven Poiseuille flow and Fourier flow with those from molecular dynamics and Monte Carlo simulations. Although it is phenomenological, the generalized Enskog equation is capable of capturing the flow dynamics of dense granular gases, and the fast spectral method is accurate and efficient. As example applications, Fourier and Couette flows of a dense granular gas are investigated. In addition to the temperature profile, both the density and the high-energy tails in the velocity distribution functions are found to be strongly influenced by the restitution coefficient.

### MSC:

 76N15 Gas dynamics (general theory) 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 76M22 Spectral methods applied to problems in fluid mechanics 76T15 Dusty-gas two-phase flows
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### References:

 [1] Chapman, S.; Cowling, T. G., The mathematical theory of non-uniform gases, (1970), Cambridge University Cambridge · Zbl 0098.39702 [2] van Beijeren, H.; Ernst, M. H., Kinetic theory of hard spheres, J. Stat. Phys., 21, 125-167, (1979) [3] Esteban, M. J.; Perthame, B., On the modified Enskog equation for elastic and inelastic collisions. models with spin, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 8, 289-308, (1991) · Zbl 0850.70141 [4] Brey, J. J.; Dufty, J. W.; Santos, A., Dissipative dynamics for hard spheres, J. Stat. Phys., 87, 1051-1066, (1997) · Zbl 0945.82562 [5] Bobylev, A. V.; Carrillo, J. A.; Gamba, I. M., On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Stat. Phys., 98, 743-773, (2000) · Zbl 1056.76071 [6] Frezzotti, A.; Gibelli, L.; Lorenzani, S., Mean field kinetic theory description of evaporation of a fluid into vacuum, Phys. Fluids, 17, (2005) · Zbl 1187.76165 [7] Barbante, P.; Frezzotti, A.; Gibelli, L., A kinetic theory description of liquid menisci at the microscale, Kinet. Relat. Models, 8, 235-254, (2015) · Zbl 1362.82044 [8] Frezzotti, A.; Sgarra, C., Numerical analysis of a shock wave solution of the Enskog equation obtained via a Monte Carlo method, J. Stat. Phys., 73, 193-207, (1993) · Zbl 1101.82335 [9] Frezzotti, A., Molecular dynamics and Enskog theory calculation of shock profiles in a dense hard sphere gas, Comput. Math. Appl., 35, 103-112, (1998) · Zbl 0907.35101 [10] Bird, G. A., Molecular gas dynamics and the direct simulation of gas flows, (1994), Clarendon Press Oxford [11] Nanbu, K., Direct simulation scheme derived from the Boltzmann equation. I. monocomponent gases, J. Phys. Soc. Jpn., 52, 2042-2049, (1983) [12] Alexander, F. J.; Garcia, A. L.; Alder, B. J., A consistent Boltzmann algorithm, Phys. Rev. Lett., 74, 5212-5215, (1995) [13] Montanero, J. M.; Santos, A., Monte Carlo simulation method for the Enskog equation, Phys. Rev. E, 54, 438-444, (1996) [14] Frezzotti, A., A particle scheme for the numerical solution of the Enskog equation, Phys. Fluids, 9, 1329-1335, (1997) · Zbl 1185.76835 [15] Popken, L., Grid-free particle method for the inhomogeneous Enskog equation and its application to a Riemann-problem, Eur. J. Mech. B, Fluids, 17, 255-265, (1998) · Zbl 0922.76268 [16] Nedea, S. V.; Frijns, A. J.H.; van Steenhoven, A. A.; Jansen, A. P.J.; Markvoort, A. J.; Hilbers, P. A.J., Density distribution for a dense hard-sphere gas in micro/nano-channels: analytical and simulation results, J. Comput. Phys., 219, 532-552, (2006) · Zbl 1189.82101 [17] Montanero, J. M.; Santos, A., Computer simulation of uniformly heated granular fluids, Granul. Matter, 2, 53-64, (2000) [18] Gamba, I. M.; Rjasanow, S.; Wagner, W., Direct simulation of the uniformly heated granular Boltzmann equation, Math. Comput. Model., 42, 683-700, (2005) · Zbl 1088.35049 [19] Filbet, F.; Pareschi, L.; Toscani, G., Accurate numerical methods for the collisional motion of (heated) granular flows, J. Comput. Phys., 202, 216-235, (2005) · Zbl 1288.76056 [20] Mouhot, C.; Pareschi, L., Fast algorithms for computing the Boltzmann collision operator, Math. Comput., 75, 1833-1852, (2006) · Zbl 1105.76043 [21] Filbet, F.; Mouhot, C.; Pareschi, L., Solving the Boltzmann equation in $$N \log_2 N$$, SIAM J. Sci. Comput., 28, 1029-1053, (2006) · Zbl 1174.82012 [22] Wu, L.; White, C.; Scanlon, T. J.; Reese, J. M.; Zhang, Y., Deterministic numerical solutions of the Boltzmann equation using the fast spectral method, J. Comput. Phys., 250, 27-52, (2013) · Zbl 1349.76790 [23] Wu, L.; Reese, J. M.; Zhang, Y., Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows, J. Fluid Mech., 746, 53-84, (2014) [24] Wu, L.; Reese, J. M.; Zhang, Y., Oscillatory rarefied gas flow inside rectangular cavities, J. Fluid Mech., 748, 350-367, (2014) [25] Wu, L.; Zhang, J.; Reese, J. M.; Zhang, Y., A fast spectral method for the Boltzmann equation for monatomic gas mixtures, J. Comput. Phys., 298, 602-621, (2015) · Zbl 1349.76575 [26] Santos, A., Transport coefficients of d-dimensional inelastic Maxwell models, Physica A, 321, 442-466, (2003) · Zbl 1011.82020 [27] van Noije, T. P.C.; Ernst, M. H., Velocity distributions in homogeneous granular fluids: the free and the heated case, Granul. Matter, 1, 57-64, (1998) [28] Ernst, M. H.; Brito, R., High-energy tails for inelastic Maxwell models, Europhys. Lett., 58, 182-187, (2002) [29] Bobylev, A. V.; Cercignani, C., Moment equations for a granular material in a thermal Bath, J. Stat. Phys., 106, 547-567, (2002) · Zbl 1001.82092 [30] Gamba, I. M.; Panferov, V.; Villani, C., On the Boltzmann equation for diffusively excited granular media, Commun. Math. Phys., 246, 503-541, (2004) · Zbl 1106.82031 [31] Ernst, M. H.; Trizac, E.; Barrat, A., The rich behavior of the Boltzmann equation for dissipative gases, Europhys. Lett., 76, 56-62, (2006) [32] Ernst, M. H.; Trizac, E.; Barrat, A., The Boltzmann equation for driven systems of inelastic soft spheres, J. Stat. Phys., 124, 549-586, (2006) · Zbl 1134.82034 [33] Trizac, E.; Barrat, A.; Ernst, M. H., Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction, Phys. Rev. E, 76, (2007) · Zbl 1189.82041 [34] Sanchez, I. C., Virial coefficients and close-packing of hard spheres and discs, J. Chem. Phys., 101, 7003, (1994) [35] Henderson, D., Simple equation of state for hard disks, Mol. Phys., 30, 971-972, (1975) [36] Baus, M.; Colot, J. L., Thermodynamics and structure of a fluid of hard rods, disks, spheres, or hyperspheres from rescaled virial expansions, Phys. Rev. A, 36, 3912-3925, (1987) [37] Shen, G.; Ge, W., Simulation of hard-disk flow in microchannels, Phys. Rev. E, 81, (2010) [38] Frezzotti, A., Monte Carlo simulation of the heat flow in a dense hard sphere gas, Eur. J. Mech. B, Fluids, 18, 103-119, (1999) · Zbl 0935.76069 [39] Carnahan, N. F.; Starling, K. E., Equation of state for nonattracting rigid spheres, J. Chem. Phys., 51, 635-636, (1969) [40] Reyes, F. V.; Santos, A.; Garzó, V., Non-Newtonian granular hydrodynamics. what do the inelastic simple shear flow and the elastic Fourier flow have in common?, Phys. Rev. Lett., 104, (2010) [41] Reyes, F. V.; Garzó, V.; Santos, A., Class of dilute granular Couette flows with uniform heat flux, Phys. Rev. E, 83, (2011) [42] Reyes, F. V.; Santos, A.; Garzó, V., Steady base states for non-Newtonian granular hydrodynamics, J. Fluid Mech., 719, 431-464, (2013) · Zbl 1284.76391 [43] Filbet, F.; Ray, T., A rescaling velocity method for dissipative kinetic equations. applications to granular media, J. Comput. Phys., 248, 177-199, (2013) · Zbl 1349.76330
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