Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas

Abstract

Motivated by the disagreement found at high dissipation between simulation data for the heat flux transport coefficients and the expressions derived from the Boltzmann equation by the standard first Sonine approximation [Brey et al., Phys. Rev. E 70, 051301 (2004); J. Phys.: Condens. Matter 17, S2489 (2005)], we implement in this paper a modified version of the first Sonine approximation in which the Maxwell-Boltzmann weight function is replaced by the homogeneous cooling state distribution. The structure of the transport coefficients is common in both approximations, the distinction appearing in the coefficient of the fourth cumulant a2. Comparison with computer simulations shows that the modified approximation significantly improves the estimates for the heat flux transport coefficients at strong dissipation. In addition, the slight discrepancies between simulation and the standard first Sonine estimates for the shear viscosity and the self-diffusion coefficient are also partially corrected by the modified approximation. Finally, the extension of the modified first Sonine approximation to the transport coefficients of the Enskog kinetic theory is presented.

Higher-order moment theories for dilute granular gases of smooth hard spheres

“…As the reduced transport coefficients obtained via Grad's method of moments in Garzó (2013) coincide with those in Brey et al (1998a), the dashed (green) lines also display the reduced transport coefficients obtained in Garzó (2013). The dash-dotted (magenta) lines display the results obtained with the theoretical expressions for the reduced transport coefficients deduced through the modified version of the first Sonine approximation in Garzó et al (2007). The squares are the results obtained with the theoretical expressions derived via the so-called computer-aided method devised by Noskowicz et al (2007) while the triangles denote the numerical solution of the Boltzmann equation obtained through Green-Kubo (GK) relations by means of the direct-simulation Monte Carlo (DSMC) method (Bird 1994) in Brey et al (2005).…”

“…Although, the modified Sonine approach (Garzó et al 2007) is an excellent theoretical approach whose predictions for all the transport coefficients (η * , κ * and λ * ) are in very good agreement with the simulation results, the G26 theory slightly improves the results for the reduced shear viscosity coefficient η * . However, the latter could not yield the other two transport coefficients related to the heat flux (κ * and λ * ) correctly for e 0.7, exactly in the same way as the first Sonine approach (Brey et al 1998a).…”

“…Nevertheless, the results from the DSMC simulations with the implementation of external force (shown by circles) differ from those obtained with the GK relations (shown by triangles), especially for e 0.7. The difference between the results from the two simulations could be due to velocity correlations in the correlation function present in the GK relation for the shear viscosity (Garzó et al 2007). In comparison to the results from the modified version of the first Sonine approximation (dash-dotted magenta line), the present results for the reduced shear viscosity (solid black line) are in better agreement with the DSMC simulations of Garzó et al (2007) in the elastic (e = 1) case, the theoretical expressions of Noskowicz et al (2007) lead to η * = 1.01205 while the DSMC simulations of Garzó et al (2007) give η * ≈ 1.016.…”

“…The difference between the results from the two simulations could be due to velocity correlations in the correlation function present in the GK relation for the shear viscosity (Garzó et al 2007). In comparison to the results from the modified version of the first Sonine approximation (dash-dotted magenta line), the present results for the reduced shear viscosity (solid black line) are in better agreement with the DSMC simulations of Garzó et al (2007) in the elastic (e = 1) case, the theoretical expressions of Noskowicz et al (2007) lead to η * = 1.01205 while the DSMC simulations of Garzó et al (2007) give η * ≈ 1.016. Thus, in the elastic (e = 1) case, η * ≈ 1.014851 from the expression (5.23) 1 obtained here is much closer to its true value in comparison to that from the theoretical expressions of Noskowicz et al (2007).…”

“…The circles in figure 5 also denote the DSMC simulation results from Montanero et al (2005) obtained with another method-by the implementation of an external force which compensates for the collisional cooling. The DSMC simulation data from this method for e = {0.6, 0.7, 0.8, 0.9, 1} were obtained by Montanero et al (2005) while those for e = {0.2, 0.3, 0.4, 0.5} were obtained by Garzó et al (2007).…”