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

Abstract

Grad's method of moments is employed to develop higher-order Grad moment equations-up to first 26-moments-for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff's law while the other higher-order moments decay on a faster time scale. The nonlinear terms of fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have negligible effect on Haff's law. Furthermore, an even larger Grad moment system which includes the fully contracted sixth moment is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of scalar sixth order moment into the Grad moment system has negligible effect on Haff's law. The constitutive relations for the stress and heat flux (i.e., the Navier-Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via CE expansion and via computer simulations. The linear stability of the homogeneous cooling state is analyzed through the Grad 26-moment system and various sub-systems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment and existing theories are also in excellent agreement.

Impact of roughness on the instability of a free-cooling granular gas

“…To the best of our knowledge, the first work where the impact of roughness on a transport coefficient (self-diffusion) was investigated for arbitrary inelasticity and roughness was carried out by Bodrova and Brilliantov [15]. On the other hand, in contrast to the case of smooth-sphere granular gases [4,9,11,12,[16][17][18], most of the attempts made for evaluating the other transport coefficients of inelastic rough hard spheres have been restricted to nearly elastic collisions (α 1) and either nearly smooth particles (β −1) [19][20][21] or nearly perfectly rough particles (β 1) [19,22]. An extension of the previous works to arbitrary values of α and β has been recently carried out by Kremer et al [23] for a dilute granular gas.…”

Grad-type fourteen-moment theory for dilute granular gases

“…Despite this difference in the values of , it turns out-but not shown here-that for any fix e, the granular temperature profiles from eq. (47) (with or without the underlined term) for the two values of (corresponding to line and symbol in the main panel of figure 1) differ from each other only negligibly, see [32]. Therefore, it is justifiable to drop Δ 2 and Δ 3 terms in eqs (39) and (42).…”

Shear-banding instability in arbitrarily inelastic granular shear flows

“…All the quantities, except µ * k , in (9) are also given in [45] for hard-disk flows (d = 2). The µ * k in the present work is twice of that of [45] but is the same as that in [53] for d = 2 in order to keep the standard form of the reduced Dufour-like coefficient (µ * = nµ/(κ 0 T )) given, e.g., in [31,39,43,53].…”