The orientational or angular correlation between the directions of the translational and rotational motions is analyzed theoretically for the homogeneous cooling state of a rough granular gas. The dynamical equations are derived using an approximate form of the single-particle distribution function that incorporates angular correlations. The goal is to assess the effects of higher-order angular corrections for which both quadratic- and quartic-order terms (in translational and rotational velocities of particles) are retained in the perturbation expansion of the distribution function. We show that higher-order corrections can markedly affect the steady-state orientational correlation when the normal restitution coefficient is moderate or small, and this effect is more prominent for nearly smooth particles. The transient evolution of orientational correlation is found to be significantly affected by higher-order terms. In particular the higher-order orientational correlations can dominate over the leading-order contribution during short times even in the quasi-elastic limit, although the steady correlation remains unaffected by such corrections in the same limit. The buildup of correlations during the transient stage seems to be closely tied to the evolution of the ratio between the rotational and translational temperatures. It is demonstrated that the transient dynamics of the temperature ratio and its steady state remain insensitive to higher-order angular correlation.
The perturbation expansion technique is employed to solve the Boltzmann equation for the acceleration-driven steady Poiseuille flow of a dilute molecular gas flowing through a planar channel. Neglecting wall effects and focusing only on the bulk hydrodynamics and rheology, the perturbation solution is sought around the channel centerline in powers of the strength of acceleration. To make analytical progress, the collision term has been approximated by the Bhatnagar-Gross-Krook kinetic model for hard spheres, and the related problem for Maxwell molecules was analyzed previously by Tij and Santos [J. Stat. Phys. 76, 1399 (1994)JSTPBS0022-471510.1007/BF02187068]. The analytical expressions for hydrodynamic (velocity, temperature, and pressure) and rheological fields (normal stress differences, shear viscosity, and heat flux) are obtained by retaining terms up to tenth order in acceleration, with one aim of the present work being to understand the convergence properties of the underlying perturbation series solutions. In addition, various rarefaction effects (e.g., the bimodal shape of the temperature profile, nonuniform pressure profile, normal-stress differences, and tangential heat flux) are also critically analyzed in the Poiseuille flow as functions of the local Froude number. The hydrodynamic and rheological fields evaluated at the channel centerline confirmed the oscillatory nature of the present series solutions (when terms of increasing order are sequentially included), signaling the well-known pitfalls of asymptotic expansion. The Padé approximation technique is subsequently applied to check the region of convergence of each series solution. It is found that the diagonal Padé approximants for rheological fields agree qualitatively with previous simulation data on acceleration-driven rarefied Poiseuille flow.
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