Asymptotic expansion and Padé approximants for gravity-driven flow of a heated granular gas: Competition between inelasticity and forcing, up to Burnett order

Abstract

The gravity-driven planar channel flow of a heated granular gas is analyzed using a kinetic model with stochastic forcing to understand the roles of inelasticity and forcing on its hydrodynamics and rheology. The closed-form analytical solutions up to the fourth order in gravitational acceleration have been determined to analyze the hydrodynamic and rheological fields as functions of the restitution coefficient (${e}_{n}$) and the Froude number ${\mathrm{Fr}}_{0}$. It is found that the excess temperature [$\mathrm{\ensuremath{\Delta}}T={T}_{max}/T(0)\ensuremath{-}1$, i.e., the deviation of the maximum temperature ${T}_{max}$ of the gas from its centerline value $T(0)$] increases monotonically with decreasing ${e}_{n}$ above a critical value of the Froude number (${\mathrm{Fr}}_{0}g{\mathrm{Fr}}_{0}^{c}$), but has a ``nonmonotonic'' dependence with ${e}_{n}$ [i.e., $\mathrm{\ensuremath{\Delta}}T$ decreases with decreasing ${e}_{n}$ for ${e}_{n}\ensuremath{\in}(1,0.5)$, but increases for ${e}_{n}l0.5$] at ${\mathrm{Fr}}_{0}l{\mathrm{Fr}}_{0}^{c}$. This changeover from nonmonotonic to monotonic dependence of $\mathrm{\ensuremath{\Delta}}T$ with ${e}_{n}$ at ${\mathrm{Fr}}_{0}={\mathrm{Fr}}_{0}^{c}$ also holds for both first (${\mathcal{N}}_{1}$) and second (${\mathcal{N}}_{2}$) normal-stress differences as well as for tangential heat flux (${q}_{x}$). Phase diagrams are constructed in the (${\mathrm{Fr}}_{0},1\ensuremath{-}{e}_{n}$) plane, demarcating two regions in which the dependencies of rarefaction effects ($\mathrm{\ensuremath{\Delta}}T, {\mathcal{N}}_{1}, {\mathcal{N}}_{2}$, and ${q}_{x}$) on ${e}_{n}$ are monotonic and nonmonotonic. The inelasticity plays a ``dual'' role of decreasing (at ${\mathrm{Fr}}_{0}l{\mathrm{Fr}}_{0}^{c}$) and increasing (at ${\mathrm{Fr}}_{0}g{\mathrm{Fr}}_{0}^{c}$) the rarefaction effects with decreasing restitution coefficient ${e}_{n}$ from the elastic limit. This finding, based on the fourth-order solution, is in variance with the leading-order solution that predicts only a nonmonotonic dependence of the above quantities on ${e}_{n}$ for all ${\mathrm{Fr}}_{0}$ [Tij and Santos, J. Stat. Phys. 117, 901 (2004)]. The role of inelasticity on the region of convergence of the asymptotic series solutions are subsequently analyzed by determining the Pad\'e approximants for rheological fields [Rongali and Alam, Phys. Rev. E 98, 012115 (2018)]; the present solution is shown to have a larger range of validity in terms of both inelasticity and Froude number than its leading-order counterpart. Lastly, it is shown that the fourth-order solutions contain all Burnett-order terms (i.e., second order in the gradients of hydrodynamic fields) that can be obtained from the standard Chapman-Enskog expansion.