在Enskog动力学方程的背景下，在$d$维驱动中等密度颗粒气体的爱因斯坦关系进行了分析。Enskog方程忽略了速度相关性，但保留了体积排阻效应产生的空间相关性。正如所预期的，存在与扩散$D$和迁移率$μ$相关的爱因斯坦关系式$ε=D/(T_0μ)\neq 1$的故障，$T_0$是杂质的温度。动力学理论的结果还表明，爱因斯坦关系的破坏仅仅是由于杂质粒子参考态的强非麦克斯韦行为。随着固体体积分数和非弹性的增加，$ε$与1的偏差变得更加显著，特别是当系统由高斯恒温器的作用驱动时。这一结论定性地与最近对稠密气体的一些模拟一致[Puglisi { \em et al.}，2007 { \em J. Stat.机械。} P08016]，尽管在计算机模拟中观察到的偏差比从Enskog动力学理论中获得的偏差更重要。理论和模拟之间的定量差异的可能原因进行了讨论。
英文摘要：
The Einstein relation for a driven moderately dense granular gas in $d$-dimensions is analyzed in the context of the Enskog kinetic equation. The Enskog equation neglects velocity correlations but retains spatial correlations arising from volume exclusion effects. As expected, there is a breakdown of the Einstein relation $ε=D/(T_0μ)\neq 1$ relating diffusion $D$ and mobility $μ$, $T_0$ being the temperature of the impurity. The kinetic theory results also show that the violation of the Einstein relation is only due to the strong non-Maxwellian behavior of the reference state of the impurity particles. The deviation of $ε$ from unity becomes more significant as the solid volume fraction and the inelasticity increase, especially when the system is driven by the action of a Gaussian thermostat. This conclusion qualitatively agrees with some recent simulations of dense gases [Puglisi {\em et al.}, 2007 {\em J. Stat. Mech.} P08016], although the deviations observed in computer simulations are more important than those obtained here from the Enskog kinetic theory. Possible reasons for the quantitative discrepancies between theory and simulations are discussed.
Mass transport of impurities in a moderately dense granular gas
通过Chapman-Enskog展开，得到了由非弹性Enskog方程描述的中等密度颗粒状气体中杂质的质量通量的输运系数。输运系数被确定为最近导出的多分散颗粒混合物的一组耦合线性积分方程的解[V. Garzó，J.W. Dufty和C. M. Hrenya，Phys. Rev. E { \bf 76}，031304（2007）]。为了得到对高度非弹性碰撞足够精确的输运系数的理论表达式，我们用第二Sonine近似求解了上述积分方程。作为补充，我们采用直接模拟蒙特卡罗方法（DSMC）数值求解非弹性Enskog方程，得到二维和三维的动力学扩散系数$D_0$。我们已经观察到，在我们所有的模拟，分歧，任意大的非弹性，在两个解决方案（DSMC和第二Sonine近似）的值小于4%。此外，我们表明，第二Sonine近似$D_0$产生了显着的改善（高达50%）比第一Sonine近似的杂质颗粒轻于周围的气体和大的非弹性的范围内。本文报道的结果是直接应用在颗粒流中的重要问题，如由重力和热梯度驱动的偏析。我们在这里分析的隔离标准，从我们的理论表达式的传输系数。
Two different kinetic theories [J. Solsvik and E. Manger (SM-theory), Phys. Fluids \textbf{33}, 043321 (2021) and V. Garzó, J. W. Dufty, and C. M. Hrenya (GDH-theory), Phys. Rev. E \textbf{76}, 031303 (2007)] are considered to determine the shear viscosity $η$ for a moderately dense granular binary mixture of smooth hard spheres. The mixture is subjected to a simple shear flow and heated by the action of an external driving force (Gaussian thermostat) that exactly compensates the energy dissipated in collisions. The set of Enskog kinetic equations is the starting point to obtain the dependence…
The Granular Integration Through Transients (GITT) formalism gives a theoretical description of the rheology of moderately dense granular flows and suspensions. In this work, we extend the GITT equations beyond the case of simple shear flows studied before. Applying this to the particular example of extensional flows, we show that the predicted behavior is somewhat different from that of the more frequently studied simple shear case, as illustrated by the possibility of non monotonous evolution of the effective friction coefficient $μ$ with the inertial number $\mathcal{I}$. By the reduction o…
The Navier--Stokes transport coefficients for a model of a confined quasi-two-dimensional granular binary mixture of inelastic hard spheres are determined from the Boltzmann kinetic equation. A normal or hydrodynamic solution to the Boltzmann equation is obtained via the Chapman--Enskog method for states near the local version of the homogeneous time-dependent state. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. As expected, they are given in terms of the solutions…
The Enskog kinetic theory for moderately dense granular suspensions is considered as a model to determine the Navier-Stokes transport coefficients. The influence of the interstitial gas on solid particles is modeled by a viscous drag force term plus a stochastic Langevin-like term. The suspension model is solved by means of the Chapman--Enskog method conveniently adapted to dissipative dynamics. The momentum and heat fluxes as well as the cooling rate are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the c…
Enskog Theory for Polydisperse Granular Mixtures. III. Comparison of dense and dilute transport coefficients and equations of state for a binary mixture
最近的计算机模拟结果[Barrat { \em et al.}，Physica A 334（2004）513]已经显示了当气体$T$的温度被通常爱因斯坦关系中的杂质$T_0$的温度代替时，扩散$D$和迁移率$λ$系数之间的爱因斯坦关系$ε\equiv D/(T_0λ)=1$的有效性。本文用Chapman-Enskog方法解析求解Boltzmann-Lorentz方程来分析这个问题。气体在外部驱动力（恒温器）的作用下被加热，该驱动力确实起作用以补偿碰撞损失的能量。考虑两种类型的恒温器：（a）与粒子速度成比例的确定性力（高斯恒温器），以及（b）白噪声外力（随机恒温器）。的扩散和迁移率系数给出的两个线性积分方程，这是近似解决的Sonine多项式展开的二阶的解决方案。结果表明，爱因斯坦关系（$ε\neq 1$）的破坏仅仅是由于杂质速度分布函数的非麦克斯韦行为（吉布斯态的缺失）。在定量水平上，动力学理论的结果还表明，$ε$从1的偏差是更显着的高斯恒温器的情况下比在随机的情况下，在这种情况下的爱因斯坦关系的偏差一般小于1%。这一结论与计算机模拟的结果吻合得很好。
