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arXiv 2107.10522cond-mat.softcond-mat.stat-mech
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简单剪切流下稀疏惯性悬浮液二元混合物的流变学

Rheology of a dilute binary mixture of inertial suspension under simple shear flow

Satoshi Takada, Hisao Hayakawa, Vicente Garzó

AI总结:

本文基于 Boltzmann 方程和 Grad 矩阵法分析了简单剪切流下稀疏惯性悬浮液二元混合物的流变学,发现温度比与粘度比随尺寸比增加在特定剪切率下发生不连续变化,并精确求解了 BGK 型动力学模型。

摘要:

在 Boltzmann 动力学方程的框架下,分析了简单剪切流下稀疏惯性悬浮液二元混合物的流变学。周围粘性气体对固体颗粒的影响通过确定性的粘性曳力加上一个用环境温度 $T_\ ext{env}$ 定义的随机类 Langevin 项来考虑。采用 Grad 矩阵法确定了温度比和压力张量,表示为恢复系数、浓度、混合物各组分质量与直径以及环境温度的函数。将解析结果与事件驱动 Langevin 仿真进行了比较,针对具有相同质量密度 $m_1/m_2=(σ^{(1)}/σ^{(2)})^3$ 的硬球混合物,其中 $m_i$ 和 $σ^{(1)}$ 分别为物种 $i$ 的质量和直径。证实了理论预测与不同尺寸比 $σ^{(1)}/σ^{(2)}$ 以及广泛参数空间内弹性与非弹性碰撞的仿真结果一致。值得注意的是,随着尺寸比增加,温度比 $T_1/T_2$ 和粘度比 $η_1/η_2$($η_i$ 为物种 $i$ 对总剪切粘度 $η=η_1+η_2$ 的部分贡献)在特定剪切率下发生不连续变化;由于系统尺寸较小,这一特征(预期在热力学极限下发生)无法被仿真完全捕捉。此外,当 $T_\ ext{env}$ 远小于动力学温度 $T$ 时,精确求解了适用于非弹性硬球混合物的 Bhatnagar--Gross--Krook (BGK) 型动力学模型。对 Grad 方法、BGK 模型和仿真得到的速度分布函数进行了比较。

英文摘要:

The rheology of a dilute binary mixture of inertial suspension under simple shear flow is analyzed in the context of the Boltzmann kinetic equation. The effect of the surrounding viscous gas on the solid particles is accounted for by means of a deterministic viscous drag force plus a stochastic Langevin-like term defined in terms of the environmental temperature $T_\text{env}$. Grad's moment method is employed to determine the temperature ratio and the pressure tensor in terms of the coefficients of restitution, concentration, the masses and diameters of the components of the mixture, and the environmental temperature. Analytical results are compared against event-driven Langevin simulations for mixtures of hard spheres with the same mass density $m_1/m_2=(σ^{(1)}/σ^{(2)})^3$, $m_i$ and $σ^{(1)}$ being the mass and diameter, respectively, of the species $i$. It is confirmed that the theoretical predictions agree with simulations of various size ratios $σ^{(1)}/σ^{(2)}$ and for elastic and inelastic collisions in the wide range of parameters' space. It is remarkable that the temperature ratio $T_1/T_2$ and the viscosity ratio $η_1/η_2$ ($η_i$ being the partial contribution of the species $i$ to the total shear viscosity $η=η_1+η_2$) discontinuously change at a certain shear rate as the size ratio increases; this feature (which is expected to occur in the thermodynamic limit) cannot be completely captured by simulations due to small system size. In addition, a Bhatnagar--Gross--Krook (BGK)-type kinetic model adapted to mixtures of inelastic hard spheres is exactly solved when $T_\text{env}$ is much smaller than the kinetic temperature $T$. A comparison between the velocity distribution functions obtained from Grad's method, BGK model, and simulations is carried out.

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