The renormalization method proposed by Batchelor is used to derive gradient diffusion coefficients in Brownian suspensions of hard spheroidal particles with aspect ratio $\unicode[STIX]{x1D706}$ in the range $1\leqslant \unicode[STIX]{x1D706}\leqslant 3.5$ . The theory is based on pairwise steric and hydrodynamic interactions, and the results are therefore valid for dilute suspensions such that $\unicode[STIX]{x1D706}^{2}\unicode[STIX]{x1D719}\ll 1$ , where $\unicode[STIX]{x1D719}$ is the particle volume fraction. The driving force for gradient diffusion, i.e. the gradient in chemical potential, is larger for suspensions of spheroidal particles than for spheres at the same volume fraction. The hydrodynamic resistance also increases with aspect ratio, but the increase is weaker than that in the driving force. Consequently, at the same particle volume fraction, the increases in rates of gradient diffusion are greater for spheroidal particles than for spheres. The concentration-dependent gradient diffusion coefficient $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ is shown to be closely approximated by $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706}-1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ , which reduces to the result for spheres when $\unicode[STIX]{x1D706}=1$ . Here, $D_{0}$ is the Stokes–Einstein diffusivity of a spherical particle with its radius equal to the longer dimension of the spheroidal particle, and $\unicode[STIX]{x1D709}_{m}D_{0}$ is the orientation-averaged diffusivity of an isolated spheroidal particle.

中文翻译：

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硬球体颗粒稀悬浮液中的梯度扩散

Batchelor 提出的重整化方法用于推导出长宽比 $\unicode[STIX]{x1D706}$ 范围为 $1\leqslant\unicode[STIX]{x1D706}\leqslant 3.5 的硬球体颗粒的布朗悬浮液中的梯度扩散系数美元。该理论基于成对空间和流体动力学相互作用，因此结果适用于稀释悬浮液，使得 $\unicode[STIX]{x1D706}^{2}\unicode[STIX]{x1D719}\ll 1$ ，其中 $ \unicode[STIX]{x1D719}$ 是粒子体积分数。梯度扩散的驱动力，即化学势梯度，在相同体积分数下，球状颗粒悬浮液比球体更大。水动力阻力也随着纵横比的增加而增加，但增加幅度弱于驱动力。最后，在相同的颗粒体积分数下，球状颗粒的梯度扩散速率增加大于球体。浓度相关梯度扩散系数 $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ 显示为与 $D(\unicode[STIX]{x1D719},\unicode [STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706} -1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ ，当 $\unicode[STIX]{x1D706}=1$ 时简化为球体的结果。这里，$D_{0}$ 是半径等于球状粒子较长尺寸的球形粒子的斯托克斯-爱因斯坦扩散系数，$\unicode[STIX]{x1D709}_{m}D_{0}$是孤立球状粒子的取向平均扩散率。球体颗粒的梯度扩散速率增加大于球体。浓度相关梯度扩散系数 $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ 显示为与 $D(\unicode[STIX]{x1D719},\unicode [STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706} -1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ ，当 $\unicode[STIX]{x1D706}=1$ 时简化为球体的结果。这里，$D_{0}$ 是半径等于球状粒子较长尺寸的球形粒子的斯托克斯-爱因斯坦扩散系数，$\unicode[STIX]{x1D709}_{m}D_{0}$是孤立球状粒子的取向平均扩散率。球体颗粒的梯度扩散速率增加大于球体。浓度相关梯度扩散系数 $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ 显示为与 $D(\unicode[STIX]{x1D719},\unicode [STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706} -1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ ，当 $\unicode[STIX]{x1D706}=1$ 时简化为球体的结果。这里，$D_{0}$ 是半径等于球状粒子较长尺寸的球形粒子的斯托克斯-爱因斯坦扩散系数，$\unicode[STIX]{x1D709}_{m}D_{0}$是孤立球状粒子的取向平均扩散率。浓度相关梯度扩散系数 $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ 显示为与 $D(\unicode[STIX]{x1D719},\unicode [STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706} -1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ ，当 $\unicode[STIX]{x1D706}=1$ 时简化为球体的结果。这里，$D_{0}$ 是半径等于球状粒子较长尺寸的球形粒子的斯托克斯-爱因斯坦扩散系数，$\unicode[STIX]{x1D709}_{m}D_{0}$是孤立球状粒子的取向平均扩散率。浓度相关梯度扩散系数 $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ 显示为与 $D(\unicode[STIX]{x1D719},\unicode [STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706} -1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ ，当 $\unicode[STIX]{x1D706}=1$ 时简化为球体的结果。这里，$D_{0}$ 是半径等于球状粒子较长尺寸的球形粒子的斯托克斯-爱因斯坦扩散系数，$\unicode[STIX]{x1D709}_{m}D_{0}$是孤立球状粒子的取向平均扩散率。\unicode[STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{ x1D706}-1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ ，当 $\unicode[STIX]{x1D706}=1$ 时简化为球体的结果。这里，$D_{0}$ 是半径等于球状粒子较长尺寸的球形粒子的斯托克斯-爱因斯坦扩散系数，$\unicode[STIX]{x1D709}_{m}D_{0}$是孤立球状粒子的取向平均扩散率。\unicode[STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{ x1D706}-1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$ ，当 $\unicode[STIX]{x1D706}=1$ 时简化为球体的结果。这里，$D_{0}$ 是半径等于球状粒子较长尺寸的球形粒子的斯托克斯-爱因斯坦扩散系数，$\unicode[STIX]{x1D709}_{m}D_{0}$是孤立球状粒子的取向平均扩散率。