含义Meaning
小球在黏性流体中低速运动时所受阻力与速度成正比（1851）。重力、浮力与阻力平衡时达到终端沉降速度，与半径平方成正比——用于粒度分析、离心分离、密立根油滴实验。
At low speed a sphere’s drag is proportional to velocity (1851). Balancing gravity, buoyancy and drag gives a terminal velocity ∝ r² — used in particle sizing, centrifugation and Millikan’s oil-drop experiment.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(F_d\) | 黏性阻力viscous drag | N |
| \(v_t\) | 终端速度terminal velocity | m·s⁻¹ |
| \(\rho_p, \rho_f\) | 颗粒、流体密度particle and fluid densities | kg·m⁻³ |
推导Derivation
求解低雷诺数（\(Re = \rho vr/\eta \ll 1\)）纳维–斯托克斯方程，得球的阻力 \(6\pi\eta rv\)。
Solving low-Re Navier–Stokes gives drag \(6\pi\eta rv\).
终端时 \(\tfrac43\pi r^3(\rho_p - \rho_f)g = 6\pi\eta rv\)，解出 \(v_t\)。
Balance net weight with drag.
离心场中把 g 换成 \(\omega^2r_\text{rot}\)。
In a centrifuge replace g by ω²r.
适用条件与局限Assumptions & limitations
只适用于 Re < 约 0.1–1 的球形颗粒；颗粒浓度高时相互干扰（受阻沉降）；极小颗粒还受布朗运动影响。
Valid for spheres at Re < ~0.1–1; concentrated suspensions hinder settling; tiny particles are dominated by Brownian motion.
例题Worked example
半径 1 μm 的石英颗粒（ρ = 2650 kg/m³）在水中：v = 2×10⁻¹²×1650×9.81/(9×8.9×10⁻⁴) = 4.0 μm/s，沉降 10 cm 约需 7 小时。
A 1 μm quartz grain settles at 4 μm/s in water — ~7 h for 10 cm.