含义Meaning
不可压缩牛顿流体在圆管中层流的体积流量与半径四次方成正比。奥氏（毛细管）黏度计据此由流出时间测黏度；血管半径的微小变化对血流影响巨大。
Laminar flow of an incompressible Newtonian fluid through a pipe scales with the fourth power of radius. Ostwald capillary viscometers use it, and small changes in blood-vessel radius greatly change flow.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(Q\) | 体积流量volumetric flow rate | m³·s⁻¹ |
| \(\Delta p\) | 管两端压差pressure drop | Pa |
| \(L\) | 管长pipe length | m |
| \(\eta\) | 动力黏度dynamic viscosity | Pa·s |
推导Derivation
半径 s 的液柱：压力 \(\Delta p\pi s^2\) 与侧面黏性力 \(-\eta\,(dv/ds)\,2\pi sL\) 平衡。
For a cylinder of radius s, pressure force balances viscous shear.
积分（壁面 v = 0）得抛物线速度分布 \(v(s) = \frac{\Delta p}{4\eta L}(r^2 - s^2)\)。
Integrate with no slip to get a parabolic profile.
\(Q = \int_0^r v\cdot2\pi s\,ds = \pi r^4\Delta p/(8\eta L)\)。
Integrate over the cross-section.
适用条件与局限Assumptions & limitations
层流（雷诺数 < 约 2000）、牛顿流体、充分发展流动；血液等非牛顿流体只是近似。
Laminar (Re < ~2000), Newtonian, fully developed flow; blood is only approximate.
例题Worked example
半径减小 10%，流量降至 0.9⁴ = 66%。
A 10 % narrower tube carries only 66 % of the flow.