输运现象

牛顿黏性定律

\[f = -\eta \frac{\mathrm{d}u}{\mathrm{d}z}A\] 其中 \(A\) 为切平面内面积.

动量流密度

\[J_p = -\eta \frac{\mathrm{d}u}{\mathrm{d}z}\] \[\Rightarrow f = J_p A\]

物体在流体中运动时, 速度不快(层流), 有: \[f = 6\pi\eta vR\]

菲克定律

\[J_N = -D \frac{\mathrm{d}n}{\mathrm{d}z}\] \[\frac{\mathrm{d}N}{\mathrm{d}z} = J_N A\]

布朗运动中有: \[D = \frac{kT}{6\pi\eta r}\]

对流、传导、辐射

傅里叶热传导定律

\[\dot{Q} = -\kappa \frac{\mathrm{d}T}{\mathrm{d}z}A = J_T A\] \[J_T = -\kappa \frac{\mathrm{d}T}{\mathrm{d}z}\]

\[\bar{\lambda} = \frac{\bar{v}}{\bar{\omega}}\] 其中 \(\bar{\omega}/\bar{Z}\) 为平均碰撞频率.

推导有: \[\bar{\lambda} = \frac{1}{\sqrt{2}n\sigma} = \frac{1}{\sqrt{2}\pi n d^2}\] 其中 \(n\) 分子数密度, \(\sigma = \pi d^2\) 为散射截面.

自由程分布

\[P(\lambda)\mathrm{d}\lambda = \frac{1}{\bar{\lambda}} \mathrm{e}^{-l/\bar{\lambda}}\mathrm{d}l, \ \mathrm{d}N = N_0 P(\lambda)\mathrm{d}\lambda\]

\[\eta = \frac{1}{3}\rho \bar{v}\bar{\lambda}\] \[ \kappa = \frac{1}{3}\rho \bar{v}\bar{\lambda} \frac{-C_{V, m}}{M} \] \[D = \frac{1}{3}\bar{v}\bar{\lambda}\]