27、对流 - 扩散 - 反应偏微分方程推导与常微分方程系统刚度及特征值分析

对流 - 扩散 - 反应偏微分方程推导与常微分方程系统刚度及特征值分析

1. 对流 - 扩散 - 反应偏微分方程推导

1.1 笛卡尔坐标系下的推导

在笛卡尔坐标系中，通过对增量体积 $\Delta x\Delta y\Delta z$ 进行质量平衡分析，得到如下方程：
$\Delta x\Delta y\Delta z \frac{\partial u}{\partial t} = -\Delta y\Delta zD_x\frac{\partial u}{\partial x}| {x} - (-\Delta y\Delta zD_x\frac{\partial u}{\partial x}| {x+\Delta x}) - \Delta x\Delta zD_y\frac{\partial u}{\partial y}| {y} - (-\Delta x\Delta zD_y\frac{\partial u}{\partial y}| {y+\Delta y}) - \Delta x\Delta yD_z\frac{\partial u}{\partial z}| {z} - (-\Delta x\Delta yD_z\frac{\partial u}{\partial z}| {z+\Delta z}) + \Delta y\Delta z(v_xu| {x} - v_xu| {x+\Delta x}) + \Delta x\Delta z(v_yu| {y} - v_yu| {y+\Delta y}) + \Delta x\Delta y(v_zu| {z}