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# 物质导数

## 定义

物质导数（Material Derivative）的定义：

$$
\underbrace{\frac{D}{D t}}*{\text {物质导数}}=\underbrace{\frac{\partial}{\partial t}}*{\text {当地导数 }}+\underbrace{(\mathbf{V} \cdot \nabla)}\_{\text {对流导数 }}
$$

物质导数的物理意义为：它是**运动的流体微团的物理量随时间的变化率，它等于该物理量由当地时间变化所引起的变化率与由流体对流引起的变化率的和**。

## 推导

选取一个运动的流体微元，则流体微元的速度矢量为 $$\textbf{v} =u\textbf{i}+v\textbf{j}+w\textbf{k}$$ ，密度为  $$\rho$$ ，由于微元随流体运动，则 $$u,v,w,\rho$$ 均是时间和空间的函数：

$$
\begin{array}{l}
u=u(x, y, z, t) \\
v=v(x, y, z, t) \\
w=w(x, y, z, t) \\
\rho=\rho(x, y, z, t)
\end{array}
$$

假设在 $$t\_1$$ 时刻，微元处于流场的1点，在这一时刻、这一点上的密度为：

$$
\rho\_1=\rho(x\_1,y\_1,z\_1,t\_1)
$$

在流场的作用下，流体运动到2点，此时微元的密度为：

$$
\rho\_2=\rho(x\_2,y\_2,z\_2,t\_2)
$$

![](/files/-MZwKU_ti1ohsTKY8c9Z)

<https://zhuanlan.zhihu.com/p/146396629>


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