热传导

热传导 #

定义

$u_t-a^2\Delta u=0$

推导 #

能量守恒定律

$u(x,y,z,t)$ 表示温度, $\vec{v}=-k\nabla u$.

$$\int_D c\rho u(x,y,z,t+\delta t)\text{d} x\text{d} y\text{d} z-\int_D c\rho u(x,y,z,t)\text{d} x\text{d} y\text{d} z=-\int_{t}^{t+\delta t}\int_{\partial D}\vec{v}\cdot\vec{n}\text{d} s \text{d} t$$ $$\Rightarrow \int_D\int_t^{t+\delta t} c\rho u_t\text{d} t \text{d} x\text{d} y \text{d} z=\int_t^{t+\delta t}\int_{\partial D}k \nabla u \cdot \vec{n}\text{d} s\text{d} t$$

由连续性和作业可知.

$$\int_D c\rho u_t\text{d} t\text{d} x\text{d} y\text{d} z=\int_{\partial D}k\nabla u\cdot \vec{n}\text{d} s\text{d} t\overunderset{Gauss}{}{=}$$