[物理学与PDEs]第2章第5节 一维流体力学方程组的 Lagrange 形式 5.4 一维粘性热传导流体力学方程组的 Lagrange 形式

1. 一维粘性热传导流体力学方程组 $$\beex \bea \cfrac{\p\rho}{\p t}+\cfrac{\p }{\p x}(\rho u)&=0,\\ \cfrac{\p u}{\p t}+u\cfrac{\p u}{\p x} +\cfrac{1}{\rho}\cfrac{\p p}{\p x} -\cfrac{1}{\rho}\cfrac{\p }{\p x}\sez{\sex{\cfrac{4\mu}{3}+\mu'}\cfrac{\p u}{\p x}}&=F,\\ \rho \cfrac{\p e}{\p t} +\rho u\cfrac{\p e}{\p x} +p\cfrac{\p u}{\p x} -\sex{\cfrac{4\mu}{3}+\mu'}\sex{\cfrac{\p u}{\p x}}^2 &=\cfrac{\p}{\p x}\sex{\kappa\cfrac{\p T}{\p x}}. \eea \eeex$$ 的 Lagrange 形式为 $$\beex \bea \cfrac{\p \tau}{\p t}-\cfrac{\p u}{\p x}&=0,\\ \cfrac{\p u}{\p t}+\cfrac{\p p}{\p x}-\cfrac{\p}{\p x}\sez{\sex{\cfrac{4}{3}\mu+\mu'}\rho \cfrac{\p u}{\p x}}&=F,\\ \cfrac{\p e}{\p t}+p\cfrac{\p u}{\p t} -\rho\sex{\cfrac{4}{3}\mu+\mu'}\sex{\cfrac{\p u}{\p x}}^2 &=\cfrac{\p}{\p x}\sex{\kappa\rho \cfrac{\p T}{\p x}}. \eea \eeex$$

2. Lagrange 坐标能将原非线性方程组的形式变得比较简单.

3. 初边值问题也可以由 Euler 坐标系 $\to$ Lagrange 坐标系.