[物理学与PDEs]第3章第5节 一维磁流体力学方程组 5.1 一维磁流体力学方程组

1.  当磁流体力学方程组中的量只依赖于 $t$ 及一个空间变量时, 该方程组称为一维的.

2.  一维磁流体力学方程组 $$\beex \bea \cfrac{\p H_2}{\p t}& +u_1\cfrac{\p H_2}{\p x} +H_2\cfrac{\p u_1}{\p x} -H_1\cfrac{\p u_2}{\p x} =\cfrac{1}{\sigma\mu_0}\cfrac{\p^2H_2}{\p x^2},\\ \cfrac{\p H_3}{\p t}&+u_1\cfrac{\p H_3}{\p x} +H_3\cfrac{\p u_1}{\p x} -H_1\cfrac{\p u_3}{\p x} =\cfrac{1}{\sigma\mu_0}\cfrac{\p^2H_3}{\p x^2},\\ \cfrac{\p \rho}{\p t}&+u_1\cfrac{\p \rho}{\p x}+\rho\cfrac{\p u_1}{\p x}=0,\\ \cfrac{\p u_1}{\p t}&+u_1\cfrac{\p u_1}{\p x} +\cfrac{1}{\rho}\cfrac{\p p}{\p x} -\cfrac{1}{\rho}\cfrac{\p}{\p x} \sez{\sex{\cfrac{4}{3}\bar \mu+\bar \mu'}\cfrac{\p u_1}{\p x}} +\cfrac{\mu_0}{\rho}\sex{H_2\cfrac{\p H_2}{\p x}+H_3\cfrac{\p H_3}{\p x}}=F_1,\\ \cfrac{\p u_2}{\p t}& +u_1\cfrac{\p u_2}{\p x} -\cfrac{1}{\rho}\cfrac{\p }{\p x}\sex{\bar \mu \cfrac{\p u_2}{\p x}} -\cfrac{\mu_0}{\rho}H_1\cfrac{\p H_2}{\p x}=F_2,\\ \cfrac{\p u_3}{\p t}&+u_1\cfrac{\p u_3}{\p x} -\cfrac{1}{\rho}\cfrac{\p}{\p x}\sex{\bar \mu\cfrac{\p u_3}{\p x}} -\cfrac{\mu_0}{\rho }H_1\cfrac{\p H_3}{\p x}=F_3,\\ \rho T\cfrac{\p S}{\p t}& +\rho T u_1\cfrac{\p S}{\p x} -\sex{\cfrac{4}{3}\bar \mu+\mu'}\sex{\cfrac{\p u_1}{\p x}}^2 -\bar \mu\sex{\cfrac{\p u_2}{\p x}}^2 -\bar\mu \sex{\cfrac{\p u_3}{\p x}}^2 =\cfrac{\p}{\p x}\sex{\kappa\cfrac{\p T}{\p x}}. \eea \eeex$$

3.  一维理想磁流体力学方程组 $$\beex \bea \cfrac{\p H_2}{\p t}& +u_1\cfrac{\p H_2}{\p x} +H_2\cfrac{\p u_1}{\p x} -H_1\cfrac{\p u_2}{\p x} =0,\\ \cfrac{\p H_3}{\p t}&+u_1\cfrac{\p H_3}{\p x} +H_3\cfrac{\p u_1}{\p x} -H_1\cfrac{\p u_3}{\p x} =0,\\ \cfrac{\p \rho}{\p t}&+u_1\cfrac{\p \rho}{\p x}+\rho\cfrac{\p u_1}{\p x}=0,\\ \cfrac{\p u_1}{\p t}&+u_1\cfrac{\p u_1}{\p x} +\cfrac{1}{\rho}\sex{\tilde c^2\cfrac{\p \rho}{\p x}+\cfrac{\p p}{\p S}\cfrac{\p S}{\p x}} +\cfrac{\mu_0}{\rho}\sex{H_2\cfrac{\p H_2}{\p x}+H_3\cfrac{\p H_3}{\p x}}=F_1,\\ \cfrac{\p u_2}{\p t}& +u_1\cfrac{\p u_2}{\p x} -\cfrac{\mu_0}{\rho}H_1\cfrac{\p H_2}{\p x}=F_2,\\ \cfrac{\p u_3}{\p t}&+u_1\cfrac{\p u_3}{\p x} -\cfrac{\mu_0}{\rho }H_1\cfrac{\p H_3}{\p x}=F_3,\\ \cfrac{\p S}{\p t}&+u_1\cfrac{\p S}{\p x}=0.  \eea \eeex$$

(1)  其为对称双曲组.

(2)  当 $H_1\neq 0$, $H_2^2+H_3^2\neq 0$ 时, 其为一维严格双曲组.

(3)  当 $H_1=0$ 或 $H_2^2+H_3^2=0$ 时, 其为一维双曲组.