张量部分
\(\varepsilon_{ijk}\varepsilon_{ist} = \delta_{js}\delta_{kt}-\delta{jt}\delta{ks}\)
\(\vec{a}\times\vec{b} = \varepsilon_{ijk}a_jb_k\) \(rot \vec{a} = \varepsilon_{ijk}\frac{\partial a_k}{\partial x_j}\) \(div(\phi vec{a}) = \phi div \vec{a} + grad\phi \cdot \vec{a}\) \(div(\vec{a} \times \vec{b}) = \vec{b} \cdot rot\vec{a} - \vec{a} \cdot rot\vec{b}\) \(rot(\phi\vec{a}) = \phi rot\vec{a} + grad\phi \times \vec{a}\)
基本概念
\[\frac{d\phi}{dt} = \frac{\partial \phi}{\partial t} + \vec{v}\nabla\phi\]\(\frac{d}{dt}\oint_l\vec{v} \cdot \delta \vec{r} = \oint_l\frac{d\vec{v}}{dt} \cdot \delta\vec{r}\) \(\frac{d}{dt}\int_S\Omega \cdot dS = \int_S(\frac{d\Omega}{dt} + \Omega div\vec{v} - (\Omega \cdot \nabla)\vec{v})\cdot \delta S\) \(\frac{d}{dt}\int_\tau\phi\delta\tau = \int_\tau(\frac{d\phi}{dt} + \phi div\vec{v})\delta\tau = \int_\tau[\frac{\partial\phi}{\partial t} + div(\phi \vec{v})]\delta\tau = \int_\tau\frac{\partial \phi}{dt} + \int_S\phi v_n \delta S\) \(\frac{d}{dt}\int_\tau\vec{a}\delta\tau = \int_\tau(\frac{d\vec{a}}{dt} + \vec{a} div\vec{v})\delta\tau = \int_\tau\frac{\partial \vec{a}}{dt} + \int_Sv_n\vec{a} \delta S\)
基本方程组
连续性方程
\[\frac{\partial \rho}{\partial t} + \rho div(\vec{v}) = 0\]运动方程
\[\rho \frac{d\vec{v}}{dt} = \rho F + divP\]能量方程
\[\rho\frac{dU}{dt} = P:S + div(kgrad T)+\rho q\]本构方程
\[P = -pI+2\mu(S-\frac{1}{3}Idiv\vec{v})+\mu^\prime Idiv\vec{v}\]状态方程
\[pV = \frac{m}{M}R_0T\]