速度勾配tensorの時間変化率

$ \frac{\mathrm D\bm l}{\mathrm Dt}=\frac{\mathrm D\bm v}{\mathrm Dt}\overleftarrow{\bm\nabla}-\bm l^2

証明

$ \frac{\partial}{\partial t}\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)}=\frac{\partial}{\partial t}\dot{\bm{F}}\cdot{\bm{F}^{-1}}

$ = \ddot{\bm F}\cdot{\bm F}^{-1}+\dot{\bm F}\cdot\dot{{\bm F}^{-1}}

$ = \ddot{\bm F}\cdot{\bm F}^{-1}-\dot{\bm F}\cdot{\bm F}^{-1}\cdot\dot{\bm F}\cdot{\bm F}^{-1}

$ = \ddot{\bm F}\cdot{\bm F}^{-1}-(\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)})^2

$ = \bm\nabla\ddot{\bm\phi}\cdot{\bm F}^{-1}-(\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)})^2

$ = \bm\nabla\ddot{\bm\phi}\cdot\left(\left.\bm\phi^{-1}\overleftarrow{\bm\nabla}\right|_{\bm x=\bm\phi(\bm X,t)}\right)-(\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)})^2

$ = \left.\left(\bm\nabla\ddot{\bm\phi}\right|_{\bm X=\bm\phi^{-1}(\bm x,t)}\cdot\left.\bm\phi^{-1}\overleftarrow{\bm\nabla}\right)\right|_{\bm x=\bm\phi(\bm X,t)}-(\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)})^2

$ = \left.\bm\nabla\ddot{\bm\phi}(\bm\phi^{-1}(\bm X,t))\right|_{\bm x=\bm\phi(\bm X,t)}-(\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)})^2

$ = \left.\bm\nabla\frac{\mathrm D\bm v}{\mathrm Dt}\right|_{\bm x=\bm\phi(\bm X,t)}-(\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)})^2

$ \iff\frac{\mathrm D\bm l}{\mathrm Dt}=\bm\nabla\frac{\mathrm D\bm v}{\mathrm Dt}-\bm l^2

もっと一般的にやったほうが楽だった

$ \frac{\mathrm D\bm\nabla\bm A}{\mathrm Dt}=\bm\nabla\frac{\partial\bm A}{\partial t}+\bm v\cdot\bm\nabla\bm\nabla\bm A

$ =\bm\nabla\frac{\partial\bm A}{\partial t}+\bm\nabla(\bm v\cdot\bm\nabla\bm A)-(\bm\nabla\bm v)\cdot\bm\nabla\bm A

$ =\bm\nabla\frac{\mathrm D\bm A}{\mathrm Dt}-(\bm\nabla\bm v)\cdot\bm\nabla\bm A