流速の物質表示と空間表示の微分

2024-04-04 22:17:00 似たようなメモを変形勾配tensorの時間微分に書いてある

$ \pmb{x}=\pmb{\phi}(\pmb{\phi}^{-1}(\pmb{x},t),t)

$ \pmb{X}=\pmb{\phi}^{-1}(\pmb{\phi}(\pmb{X},t),t)

変換函数の微分

$ \mathrm{d}\pmb{\phi}=(\pmb{\nabla}\otimes\pmb{\phi})^\top\mathrm{d}\pmb{X}+\frac{\partial \pmb{\phi}}{\partial t}\mathrm{d}t

$ \mathrm{d}\pmb{\phi}^{-1}=(\pmb{\nabla}\otimes\pmb{\phi}^{-1})^\top\mathrm{d}\pmb{x}+\frac{\partial \pmb{\phi}^{-1}}{\partial t}\mathrm{d}t

$ \mathrm{d}\pmb{x}=\left.(\pmb{\nabla}\otimes\pmb{\phi})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\mathrm{d}\pmb{\phi}^{-1}+\left.\frac{\partial \pmb{\phi}^{-1}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}}\mathrm{d}t

$ \implies\pmb{I}=\left.(\pmb{\nabla}\otimes\pmb{\phi})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}(\pmb{\nabla}\otimes\pmb{\phi}^{-1})^\top

$ \iff \left.\pmb{\nabla}\otimes\pmb{\phi}\right|_{\pmb{X}=\pmb{\phi}^{-1}}=\left(\pmb{\nabla}\otimes\pmb{\phi}^{-1}\right)^{-1}

$ \iff \left(\pmb{\nabla}\otimes\pmb{\phi}\right)^{-1}=\left.\pmb{\nabla}\otimes\pmb{\phi}^{-1}\right|_{\pmb{x}=\pmb{\phi}}

逆函数の微分を使う

$ \implies\mathrm{d}\pmb{\phi}^{-1}=\left(\left.(\pmb{\nabla}\otimes\pmb{\phi})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\right)^{-1}\left(\mathrm{d}\pmb{x}-\left.\frac{\partial \pmb{\phi}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}^{-1}}\mathrm{d}t\right)

$ \frac{\partial \pmb{\phi}^{-1}}{\partial t}=-\left(\left.(\pmb{\nabla}\otimes\pmb{\phi})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\right)^{-1}\left.\frac{\partial \pmb{\phi}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}^{-1}}=-\left.\left({(\pmb{\nabla}\otimes\pmb{\phi})^\top}^{-1}\frac{\partial \pmb{\phi}}{\partial t}\right)\right|_{\pmb{X}=\pmb{\phi}^{-1}} か？takker.icon

変換函数の2階微分

流速

物質表示：$ \pmb{v}:(\pmb{X},t)\mapsto\frac{\partial \pmb{\phi}}{\partial t}

空間表示：$ \pmb{u}:(\pmb{x},t)\mapsto\left.\frac{\partial \pmb{\phi}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}^{-1}}

空間速度と呼んでいるtakker.icon

$ \pmb{v}=\pmb{u}(\pmb{\phi}(\pmb{X},t),t)

$ \pmb{u}=\pmb{v}(\pmb{\phi}^{-1}(\pmb{x},t),t)

$ \mathrm{d}\pmb{v}=\left.(\pmb{\nabla}\otimes\pmb{u})^\top\right|_{\pmb{x}=\pmb{\phi}}\mathrm{d}\pmb{\phi}+\left.\frac{\partial \pmb{u}}{\partial t}\right|_{\pmb{x}=\pmb{\phi}}\mathrm{d}t

$ = \left.(\pmb{\nabla}\otimes\pmb{u})^\top\right|_{\pmb{x}=\pmb{\phi}}(\pmb{\nabla}\otimes\pmb{\phi})^\top\mathrm{d}\pmb{X}+\left.(\pmb{\nabla}\otimes\pmb{u})^\top\right|_{\pmb{x}=\pmb{\phi}}\frac{\partial \pmb{\phi}}{\partial t}\mathrm{d}t+\left.\frac{\partial \pmb{u}}{\partial t}\right|_{\pmb{x}=\pmb{\phi}}\mathrm{d}t

$ \implies

$ (\pmb{\nabla}\otimes\pmb{v})^\top=\left.(\pmb{\nabla}\otimes\pmb{u})^\top\right|_{\pmb{x}=\pmb{\phi}}(\pmb{\nabla}\otimes\pmb{\phi})^\top

$ \iff \pmb{\nabla}\otimes\pmb{v}=\pmb{\nabla}\otimes\pmb{\phi}\left.\pmb{\nabla}\otimes\pmb{u}\right|_{\pmb{x}=\pmb{\phi}}

$ \frac{\partial \pmb{v}}{\partial t}=\left.(\pmb{\nabla}\otimes\pmb{u})^\top\right|_{\pmb{x}=\pmb{\phi}}\frac{\partial \pmb{\phi}}{\partial t}+\left.\frac{\partial \pmb{u}}{\partial t}\right|_{\pmb{x}=\pmb{\phi}}=\left.(\pmb{\nabla}\otimes\pmb{u})^\top\right|_{\pmb{x}=\pmb{\phi}}\pmb{v}+\left.\frac{\partial \pmb{u}}{\partial t}\right|_{\pmb{x}=\pmb{\phi}}=\left.\frac{\mathrm{D} \pmb{u}}{\mathrm{D} t}\right|_{\pmb{x}=\pmb{\phi}}

$ \mathrm{d}\pmb{u}=\left.(\pmb{\nabla}\otimes\pmb{v})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\mathrm{d}\pmb{\phi}^{-1}+\left.\frac{\partial \pmb{v}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}^{-1}}\mathrm{d}t

$ \frac{\partial\pmb{u}}{\partial t}=\left.(\pmb{\nabla}\otimes\pmb{v})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\frac{\partial \pmb{\phi}^{-1}}{\partial t}+\left.\frac{\partial \pmb{v}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}^{-1}}

$ =\left.(\pmb{\nabla}\otimes\pmb{v})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\frac{\partial \pmb{\phi}^{-1}}{\partial t}+\frac{\mathrm{D}\pmb{u}}{\mathrm{D}t}

流速の物質表示と空間表示の微分#62b0fb9b1280f00000aa091eを使うと

$ =-\left.(\pmb{\nabla}\otimes\pmb{v})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\left(\left.(\pmb{\nabla}\otimes\pmb{\phi})^\top\right|_{\pmb{X}=\pmb{\phi}^{-1}}\right)^{-1}\left.\frac{\partial \pmb{\phi}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}^{-1}}+\frac{\mathrm{D}\pmb{u}}{\mathrm{D}t}

$ =-\left(\left.(\pmb{\nabla}\otimes\pmb{v})^\top{(\pmb{\nabla}\otimes\pmb{\phi})^\top}^{-1}\right)\right|_{\pmb{X}=\pmb{\phi}^{-1}}\pmb{u}+\frac{\mathrm{D}\pmb{u}}{\mathrm{D}t}

$ \implies\frac{\mathrm{D}\pmb{u}}{\mathrm{D}t}=\left(\left.(\pmb{\nabla}\otimes\pmb{v})^\top{(\pmb{\nabla}\otimes\pmb{\phi})^\top}^{-1}\right)\right|_{\pmb{X}=\pmb{\phi}^{-1}}\pmb{u}+\frac{\partial\pmb{u}}{\partial t}

となる。

流速の物質表示と空間表示の微分#62b1011b1280f00000aa092bより$ \left(\left.(\pmb{\nabla}\otimes\pmb{v})^\top{(\pmb{\nabla}\otimes\pmb{\phi})^\top}^{-1}\right)\right|_{\pmb{X}=\pmb{\phi}^{-1}}=(\pmb{\nabla}\otimes\pmb{u})^\topなので、↓と矛盾しない

実質微分と運動函数の関係

$ \frac{\mathrm{D}\pmb{u}}{\mathrm{D}t}:=(\pmb{\nabla}\otimes\pmb{u})^\top\pmb{u}+\frac{\partial \pmb{u}}{\partial t}=\left.\frac{\partial\pmb{v}}{\partial t}\right|_{\pmb{X}=\pmb{\phi}^{-1}}=\left.\frac{\partial^2\pmb{\phi}}{{\partial t}^2}\right|_{\pmb{X}=\pmb{\phi}^{-1}}

一般に、$ \frac{\mathrm{D}\pmb{g}}{\mathrm{D}t}:=(\pmb{\nabla}\otimes\pmb{g})^\top\pmb{u}+\frac{\partial \pmb{g}}{\partial t}=\left.\frac{\partial}{\partial t}\pmb{g}(\pmb{\phi}(\pmb{X},t),t)\right|_{\pmb{X}=\pmb{\phi}^{-1}}

任意の函数のpush forward/pull back

scalar函数$ f,g

vector函数

完全流体の運動方程式を物質表示で導出する

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