Farvre平均した連続体の質量保存則

$ \frac{\widetilde{\mathrm D}\overline\rho}{\widetilde{\mathrm D} t}+\overline{\rho}\bm\nabla\cdot\widetilde{\bm v}=0

導出

$ \frac{\mathrm D\rho}{\mathrm Dt}+\rho\bm\nabla\cdot\bm v=0

$ \implies\overline{\frac{\mathrm D\rho}{\mathrm Dt}+\rho\bm\nabla\cdot\bm v}=0

$ \iff\overline{\frac{\partial\rho}{\partial t}+\bm\nabla\cdot\rho\bm v}=0

$ \iff\frac{\partial\overline\rho}{\partial t}+\bm\nabla\cdot\overline{\rho\bm v}=0

$ \iff\frac{\partial\overline\rho}{\partial t}+\bm\nabla\cdot\overline{\rho}\widetilde{\bm v}=0

$ \iff\frac{\partial\overline\rho}{\partial t}+\widetilde{\bm v}\cdot\bm\nabla\overline{\rho}+\overline{\rho}\bm\nabla\cdot\widetilde{\bm v}=0

$ \underline{\iff\frac{\widetilde{\mathrm D}\overline\rho}{\widetilde{\mathrm D} t}+\overline{\rho}\bm\nabla\cdot\widetilde{\bm v}=0\quad}_\blacksquare