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A.2.2 成分の質量保存式での計算の途中式

計算1

$$\rho \bm{v} \cdot \bm{\nabla} \omega_i + \rho \omega_i \bm{\nabla} \cdot \bm{v} + \omega_i \bm{v} \cdot \bm{\nabla}\rho$$

$$= \rho \bigg( u \dfrac{\partial \omega_i}{\partial x} + v \dfrac{... ... \dfrac{\partial \rho}{\partial y} + w \dfrac{\partial \rho}{\partial z} \bigg)$$

$$= \rho \omega_i \dfrac{\partial u}{\partial x} + \rho u \dfrac{\p... ...c{\partial \omega_i}{\partial z} + \omega_i w \dfrac{\partial \rho}{\partial z}$$

$$= \dfrac{\partial (\rho \omega_i u)}{\partial x} + \dfrac{\partial (\rho \omega_i v)}{\partial y} + \dfrac{\partial (\rho \omega_i w)}{\partial z}$$

$$= \left( \begin{array}{c} \dfrac{\partial }{\partial x} \vspace{.... ...e{.5em} \\ \rho \omega_i v \vspace{.5em} \\ \rho \omega_i w \end{array} \right)$$

$$= \bm{\nabla} \cdot (\rho \omega_i \bm{v})$$ (A.10)

計算2

$$\rho \bm{\nabla} \cdot \bm{\nabla} \omega_i + \bm{\nabla} \rho \cdot \bm{\nabla} \omega_i$$

$$= \rho \left( \begin{array}{c} \dfrac{\partial }{\partial x} \vsp... ...l y} \vspace{.5em} \\ \dfrac{\partial \omega_i}{\partial z} \end{array} \right)$$

$$= \dfrac{\partial \rho}{\partial x} \dfrac{\partial \omega_i}{\pa... ...l z} + \rho \dfrac{\partial }{\partial z} \dfrac{\partial \omega_i}{\partial z}$$

$$= \dfrac{\partial }{\partial x}\bigg( \rho \dfrac{\partial \omega... ...{\partial }{\partial z}\bigg( \rho \dfrac{\partial \omega_i}{\partial z} \bigg)$$

$$= \left( \begin{array}{c} \dfrac{\partial }{\partial x} \vspace{.... ...\vspace{.5em} \\ \rho \dfrac{\partial \omega_i}{\partial z} \end{array} \right)$$

$$= \bm{\nabla} \cdot \big( \rho \bm{\nabla} \omega_i \big)$$ (A.11)

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