2.4.4.1 圧縮性流体（密度 [kg/m ]は変化する）

xyz軸での出入の総和となる式(2.181) $^{\text{p.\pageref{eq-eneworXYZ}}}$ へ面に垂直な $\tau$ の式(2.180) $^{\text{p.\pageref{eq-sigmatau-com}}}$ と面に平行な $\tau$ の式(2.66) $^{\text{p.\pageref{eq-tau}}}$ を代入すると、コントロールボリューム全体での時間あたりの仕事が次式で求められる。

$\displaystyle \bigg($	$\displaystyle - P \dfrac{\partial u}{\partial x} - P \dfrac{\partial v}{\partia... ...artial x} - v \dfrac{\partial P}{\partial y} - w \dfrac{\partial P}{\partial z}$
	$\displaystyle + u \dfrac{\partial \tau_{xx}}{\partial x} + v \dfrac{\partial \t... ...frac{\partial \tau_{xz}}{\partial x} + w \dfrac{\partial \tau_{yz}}{\partial y}$
	$\displaystyle + \underbrace{ \tau_{xx} \dfrac{\partial u}{\partial x} + \tau_{y... ...\partial z} + \tau_{xz} \dfrac{\partial w}{\partial x} }_{散逸項} \bigg) dxdydz$
$\displaystyle =$	$\displaystyle \bigg[ - P \bigg( \frac{\partial u}{\partial x} + \frac{\partial ... ...l x} + v \frac{\partial P}{\partial y} + w \frac{\partial P}{\partial z} \bigg)$
	$\displaystyle + u \frac{\partial }{\partial x} \bigg\{ \mu \bigg( 2 \dfrac{\par... ...{\partial w}{\partial z} - \dfrac{2}{3} \bm{\nabla} \cdot \bm{v} \bigg) \bigg\}$
	$\displaystyle + \mu u \bigg\{ \frac{\partial }{\partial y} \bigg( \frac{\partia... ...g( \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \bigg) \bigg\}$
	$\displaystyle + \mu w \bigg\{ \frac{\partial }{\partial x} \bigg( \frac{\partia... ...g( \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} \bigg) \bigg\}$
	$\displaystyle + \underbrace{ \mu \bigg( 2 \dfrac{\partial u}{\partial x} - \dfr... ...{2}{3} \bm{\nabla} \cdot \bm{v} \bigg) \frac{\partial w}{\partial z} }_{散逸項}$
	$\displaystyle \underbrace{ + \mu \bigg( \frac{\partial v}{\partial x} + \frac{\... ...}{\partial z} + \frac{\partial w}{\partial x} \bigg)^2 }_{散逸項} \bigg] dxdydz$
$\displaystyle =$	$\displaystyle \bigg( - P \bigg( \frac{\partial u}{\partial x} + \frac{\partial ... ...l x} + v \frac{\partial P}{\partial y} + w \frac{\partial P}{\partial z} \bigg)$
	$\displaystyle + \mu \bigg[ u \bigg\{ \frac{\partial ^2 u}{\partial x^2 } + \fra... ... + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \bigg) \bigg\}$
	$\displaystyle + v \bigg\{ \frac{\partial ^2 v}{\partial x^2 } + \frac{\partial ... ... + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \bigg) \bigg\}$
	$\displaystyle + w \bigg\{ \frac{\partial ^2 w}{\partial x^2 } + \frac{\partial ... ... + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \bigg) \bigg\}$
	$\displaystyle + \underbrace{ 2 \bigg( \dfrac{\partial u}{\partial x} \bigg)^2 +... ...rtial y} \bigg)^2 + 2 \bigg( \dfrac{\partial w}{\partial z} \bigg)^2 }_{散逸項}$
	$\displaystyle \underbrace{ - \dfrac{2}{3} \bigg\{ \dfrac{\partial u}{\partial x... ...rtial v}{\partial y} + \dfrac{\partial w}{\partial z} \bigg) \bigg\} }_{散逸項}$
	$\displaystyle \underbrace{ + \bigg( \dfrac{\partial v}{\partial x} + \dfrac{\pa... ...al z} + \dfrac{\partial w}{\partial x} \bigg)^2 }_{散逸項} \bigg] \bigg) dxdydz$
$\displaystyle =$	$\displaystyle \bigg( - P \bigg( \frac{\partial u}{\partial x} + \frac{\partial ... ...l x} + v \frac{\partial P}{\partial y} + w \frac{\partial P}{\partial z} \bigg)$
	$\displaystyle + \mu \bigg[ u \bigg\{ \frac{\partial ^2 u}{\partial x^2 } + \fra... ... + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \bigg) \bigg\}$
	$\displaystyle + v \bigg\{ \frac{\partial ^2 v}{\partial x^2 } + \frac{\partial ... ... + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \bigg) \bigg\}$
	$\displaystyle + w \bigg\{ \frac{\partial ^2 w}{\partial x^2 } + \frac{\partial ... ... + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \bigg) \bigg\}$
	$\displaystyle + \underbrace{ \dfrac{2}{3} \bigg\{ 3 \bigg( \dfrac{\partial u}{\... ...rtial y} \bigg)^2 + 3 \bigg( \dfrac{\partial w}{\partial z} \bigg)^2 }_{散逸項}$
	$\displaystyle \underbrace{ - \dfrac{\partial u}{\partial x} \bigg( \dfrac{\part... ...rtial v}{\partial y} + \dfrac{\partial w}{\partial z} \bigg) \bigg\} }_{散逸項}$
	$\displaystyle \underbrace{ + \bigg( \dfrac{\partial v}{\partial x} + \dfrac{\pa... ...al z} + \dfrac{\partial w}{\partial x} \bigg)^2 }_{散逸項} \bigg] \bigg) dxdydz$
$\displaystyle =$	$\displaystyle \bigg( - \bigg( u \frac{\partial P}{\partial x} + v \frac{\partia... ...ial x} + \dfrac{\partial v}{\partial y} + \dfrac{\partial w}{\partial z} \bigg)$
	$\displaystyle + \mu \bigg[ u \bigg( \frac{\partial ^2 u}{\partial x^2 } + \frac... ...frac{\partial ^2 w}{\partial y^2 } + \frac{\partial ^2 w}{\partial z^2 } \bigg)$
	$\displaystyle + \underbrace{ \dfrac{2}{3} \bigg\{ 2 \bigg( \dfrac{\partial u}{\... ...dfrac{\partial w}{\partial z} \dfrac{\partial u}{\partial x} \bigg\} }_{散逸項}$
	$\displaystyle \underbrace{ + \bigg( \dfrac{\partial v}{\partial x} + \dfrac{\pa... ...al z} + \dfrac{\partial w}{\partial x} \bigg)^2 }_{散逸項} \bigg] \bigg) dxdydz$
$\displaystyle =$	$\displaystyle \bigg( - \bm{v} \cdot \bm{\nabla} P + \bigg( - P + \dfrac{1}{3} \... ...u + v \bm{\nabla} \cdot \bm{\nabla} v + w \bm{\nabla} \cdot \bm{\nabla} w \big)$
	$\displaystyle + \dfrac{2}{3} \bigg\{ \underbrace{ \bigg( \dfrac{\partial u}{\pa... ...artial w}{\partial x} \bigg)^2 }_{散逸項、 \varPhi と置く} \bigg] \bigg) dxdydz$
$\displaystyle =$	$\displaystyle \bigg( - \bm{v} \cdot \bm{\nabla} P + \bigg( - P + \dfrac{1}{3} \... ...+ \mu \bm{v} \cdot \bm{\nabla} \cdot \bm{\nabla} \bm{v} + \varPhi \bigg) dxdydz$	(2.181)