2.4.4.2 非圧縮性流体（密度 [kg/m ]は一定）

密度 $\rho$ [kg/m

]が一定の非圧縮性流体の場合は面に垂直な $\tau$ の式(2.180) $^{\text{p.\pageref{eq-sigmatau-com}}}$ は質量保存の式(2.38)

$\bm{\nabla} \cdot \bm{v} = 0$ から次式のようになる。

$\displaystyle \bigg($	$\displaystyle - P \dfrac{\partial u}{\partial x} - P \dfrac{\partial v}{\partia... ...artial x} - v \dfrac{\partial P}{\partial y} - w \dfrac{\partial P}{\partial z}$
	$\displaystyle + u \dfrac{\partial \tau_{xx}}{\partial x} + v \dfrac{\partial \t... ...frac{\partial \tau_{xz}}{\partial x} + w \dfrac{\partial \tau_{yz}}{\partial y}$
	$\displaystyle + \underbrace{ \tau_{xx} \dfrac{\partial u}{\partial x} + \tau_{y... ...\partial z} + \tau_{xz} \dfrac{\partial w}{\partial x} }_{散逸項} \bigg) dxdydz$
$\displaystyle =$	$\displaystyle \bigg[ - P \bigg( \frac{\partial u}{\partial x} + \frac{\partial ... ...l x} + v \frac{\partial P}{\partial y} + w \frac{\partial P}{\partial z} \bigg)$
	$\displaystyle + 2 \mu \bigg( u \frac{\partial }{\partial x} \dfrac{\partial u}{... ...rtial y} + w \frac{\partial }{\partial z} \dfrac{\partial w}{\partial z} \bigg)$
	$\displaystyle + \mu u \bigg\{ \frac{\partial }{\partial y} \bigg( \frac{\partia... ...g( \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \bigg) \bigg\}$
	$\displaystyle + \mu w \bigg\{ \frac{\partial }{\partial x} \bigg( \frac{\partia... ...g( \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} \bigg) \bigg\}$
	$% latex2html id marker 23883 $\displaystyle + \underbrace{ \mu \bigg( 2 \dfrac{... ...bm{v}}_{式(\ref{eq-mass})より0} \bigg) \frac{\partial w}{\partial z} }_{散逸項}$$
	$\displaystyle \underbrace{ + \mu \bigg( \frac{\partial v}{\partial x} + \frac{\... ...}{\partial z} + \frac{\partial w}{\partial x} \bigg)^2 }_{散逸項} \bigg] dxdydz$
$\displaystyle =$	$% latex2html id marker 23886 $\displaystyle \Bigg( - P \underbrace{ \bigg( \fra... ...l x} + v \frac{\partial P}{\partial y} + w \frac{\partial P}{\partial z} \bigg)$$
	$% latex2html id marker 23887 $\displaystyle + \mu \bigg[ u \bigg\{ \frac{\parti... ...ial y} + \frac{\partial w}{\partial z} \bigg)}_{式(\ref{eq-mass})より0} \bigg\}$$
	$% latex2html id marker 23888 $\displaystyle + v \bigg\{ \frac{\partial ^2 v}{\p... ...ial y} + \frac{\partial w}{\partial z} \bigg)}_{式(\ref{eq-mass})より0} \bigg\}$$
	$% latex2html id marker 23889 $\displaystyle + w \bigg\{ \frac{\partial ^2 w}{\p... ...ial y} + \frac{\partial w}{\partial z} \bigg)}_{式(\ref{eq-mass})より0} \bigg\}$$
	$\displaystyle + \underbrace{ 2 \bigg( \dfrac{\partial u}{\partial x} \bigg)^2 +... ...frac{\partial w}{\partial z} \bigg)^2 }_{散逸項、次項と合わせて \varPhi と置く}$
	$\displaystyle \underbrace{ + \bigg( \dfrac{\partial v}{\partial x} + \dfrac{\pa... ...tial x} \bigg)^2 }_{散逸項、前項と合わせて \varPhi と置く} \bigg] \Bigg) dxdydz$
	$\displaystyle = \bigg[ - \bigg( u \frac{\partial P}{\partial x} + v \frac{\partial P}{\partial y} + w \frac{\partial P}{\partial z} \bigg)$
	$\displaystyle + \mu \bigg\{ u \bigg( \frac{\partial ^2 u}{\partial x^2 } + \fra... ... } + \frac{\partial ^2 w}{\partial z^2 } \bigg) \bigg\} + \varPhi \bigg] dxdydz$
	$\displaystyle = \Big\{ - \bm{v} \cdot \bm{\nabla} P + \mu \big( u \bm{\nabla} \... ...\bm{\nabla} v + w \bm{\nabla} \cdot \bm{\nabla} w \big) + \varPhi \Big\} dxdydz$
	$\displaystyle = \left( - \bm{v} \cdot \bm{\nabla} P + \mu \bm{v} \cdot \bm{\nabla} \cdot \bm{\nabla} \bm{v} + \varPhi \right) dxdydz$	(2.183)