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Next: 2.4.4.2 非圧縮性流体(密度 [kg/m ]は一定) Up: 2.4.4 時間あたりの仕事 Previous: 2.4.4 時間あたりの仕事

2.4.4.1 圧縮性流体(密度$ \rho $ [kg/m$ ^3$ ]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)

ここで$ \varPhi$ [W/m$ ^3$ ]はエネルギー散逸関数であり次式で表される。

$\displaystyle \varPhi = \mu \bigg[\dfrac{2}{3} \bigg\{ \underbrace{ \bigg( \dfr...
... \dfrac{\partial w}{\partial x} \bigg)^2 }_{流体の剪断変形(ずり変形)} \bigg]
$


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Next: 2.4.4.2 非圧縮性流体(密度 [kg/m ]は一定) Up: 2.4.4 時間あたりの仕事 Previous: 2.4.4 時間あたりの仕事


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