2.4.4 時間あたりの仕事

$\displaystyle \left( \begin{array}{c} \sigma_{x, {x -}} \tau_{xy, {x -}} ... ...ma_{x, {x -}} u_{x -}+ \tau_{xy, {x -}} v_{x -}+ \tau_{xz, {x -}} w_{x -}) dydz$

(2.165)

$\displaystyle \left( \begin{array}{c} \sigma_{x, {x +}} \tau_{xy, {x +}} \tau_{xz, {x +}} \end{array} \right) \cdot$	$\displaystyle \left( \begin{array}{c} u_{x +} v_{x +} w_{x +} \end{array} \... ...frac{\partial w}{\partial x} \right\vert _ {{x -}} d x \end{array} \right) dydz$
	$\displaystyle = \bigg(\sigma_{x, {x -}} u_{x -}+ \sigma_{x, {x -}} \left. \dfra... ...\partial u}{\partial x} \right\vert _ {{x -}} dx^2 }_{十分に小さいため無視する}$
	$\displaystyle + \tau_{xy, {x -}} v_{x -}+ \tau_{xy, {x -}} \left. \dfrac{\parti... ...\partial v}{\partial x} \right\vert _ {{x -}} dx^2 }_{十分に小さいため無視する}$
	$\displaystyle + \tau_{xz, {x -}} w_{x -}+ \tau_{xz, {x -}} \left. \dfrac{\parti... ...{\partial x} \right\vert _ {{x -}} dx^2 }_{十分に小さいため無視する}\bigg) dydz$	(2.166)

$\displaystyle \left( \begin{array}{c} \tau_{yx, {y -}} \sigma_{x, {x -}} ... ..._{yx, {y -}} u_{y -}+ \sigma_{y, {y -}} v_{y -}+ \tau_{yz, {y -}} w_{y -}) dzdx$

(2.167)

$\displaystyle \left( \begin{array}{c} \tau_{yx, {y +}} \sigma_{y, {y +}} \tau_{yz, {y +}} \end{array} \right) \cdot$	$\displaystyle \left( \begin{array}{c} u_{y +} v_{y +} w_{y +} \end{array} \... ...frac{\partial w}{\partial y} \right\vert _ {{y -}} d y \end{array} \right) dzdx$
	$\displaystyle = \bigg(\tau_{yx, {y -}} u_{y -}+ \tau_{yx, {y -}} \left. \dfrac{... ...\partial u}{\partial y} \right\vert _ {{y -}} dy^2 }_{十分に小さいため無視する}$
	$\displaystyle + \sigma_{y, {y -}} v_{y -}+ \sigma_{y, {y -}} \left. \dfrac{\par... ...\partial v}{\partial y} \right\vert _ {{y -}} dy^2 }_{十分に小さいため無視する}$
	$\displaystyle + \tau_{yz, {y -}} w_{y -}+ \tau_{yz, {y -}} \left. \dfrac{\parti... ...{\partial y} \right\vert _ {{y -}} dy^2 }_{十分に小さいため無視する}\bigg) dzdx$	(2.168)

$\displaystyle \left( \begin{array}{c} \tau_{zx, {z -}} \tau_{zy, {z -}} \... ..._{zx, {z -}} u_{z -}+ \tau_{zy, {z -}} v_{z -}+ \sigma_{z, {z -}} w_{z -}) dxdy$

(2.169)

$\displaystyle \left( \begin{array}{c} \tau_{zx, {z +}} \tau_{zy, {z +}} \sigma_{z, {z +}} \end{array} \right) \cdot$	$\displaystyle \left( \begin{array}{c} u_{z +} v_{z +} w_{z +} \end{array} \... ...frac{\partial w}{\partial z} \right\vert _ {{z -}} d z \end{array} \right) dxdy$
	$\displaystyle = \bigg(\tau_{zx, {z -}} u_{z -}+ \tau_{zx, {z -}} \left. \dfrac{... ...\partial u}{\partial z} \right\vert _ {{z -}} dz^2 }_{十分に小さいため無視する}$
	$\displaystyle + \tau_{zy, {z -}} v_{z -}+ \tau_{zy, {z -}} \left. \dfrac{\parti... ...\partial v}{\partial z} \right\vert _ {{z -}} dz^2 }_{十分に小さいため無視する}$
	$\displaystyle + \sigma_{z, {z -}} w_{z -}+ \sigma_{z, {z -}} \left. \dfrac{\par... ...{\partial z} \right\vert _ {{z -}} dz^2 }_{十分に小さいため無視する}\bigg) dxdy$	(2.170)

$\displaystyle \bigg( \sigma_{x, {x -}} \left. \dfrac{\partial u}{\partial x} \r... ...t. \dfrac{\partial {\tau_{xz}}}{\partial x} \right\vert _ {{x -}} \bigg) dxdydz$

(2.171)

$\displaystyle \bigg( \tau_{yx, {y -}} \left. \dfrac{\partial u}{\partial y} \ri... ...eft. \dfrac{\partial \tau_{yz}}{\partial y} \right\vert _ {{y -}} \bigg) dxdydz$

(2.172)

$\displaystyle \bigg( \tau_{zx, {z -}} \left. \dfrac{\partial u}{\partial z} \ri... ...left. \dfrac{\partial \sigma_z}{\partial z} \right\vert _ {{z -}} \bigg) dxdydz$

(2.173)

$\displaystyle \left. \begin{array}{ccc} \sigma_x & = & - P + \mu \bigg( 2 \dfra... ... \dfrac{2}{3} \nabla \cdot \bm{v} \bigg) = - P + \tau_{zz} \end{array} \right\}$

(2.174)

$\displaystyle \bigg\{$	$\displaystyle \big(- P + \tau_{xx} \big) \dfrac{\partial u}{\partial x} + \big(... ...tial v}{\partial y} + \big(- P + \tau_{zz} \big) \dfrac{\partial w}{\partial z}$
	$\displaystyle + u \dfrac{\partial }{\partial x} \big(- P + \tau_{xx} \big) + v ... ... + \tau_{yy} \big) + w \dfrac{\partial }{\partial z} \big(- P + \tau_{zz} \big)$
	$\displaystyle + \tau_{xy} \dfrac{\partial v}{\partial x} + \tau_{yx} \dfrac{\pa... ..._{zx} \dfrac{\partial u}{\partial z} + \tau_{xz} \dfrac{\partial w}{\partial x}$
	$\displaystyle + u \dfrac{\partial \tau_{yx}}{\partial y} + u \dfrac{\partial \t... ...tau_{xz}}{\partial x} + w \dfrac{\partial \tau_{yz}}{\partial y} \bigg\} dxdydz$
$\displaystyle =$	$\displaystyle \bigg( - P \dfrac{\partial u}{\partial x} - P \dfrac{\partial v}{... ...artial x} - v \dfrac{\partial P}{\partial y} - w \dfrac{\partial P}{\partial z}$
	$% latex2html id marker 21240 $\displaystyle \underbrace{ + \tau_{xx} \dfrac{\pa... ...については詳細を付録\ref{sec-Dissipation}(p.\pageref{sec-Dissipation})に記す。}$$
	$\displaystyle + u \dfrac{\partial \tau_{xx}}{\partial x} + v \dfrac{\partial \t... ...\tau_{xz}}{\partial x} + w \dfrac{\partial \tau_{yz}}{\partial y} \bigg) dxdydz$
$\displaystyle =$	$\displaystyle \bigg( - P \dfrac{\partial u}{\partial x} - P \dfrac{\partial v}{... ...artial x} - v \dfrac{\partial P}{\partial y} - w \dfrac{\partial P}{\partial z}$
	$\displaystyle + u \dfrac{\partial \tau_{xx}}{\partial x} + v \dfrac{\partial \t... ...\tau_{xz}}{\partial x} + w \dfrac{\partial \tau_{yz}}{\partial y} \bigg) dxdydz$	(2.175)