A.2 速度と渦度の関係

$\displaystyle \nabla \times$ $\displaystyle \mbox{\boldmath$\omega $}$ $\displaystyle =\nabla \times (\nabla \times$ $\displaystyle \mbox{\boldmath$v$}$ $\displaystyle )=\nabla \,(\nabla \cdot$ $\displaystyle \mbox{\boldmath$v$}$ $\displaystyle )-\nabla ^2$ $\displaystyle \mbox{\boldmath$v$}$ $\displaystyle =-\nabla ^2$ $\displaystyle \mbox{\boldmath$v$}$

(38)

$\displaystyle -\left(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}\right)\,u$	$\displaystyle =$	$\displaystyle \frac{\partial \omega _3}{\partial y}-\frac{\partial \omega _2}{\partial z}$	(39)
$\displaystyle -\left(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}\right)\,v$	$\displaystyle =$	$\displaystyle \frac{\partial \omega _1}{\partial z}-\frac{\partial \omega _3}{\partial x}$	(40)
$\displaystyle -\left(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2}\right)\,w$	$\displaystyle =$	$\displaystyle \frac{\partial \omega _2}{\partial x}-\frac{\partial \omega _1}{\partial y}$	(41)

$\displaystyle -\{(ir_xl)^2+(ir_ym)^2-(r_zn)^2\}\,\hat{u}$	$\displaystyle =$	$\displaystyle ir_ym\,\hat{\omega }_3-r_zn\,\hat{\omega }_2$
	$\displaystyle =$	$\displaystyle ir_ym\cdot \frac{i\,(r_xl\,\hat{\omega }_1+r_ym\,\hat{\omega }_2)}{r_zn}-r_zn\,\hat{\omega }_2$
	$\displaystyle =$	$\displaystyle -\,\frac{(r_xl)\,(r_ym)\,\hat{\omega }_1+\{(r_ym)^2+(r_zn)^2\}\,\hat{\omega }_2}{r_zn}$

$\displaystyle \hat{u}=-\,\frac{(r_xl)\,(r_ym)\,\hat{\omega }_1+\{(r_ym)^2+(r_zn)^2\}\,\hat{\omega }_2}{r_zn\,\{(r_xl)^2+(r_ym)^2+(r_zn)^2\}}$

(42)

$\displaystyle -\{(ir_xl)^2+(ir_ym)^2\}\,\hat{u}$

$\displaystyle =$

$\displaystyle ir_ym\,\hat{\omega }_3$

$\displaystyle \hat{u}=\frac{ir_ym\,\hat{\omega }_3}{(r_xl)^2+(r_ym)^2}$

(43)

$\displaystyle \hat{v}$	$\displaystyle =$	$\displaystyle \left\{\begin{array}{l} \displaystyle \frac{\{(r_xl)^2+(r_zn)^2\}... ...2} \qquad \qquad \quad \qquad \qquad \quad \,\,\,\,(\,n=0\,) \end{array}\right.$	(44)
$\displaystyle \quad$
$\displaystyle \hat{w}$	$\displaystyle =$	$\displaystyle \left\{\begin{array}{l} \displaystyle \frac{i\,(r_xl\,\hat{\omega... ...ad \qquad \qquad \qquad \qquad \qquad \qquad \quad (\,n=0\,) \end{array}\right.$	(45)