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: $B$3$NJ8=q$K$D$$$F(B...

$B?eM}3X(BI $B>.%F%9%H(B (2002.11.12)

$BLdBj#1(B   $B0J2<$N(B \fbox {A} $\sim$ \fbox {L} $B$rE,@Z$KKd$a$k8@MU$"$k$$$O<0$rEz$($h!#(B

$B6u4V:BI8$r!"(B($x$,$y$,$z$) $B$ND>8r:BI8$G$l$NJ}8~$NB.EY$r(B ($u$, $v$, $w$) $B$H$7$?>l9g$K!"(B $BHs05=L@-N.BN$NO"B3<0$O(B

\begin{displaymath} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \end{displaymath}

$B$HI=$5$l$k!#$^$?!"(B$z$ $B:BI8$r1tD>J}8~$H$7$?;~$K!"0lDj$N=ENO2CB.EY(B $g$ $B$N(B $B$b$H$G$N1?F0J}Dx<0$O!"G4@-$rL5;k$9$k$H(B

\begin{displaymath} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial... ...{\partial z} = - \frac{1}{\rho} \frac{\partial p}{\partial x} \end{displaymath}


\begin{displaymath} \frac{\partial v}{\partial t} + u \frac{\partial v}{\partia... ...{\partial z} = - \frac{1}{\rho} \frac{\partial p}{\partial y} \end{displaymath}


\begin{displaymath} \frac{\partial w}{\partial t} + u \frac{\partial w}{\partia... ...rtial z} = - \frac{1}{\rho} \frac{\partial p}{\partial z} - g \end{displaymath}

$BB.EY%]%F%s%7%c%k(B $\phi$ $B$O!"B.EY(B ($u$, $v$, $w$) $B$,(B $B12EY(B $B$r(B $B;}$?$J$$;~$KDj5A$5$l!"(B$\phi$ $B$r;H$C$F(B ($u$, $v$, $w$) $B$O!"(B

\begin{displaymath} u = \frac{\partial \phi}{\partial x}, \,\,\,\, v = \frac{\... ...i}{\partial y }, \,\,\,\, w = \frac{\partial \phi}{\partial z} \end{displaymath}

$B$HI=$5$l$k!#(B

$B12EY(B $\omega$ $B$O(B $\mbox{rot}\mathbf{u}$ $B$HDj5A$5$l$k!#(B $\omega$ $B$N(B $z$ $B@.J,(B $\omega_{z}$ $B$O!"(B$u$, $v$ $B$r;H$C$F(B $\frac{\partial v}{\partial x} - \frac{\partial u}{\partial x}$ $B$H(B $BI=$5$l$k!#(B

$B12L5$7$NN.$l$N>l9g$K$O!"%Y%k%L!<%$$NDjM}$O;~4VJQ2=$r(B $B4^$`7A$K3HD%$5$l!"B.EY%]%F%s%7%c%k(B $\phi$ $B$r;H$C$F!"(B

\begin{displaymath}\frac{\partial \phi}{\partial t} + \frac{1}{2}\vert\mbox{grad}\phi\vert^2 + \frac{p}{\rho} + gz = \mbox{$B0lDj(B} \end{displaymath}

$B$H$J$k!#(B

$BN.@~4X?t(B $\psi$ $B$O!"#2

\begin{displaymath} u = \frac{\partial \psi}{\partial y}, \,\,\,\, v = -\frac{\partial \psi}{\partial x} \end{displaymath}

$B$HDj5A$5$l!"(B $\psi = \mbox{$B0lDjCM(B} $$B$rK~$?$9(B ($x$, $y$) $B$+$i$J$k@~$ON.@~$H$J$k!#(B

$BJ#AGB.EY%]%F%s%7%c%k(B$W$$B$OHs05=LHsG4@-N.BN$N#2N.$KBP$7$F(B

\begin{displaymath} W = \phi + i \psi \end{displaymath}

$B$HDj5A$5$l!"%3!<%7!


$BLdBj#2(B $B0J2<$NLd$GM?$($i$l$kB.EY%]%F%s%7%c%k!"$"$k$$$ON.@~4X?t!"(B $BJ#AGB.EY%]%F%s%7%c%k$NI=$9N.$l$N>l(B ($B#2, $y$) $B$K(B $B$*$1$kN.B.(B ($u$, $v$) $B$NCM$r5a$a!"4JC1$KN.$l$NMM;R$r?^<($;$h!#(B

  1. $ \phi = 2 s$, $B$?$@$7(B $ s = \sqrt{(x-1)^2+(y+2)^2} $
    \fbox{\begin{minipage}{10cm} $B>eLd$N<0$KF~$l$F7W;;$9$k$@$1$J$N$G>JN,!#(B $BN.$l>l$O!u$K=P$F9T$/7A$K$J$k(B \end{minipage}}">

  2. $ \psi = x$
    \fbox{\begin{minipage}{10cm} $B>eLd$N<0$KF~$l$F7W;;$9$k$@$1$J$N$G>JN,!#(B $BN.$l>l$O!
  3. $ W = z^3 $, $B$3$3$G(B $z = x + iy$
    \fbox{\begin{minipage}{10cm} $B>eLd$N<0$KF~$l$F7W;;$9$k$@$1$J$N$G>JN,!#(B $BN.$l>l$O(B... ...$BG8r:5$9$kD>@~$0O$^$l$?$=$l$>$l$NHO0O$G!
  4. $ W = 2 \left( z + \frac{1}{z} \right)$
    \fbox{\begin{minipage}{10cm} $B>eLd$N<0$KF~$l$F7W;;$9$k$@$1$J$N$G>JN,!#(B $BN.$l>l$O!



I. Tamagawa $BJ?@.(B15$BG/(B3$B7n(B6$BF|(B