<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% è¡¨é¡Œ  3 æ¬¡å…ƒå›žè»¢çƒæ®»å†…ã§ã®ç²˜æ€§æµä½“(éžåœ§ç¸®ãƒŠãƒ“ã‚¨ãƒ¼ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹æ–¹ç¨‹å¼)ã®å®šå¼åŒ–
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\Dtitle[3 æ¬¡å…ƒå›žè»¢çƒæ®»å†…ã§ã®ç²˜æ€§æµä½“]
       {3 æ¬¡å…ƒå›žè»¢çƒæ®»å†…ã§ã®ç²˜æ€§æµä½“\\
        (éžåœ§ç¸®ãƒŠãƒ“ã‚¨ãƒ¼ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹æ–¹ç¨‹å¼)\\
        ã®å®šå¼åŒ–}
\Dauthor{ç«¹åºƒ çœŸä¸€}
\Ddate{å¹³æˆ 17 å¹´ 6 æœˆ 14 æ—¥} 
\pagestyle{Dheadings}
\begin{document}
% psfrag ç”¨ã®è¨­å®š
\psfrag{F}{$\varphi$}
\psfrag{L}{$\lambda$}
\psfrag{R}{$r$}
\psfrag{ri}{$r_{i}$}
\psfrag{ro}{$r_{o}$}
\psfrag{O}{$\Omega$}
\psfrag{x}{$x$}
\psfrag{y}{$y$}
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\section{ã¯ã˜ã‚ã«}

ã“ã“ã§ã¯ä¸­å¿ƒãŒåŒã˜ã§å›žè»¢ã—ã¦ã„ã‚‹ 2 ã¤ã®çƒé¢ã®é–“ã«ã¯ã•ã¾ã‚ŒãŸ
çƒæ®»é&nbsp;˜åŸŸã®éžåœ§ç¸®ç²˜æ€§æµä½“ã®åŸºç¤Žæ–¹ç¨‹å¼ã®å®šå¼åŒ–ã«ã¤ã„ã¦è¨˜è¿°ã™ã‚‹.
é€Ÿåº¦å&nbsp;´ã‚’ãƒˆãƒ­ã‚¤ãƒ€ãƒ«--ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¸åˆ†è§£ã—,
ãã®æ™‚é–“ç™ºå±•æ–¹ç¨‹å¼ã¨å¢ƒç•Œæ¡ä»¶ã‚’å®šå¼åŒ–ã™ã‚‹
(Zhang and Busse,1987, Chandrasekhar, 1961).

%--------------------------------------------------------------------------
\section{æ…£æ€§ç³»ã§ã®å®šå¼åŒ–}

  \subsection{æ”¯é…æ–¹ç¨‹å¼}

    ä¸­å¿ƒã‚’å…±æœ‰ã™ã‚‹åŠå¾„ $r_{i}, r_{o}$ ã®çƒé¢ã«æŒŸã¾ã‚ŒãŸ
    çƒæ®»å†…ã«ã‚ã‚‹éžåœ§ç¸®ç²˜æ€§æµä½“ã«ã¤ã„ã¦è€ƒãˆã‚‹.
    å†…å¤–ã®çƒé¢ã®å›žè»¢è§’é€Ÿåº¦ã‚’ãã‚Œãžã‚Œ 
    $\Dvect{\Omega}_i$, $\Dvect{\Omega}_o$ ã¨ã™ã‚‹. 
    åº§æ¨™ç³»ã‚’çƒæ®»ã®ä¸­å¿ƒã‚’åŽŸç‚¹ã—, 
    ç·¯åº¦çµŒåº¦åº§æ¨™ç³» $(\lambda, \varphi, r)$ ã‚’å¼µã‚‹. 
  %  ã“ã‚Œã‚’ \Dfigref{æ…£æ€§ç³»ã®å›³} ã«ç¤ºã™.

    æ”¯é…æ–¹ç¨‹å¼ã¯éžåœ§ç¸®ç²˜æ€§æµä½“ã®ãƒŠãƒ“ã‚¨ãƒ¼ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹æ–¹ç¨‹å¼ã§ã‚ã‚‹. 
    %
    \begin{eqnarray} 
     &amp;&amp;\Deqlab{æ…£æ€§ç³»é€£ç¶šã®å¼}
      \Ddiv{\Dvect{u}}=0, \\
     &amp;&amp;\Deqlab{æ…£æ€§ç³»é‹å‹•æ–¹ç¨‹å¼}
      \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
      = -\frac{1}{\rho}\Dgrad p +\nu \Dlapla \Dvect{u} + \Dvect{\mathcal{F}}
    \end{eqnarray}
    %
    å„ã€…ã®è¨˜å·ã¯æ¬¡ã®é€šã‚Šã§ã‚ã‚‹:
    \begin{quote}
     \begin{tabular}{ll}
      $\Dvect{u}$  
                     &amp; æµé€Ÿãƒ™ã‚¯ãƒˆãƒ«. 
                       æˆåˆ†æ¯Žã«æ›¸ãå&nbsp;´åˆã«ã¯,
                       $\Dvect{u}=(u_\lambda,u_\varphi,u_r)$ \\
      $\rho$         &amp; æµä½“ã®å¯†åº¦. éžåœ§ç¸®ã‚’ä»®å®šã—ã¦ã„ã‚‹ã®ã§ä¸€å®šã§ã‚ã‚‹. \\
      $p$       
                     &amp; æµä½“ã®å…¨åœ§åŠ›. \\
      $\nu$          &amp; å‹•ç²˜æ€§çŽ‡. ä¸€æ§˜ä¸€å®šã§ã‚ã‚‹ã¨ã™ã‚‹. \\
      $\mathcal{F}$  &amp; å¤–åŠ›. 
     \end{tabular}
    \end{quote}  
    %
    ç·¯åº¦çµŒåº¦åº§æ¨™ã§ã®æˆåˆ†ã§è¡¨ç¤ºã™ã‚Œã°(c.f. Landau and Lifshitz, 1987)
    \footnote{
      ãƒ©ãƒ³ãƒ€ã‚¦ã®æ•™ç§‘æ›¸ã§ã®æ¥µåº§æ¨™è¡¨ç¾ã§
      $\theta=\pi-\varphi, \phi=\lambda$, 
      $v_\theta=u_\varphi, u_\phi=u_\lambda, v_r=u_r$, 
      $\partial_\theta = -\partial_\varphi$, 
      $\sigma_{\lambda\lambda}=\sigma_{\lambda\lambda}$,
      $\sigma_{\theta\theta}=\sigma_{\varphi\varphi}$,
      $\sigma_{r\phi}=\sigma_{r\lambda}$,
      $\sigma_{r\theta}=-\sigma_{r\varphi}$,
      $\sigma_{\theta\phi}=-\sigma_{\lambda\varphi}$
      ã¨å¤‰æ›ã™ã‚Œã°è‰¯ã„. 
      }
    %
    \begin{eqnarray}
      \Deqlab{æ…£æ€§ç³»é€£ç¶šã®å¼æˆåˆ†è¡¨ç¤º}
       &amp;&amp;  \frac{1}{r\cos\varphi}\DP{u_\lambda}{\lambda}
         + \frac{1}{r\cos\varphi}\DP{(u_\varphi\cos\varphi)}{\varphi}
         + \frac{1}{r^2}\DP{u_r}{r}=0, \\
      \Deqlab{æ…£æ€§ç³»é‹å‹•æ–¹ç¨‹å¼çµŒåº¦æˆåˆ†}
       &amp;&amp;
         \DP{u_\lambda}{t} + (\Dvect{u}\cdot\Dgrad)u_\lambda 
           +\frac{u_r u_\lambda}{r}-\frac{u_\varphi u_\lambda \tan\varphi}{r}
           \nonumber\\
       &amp;&amp;  = -\frac{1}{\rho r\cos\varphi}\DP{p}{\lambda}
           + \nu \left[ \Dlapla u_\lambda 
                       +\frac{2}{r^2\cos\varphi}\DP{u_r}{\lambda}
                       -\frac{2\sin\varphi}{r^2\cos^2\varphi}\DP{u_\varphi}{\lambda}
                       -\frac{u_\lambda}{r^2\cos^2\varphi}\right]
           + \mathcal{F}_\lambda,\\
      \Deqlab{æ…£æ€§ç³»é‹å‹•æ–¹ç¨‹å¼ç·¯åº¦æˆåˆ†}
       &amp;&amp;
         \DP{u_\varphi}{t} + (\Dvect{u}\cdot\Dgrad)u_\varphi
           +\frac{u_r u_\varphi}{r}+\frac{u_\lambda^2 \tan\varphi}{r}
           \nonumber\\
       &amp;&amp; =-\frac{1}{\rho r}\DP{p}{\varphi}
           + \nu \left[ \Dlapla u_\varphi
                       +\frac{2\sin\varphi}{r^2\cos^2\varphi}
                            \DP{u_\lambda}{\lambda}
                       +\frac{2}{r^2}\DP{u_r}{\varphi}
                       -\frac{u_\varphi}{r^2\cos^2\varphi}\right]
           + \mathcal{F}_\varphi,\\
      \Deqlab{æ…£æ€§ç³»é‹å‹•æ–¹ç¨‹å¼å‹•å¾„æˆåˆ†}
       &amp;&amp;
         \DP{u_r}{t} + (\Dvect{u}\cdot\Dgrad)u_r
           -\frac{u_\varphi^2+ u_\lambda^2}{r}
           \nonumber\\
       &amp;&amp; =-\frac{1}{\rho}\DP{p}{r}
           + \nu \left[ \Dlapla u_r
                       -\frac{2}{r^2\cos\varphi}
                            \DP{(u_\varphi\cos\varphi)}{\varphi}
                       -\frac{2}{r^2\cos\varphi}\DP{u_\lambda}{\lambda}
                       -\frac{2u_r}{r^2}\right]
           + \mathcal{F}_r,\\
    \end{eqnarray}
    %
    ã“ã“ã§
    \begin{eqnarray}
      \Dvect{u}\cdot\Dgrad &amp;=&amp;
           \frac{u_\lambda}{r\cos\varphi}\DP{}{\lambda}
         + \frac{u_\varphi}{r}\DP{}{\varphi}
         + u_r\DP{}{r}, \\
      \Dlapla &amp;=&amp;   \frac{1}{r^2\cos^2\varphi}\DP[2]{}{\lambda}
                  + \frac{1}{r^2\cos\varphi}\DP[]{}{\varphi}
                        \left(\cos\varphi\DP{}{\varphi}\right)
                  + \frac{1}{r^2}\DP{}{r}\left(r^2\DP{}{r}\right),
    \end{eqnarray}
    %
    ã§ã‚ã‚‹. ã¾ãŸ, å„å¿œåŠ›æˆåˆ†ã¯
    \begin{eqnarray}
      \sigma_{\lambda\lambda} 
          &amp;=&amp; -p + 2\eta 
                    \left( \frac{1}{r\cos\varphi}\DP{u_\lambda}{\lambda}
                           +\frac{u_r}{r}
                           -\frac{u_\varphi\tan\varphi}{r}  \right),\\
      \sigma_{\varphi\varphi} 
          &amp;=&amp; -p + 2\eta\left( \frac{1}{r}\DP{u_\varphi}{\varphi}
                                +\frac{u_r}{r}  \right) \\
      \sigma_{rr}
          &amp;=&amp; -p + 2 \eta \DP{u_r}{r}, \\
      \sigma_{\lambda\varphi}
          &amp;=&amp; \eta\left(  \frac{1}{r\cos\varphi}\DP{u_\varphi}{\lambda}
                         +\frac{1}{r}\DP{u_\lambda}{\varphi}
                         +\frac{u_\lambda\tan\varphi}{r} \right),\\
      \sigma_{\varphi r}
          &amp;=&amp; \eta\left(  \frac{1}{r}\DP{u_r}{\varphi}
                         +\DP{u_\varphi}{r}-\frac{u_\varphi}{r} \right),\\
      \sigma_{r \lambda}
          &amp;=&amp; \eta\left(  \DP{u_\lambda}{r}
                         +\frac{1}{r\cos\varphi}\DP{u_r}{\lambda}
                         - \frac{u_\lambda}{r} \right).
    \end{eqnarray}

    é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ã¯, å¢ƒç•Œã‚’æ¨ªåˆ‡ã‚‹æµã‚ŒãŒå­˜åœ¨ã—ãªã„, ã§ã‚ã‚‹.
    ã™ãªã‚ã¡, 
    \begin{equation}
      \Deqlab{æ…£æ€§ç³»é‹å‹•å¢ƒç•Œæ¡ä»¶}
      u_r = 0, \qquad \mbox{at} \quad r=r_i,\ r_o. 
    \end{equation}
    %
    åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ã¨ã—ã¦ã¯,
    å¿œåŠ›ãªã—(stress-free), ã‚‚ã—ãã¯ç²˜ç€æ¡ä»¶(rigid)ã‚’è€ƒãˆã‚‹.

    å¿œåŠ›ãªã—æ¡ä»¶ã®å&nbsp;´åˆã«ã¯ $\sigma_{r \lambda}=\sigma_{\varphi r}=0$ ã¨
    $u_r=0$ ã‚ˆã‚Š,
    \begin{equation}
      \Deqlab{æ…£æ€§ç³»å¿œåŠ›ãªã—æ¡ä»¶}
       \DP{}{r}\left(\frac{u_\lambda}{r}\right) 
       = \DP{}{r}\left(\frac{u_\varphi}{r}\right) 
        = 0, 
        \quad \mbox{at} \quad r=r_i,\ r_o. 
    \end{equation}

    ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, é‹å‹•å­¦çš„æ¡ä»¶ã¨åˆã‚ã›ã¦, 
    \begin{equation}
     \Dvect{u} = \Dvect{\Omega}_i\times\Dvect{r} 
      \quad \mbox{at} \quad r=r_i, 
     \qquad
     \Dvect{u} = \Dvect{\Omega}_o\times\Dvect{r}
      \quad \mbox{at} \quad r=r_o. 
    \end{equation}
    %
    ã“ã‚Œã‚‰ã‚’æˆåˆ†ã§æ›¸ãä¸‹ã™ã¨\footnotemark[2]
    %
    \begin{eqnarray}
      \Deqlab{æ…£æ€§ç³»ç²˜ç€æ¡ä»¶}
       &amp; &amp; u_\lambda =\Omega_i r_i 
                   [ -\cos\varphi_i\sin\varphi\cos(\lambda_i-\lambda)
                  + \sin\varphi_i\cos\varphi ], \\
       &amp; &amp; u_\varphi = \Omega_i r_i 
               [ \cos\varphi_i\sin(\lambda-\lambda_i)],
            \qquad \mbox{at}~~ r=r_i, \\
       &amp; &amp; u_\lambda = \Omega_o r_o 
                   [ -\cos\varphi_o\sin\varphi\cos(\lambda_o-\lambda)
                  + \sin\varphi_o\cos\varphi ], \\
       &amp; &amp; u_\varphi = \Omega_o r_o 
               [ \cos\varphi_o\sin(\lambda-\lambda_o)],
           \qquad \mbox{at}~~ r=r_o. 
    \end{eqnarray}
    %
    ã“ã“ã§ $\Omega_i=|\Dvect{\Omega_i}|$,  $\Omega_o=|\Dvect{\Omega_o}|$ 
    ã¯å†…å¤–çƒé¢ã®å›žè»¢è§’é€Ÿåº¦ã®å¤§ãã•, 
    $(\lambda_i,\varphi_i)$, $(\lambda_o,\varphi_o)$ ã¯
    å›žè»¢è»¸ã®æ–¹å‘ã‚’è¡¨ã™çµŒåº¦ç·¯åº¦ã§ã‚ã‚‹. 

    \footnotetext[2]{
      çµŒåº¦ $\lambda$, ç·¯åº¦ $\varphi$, å‹•å¾„ $r$ ã§ã®ç·¯åº¦çµŒåº¦åº§æ¨™ç³»ã®
      å˜ä½ãƒ™ã‚¯ãƒˆãƒ«ã‚’ç›´ç·šç›´äº¤åº§æ¨™$(x,y,z)$ ã§æˆåˆ†è¡¨ç¤ºã™ã‚‹ã¨, 
      $\Dvect{e}_\lambda(\lambda,\varphi,r)
          =(-\sin\lambda,\cos\lambda,0)$, 
      $\Dvect{e}_\varphi(\lambda,\varphi,r)
          =(-\cos\lambda\sin\varphi,-\sin\lambda\sin\varphi,\cos\varphi)$, 
      $\Dvect{e}_r(\lambda,\varphi,r)
          =(\cos\lambda\cos\varphi,\sin\lambda\cos\varphi,\sin\varphi)$
      ã§ã‚ã‚Š, $\Dvect{r}=r\Dvect{e}_r$ ã§ã‚ã‚‹. 
      ä¸€æ–¹ $\lambda_i,\varphi_i$ ã‚’å†…çƒé¢ã®å›žè»¢è»¸æ–¹å‘ã‚’è¡¨ã™çµŒåº¦ç·¯åº¦ã¨ã™ã‚‹ã¨
      $\Dvect{k}_i = \Dvect{e}_r(\lambda_i,\varphi_i,r_i)$ ã§ã‚ã‚‹. 
      ã—ãŸãŒã£ã¦, 
      \begin{displaymath}
        \Dvect{\Omega}_i\times\Dvect{r}=
          \Omega_i\Dvect{e}_r(\lambda_i,\varphi_i,r_i)
          \times r_i\Dvect{e}_r(\lambda,\varphi,r_i)
      \end{displaymath}
      ã“ã®å„æˆåˆ†ã‚’è¨ˆç®—ã™ã‚‹ã¨
      \begin{eqnarray*}
        u_\lambda &amp;=&amp; \Dvect{e}_\lambda
                       \cdot(\Dvect{\Omega}_i\times\Dvect{r})\\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_\lambda
                \cdot(\Dvect{e}_r(\lambda_i,\varphi_i,r_i)\times\Dvect{e}_r) \\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_r(\lambda_i,\varphi_i,r_i)
                \cdot(\Dvect{e}_r\times\Dvect{e}_\lambda)  \\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_r(\lambda_i,\varphi_i,r_i)
                \cdot\Dvect{e}_\varphi \\
           &amp;=&amp; \Omega_i r_i 
               [   \cos\lambda_i\cos\varphi_i (-\cos\lambda\sin\varphi)
                  + \sin\lambda_i\cos\varphi_i (-\sin\lambda\sin\varphi)
                  + \sin\varphi_i\cos\varphi ]\\
           &amp;=&amp; \Omega_i r_i 
               [ -\cos\varphi_i\sin\varphi
                   (\cos\lambda_i\cos\lambda+\sin\lambda_i\sin\lambda)
                  + \sin\varphi_i\cos\varphi ]\\
           &amp;=&amp; \Omega_i r_i 
               [ -\cos\varphi_i\sin\varphi\cos(\lambda_i-\lambda)
                  + \sin\varphi_i\cos\varphi ], \\
         u_\varphi &amp;=&amp; \Dvect{e}_\varphi
                \cdot(\Dvect{\Omega}_i\times\Dvect{r}) \\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_\varphi
                \cdot(\Dvect{e}_r(\lambda_i,\varphi_i,r_i)\times\Dvect{e}_r)\\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_r(\lambda_i,\varphi_i,r_i)
                \cdot(\Dvect{e}_r\times\Dvect{e}_\varphi) \\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_r(\lambda_i,\varphi_i,r_i)
                \cdot(-\Dvect{e}_\lambda)  \\
           &amp;=&amp; -\Omega_i r_i \Dvect{e}_r(\lambda_i,\varphi_i,r_i)
                \cdot\Dvect{e}_\lambda \\
           &amp;=&amp; -\Omega_i r_i 
               [  \cos\lambda_i\cos\varphi_i (-\sin\lambda)
                 + \sin\lambda_i\cos\varphi_i (\cos\lambda)
                 + \sin\varphi_i (0)] \\
           &amp;=&amp; -\Omega_i r_i 
               [ \cos\varphi_i
                 (-\cos\lambda_i\sin\lambda+\sin\lambda_i\cos\lambda)]\\
           &amp;=&amp; -\Omega_i r_i 
               [ \cos\varphi_i\sin(\lambda_i-\lambda)],\\
           &amp;=&amp; \Omega_i r_i 
               [ \cos\varphi_i\sin(\lambda-\lambda_i)],\\
        u_r &amp;=&amp; \Dvect{e}_r
                \cdot(\Dvect{\Omega}_i\times\Dvect{r})\\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_r
                \cdot(\Dvect{e}_r(\lambda_i,\varphi_i,r_i)\times\Dvect{e}_r)\\
           &amp;=&amp; \Omega_i r_i \Dvect{e}_r(\lambda_i,\varphi_i,r_i)
                \cdot(\Dvect{e}_r\times\Dvect{e}_r)=0.
       \end{eqnarray*}
    }

  \subsection{ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°}

    \subsubsection{çƒæ®»ã®åŽšã•ã§ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã™ã‚‹å&nbsp;´åˆ}

      ç†±å¯¾æµå•é¡Œã¨ã®æ¯”è¼ƒãƒ»å¯¾å¿œã‚’è€ƒå¯Ÿã™ã‚‹å&nbsp;´åˆã«ã¯, 
      é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã‚’çƒæ®»ã®åŽšã•ã§ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã™ã‚‹ã®ãŒã‚ˆã„. 
      ãã®å&nbsp;´åˆã«ã¯å„ã‚¹ã‚±ãƒ¼ãƒ«ã¯,
      %
      \begin{quote}
      \begin{tabular}{ll}
         é•·ã•ã‚¹ã‚±ãƒ¼ãƒ« &amp; çƒæ®»ã®åŽšã• $D = r_{o} - r_{i}$      \\
         é€Ÿåº¦ã‚¹ã‚±ãƒ¼ãƒ« &amp; åˆæœŸå€¤ã®å…¸åž‹çš„ãªé€Ÿåº¦ $U$, ã‚ã‚‹ã„ã¯å¤–åŠ› $\sqrt{D\mathcal{F}}$ \\
         æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ« &amp; $D/U$ ã‚ã‚‹ã„ã¯ $\sqrt{D/\mathcal{F}}$\\
         åœ§åŠ›ã®ã‚¹ã‚±ãƒ¼ãƒ« &amp; $\rho U^2$ ã‚ã‚‹ã„ã¯ $\rho D\mathcal{F}$ 
      \end{tabular}
      \end{quote}
       %
      ãã®çµæžœ, æ–¹ç¨‹å¼ç³»ã¯
      %
      \begin{eqnarray}
       &amp;&amp; \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒé€£ç¶šã®å¼}
        \Ddiv\Dvect{u} = 0, \\
       &amp;&amp;\Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼}
        \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
        = - \Dgrad \pi  + \frac{1}{R_e}\Dlapla \Dvect{u} + \Dvect{\mathcal{F}}
      \end{eqnarray}
      %
      ã¨ãªã‚‹.
      ã“ã“ã§ $\Dvect{u},\pi,\Dvect{\mathcal{F}}$ ã¯
      å„ã€…ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸé€Ÿåº¦, åœ§åŠ›, å¤–åŠ›ã¨ã—ã¦ã‚ã‚‰ãŸã‚ã¦
      å®šç¾©ã—ãªãŠã—ãŸã‚‚ã®ã§ã‚ã‚‹.

      ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã«ã‚ˆã£ã¦ç¾ã‚ŒãŸç„¡æ¬¡å…ƒæ•°ã¯ãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•°, 
      \begin{equation}
         R_e \equiv \frac{\nu}{UD} \quad \mbox{or} \quad \frac{\nu}{\mathcal{F}},
      \end{equation}
      ã§ã‚ã‚‹. 
      ã“ã‚Œã«åŠ&nbsp;ãˆã¦ç³»ã«ç¾ã‚Œã‚‹é‡è¦ãªãƒ‘ãƒ©ãƒ¡ãƒ¼ã‚¿ã¨ã—ã¦
      å†…å¾„å¤–å¾„æ¯” $\zeta \equiv r_{i}/r_{o}$ ãŒæš—ã«å­˜åœ¨ã™ã‚‹.

      ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸå¢ƒç•Œæ¡ä»¶ã¯, 
      é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒ
      %
      \begin{equation}
        \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒé‹å‹•å¢ƒç•Œæ¡ä»¶}
        u_r = 0, 
        \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}.
      \end{equation}
      %
      å¿œåŠ›ãªã—æ¡ä»¶ã¯
      %
      \begin{equation}
        \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}
         \DP{}{r}\left(\frac{u_\lambda}{r}\right) 
         = \DP{}{r}\left(\frac{u_\varphi}{r}\right) = 0, 
           \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}.
      \end{equation}

      ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã¯
      %
      \begin{equation}
       \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
       \Dvect{u} = \Dvect{\Omega_i}\times\Dvect{r} 
         \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \\
       \qquad
       \Dvect{u} = \Dvect{\Omega_o}\times\Dvect{r}
         \quad \mbox{at} \quad r=\frac{1}{1-\zeta}. 
      \end{equation}
      %
      æˆåˆ†ã§æ›¸ã‘ã°
      \begin{eqnarray}
       \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶æˆåˆ†}
         &amp; &amp; u_\lambda =\Omega_i \left(\frac{\zeta}{1-\zeta}\right)
                     [ -\cos\varphi_i\sin\varphi\cos(\lambda_i-\lambda)
                    + \sin\varphi_i\cos\varphi ], \\
         &amp; &amp; u_\varphi = \Omega_i \left(\frac{\zeta}{1-\zeta}\right)
                 [ \cos\varphi_i\sin(\lambda-\lambda_i)],
              \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \\
         &amp; &amp; u_\lambda = \Omega_o \left(\frac{1}{1-\zeta}\right)
                     [ -\cos\varphi_o\sin\varphi\cos(\lambda_o-\lambda)
                    + \sin\varphi_o\cos\varphi ], \\
         &amp; &amp; u_\varphi = \Omega_o \left(\frac{1}{1-\zeta}\right)
                 [ \cos\varphi_o\sin(\lambda-\lambda_o)],
              \quad \mbox{at} \quad r=\frac{1}{1-\zeta}, \\
      \end{eqnarray}
      %
      ã“ã“ã§ $\Dvect{\Omega_i},\Dvect{\Omega_o}$ ãŠã‚ˆã³ $\Omega_i,\Omega_o$ ã¯
      æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ«åˆ†ã® 1 $U/D$ 
      ã‚ã‚‹ã„ã¯ $\sqrt{\mathcal{F}/D}$ ã§ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸã‚‚ã®ã§ã‚ã‚‹ã“ã¨ã«
      æ³¨æ„ã•ã‚ŒãŸã„. 
      ã‚‚ã—ã‚‚æµã‚ŒãŒå¢ƒç•Œé¢ã®å›žè»¢ã§åŠ±èµ·ã•ã‚Œã‚‹çŠ¶æ³ã‚’è€ƒãˆã‚‹å&nbsp;´åˆã«ã¯
      $\Omega_i,\Omega_o$ ã®ã©ã¡ã‚‰ã‹ãŒ 1 ã¨ãªã‚‹ã‚ˆã†ã«
      é€Ÿåº¦ã‚¹ã‚±ãƒ¼ãƒ«ã‚’(ä¾‹ãˆã° $\Omega_i D$ ã«)é¸ã¹ã°è‰¯ã„. 

    \subsubsection{å¤–å´çƒæ®»åŠå¾„ã§ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã™ã‚‹å&nbsp;´åˆ}

      ä¸€æ–¹ã§, çƒé¢ 2 æ¬¡å…ƒç³»ã¨ã®å¯¾å¿œã‚’è€ƒå¯Ÿã™ã‚‹å&nbsp;´åˆã«ã¯, 
      é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ã¦å¤–å´çƒæ®»åŠå¾„ã‚’é¸ã¶ã®ãŒåˆ†ã‹ã‚Šã‚„ã™ã„. 
      ãã®å&nbsp;´åˆã«ã¯å„ã‚¹ã‚±ãƒ¼ãƒ«ã¯,
      %
      \begin{quote}
      \begin{tabular}{ll}
         é•·ã•ã‚¹ã‚±ãƒ¼ãƒ« &amp; å¤–å´çƒæ®»ã®åŠå¾„ $r_{o}$      \\
         é€Ÿåº¦ã‚¹ã‚±ãƒ¼ãƒ« &amp; åˆæœŸå€¤ã®å…¸åž‹çš„ãªé€Ÿåº¦ $U$, ã‚ã‚‹ã„ã¯å¤–åŠ› $\sqrt{D\mathcal{F}}$ \\
         æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ« &amp; $r_{o}/U$ ã‚ã‚‹ã„ã¯ $\sqrt{r_{o}/\mathcal{F}}$\\
         åœ§åŠ›ã®ã‚¹ã‚±ãƒ¼ãƒ« &amp; $\rho U^2$ ã‚ã‚‹ã„ã¯ $\rho r_{o}\mathcal{F}$ 
      \end{tabular}
      \end{quote}
       %
      ãã®çµæžœ, æ–¹ç¨‹å¼ç³»ã¯
      %
      \begin{eqnarray}
       &amp;&amp; 
        \Ddiv\Dvect{u} = 0, \\
       &amp;&amp;
        \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
        = - \Dgrad \pi  + \frac{1}{R_e^*}\Dlapla \Dvect{u} 
          + \Dvect{\mathcal{F}}
      \end{eqnarray}
      %
      ã¨ãªã‚‹.
      $\Dvect{u},\pi,\Dvect{\mathcal{F}}$ ã¯å…ˆã¨åŒæ§˜, 
      å„ã€…ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸé€Ÿåº¦, åœ§åŠ›, å¤–åŠ›ã¨ã—ã¦ã‚ã‚‰ãŸã‚ã¦
      å®šç¾©ã—ãªãŠã—ãŸã‚‚ã®ã§ã‚ã‚‹.
      ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã«ã‚ˆã£ã¦ç¾ã‚ŒãŸç„¡æ¬¡å…ƒæ•°ã¯
      é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã®ç•°ãªã‚‹ãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•°, 
      \begin{equation}
         R_e^* \equiv \frac{\nu}{U r_o} 
             \quad \mbox{or} \quad \frac{\nu}{\mathcal{F}},
      \end{equation}
      ã§ã‚ã‚‹. 
      ã“ã‚Œã«åŠ&nbsp;ãˆã¦ç³»ã«ç¾ã‚Œã‚‹ãƒ‘ãƒ©ãƒ¡ãƒ¼ã‚¿ã¨ã—ã¦
      å†…å¾„å¤–å¾„æ¯” $\zeta \equiv r_{i}/r_{o}$ ãŒã‚„ã¯ã‚Šå­˜åœ¨ã™ã‚‹.

      ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸå¢ƒç•Œæ¡ä»¶ã¯, 
      é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒ
      %
      \begin{equation}
        \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒé‹å‹•å¢ƒç•Œæ¡ä»¶}
        u_r = 0, 
        \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}.
      \end{equation}
      %
      å¿œåŠ›ãªã—æ¡ä»¶ã¯
      %
      \begin{equation}
        \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}
         \DP{}{r}\left(\frac{u_\lambda}{r}\right) 
         = \DP{}{r}\left(\frac{u_\varphi}{r}\right) = 0, 
           \quad \mbox{at} \quad r=\zeta, 1.
      \end{equation}

      ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã¯
      %
      \begin{equation}
       \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
       \Dvect{u} = \Dvect{\Omega_i}\times\Dvect{r} 
         \quad \mbox{at} \quad r=\zeta, \\
       \qquad
       \Dvect{u} = \Dvect{\Omega_o}\times\Dvect{r}
         \quad \mbox{at} \quad r=1. 
      \end{equation}
      %
      æˆåˆ†ã§æ›¸ã‘ã°
      \begin{eqnarray}
       \Deqlab{æ…£æ€§ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶æˆåˆ†}
         &amp; &amp; u_\lambda =\Omega_i \left(\frac{\zeta}{1-\zeta}\right)
                     [ -\cos\varphi_i\sin\varphi\cos(\lambda_i-\lambda)
                    + \sin\varphi_i\cos\varphi ], \\
         &amp; &amp; u_\varphi = \Omega_i \left(\frac{\zeta}{1-\zeta}\right)
                 [ \cos\varphi_i\sin(\lambda-\lambda_i)],
              \quad \mbox{at} \quad r=\zeta, \\
         &amp; &amp; u_\lambda = \Omega_o \left(\frac{1}{1-\zeta}\right)
                     [ -\cos\varphi_o\sin\varphi\cos(\lambda_o-\lambda)
                    + \sin\varphi_o\cos\varphi ], \\
         &amp; &amp; u_\varphi = \Omega_o \left(\frac{1}{1-\zeta}\right)
                 [ \cos\varphi_o\sin(\lambda-\lambda_o)],
              \quad \mbox{at} \quad r=1. \\
      \end{eqnarray}
      %
      ã“ã“ã§ $\Dvect{\Omega_i},\Dvect{\Omega_o}$ ãŠã‚ˆã³ $\Omega_i,\Omega_o$ ã¯
      æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ«åˆ†ã® 1 $U/r_o$ 
      ã‚ã‚‹ã„ã¯ $\sqrt{\mathcal{F}/r_o}$ ã§ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸã‚‚ã®ã§ã‚ã‚‹ã“ã¨ã«
      æ³¨æ„ã•ã‚ŒãŸã„. 
      ã‚‚ã—ã‚‚æµã‚ŒãŒå¢ƒç•Œé¢ã®å›žè»¢ã§åŠ±èµ·ã•ã‚Œã‚‹çŠ¶æ³ã‚’è€ƒãˆã‚‹å&nbsp;´åˆã«ã¯
      $\Omega_i,\Omega_o$ ã®ã©ã¡ã‚‰ã‹ãŒ 1 ã¨ãªã‚‹ã‚ˆã†ã«
      é€Ÿåº¦ã‚¹ã‚±ãƒ¼ãƒ«ã‚’(ä¾‹ãˆã° $\Omega_i r_o$ ã«)é¸ã¹ã°è‰¯ã„. 

    \subsection{å„ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°æ–¹ç¨‹å¼ç³»ã®å¯¾å¿œé–¢ä¿‚}

      çƒæ®»ã®åŽšã•ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ã¦é¸ã‚“ã&nbsp;å&nbsp;´åˆã¨ã®æ–¹ç¨‹å¼ç³»ã¨æ¯”è¼ƒã™ã‚‹ã¨, 
      ç„¡æ¬¡å…ƒãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•°ã®å®šç¾©ã¨, å¢ƒç•Œã§ã®å‹•å¾„åº§æ¨™ãŒç•°ãªã‚‹ã&nbsp;ã‘ã§ã‚ã‚‹. 
      ãã“ã§ä»¥ä¸‹ã§ã¯çƒæ®»ã®åŽšã•ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ã¦é¸ã‚“ã&nbsp;å&nbsp;´åˆã®
      å®šå¼åŒ–ã®ã¿è¨˜ã™. 
      å¤–å´å¢ƒç•Œã®åŠå¾„ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ãŸå&nbsp;´åˆã®çµæžœã¯, ä»¥ä¸‹ã§ç™»å&nbsp;´ã™ã‚‹
      ãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•° $R_e$ ã‚’ $R_e^*$ ã«, å¢ƒç•Œã§ã®å‹•å¾„åº§æ¨™
      $r=\zeta/1-\zeta,1/1-\zeta$ ã‚’ $r=\zeta,1$ ã«èª­ã¿ã‹ãˆã‚Œã°è‰¯ã„. 

      å„ç‰©ç†é‡ã®å€¤ã‚’å¤‰æ›ã—ãŸã„å&nbsp;´åˆã«ã¯
      ä¸Šä»˜ã * å°ã‚’å¤–å´å¢ƒç•Œã®åŠå¾„ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ãŸå&nbsp;´åˆã®å¤‰æ•°ã¨ã—ã¦
      %
      \begin{eqnarray*}
        t^*      &amp;=&amp; (D/r_o)t = (1-\zeta)t, \\
        \Omega^* &amp;=&amp; (r_o/D)\Omega = (1/1-\zeta)\Omega, \\
        R_e^*    &amp;=&amp; (D/r_o)R_e = (1-\zeta)R_e.
      \end{eqnarray*}
      %
      ã¨è¨ˆç®—ã™ã‚Œã°è‰¯ã„. 


  \subsection{ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«ã‚ˆã‚‹æ–¹ç¨‹å¼ç³»ã®è¡¨ç¾} 

    \subsubsection{æ”¯é…æ–¹ç¨‹å¼}

       é€£ç¶šã®å¼ã®ã‚½ãƒ¬ãƒŽã‚¤ãƒ€ãƒ«æ¡ä»¶\Deqref{æ…£æ€§ç³»ç„¡æ¬¡å…ƒé€£ç¶šã®å¼}ã‚ˆã‚Š, 
       é€Ÿåº¦å&nbsp;´ã‚’ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒ»ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’ç”¨ã„ã¦
       è¡¨ã™ã“ã¨ãŒã§ãã‚‹:
       %
       \begin{eqnarray}
        \Deqlab{é€Ÿåº¦åˆ†è§£}
        \Dvect{u} &amp;=&amp; \Drot(\psi \Dvect{r}) + \Drot\Drot(\phi\Dvect{r}), \\
       \end{eqnarray}
       %
       ã“ã‚Œã‚‰ã®ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ« $\psi, \phi $ ã‚’ç”¨ã„ã¦æ–¹ç¨‹å¼ç³»ã‚’è¡¨ã™. 
       é‹å‹•æ–¹ç¨‹å¼ \Deqref{æ…£æ€§ç³»ç„¡æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼}ã«
       $\Dvect{r}\cdot\Drot$, $\Dvect{r}\cdot\Drot\Drot$ ã‚’ä½œç”¨ã™ã‚‹ã“ã¨ã§,
       \footnotemark[3]
       \begin{eqnarray}
        \Deqlab{æ…£æ€§ç³»ãƒˆãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}
            \DP{}{t} (L_2 \psi)
         &amp;=&amp; \frac{1}{R_e} L_2\Dlapla \psi 
             - \Dvect{r}\cdot\Drot [(\Dvect{u}\cdot\Dgrad)\Dvect{u}]
             + \Dvect{r}\cdot\Drot\Dvect{\mathcal{F}}, \\
        \Deqlab{æ…£æ€§ç³»ãƒãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}
           \DP{}{t} (L_2 \Dlapla \phi)
         &amp;=&amp; \frac{1}{R_e} L_2\Dlapla\Dlapla \phi 
           + \Dvect{r}\cdot\Drot\Drot [(\Dvect{u}\cdot\Dgrad)\Dvect{u}]
           - \Dvect{r}\cdot\Drot\Drot\Dvect{\mathcal{F}}, 
       \end{eqnarray}
       %
       ãŒå¾—ã‚‰ã‚Œã‚‹. ã‚ã‚‹ã„ã¯\Deqref{æ…£æ€§ç³»ãƒˆãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}, 
       \Deqref{æ…£æ€§ç³»ãƒãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}ã®å³è¾ºéžç·šå½¢é&nbsp;…ã‚’æ’ç­‰å¼
       \begin{displaymath}
        \Dvect{u}\cdot\Dgrad\Dvect{u} 
        = \frac{1}{2}\Dgrad |\Dvect{u}|^2 - \Dvect{u}\times(\Drot\Dvect{u})
        = \frac{1}{2}\Dgrad |\Dvect{u}|^2 + (\Drot\Dvect{u})\times\Dvect{u}
       \end{displaymath}
       ã‚’ç”¨ã„ã¦å¤‰å½¢ã—ãŸå¼,
       % 
       \begin{eqnarray}
        \Deqlab{æ…£æ€§ç³»ãƒˆãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•ZxV}
            \DP{}{t} (L_2 \psi)
         &amp;=&amp; \frac{1}{R_e} L_2\Dlapla \psi 
             - \Dvect{r}\cdot\Drot [(\Drot\Dvect{u})\times\Dvect{u}]
             + \Dvect{r}\cdot\Drot\Dvect{\mathcal{F}}, \\
        \Deqlab{æ…£æ€§ç³»ãƒãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•ZxV}
           \DP{}{t} (L_2 \Dlapla \phi)
         &amp;=&amp; \frac{1}{R_e} L_2\Dlapla\Dlapla \phi 
           + \Dvect{r}\cdot\Drot\Drot [(\Drot\Dvect{u})\times\Dvect{u}]
           - \Dvect{r}\cdot\Drot\Drot\Dvect{\mathcal{F}}, 
       \end{eqnarray}
       %
       ãŒå¾—ã‚‰ã‚Œã‚‹. 
       ã“ã“ã§å¼ä¸­ã®æ¼”ç®—å­ $L_2$ ã¯
       åŠå¾„ 1 ã®çƒé¢ä¸Šã® 2 æ¬¡å…ƒãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³ã®é€†ç¬¦å·ã®ã‚‚ã®ã§ã‚ã‚Š, 
       %
       \begin{eqnarray}
         L_{2} &amp;\equiv&amp;  -r^{2}\Dlapla +\DP{}{r}r^{2}\DP{}{r}  \nonumber\\
               &amp;=&amp; -\left[
                     \frac{1}{\cos^{2}\varphi}\DP[2]{}{\lambda} 
                   +
                     \frac{1}{\cos\varphi}\DP{}{\varphi}
                     \left(\cos\varphi\right)\DP{}{\varphi} \right].
       \end{eqnarray}
       %
       \footnotetext[3]{
         ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒ»ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®è¨ˆç®—ã§ã®
         ä¾¿åˆ©ãªå…¬å¼ã‚’å°Žå‡ºã—ã¦ãŠã. 
         ç·¯åº¦çµŒåº¦åº§æ¨™ç³»ã§ã®å›žè»¢ã¯
         \begin{displaymath}
           \Drot\Dvect{A}
           =\left(\frac{1}{r}\DP{A_r}{\varphi}
                     -\frac{1}{r}\DP{}{r}(rA_\varphi),
                  \frac{1}{r}\DP{}{r}(rA_\lambda)
                     -\frac{1}{r\cos\varphi}\DP{A_r}{\lambda},
                  \frac{1}{r\cos\varphi}\DP{A_\varphi}{\lambda}
                     - \frac{1}{r\cos\varphi}
                         \DP{(A_\lambda\cos\varphi)}{\varphi}
            \right),
         \end{displaymath}
         %
         ã§ã‚ã‚‹. ã—ãŸãŒã£ã¦
         %
         \begin{eqnarray*}
           \psi\Dvect{r} &amp;=&amp; (0,0, r\psi), \\
           \Drot(\psi\Dvect{r}) &amp;=&amp;
              \left(  \DP{\psi}{\varphi},
                     -\frac{1}{\cos\varphi}\DP{\psi}{\lambda},
                     0
              \right) \\
           \Drot\Drot(\psi\Dvect{r}) &amp;=&amp; 
              \left( \frac{1}{r\cos\varphi}
                         \DP{}{r}\left(r\DP{\psi}{\lambda}\right),
                     \frac{1}{r}\DP{}{r}\left(r\DP{\psi}{\varphi}\right),
                     \frac{L_2\psi}{r}
              \right), \\
           \Drot\Drot\Drot(\psi\Dvect{r}) &amp;=&amp; 
              \left( -\DP{(\Dlapla\psi)}{\varphi},
                      \frac{1}{\cos\varphi}\DP{(\Dlapla\psi)}{\lambda},
                     0
              \right), \\
           \Drot\Drot\Drot\Drot(\psi\Dvect{r}) &amp;=&amp; 
              \left( -\frac{1}{r\cos\varphi}
                         \DP{}{r}\left(r\DP{(\Dlapla\psi)}{\lambda}\right),
                     -\frac{1}{r}\DP{}{r}
                         \left(r\DP{(\Dlapla\psi)}{\varphi}\right),
                     -\frac{L_2\Dlapla\psi}{r}
              \right), \\
           \Drot\Drot\Drot\Drot\Drot(\psi\Dvect{r}) &amp;=&amp; 
              \left( \DP{(\Dlapla\Dlapla\psi)}{\varphi},
                    -\frac{1}{\cos\varphi}\DP{(\Dlapla\Dlapla\psi)}{\lambda},
                     0
              \right), \\
           \Drot\Drot\Drot\Drot\Drot\Drot(\psi\Dvect{r}) &amp;=&amp; 
              \left( \frac{1}{r\cos\varphi}\DP{}{r}
                          \left(r\DP{(\Dlapla\Dlapla\psi)}{\lambda}\right),
                     \frac{1}{r}\DP{}{r}
                         \left(r\DP{(\Dlapla\Dlapla\psi)}{\varphi}\right),
                     \frac{L_2\Dlapla\Dlapla\psi}{r}
              \right), 
         \end{eqnarray*}
         %
         ã“ã‚Œã‚‰ã¨ $\Dvect{r}$ ã®å†…ç©ã‚’ã¨ã‚‹ã“ã¨ã«ã‚ˆã‚Š, 
         ä»¥ä¸‹ã®å…¬å¼ãŒå¾—ã‚‰ã‚Œã‚‹. 
         %
         \begin{eqnarray*}
           &amp;&amp; \Dvect{r}\cdot(\psi\Dvect{r}) = r^2\psi \\
           &amp;&amp; \Dvect{r}\cdot\Drot\Drot(\psi\Dvect{r}) = L_2\psi \\
           &amp;&amp; \Dvect{r}\cdot\Drot\Drot\Drot\Drot(\psi\Dvect{r}) = 
                     -L_2\Dlapla\psi,\\
           &amp;&amp; \Dvect{r}\cdot\Drot\Drot\Drot\Drot\Drot\Drot(\psi\Dvect{r}) =
                     L_2\Dlapla\Dlapla\psi,\\
           &amp;&amp; \Dvect{r}\cdot\Drot(\psi\Dvect{r}) 
                 = \Dvect{r}\cdot\Drot\Drot\Drot(\psi\Dvect{r})
                 =  \Dvect{r}\cdot\Drot\Drot\Drot\Drot\Drot(\psi\Dvect{r})
                 = 0, 
         \end{eqnarray*}
         }

      $\Dvect{r}\cdot\Drot, \Dvect{r}\cdot\Drot\Drot$ ã®åº§æ¨™è¡¨ç¾ã¯
      \begin{eqnarray*}
        \Dvect{r}\cdot\Drot\Dvect{\mathcal{F}}
          &amp;=&amp; \frac{1}{\cos\varphi}\DP{\mathcal{F}_\varphi}{\lambda}
            - \frac{1}{\cos\varphi}\DP{}{\varphi}(\mathcal{F}_\lambda\cos\varphi),\\
        \Dvect{r}\cdot\Drot\Drot\Dvect{\mathcal{F}}
          &amp;=&amp; \frac{1}{\cos\varphi}\DP{(\Drot\Dvect{\mathcal{F}})_\varphi}{\lambda}
            - \frac{1}{\cos\varphi}\DP{}{\varphi}
                 [(\Drot\Dvect{\mathcal{F}})_\lambda\cos\varphi]\\
          &amp;=&amp; \frac{1}{\cos\varphi}\DP{}{\lambda}
                    \left(  \frac{1}{r}\DP{}{r}(r\mathcal{F}_\lambda)
                           -\frac{1}{r\cos\varphi}\DP{\mathcal{F}_r}{\lambda} \right)\\
          &amp; &amp;  - \frac{1}{\cos\varphi}\DP{}{\varphi}
                 \left[\cos\varphi 
                        \left( \frac{1}{r}\DP{\mathcal{F}_r}{\varphi} 
                              - \frac{1}{r}\DP{}{r}(r\mathcal{F}_\varphi)\right)\right]\\
          &amp;=&amp; \frac{1}{\cos\varphi}\DP{}{\lambda}
                     \frac{1}{r}\DP{}{r}(r \mathcal{F}_\lambda)
               + \frac{1}{\cos\varphi}\DP{}{\varphi}
                     \frac{\cos\varphi}{r}\DP{}{r}(r\mathcal{F}_\varphi)\\
          &amp; &amp;  - \frac{1}{r\cos^2\varphi}\DP[2]{\mathcal{F}_r}{\lambda}
               - \frac{1}{r\cos\varphi}\DP{}{\varphi}
                     \left(\cos\varphi\DP{\mathcal{F}_r}{\varphi}\right)\\
          &amp;=&amp; \frac{1}{r}\DP{}{r}\left[r \left(
                   \frac{1}{\cos\varphi}\DP{\mathcal{F}_\lambda}{\lambda}
                  + \frac{1}{\cos\varphi}\DP{}{\varphi}(\mathcal{F}_\varphi\cos\varphi)
                  \right)\right]
                + \frac{L_2 \mathcal{F}_r}{r}.
      \end{eqnarray*}


    \subsubsection{é€Ÿåº¦å&nbsp;´ã¨ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¨ã®é–¢ä¿‚å¼}

      ãƒˆãƒ­ã‚¤ãƒ€ãƒ«, ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‹ã‚‰
      é€Ÿåº¦å&nbsp;´ã‚’æ±‚ã‚ã‚‹å¼ã¯æ¬¡ã®é€šã‚Šã§ã‚ã‚‹:
      \begin{eqnarray}
         u_\lambda  &amp; = &amp; \DP[]{\psi}{\varphi} 
               + \Dinv{r\cos\varphi}
                    \DP[]{}{r}\left(r\DP[]{\phi}{\lambda}\right),\\
         u_\varphi &amp; = &amp;  - \Dinv{\cos\varphi}\DP[]{\psi}{\lambda}
               + \Dinv{r}
                    \DP[]{}{r}\left(r\DP[]{\phi}{\varphi} \right),\\
         u_r  &amp; = &amp; \frac{L_2 ~\phi}{r}. 
      \end{eqnarray}
      %
      é€†ã«, é€Ÿåº¦å&nbsp;´ã®å„æˆåˆ†ã‹ã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’æ±‚ã‚ã‚‹ã«ã¯, 
      \Deqref{é€Ÿåº¦åˆ†è§£}ã«
      $\Dvect{r}\cdot$, $\Dvect{r}\cdot\Drot$ ã‚’ä½œç”¨ã•ã›ã‚Œã°ã‚ˆã„. 
      %
      \begin{eqnarray*}
        &amp; &amp; \Dvect{r}\cdot\Dvect{u} 
            = \Dvect{r}\cdot[\Drot(\psi\Dvect{r}) + \Drot\Drot(\phi\Dvect{r}) ]
            = \Dvect{r}\cdot\Drot\Drot(\phi\Dvect{r}) 
            = L_2\phi,\\
        &amp; &amp; \Dvect{r}\cdot\Drot\Dvect{u} 
            = \Dvect{r}\cdot\Drot[\Drot(\psi\Dvect{r}) + \Drot\Drot(\phi\Dvect{r}) ]
            = \Dvect{r}\cdot\Drot\Drot(\psi\Dvect{r}) = L_2\psi,
      \end{eqnarray*}
      %
      ã—ãŸãŒã£ã¦, 
      \begin{equation}
        \Deqlab{é€Ÿåº¦å&nbsp;´ã‹ã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}
            \phi = L_2^{-1}(ru_r), \quad
            \psi = L_2^{-1}[\Dvect{r}\cdot(\Drot\Dvect{u})]. 
      \end{equation}

    \subsubsection{å¢ƒç•Œæ¡ä»¶}

       é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶\Deqref{æ…£æ€§ç³»ç„¡æ¬¡å…ƒé‹å‹•å¢ƒç•Œæ¡ä»¶}ã‚’
       ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã™ã‚Œã°,
       \begin{equation}
         \phi = 0, 
           \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}. 
       \end{equation}

       åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒå¿œåŠ›ãªã—ã®å&nbsp;´åˆã¯\Deqref{æ…£æ€§ç³»ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}ã‚’
       ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã—ã¦
       \begin{equation}
         \Deqlab{æ…£æ€§ç³»ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}
         \DP[2]{\phi}{r} = \DP{}{r}\left(\frac{\psi}{r}\right) = 0, 
         \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \frac{1}{1-\zeta}. 
       \end{equation}

       ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, 
       \Deqref{é€Ÿåº¦å&nbsp;´ã‹ã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}ã‚’ç”¨ã„ã¦
       \Deqref{æ…£æ€§ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}ã‚’ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã—ã¦
       \footnotemark[4]
       \begin{eqnarray*}
         \Deqlab{æ…£æ€§ç³»ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
            \DP{\phi}{r} = 0, \quad
            \psi = (\Dvect{r}\cdot\Dvect{\Omega_i}),
            \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \\
            \DP{\phi}{r} = 0, \quad
            \psi = (\Dvect{r}\cdot\Dvect{\Omega_o}),
            \quad \mbox{at} \quad r=\frac{1}{1-\zeta}, \\
       \end{eqnarray*}

       \footnotetext[4]{
         ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®æ¡ä»¶ã¯, 
         å¢ƒç•Œçƒé¢ä¸Šã§ã®æ°´å¹³ç™ºæ•£ãŒ 0 ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰å°Žã‹ã‚Œã‚‹. 
         %
         \begin{displaymath}
           \frac{1}{r}\DP{(r u_r)}{r}=0, 
            \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta},
         \end{displaymath}
         %
         ã“ã“ã§ $u_r = L_2\phi/r$ ã‚ˆã‚Š, 
         \begin{displaymath}
           \frac{1}{r}\DP{(L_2 \phi)}{r}=0, 
            \quad \mbox{\it i.e.} \quad \DP{\phi}{r}=0, 
            \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}.
         \end{displaymath}
         %
         ä¸€æ–¹ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¯
         \begin{eqnarray*}
            \psi &amp;=&amp; L_2^{-1}
                 [\Dvect{r}\cdot\Drot(\Dvect{\Omega_i}\times\Dvect{r})]\\
              &amp;=&amp; L_2^{-1}\Dvect{r}\cdot
                  [  (\Ddiv\Dvect{r})\Dvect{\Omega_i}
                   - (\Ddiv\Dvect{\Omega_i})\Dvect{r}
                   + (\Dvect{r}\cdot\Dgrad)\Dvect{\Omega_i}
                   -(\Dvect{\Omega_i}\cdot\Ddiv)\Dvect{r}  ]\\
              &amp;=&amp; L_2^{-1}\Dvect{r}\cdot [3\Dvect{\Omega_i} -\Dvect{\Omega_i} ]
               =  L_2^{-1}2(\Dvect{r}\cdot\Dvect{\Omega_i})
               =  (\Dvect{r}\cdot\Dvect{\Omega_i}).
            \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}.
         \end{eqnarray*}
         ä¸Šã® $L_2^{-1}2(\Dvect{r}\cdot\Dvect{\Omega_i})$ ã®è¨ˆç®—ã¯, 
         $(\Dvect{r}\cdot\Dvect{\Omega})$ ã‚’ç›´äº¤ç›´ç·šåº§æ¨™æˆåˆ†ã§æ›¸ã‘ã°ã‚ˆã, 
         \begin{displaymath}
           (\Dvect{r}\cdot\Dvect{\Omega})
           = \Omega_x r\cos\varphi\cos\lambda
            +\Omega_y r\cos\varphi\sin\lambda
            +\Omega_z r\sin\varphi.
         \end{displaymath}
         %
         ã“ã“ã§ $\cos\varphi\cos\lambda, \cos\varphi\sin\lambda,
         \sin\varphi$ ã¯ãã‚Œãžã‚Œçƒé¢èª¿å’Œå‡½æ•° $Y_1^1, Y_1^0$ ã®
         æˆåˆ†ã§ã‚ã‚‹ã‹ã‚‰ $L_2$ ã®å›ºæœ‰é–¢æ•°ã§ã‚ã‚Š, 
         ãã®å›ºæœ‰å€¤ã¯ $1\times(1+1)=2$ ã§ã‚ã‚‹
         ($L_2$ ã®å›ºæœ‰é–¢æ•°ã¯ $Y_n^m$ ã§å›ºæœ‰å€¤ã¯ $n(n+1)$). 
         é€†ã« $L_2^{^-1}$ ã«å¯¾ã—ã¦ã¯
         $(\Dvect{r}\cdot\Dvect{\Omega})$ ãŒã‚„ã¯ã‚Šå›ºæœ‰é–¢æ•°ã§ã‚ã‚Š, 
         ãã®å›ºæœ‰å€¤ã¯ $1/2$ ã§ã‚ã‚‹. 
         ã™ãªã‚ã¡
         \begin{displaymath}
           L_2(\Dvect{r}\cdot{\Omega}) = 2(\Dvect{r}\cdot{\Omega}), 
           \quad
           L_2^{-1}(\Dvect{r}\cdot{\Omega}) 
             = \frac{1}{2}(\Dvect{r}\cdot{\Omega}) 
         \end{displaymath}
         ã¨ãªã‚‹. 
         }

       æˆåˆ†ã§è¡¨ã›ã°\footnotemark[6]
       %
       \begin{eqnarray}
         \Deqlab{æ…£æ€§ç³»ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶æˆåˆ†}
         &amp; &amp; \DP{\phi}{r} = 0, \\
         &amp; &amp; \psi = \Omega_i \left(\frac{\zeta}{1-\zeta}\right)
                     [ \cos\varphi_i\cos\varphi\cos(\lambda_i-\lambda)
                      + \sin\varphi_i\sin\varphi ]
             \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \\
         &amp; &amp; \DP{\phi}{r} = 0, \\
         &amp; &amp; \psi = \Omega_o \left(\frac{1}{1-\zeta}\right)
                     [ \cos\varphi_o\cos\varphi\cos(\lambda_o-\lambda)
                      + \sin\varphi_o\sin\varphi ]
             \quad \mbox{at} \quad \frac{1}{1-\zeta}. 
       \end{eqnarray}
       %

       \footnotetext[6]{
         $\Dvect{\Omega_i},\Dvect{r}$ ã‚’ç›´äº¤ç›´ç·šåº§æ¨™æˆåˆ†ã§è¡¨ã—
         å†…ç©ã‚’è¨ˆç®—ã™ã‚Œã°
         \begin{eqnarray*}
           \Dvect{r}
             &amp;=&amp;r(\cos\varphi\cos\lambda,\cos\varphi\sin\lambda,\sin\varphi),
           \quad
           \Dvect{\Omega_i}
              =\Omega_i(\cos\varphi_i\cos\lambda_i,
                 \cos\varphi_i\sin\lambda_i,
                 \sin\varphi_i),\\
           (\Dvect{r}\cdot\Dvect{\Omega_i})
             &amp;=&amp; \Omega_i r
                 ( \cos\varphi\cos\lambda
                    \cos\varphi_i\cos\lambda_i
                 +\cos\varphi\sin\lambda
                    \cos\varphi_i\sin\lambda_i
                 +\sin\varphi\sin\varphi_i)\\
             &amp;=&amp; \Omega_i r
                 ( \cos\varphi\cos\varphi_i
                   (\cos\lambda\cos\lambda_i
                    + \sin\lambda\sin\lambda_i)
                  +\sin\varphi\sin\varphi_i)\\
             &amp;=&amp; \Omega_i r
                 ( \cos\varphi\cos\varphi_i
                     \cos(\lambda-\lambda_i)
                  +\sin\varphi\sin\varphi_i).
         \end{eqnarray*}
         ãŸã&nbsp;ã— $r\equiv|\Dvect{r}|,\Omega_i \equiv |\Dvect{\Omega_i}|$ ã§ã‚ã‚‹.
         %
         ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®å¢ƒç•Œæ¡ä»¶ã¯å¤‰å½¢ã‚’é€”ä¸­ã§æ­¢ã‚ãŸå¼, 
         \begin{displaymath}
           \psi = \Omega_i \left(\frac{\zeta}{1-\zeta}\right)
                   [ \cos\varphi_i\cos\lambda_i\cos\varphi\cos\lambda
                    +\cos\varphi_i\sin\lambda_i\cos\varphi\sin\lambda
                    + \sin\varphi_i\sin\varphi ]
         \end{displaymath}
         %
         ã‹ã‚‰, çƒé¢èª¿å’Œå‡½æ•° 
         $Y_n^m(\lambda,\varphi)=P_n^m(\sin\varphi)\exp(im\lambda)$ 
         ã® $n=1$ æˆåˆ†ã§è¡¨ã•ã‚Œã‚‹ã“ã¨ãŒã‚ã‹ã‚‹. 
         ãã“ã§
         $\psi(r=\zeta/(1-\zeta))$ ã‚’
         \begin{displaymath}
           \psi(\lambda,\varphi,r=\zeta/(1-\zeta))=
            \sum_{n=1}^\infty\left[
                 a_n^0P_n^0(\sin\varphi) + 
                 \sum_{m=1}^n
                   P_n^m(\sin\varphi)[a_n^m\sqrt{2}\cos(m\lambda)
                                      -b_n^m\sqrt{2}\sin(m\lambda)]
              \right]
         \end{displaymath}
         %
         ã¨å±•é–‹ã™ã‚‹ã¨, $P_1^0(\sin\varphi)=\sin\varphi$,
         $P_1^1(\sin\varphi)=\cos\varphi$ ã§ã‚ã‚‹ã‹ã‚‰
         %
         \begin{eqnarray*}
           a_1^0 &amp;=&amp; \Omega_i\left(\frac{\zeta}{1-\zeta}\right)
                     \sin\varphi_i,\\
           a_1^1 &amp;=&amp; \Omega_i\left(\frac{\zeta}{1-\zeta}\right)
                       \frac{1}{\sqrt{2}}\cos\varphi_i\cos\lambda_i,\\
           b_1^1 &amp;=&amp; -\Omega_i\left(\frac{\zeta}{1-\zeta}\right)
                       \frac{1}{\sqrt{2}}\cos\varphi_i\sin\lambda_i
         \end{eqnarray*}
         %
         ã¨å±•é–‹ä¿‚æ•°ãŒä¸Žãˆã‚‰ã‚Œã‚‹. 
         }

  \subsection{è§£æžçš„ãªè§£ã®ä¾‹}

    \subsubsection{å¿œåŠ›ãªã—æ¡ä»¶ä¸‹ã§ã®å‰›ä½“å›žè»¢è§£}

       å¤–åŠ›ãŒãªã„($\mathcal{F}=0$)å¿œåŠ›ãªã—ã®æ¡ä»¶ä¸‹ã§ã¯
       å‰›ä½“å›žè»¢ã™ã‚‹è§£ãŒå­˜åœ¨ã§ãã‚‹:
       %
       \begin{equation}
         \Dvect{u} = \Dvect{\Omega}\times\Dvect{r}, 
       \end{equation}
       %
       ã“ã“ã§ $\Dvect{\Omega}=\Omega\Dvect{e_\Omega}$ ã¯
       å‰›ä½“å›žè»¢ã‚’è¡¨ã™å›žè»¢ãƒ™ã‚¯ãƒˆãƒ«ã§ã‚ã‚Š, 
       $\Omega$ ã¯å›žè»¢ã®å¤§ãã•, 
       $\Dvect{e_\Omega}$ ã¯å›žè»¢è»¸æ–¹å‘ã®å˜ä½ãƒ™ã‚¯ãƒˆãƒ«ã§ã‚ã‚‹. 
       å…·ä½“çš„ãªé€Ÿåº¦æˆåˆ†ã¯
       \begin{eqnarray*}
         u_\lambda(\lambda,\varphi,r) &amp;=&amp;\Omega r
                   [ -\cos\varphi_\Omega\sin\varphi\cos(\lambda_\Omega-\lambda)
                  + \sin\varphi_\Omega\cos\varphi ], \\
         u_\varphi(\lambda,\varphi,r) &amp;=&amp; \Omega r
               [ \cos\varphi_\Omega\sin(\lambda-\lambda_\Omega)],\\
         u_r(\lambda,\varphi,r) &amp;=&amp; 0. 
       \end{eqnarray*}
       ãŸã&nbsp;ã—, $(\lambda_\Omega,\varphi_\Omega)$ ã¯
       å‰›ä½“å›žè»¢ã®è»¸ã®æ–¹å‘ã‚’è¡¨ã™çµŒåº¦ãƒ»ç·¯åº¦ã§ã‚ã‚‹. 
       ã“ã®æµã‚Œã‚’ä¸Žãˆã‚‹ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒ»ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¯
       \Deqref{é€Ÿåº¦å&nbsp;´ã‹ã‚‰ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«}ã‚ˆã‚Š
       \footnotemark[7]
       \footnotetext[7]{ã“ã®ã‚ãŸã‚Šã®å¼å¤‰å½¢ã¯
         ç²˜ç€å¢ƒç•Œæ¡ä»¶ã‚’ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã™ã‚‹ã¨ã“ã‚ã®å°Žå‡ºã¨
         å…¨ãåŒã˜ã§ã‚ã‚‹. }
      \begin{eqnarray*}
            \phi &amp;=&amp; L_2^{-1}(ru_r) = 0, \\
            \psi &amp;=&amp; L_2^{-1}
                 [\Dvect{r}\cdot\Drot(\Dvect{\Omega}\times\Dvect{r})]\\
              &amp;=&amp; L_2^{-1}\Dvect{r}\cdot
                  [  (\Ddiv\Dvect{r})\Dvect{\Omega}
                   - (\Ddiv\Dvect{\Omega})\Dvect{r}
                   + (\Dvect{r}\cdot\Dgrad)\Dvect{\Omega}
                   -(\Dvect{\Omega}\cdot\Ddiv)\Dvect{r}  ]\\
              &amp;=&amp; L_2^{-1}\Dvect{r}\cdot [3\Dvect{\Omega} -\Dvect{\Omega} ]
               =  L_2^{-1}2(\Dvect{r}\cdot\Dvect{\Omega})
               =  (\Dvect{r}\cdot\Dvect{\Omega}).
      \end{eqnarray*}
      %
       æˆåˆ†è¡¨ç¤ºã™ã‚Œã°
       %
       \begin{displaymath}
         \phi=0, \quad 
         \psi(\lambda,\varphi,r) = \Omega r
                   [ \cos\varphi_\Omega\cos\varphi\cos(\lambda_\Omega-\lambda)
                    + \sin\varphi_\Omega\sin\varphi ]. 
       \end{displaymath}
       %
       ã—ãŸãŒã£ã¦, å¿œåŠ›ãªã—æ¡ä»¶
       \Deqref{æ…£æ€§ç³»ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}ã‚’
       æº€ãŸã—ã¦ã„ã‚‹. 

       ä¸Šã®é€Ÿåº¦å&nbsp;´ãŒæ”¯é…æ–¹ç¨‹å¼æº€ãŸã—ã¦ã„ã‚‹ã“ã¨ã‚’ç¢ºã‹ã‚ã‚‹. 
       é€£ç¶šã®å¼ã¯
       \begin{eqnarray*}
         \Ddiv\Dvect{u}=\Ddiv(\Dvect{\Omega}\times\Dvect{r})
         = (\Drot\Dvect{\Omega})\cdot\Dvect{r}
           -(\Drot\Dvect{r})\cdot\Dvect{\Omega} = 0.
       \end{eqnarray*}
       %
       é‹å‹•æ–¹ç¨‹å¼ã®ç²˜æ€§é&nbsp;…ã¯ãƒ™ã‚¯ãƒˆãƒ«å…¬å¼ã¨é€£ç¶šã®å¼ã‚’ç”¨ã„ã¦
       \begin{eqnarray*}
         \Dlapla\Dvect{u}=\Dgrad(\Ddiv\Dvect{u})-\Drot\Drot\Dvect{u}
         =\Drot\Drot(\Dvect{\Omega\times\Dvect{r}})
         =\Drot(2\Dvect{\Omega})=0. 
       \end{eqnarray*}
       %
       æ…£æ€§é&nbsp;…ã¯ãƒ™ã‚¯ãƒˆãƒ«ã®å…¬å¼ã‚’ç”¨ã„ã¦å¤‰å½¢ã™ã‚‹ã¨
       \begin{eqnarray*}
         \Dvect{u}\cdot\Dgrad\Dvect{u} 
         &amp;=&amp;\frac{1}{2}\Dgrad|\Dvect{u}|^2-\Dvect{u}\times\Drot\Dvect{u}\\
         &amp;=&amp;\frac{1}{2}\Dgrad|\Dvect{\Omega}\times\Dvect{r}|^2
            -(\Dvect{\Omega}\times\Dvect{r})
              \times\Drot(\Dvect{\Omega}\times\Dvect{r})
       \end{eqnarray*}
       ã“ã“ã§
       \begin{displaymath}
         \Drot\Dvect{\Omega}\times\Dvect{r}
          = (\Ddiv\Dvect{r})\Dvect{\Omega}
           -(\Ddiv\Dvect{\Omega})\Dvect{r} 
           +(\Dvect{r}\cdot\Dgrad)\Dvect{\Omega}
           -(\Dvect{\Omega}\cdot\Dgrad)\Dvect{r}
         = 3\Dvect{\Omega}-\Dvect{\Omega}=2\Dvect{\Omega}
       \end{displaymath}
       %
       ã§ã‚ã‚‹ã‹ã‚‰, 
       \begin{eqnarray*}
         \Dvect{u}\cdot\Dgrad\Dvect{u} 
         &amp;=&amp;\frac{1}{2}\Dgrad|\Dvect{\Omega}\times\Dvect{r}|^2
            -(\Dvect{\Omega}\times\Dvect{r})\times 2\Dvect{\Omega}
         =\frac{1}{2}\Dgrad|\Dvect{\Omega}\times\Dvect{r}|^2
            -(2\Dvect{\Omega}\cdot\Dvect{\Omega})\Dvect{r}
            +(2\Dvect{\Omega}\cdot\Dvect{r})\Dvect{\Omega}\\
         &amp;=&amp;\frac{1}{2}\Dgrad|\Dvect{\Omega}\times\Dvect{r}|^2
            -2|\Dvect{\Omega}|^2\Dvect{r}
            +\Dgrad(\Dvect{\Omega}\cdot\Dvect{r})^2
          = \frac{1}{2}\Dgrad|\Dvect{\Omega}\times\Dvect{r}|^2
            -\Dgrad|\Dvect{\Omega}|^2|\Dvect{r}|^2
            +\Dgrad(\Dvect{\Omega}\cdot\Dvect{r})^2\\
         &amp;=&amp; \Dgrad\left[\frac{1}{2}|\Dvect{\Omega}\times\Dvect{r}|^2
            -|\Dvect{\Omega}|^2|\Dvect{r}|^2
            +(\Dvect{\Omega}\cdot\Dvect{r})^2\right].
       \end{eqnarray*}
       %
       ã—ãŸãŒã£ã¦ã“ã‚Œã«ãƒãƒ©ãƒ³ã‚¹ã™ã‚‹åœ§åŠ›ãŒ
       \begin{displaymath}
         p= -\rho\left[\frac{1}{2}|\Dvect{\Omega}\times\Dvect{r}|^2
            -|\Dvect{\Omega}|^2|\Dvect{r}|^2
            +(\Dvect{\Omega}\cdot\Dvect{r})^2
            \right]
       \end{displaymath}
       %
       ã¨å®šã‚ã‚‰ã‚Œã‚‹. ã—ãŸãŒã£ã¦é€£ç¶šã®å¼ã‚’æº€ãŸã™ã“ã¨ãŒã§ãã‚‹. 


%        é€Ÿåº¦ã®å„æˆåˆ†ã‚’é‹å‹•æ–¹ç¨‹å¼ã®éžç·šå½¢é&nbsp;…ã«ä»£å…¥ã™ã‚‹ã¨, 
%        %
%        \begin{eqnarray*}
%          &amp; &amp; (\Dvect{u}\cdot\Dgrad\Dvect{u})_\lambda \\
%          &amp;=&amp; \Dvect{u}\cdot\Dgrad u_\lambda 
%                +\frac{u_r u_\lambda}{r}
%                -\frac{u_\varphi u_\lambda \tan\varphi}{r} \\
%          &amp;=&amp; \frac{u_\lambda}{r\cos\varphi}\DP{u_\lambda}{\lambda}
%                + \frac{u_\varphi}{r}\DP{u_\lambda}{\varphi}
%                -\frac{u_\varphi u_\lambda \tan\varphi}{r}\\
%          &amp;=&amp; \frac{\Omega r
%                        [ -\cos\varphi_\Omega\sin\varphi
%                             \cos(\lambda_\Omega-\lambda)
%                         + \sin\varphi_\Omega\cos\varphi ]}
%                     {r\cos\varphi}
%                 \DP{}{\lambda}
%                    (\Omega r[ -\cos\varphi_\Omega\sin\varphi
%                                  \cos(\lambda_\Omega-\lambda)
%                          + \sin\varphi_\Omega\cos\varphi ])\\
%            &amp; &amp; + \frac{\Omega r[ \cos\varphi_\Omega
%                                    \sin(\lambda-\lambda_\Omega)]}
%                       {r}
%                    \DP{}{\varphi}
%                     (\Omega r [ -\cos\varphi_\Omega
%                                    \sin\varphi\cos(\lambda_\Omega-\lambda)
%                                + \sin\varphi_\Omega\cos\varphi ])\\
%            &amp; &amp;  -\frac{\Omega r
%                         [ -\cos\varphi_\Omega\sin\varphi
%                                 \cos(\lambda_\Omega-\lambda)
%                           + \sin\varphi_\Omega\cos\varphi ]
%                   \Omega r [ \cos\varphi_\Omega\sin(\lambda-\lambda_\Omega)]
%                   \tan\varphi}{r}\\
%          &amp;=&amp;  \frac{\Omega^2 r^2
%                        [ -\cos\varphi_\Omega\sin\varphi
%                             \cos(\lambda_\Omega-\lambda)
%                         + \sin\varphi_\Omega\cos\varphi ]
%                        [ -\cos\varphi_\Omega\sin\varphi
%                                  \sin(\lambda_\Omega-\lambda)]}
%                      {r\cos\varphi}\\
%             &amp; &amp; + \frac{\Omega^2 r^2
%                           [ \cos\varphi_\Omega
%                                    \sin(\lambda-\lambda_\Omega)]
%                           [ -\cos\varphi_\Omega
%                                    \cos\varphi\cos(\lambda_\Omega-\lambda)
%                                - \sin\varphi_\Omega\sin\varphi ]}
%                      {r}\\
%             &amp; &amp; - \frac{\Omega^2 r^2
%                         [ -\cos\varphi_\Omega\sin\varphi
%                                 \cos(\lambda_\Omega-\lambda)
%                           + \sin\varphi_\Omega\cos\varphi ]
%                         [ \cos\varphi_\Omega\sin(\lambda-\lambda_\Omega)]
%                   \sin\varphi}{r\cos\varphi}\\
%          &amp;=&amp;  \frac{\Omega^2 r}{\cos\varphi}
%                  \{
%                    [ -\cos\varphi_\Omega\sin\varphi
%                          \cos(\lambda_\Omega-\lambda)
%                      + \sin\varphi_\Omega\cos\varphi ]
%                    [ -\cos\varphi_\Omega\sin\varphi
%                            \sin(\lambda_\Omega-\lambda)]\\
%             &amp; &amp; +  [ \cos\varphi_\Omega\cos\varphi
%                          \sin(\lambda-\lambda_\Omega)]
%                    [ -\cos\varphi_\Omega
%                           \cos\varphi\cos(\lambda_\Omega-\lambda)
%                             - \sin\varphi_\Omega\sin\varphi ]\\
%             &amp; &amp; +  [ -\cos\varphi_\Omega\sin\varphi
%                            \cos(\lambda_\Omega-\lambda)
%                      + \sin\varphi_\Omega\cos\varphi ]
%                    [ \cos\varphi_\Omega\sin\varphi\sin(\lambda-\lambda_\Omega)]
%                 \}\\
%          &amp;=&amp;  \frac{\Omega^2 r}{\cos\varphi}
%                  \{
%                    [ -\cos\varphi_\Omega\sin\varphi
%                          \cos(\lambda_\Omega-\lambda)
%                      + \sin\varphi_\Omega\cos\varphi ]
%                    [ -\cos\varphi_\Omega\sin\varphi
%                            \sin(\lambda_\Omega-\lambda)]\\
%             &amp; &amp; +  [ \cos\varphi_\Omega\cos\varphi
%                          \sin(\lambda-\lambda_\Omega)]
%                    [ -\cos\varphi_\Omega
%                           \cos\varphi\cos(\lambda_\Omega-\lambda)
%                             - \sin\varphi_\Omega\sin\varphi ]\\
%             &amp; &amp; +  [ -\cos\varphi_\Omega\sin\varphi
%                            \cos(\lambda_\Omega-\lambda)
%                      + \sin\varphi_\Omega\cos\varphi ]
%                    [ \cos\varphi_\Omega\sin\varphi\sin(\lambda-\lambda_\Omega)]
%                 \}\\
%          &amp;=&amp;  \frac{\Omega^2 r}{\cos\varphi}
%                  \{
%                      \cos^2\varphi_\Omega\sin^2\varphi
%                         \sin(\lambda_\Omega-\lambda)
%                         \cos(\lambda_\Omega-\lambda)
%                     -\sin\varphi_\Omega\cos\varphi_\Omega\sin\varphi\cos\varphi
%                            \sin(\lambda_\Omega-\lambda)\\
%             &amp; &amp;     -\cos^2\varphi_\Omega\cos\varphi^2
%                          \sin(\lambda-\lambda_\Omega)
%                          \cos(\lambda_\Omega-\lambda)
%                     - \sin\varphi_\Omega\cos\varphi_\Omega
%                           \sin\varphi\cos\varphi
%                           \sin(\lambda-\lambda_\Omega)\\
%             &amp; &amp;     -\cos^2\varphi_\Omega\sin^2\varphi
%                        \sin(\lambda-\lambda_\Omega)\cos(\lambda_\Omega-\lambda)
%                      + \sin\varphi_\Omega\cos\varphi_\Omega
%                            \sin\varphi\cos\varphi 
%                            \sin(\lambda-\lambda_\Omega)
%                 \}\\
%          &amp;=&amp;  \frac{\Omega^2 r}{\cos\varphi}
%                  \{ -\sin\varphi_\Omega\cos\varphi_\Omega\sin\varphi\cos\varphi
%                            \sin(\lambda_\Omega-\lambda)
%                    -\cos^2\varphi_\Omega\cos\varphi^2
%                          \sin(\lambda-\lambda_\Omega)
%                          \cos(\lambda_\Omega-\lambda)
%                  \}\\
%          &amp;=&amp;  \Omega^2 r
%                  \{ -\sin\varphi_\Omega\cos\varphi_\Omega\sin\varphi
%                            \sin(\lambda_\Omega-\lambda)
%                    -\cos^2\varphi_\Omega\cos\varphi
%                          \sin(\lambda-\lambda_\Omega)
%                          \cos(\lambda_\Omega-\lambda)
%                  \},\\
% %
%          &amp; &amp; (\Dvect{u}\cdot\Dgrad\Dvect{u})_\varphi \\
%          &amp;=&amp; \Dvect{u}\cdot\Dgrad u_\varphi
%            +\frac{u_r u_\varphi}{r}+\frac{u_\lambda^2 \tan\varphi}{r}\\
%          &amp;=&amp; \frac{u_\lambda}{r\cos\varphi}\DP{u_\varphi}{\lambda}
%             + \frac{u_\varphi}{r}\DP{u_\varphi}{\varphi}
%             +\frac{u_\lambda^2 \tan\varphi}{r}\\
%          &amp;=&amp; \frac{\Omega r
%                    [ -\cos\varphi_\Omega\sin\varphi\cos(\lambda_\Omega-\lambda)
%                   + \sin\varphi_\Omega\cos\varphi ]}{r\cos\varphi}
%                 \DP{}{\lambda}
%                  (\Omega r
%                    [ \cos\varphi_\Omega\sin(\lambda-\lambda_\Omega)])\\
%          &amp; &amp;  + \frac{\Omega r
%                         [ \cos\varphi_\Omega\sin(\lambda-\lambda_\Omega)]}{r}
%                \DP{}{\varphi}
%                  (\Omega r
%                      [ \cos\varphi_\Omega\sin(\lambda-\lambda_\Omega)])\\
%          &amp; &amp;  +\frac{\Omega^2 r^2
%                    [ -\cos\varphi_\Omega\sin\varphi\cos(\lambda_\Omega-\lambda)
%                   + \sin\varphi_\Omega\cos\varphi ]^2
%                   \tan\varphi}{r}\\
%          &amp;=&amp; \frac{\Omega^2 r
%                      [ -\cos\varphi_\Omega\sin\varphi
%                             \cos(\lambda_\Omega-\lambda)
%                        + \sin\varphi_\Omega\cos\varphi ]
%                      [ \cos\varphi_\Omega\cos(\lambda-\lambda_\Omega)]}
%                   {\cos\varphi}\\
%          &amp; &amp;  +\frac{\Omega^2 r
%                    [ -\cos\varphi_\Omega\sin\varphi\cos(\lambda_\Omega-\lambda)
%                   + \sin\varphi_\Omega\cos\varphi ]^2
%                   \sin\varphi}{\cos\varphi}\\
%          &amp;=&amp; \frac{\Omega^2 r}{\cos\varphi}
%               \{  -\cos\varphi_\Omega^2\sin\varphi
%                           \cos^2(\lambda_\Omega-\lambda)
%                       + \sin\varphi_\Omega\cos\varphi_\Omega\cos\varphi 
%                         \cos(\lambda-\lambda_\Omega)
%                   \\
%          &amp; &amp;  + \cos^2\varphi_\Omega\sin^3\varphi\cos^2(\lambda_\Omega-\lambda)
%                -2\cos\varphi_\Omega\sin\varphi_\Omega\sin^2\varphi\cos\varphi
%                    \cos(\lambda_\Omega-\lambda)
%                    + \sin^2\varphi_\Omega\sin\varphi\cos^2\varphi
%                 \}
%        \end{eqnarray*}


    \subsubsection{ç²˜ç€æ¡ä»¶ä¸‹ã§ã®å·®åˆ†å›žè»¢è§£(æµã‚ŒãŒååˆ†ã«é…ã„å&nbsp;´åˆ)}

       å†…å¤–çƒæ®»ã®å›žè»¢ã«ã‚ˆã‚Šå¼•ãèµ·ã“ã•ã‚Œã‚‹çƒæ®»ä¸­ã®æµã‚Œã«ã¤ã„ã¦, 
       å›žè»¢è§’é€Ÿåº¦ãŒååˆ†ã«å°ã•ãç²˜æ€§é&nbsp;…ã«æ¯”ã¹ã¦æ…£æ€§é&nbsp;…ãŒç„¡è¦–ã§ãã‚‹
       çŠ¶æ³ã§ã®è§£ã‚’è¦‹å‡ºã™\footnotemark[6]
       ($Re=\Omega_i D^2/\nu, \Omega_o D^2/\nu \ll 1)$. 

       \footnotetext[6]
          {Landau and Lifsiz, Vol.6, Fluid Mechanics, Section 20, 
           Problem 1. ã‚’å‚ç…§ã®ã“ã¨. }

       å†…å¤–çƒæ®»ãŒãã‚Œãžã‚Œä¸€å®šã®è§’é€Ÿåº¦ $\Dvect{\Omega_i},\Dvect{\Omega_o}$
       ã§å›žè»¢ã—ã¦ã„ã‚‹ã¨ã™ã‚‹. 
       æº€ãŸã™ã¹ãæ–¹ç¨‹å¼ã¯, å¤–åŠ›ãŒãªãæ…£æ€§é&nbsp;…ã‚’ç„¡è¦–ã—ãŸå¼, 
       %
       \begin{displaymath}
          L_2\Dlapla \psi = 0,   L_2\Dlapla\Dlapla \phi =0, 
       \end{displaymath}
       %
       å¢ƒç•Œã§ã®æ¡ä»¶ã¯ç²˜ç€æ¡ä»¶ã§ã‚ã‚‹. 
       %
       \begin{eqnarray*}
         &amp; &amp; \phi = \DP{\phi}{r} = 0, \quad
             \psi = (\Dvect{r}\cdot\Dvect{\Omega_i})
             \quad \mbox{at} \quad r=r_i=\frac{\zeta}{1-\zeta}, \\
         &amp; &amp; \phi = \DP{\phi}{r} = 0, \quad
             \psi = (\Dvect{r}\cdot\Dvect{\Omega_o})
             \quad \mbox{at} \quad r=r_o=\frac{1}{1-\zeta}. 
       \end{eqnarray*}
       %
       ãã“ã§
       \begin{displaymath}
         \phi=0, \quad
         \psi= A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})
               +B(r)(\Dvect{r}\cdot\Dvect{\Omega_o})
       \end{displaymath}
       %
       ã®å½¢ã§è§£ã‚’æ±‚ã‚ã‚‹. ã“ã“ã§ $A(r),B(r)$ ã¯å¢ƒç•Œæ¡ä»¶
       %
       \begin{displaymath}
          A(r=r_i) = 1, \quad  A(r=r_o) = 0, \quad
         B(r=r_i) = 0, \quad   B(r=r_o) = 1,
       \end{displaymath}
       %
       ã‚’æº€ãŸã™é–¢æ•°ã§ã‚ã‚‹. 
       %
       ä»®å®šã—ãŸè§£ã®å½¢ã‚’æ”¯é…æ–¹ç¨‹å¼ã«ä»£å…¥ã™ã‚‹ã¨, $A(r), B(r)$ ã®
       æº€ãŸã™ã¹ãæ–¹ç¨‹å¼ãŒ\footnotemark[9]
       \footnotetext[9]{
         ç¬¬ 1 é&nbsp;…ç›®ã&nbsp;ã‘ä»£å…¥ã™ã‚‹ã¨
         \begin{eqnarray*}
           &amp; &amp; L_2\Dlapla A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})
             = L_2\left(\frac{1}{r^2}\DP{}{r}r^2\DP{}{r}- \frac{L_2}{r^2}\right)
                 A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})\\
           &amp;=&amp; L_2\frac{1}{r^2}\DP{}{r}
                 \left(r^2\DP{}{r}[A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
              - \frac{L_2 L_2 A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
           &amp;=&amp; \frac{1}{r^2}\DP{}{r}
               \left(r^2\DP{}{r}[A(r)L_2(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
              - \frac{A(r)}{r^2}L_2 L_2(\Dvect{r}\cdot\Dvect{\Omega_i})\\
           &amp;=&amp; \frac{1}{r^2}\DP{}{r}
               \left(r^2\DP{}{r}[A(r)2(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
              - \frac{A(r)}{r^2}4(\Dvect{r}\cdot\Dvect{\Omega_i})\\
           &amp;=&amp; \frac{2}{r^2}\DP{}{r}
               \left(r^2\DP{}{r}[A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]\right)
              - \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
           &amp;=&amp; \frac{2}{r^2}\DP{}{r}
               \left(r^2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP{A(r)}{r}
               + r^2A(r)\frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\right)
              - \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
           &amp;=&amp; \frac{2}{r^2}\DP{}{r}
                 \left(r^2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP{A(r)}{r}\right)
              +\frac{2}{r^2}\DP{}{r}
                 [rA(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]
              - \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
           &amp;=&amp; \frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\DP{}{r}
                 \left(r^2\DP{A(r)}{r}\right)
               +2\DP{A(r)}{r}
                  \frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}
               +\frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}
                 \DP{}{r}[rA(r)]
               +2\frac{A(r)}{r}\frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\\
             &amp;&amp;- \frac{4A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
           &amp;=&amp; 2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP[2]{A(r)}{r}
              + \frac{4(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\DP{A(r)}{r}
              +2\DP{A(r)}{r}\frac{(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}
              +\frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}
                 \DP{A(r)}{r}
              +\frac{2(\Dvect{r}\cdot\Dvect{\Omega_i})A(r)}{r^2}\\
             &amp;&amp;- \frac{2A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})}{r^2}\\
           &amp;=&amp; 2(\Dvect{r}\cdot\Dvect{\Omega_i})\DP[2]{A(r)}{r}
              + \frac{8(\Dvect{r}\cdot\Dvect{\Omega_i})}{r}\DP{A(r)}{r}\\
           &amp;=&amp; 2(\Dvect{r}\cdot\Dvect{\Omega_i})
                 \left(\DP[2]{A(r)}{r}+\frac{4}{r}\DP{A(r)}{r}\right)
         \end{eqnarray*}
         ç¬¬ 2 é&nbsp;…ç›®ã«ã¤ã„ã¦ã‚‚åŒæ§˜ã«è¨ˆç®—ã§ãã‚‹. 
         $\Dvect{\Omega_i}$ ã¨ $\Dvect{\Omega_o}$ ãŒç•°ãªã‚‹å&nbsp;´åˆã«é™ã‚Š
         $(\Dvect{r}\cdot\Dvect{\Omega_i})$ ã¨
         $(\Dvect{r}\cdot\Dvect{\Omega_o})$ ãŒæ’ç­‰çš„ã« 0 ã¨ãªã‚‰ãªã„ã®ã§
         ãã‚Œã‚‰ã®ä¿‚æ•°ã‹ã‚‰ $A, B$ ã®å¾®åˆ†æ–¹ç¨‹å¼ãŒæ±‚ã‚ã‚‰ã‚Œã‚‹. 
       }

       \begin{displaymath}
          \DD[2]{A(r)}{r}+\frac{4}{r}\DD{A(r)}{r}=0, \quad
          \DD[2]{B(r)}{r}+\frac{4}{r}\DD{B(r)}{r}=0. 
       \end{displaymath}
       %
       $A(r),B(r)=r^n$ ã®å½¢ã‚’ä»®å®šã™ã‚‹ã¨, 
       \begin{displaymath}
         n(n-1)r^{n-2} + 4n r^{n-2} =0, \quad n(n+3)=0, 
         \quad \mbox{i.e.} \quad n=0,-3
       \end{displaymath}
       %
       ã—ãŸãŒã£ã¦, 
       \begin{displaymath}
         A(r) = C_1 + \frac{C_2}{r^3}, \quad
         B(r) = C_3 + \frac{C_4}{r^3}
       \end{displaymath}
       ã¨ãªã‚‹. ã“ã“ã§ $C_1,C_2,C_3,C_4$ ã¯å®šæ•°ã§ã‚ã‚‹. 
       %
       å¢ƒç•Œæ¡ä»¶ã‚ˆã‚Š
       \begin{displaymath}
           C_1 + \frac{C_2}{r_i^3}=1, \quad
           C_1 + \frac{C_2}{r_o^3}=0, \quad
           C_3 + \frac{C_4}{r_i^3}=0, \quad
           C_3 + \frac{C_4}{r_o^3}=1,
       \end{displaymath}
       %
       \begin{eqnarray*}
         &amp; &amp; C_1 = \frac{1}{1-r_o^3/r_i^3}= -\frac{r_i^3}{r_o^3-r_i^3}, \quad
             C_2 = \frac{r_i^3r_o^3}{r_o^3-r_i^3}, \\
         &amp; &amp; C_3 = \frac{1}{(1-r_i^3/r_o^3)}= \frac{r_o^3}{r_o^3-r_i^3}, \quad
             C_4 = -\frac{r_i^3r_o^3}{r_o^3-r_i^3}.
       \end{eqnarray*}
       %
       ã‚ˆã£ã¦
       \begin{eqnarray*}
         \phi&amp;=&amp;0, \\
         \psi&amp;=&amp; \left( -\frac{r_i^3}{r_o^3-r_i^3}
                        +\frac{r_i^3r_o^3}{r_o^3-r_i^3}\frac{1}{r^3}\right)
                    (\Dvect{r}\cdot\Dvect{\Omega_i})
               +\left( \frac{r_o^3}{r_o^3-r_i^3}
                      -\frac{r_i^3r_o^3}{r_o^3-r_i^3}\frac{1}{r^3}\right)
                    (\Dvect{r}\cdot\Dvect{\Omega_o})\\
            &amp;=&amp; -\frac{r_i^3}{r_o^3-r_i^3}\left(1-\frac{r_o^3}{r^3}\right)
                   (\Dvect{r}\cdot\Dvect{\Omega_i})
               +\frac{r_o^3}{r_o^3-r_i^3}\left(1-\frac{r_i^3}{r^3}\right)
                    (\Dvect{r}\cdot\Dvect{\Omega_o}).
       \end{eqnarray*}
       %
       ç„¡æ¬¡å…ƒé‡ã§ $r_i,r_o$ ã‚’æ›¸ã‘ã°
       \begin{eqnarray*}
         \phi&amp;=&amp;0, \\
         \psi&amp;=&amp; -\frac{\zeta^3}{1-\zeta^3}
                \left(1-\frac{1}{(1-\zeta)^3 r^3}\right)
                    (\Dvect{r}\cdot\Dvect{\Omega_i})
               +\frac{1}{1-\zeta^3}
                  \left(1-\frac{\zeta^3}{(1-\zeta)^3 r^3}\right)
                    (\Dvect{r}\cdot\Dvect{\Omega_o})
       \end{eqnarray*}
       %
       ç‰¹ã«å†…çƒãŒå­˜åœ¨ã—ãªã„å&nbsp;´åˆã«ã¯ $r_i\longrightarrow 0$ ã¨ã—ã¦, 
       %
       \begin{displaymath}
         \phi=0, \quad  \psi=(\Dvect{r}\cdot\Dvect{\Omega_o}).
       \end{displaymath}
       %
       çƒå†…å…¨ä½“ãŒå‰›ä½“å›žè»¢ã®è§£ã¨ãªã‚‹. 
       %
       é€†ã«å¤–å´ã®çƒãŒå­˜åœ¨ã—ãªã„æ¥µé™ã§ã¯ 
       $\Dvect{\Omega_o}=0, r_o\longrightarrow\infty$ ã¨ã—ã¦
       %
       \begin{displaymath}
         \phi = 0, \quad
         \psi = \frac{1}{r^3}(\Dvect{r}\cdot\Dvect{\Omega_i}).
       \end{displaymath}

       é€Ÿåº¦å&nbsp;´ã¯ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã®å®šç¾©\Deqref{é€Ÿåº¦åˆ†è§£}ã‚ˆã‚Š
       %
       \begin{displaymath}
         \Dvect{u}=\Drot(\psi\Dvect{r})
                  =\Dgrad\psi\times\Dvect{r} +\psi\Drot\Dvect{r}
                  =\Dgrad\psi\times\Dvect{r}.
       \end{displaymath}
       %
       ã“ã“ã§ 
       \begin{eqnarray*}
         \Dgrad [A(r)(\Dvect{r}\cdot\Dvect{\Omega_i})]\times\Dvect{r}
         &amp;=&amp; (\Dvect{r}\cdot\Dvect{\Omega_i})[\Dgrad A(r)] \times\Dvect{r}
              A(r)[\Dgrad (\Dvect{r}\cdot\Dvect{\Omega_i})]\times\Dvect{r}\\
         &amp;=&amp; A(r)\Dvect{\Omega_i}\times\Dvect{r}
       \end{eqnarray*}
       %
       $A(r)$ ã¯ $r$ ã&nbsp;ã‘ã®é–¢æ•°ãªã®ã§ $\Dgrad A(r)$ ã¯ $\Dvect{r}$ ã¨
       ä¸¦è¡Œã§ã‚ã‚Š, $[\Dgrad A(r)] \times\Dvect{r}=0 $ ã§ã‚ã‚‹ã“ã¨ã«
       æ³¨æ„ã•ã‚ŒãŸã„. 
       ã—ãŸãŒã£ã¦, 
       \begin{displaymath}
         \Dvect{u}
             = -\frac{r_i^3}{r_o^3-r_i^3}\left(1-\frac{r_o^3}{r^3}\right)
                   (\Dvect{\Omega_i}\times\Dvect{r})
               +\frac{r_o^3}{r_o^3-r_i^3}\left(1-\frac{r_i^3}{r^3}\right)
                    (\Dvect{\Omega_o}\times\Dvect{r}).
       \end{displaymath}

%--------------------------------------------------------------------------

\section{å›žè»¢ç³»ã§ã®å®šå¼åŒ–}

  \subsection{æ”¯é…æ–¹ç¨‹å¼}

    ä¸­å¿ƒã‚’å…±æœ‰ã™ã‚‹åŠå¾„ $r_{i}, r_{o}$ ã®çƒé¢ã«æŒŸã¾ã‚ŒãŸ
    çƒæ®»å†…ã«ã‚ã‚‹éžåœ§ç¸®ç²˜æ€§æµä½“ã«ã¤ã„ã¦è€ƒãˆã‚‹.
    åº§æ¨™ç³»ã‚’çƒæ®»ã®ä¸­å¿ƒã‚’åŽŸç‚¹ã—, å¤–å´çƒé¢ã®å›žè»¢è§’é€Ÿåº¦ã®å‘ãã‚’åŒ—æ¥µæ–¹å‘ã¨ã™ã‚‹
    ç·¯åº¦çµŒåº¦åº§æ¨™ç³» $(\lambda, \varphi, r)$ ã¨ã™ã‚‹. 
%    ã“ã‚Œã‚’ \Dfigref{ç³»ã®å›³} ã«ç¤ºã™.
    å¤–å´ã®çƒé¢ã®å›žè»¢è§’é€Ÿåº¦ã‚’ $\Dvect{\Omega_o}=\Omega\Dvect{k}$, 
    ãã‚Œã«ç›¸å¯¾çš„ãªå†…å´çƒé¢ã®å›žè»¢è§’é€Ÿåº¦ãƒ™ã‚¯ãƒˆãƒ«ã‚’
    $\Dvect{\tilde{\Omega}}_i = \Dvect{\Omega_i}-\Dvect{\Omega_o}
      \equiv \tilde{\Omega}_i\Dvect{k}_i$ ã¨ã™ã‚‹. 
    ã“ã“ã§ $\Dvect{k}_i$, $\Dvect{k}$  ã¯ãã‚Œãžã‚Œå†…å¤–çƒé¢ã®
    å›žè»¢è»¸æ–¹å‘ã®å˜ä½ãƒ™ã‚¯ãƒˆãƒ«ã‚’è¡¨ã—ã¦ã„ã‚‹. 
    ä»¥ä¸‹ã§ã¯ãƒãƒ«ãƒ€è¨˜å·ã‚’çœç•¥ã—, ã‚ã‚‰ãŸã‚ã¦
    $\Dvect{\tilde{\Omega}_i} \rightarrow \Dvect{\Omega_i}$
    $\tilde{\Omega}_i \rightarrow \Omega_i$ ã¨å®šç¾©ã—ãªãŠã™ã“ã¨ã«ã™ã‚‹. 

%     \begin{figure}[htbp!]
%      \begin{center}
%       \Depsf[13cm]{./img/cordinate.eps}
%       \caption{è€ƒãˆã¦ã„ã‚‹ç³»ã®æ¨¡å¼å›³} \Dfiglab{ç³»ã®å›³}
%      \end{center}
%     \end{figure}

    æ”¯é…æ–¹ç¨‹å¼ã¯å¤–å´ã®çƒé¢ã¨ã¨ã‚‚ã«ã¾ã‚ã£ã¦ã„ã‚‹å›žè»¢ç³»ã§ã®éžåœ§ç¸®ç²˜æ€§æµä½“ã®
    ãƒŠãƒ“ã‚¨ãƒ¼ã‚¹ãƒˆãƒ¼ã‚¯ã‚¹æ–¹ç¨‹å¼ã§ã‚ã‚‹. 
    \begin{eqnarray} 
     &amp;&amp;\Deqlab{å›žè»¢ç³»é€£ç¶šã®å¼}
      \Ddiv{\Dvect{u}}=0, \\
     &amp;&amp;\Deqlab{å›žè»¢ç³»é‹å‹•æ–¹ç¨‹å¼}
      \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
        + 2\Omega \Dvect{k}\times\Dvect{u}
      = -\frac{1}{\rho}\Dgrad \tilde{p} +\nu \Dlapla \Dvect{u}. 
    \end{eqnarray}

    å„ã€…ã®è¨˜å·ã¯æ¬¡ã®é€šã‚Šã§ã‚ã‚‹:
    \begin{quote}
     \begin{tabular}{ll}
      $\Dvect{u}$  
                     &amp; å›žè»¢ç³»ã§ã®æµé€Ÿãƒ™ã‚¯ãƒˆãƒ«. \\
                     &amp; æˆåˆ†æ¯Žã«æ›¸ãå&nbsp;´åˆã«ã¯,
                     $\Dvect{u}=(u_\lambda,u_\varphi,u_r)$ \\
      $\rho$         &amp; æµä½“ã®å¯†åº¦. éžåœ§ç¸®ã‚’ä»®å®šã—ã¦ã„ã‚‹ã®ã§ä¸€å®šã§ã‚ã‚‹. \\
      $\tilde{p}$       
                     &amp; æµä½“ã®åœ§åŠ›ã¨é&nbsp;å¿ƒåŠ›. \\
                     &amp; é&nbsp;å¿ƒåŠ›ã¯ 
                     $\Dgrad\frac{1}{2}|\Dvect{\Omega}\times\Dvect{r}|^2$
                     ã®å½¢ã§ã‹ã‘ã‚‹ã®ã§ã“ã®åœ§åŠ›é&nbsp;…ã«å«ã‚ã‚‹ã“ã¨ãŒã§ãã‚‹. \\
      $\Dvect{k}$  
                     &amp; å¤–å´çƒé¢ã®å›žè»¢è»¸æ–¹å‘ã®å˜ä½ãƒ™ã‚¯ãƒˆãƒ«
%                       ( \Dfigref{ç³»ã®å›³}ã§ã®$z$ è»¸æ–¹å‘). \\
     \end{tabular}
    \end{quote}  
    ã¾ãŸ, å‹•ç²˜æ€§çŽ‡ $\nu$ ã¯ä¸€å®šã§ã‚ã‚‹ã¨ã™ã‚‹.

    é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ã¯, å¢ƒç•Œã‚’æ¨ªåˆ‡ã‚‹æµã‚ŒãŒå­˜åœ¨ã—ãªã„, ã§ã‚ã‚‹.
    ã™ãªã‚ã¡, 
    \begin{equation}
     u_r = 0, ~~~\mbox{at}~~~ r=r_i, \quad r_o. 
    \end{equation}

    åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ã¨ã—ã¦ã¯,
    å¿œåŠ›ãªã—(stress-free), ã‚‚ã—ãã¯ç²˜ç€æ¡ä»¶(rigid)ã‚’è€ƒãˆã‚‹.

    å¿œåŠ›ãªã—æ¡ä»¶ã®å&nbsp;´åˆã«ã¯
    \begin{equation}
     \DP{}{r}\biggl(\frac{u_\lambda}{r}\biggr) 
     = \DP{}{r}\biggl(\frac{u_\varphi}{r}\biggr) 
      = 0, ~~~\mbox{at}~~~ r=r_i, \quad r_o. 
    \end{equation}

    ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯,
    \begin{equation}
     u_\lambda = u_\varphi = 0,~~~\mbox{at}~~~ r=r_o, \quad, 
     \Dvect{u} = \Dvect{\Omega_i}\times\Dvect{r} ~~~\mbox{at}~~~ r_i.
    \end{equation}
    $\Dvect{r}$ ã¯ä¸­å¿ƒã‚’åŽŸç‚¹ã¨ã™ã‚‹ä½ç½®ãƒ™ã‚¯ãƒˆãƒ«ã§ã‚ã‚‹. 
    æˆåˆ†ã§æ›¸ãä¸‹ã™ã¨
    %
    \begin{eqnarray}
     &amp; &amp;u_\lambda = u_\varphi = 0,~~\mbox{at}~~ r=r_o, \\
     &amp; &amp; u_\lambda =\Omega_i r_i 
                 [ -\cos\varphi_i\sin\varphi\cos(\lambda_i-\lambda)
                + \sin\varphi_i\cos\varphi ], \\
     &amp; &amp; u_\varphi = \Omega_i r_i 
             [ \cos\varphi_i\sin(\lambda-\lambda_i)],
         ~~\mbox{at}~~ r=r_i. 
    \end{eqnarray}
    %
    ã“ã“ã§ $\lambda_i,\varphi_i$ ã¯å†…çƒé¢ã®å›žè»¢è»¸æ–¹å‘ã‚’è¡¨ã™çµŒåº¦ç·¯åº¦ã§ã‚ã‚‹. 

  \subsection{ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°}

    æ…£æ€§ç³»ã®å&nbsp;´åˆã¨åŒæ§˜ã«, é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ã¦ 2 é€šã‚Šã®å&nbsp;´åˆã‚’æ‰±ã†. 

    \subsubsection{çƒæ®»ã®åŽšã•ã§ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã™ã‚‹å&nbsp;´åˆ}

      ç†±å¯¾æµå•é¡Œã¨ã®æ¯”è¼ƒãƒ»å¯¾å¿œã‚’è€ƒå¯Ÿã™ã‚‹å&nbsp;´åˆã«ã¯, 
      é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã‚’çƒæ®»ã®åŽšã•ã§ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã™ã‚‹ã®ãŒã‚ˆã„. 
      ãã®å&nbsp;´åˆã«ã¯å„ã‚¹ã‚±ãƒ¼ãƒ«ã¯,
      %
      \begin{quote}
      \begin{tabular}{ll}
         é•·ã•ã‚¹ã‚±ãƒ¼ãƒ« &amp; çƒæ®»ã®åŽšã• $D = r_{o} - r_{i}$      \\
         é€Ÿåº¦ã‚¹ã‚±ãƒ¼ãƒ« &amp; åˆæœŸå€¤ã®å…¸åž‹çš„ãªé€Ÿåº¦ $U$, ã‚ã‚‹ã„ã¯å¤–åŠ› $\sqrt{D\mathcal{F}}$ \\
         æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ« &amp; $D/U$ ã‚ã‚‹ã„ã¯ $\sqrt{D/\mathcal{F}}$\\
         åœ§åŠ›ã®ã‚¹ã‚±ãƒ¼ãƒ« &amp; $\rho U^2$ ã‚ã‚‹ã„ã¯ $\rho D\mathcal{F}$ 
      \end{tabular}
      \end{quote}
       %
      ãã®çµæžœ, æ–¹ç¨‹å¼ç³»ã¯
      %
      \begin{eqnarray}
       &amp;&amp; \Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒé€£ç¶šã®å¼}
        \Ddiv\Dvect{u} = 0, \\
       &amp;&amp;\Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼}
        \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
          + \frac{1}{R_o}\Dvect{k}\times\Dvect{u}
        = - \Dgrad \pi  + \frac{1}{R_e}\Dlapla \Dvect{u} + \Dvect{\mathcal{F}}
      \end{eqnarray}
      %
      ã¨ãªã‚‹.
      ã“ã“ã§ $\Dvect{u},\pi,\Dvect{\mathcal{F}}$ ã¯
      å„ã€…ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸé€Ÿåº¦, åœ§åŠ›, å¤–åŠ›ã¨ã—ã¦ã‚ã‚‰ãŸã‚ã¦
      å®šç¾©ã—ãªãŠã—ãŸã‚‚ã®ã§ã‚ã‚‹.

      ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã«ã‚ˆã£ã¦ç¾ã‚ŒãŸç„¡æ¬¡å…ƒæ•°ã¯ãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•°ã¨ãƒ­ã‚¹ãƒ“ãƒ¼æ•°, 
      \begin{eqnarray}
         R_e &amp;\equiv&amp; \frac{\nu}{UD} \quad \mbox{or} \quad \frac{\nu}{\mathcal{F}},\\
         R_o &amp;\equiv&amp; \frac{U}{2\Omega D}\quad \mbox{or} \quad \frac{\mathcal{F}}{2\Omega},
      \end{eqnarray}
      ã§ã‚ã‚‹. 
      ã“ã‚Œã«åŠ&nbsp;ãˆã¦ç³»ã«ç¾ã‚Œã‚‹é‡è¦ãªãƒ‘ãƒ©ãƒ¡ãƒ¼ã‚¿ã¨ã—ã¦
      å†…å¾„å¤–å¾„æ¯” $\zeta \equiv r_{i}/r_{o}$ ãŒæš—ã«å­˜åœ¨ã™ã‚‹.

      ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸå¢ƒç•Œæ¡ä»¶ã¯, 
      é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒ
      %
      \begin{equation}
        \Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒé‹å‹•å¢ƒç•Œæ¡ä»¶}
        u_r = 0, 
        \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}.
      \end{equation}
      %
      å¿œåŠ›ãªã—æ¡ä»¶ã¯
      %
      \begin{equation}
        \Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}
         \DP{}{r}\left(\frac{u_\lambda}{r}\right) 
         = \DP{}{r}\left(\frac{u_\varphi}{r}\right) = 0, 
           \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}.
      \end{equation}

      ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã¯
      %
      \begin{eqnarray}
        \Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
         &amp; &amp; u_\lambda =0, \quad u_\varphi =0, 
              \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \\
         &amp; &amp; u_\lambda = \Omega_i \left(\frac{1}{1-\zeta}\right)
                     [ -\cos\varphi_o\sin\varphi\cos(\lambda_o-\lambda)
                    + \sin\varphi_o\cos\varphi ], \\
         &amp; &amp; u_\varphi = \Omega_i \left(\frac{1}{1-\zeta}\right)
                 [ \cos\varphi_o\sin(\lambda-\lambda_o)],
              \quad \mbox{at} \quad r=\frac{1}{1-\zeta}, \\
      \end{eqnarray}
      %
      ã“ã“ã§ $\Omega_i$ ã¯æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ«åˆ†ã® 1 $U/D$ 
      ã‚ã‚‹ã„ã¯ $\mathcal{F}/D$ ã§ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸã‚‚ã®ã§ã‚ã‚‹ã“ã¨ã«
      æ³¨æ„ã•ã‚ŒãŸã„. 

    \subsubsection{å¤–å´çƒæ®»åŠå¾„ã§ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã™ã‚‹å&nbsp;´åˆ}

      ä¸€æ–¹ã§, çƒé¢ 2 æ¬¡å…ƒç³»ã¨ã®å¯¾å¿œã‚’è€ƒå¯Ÿã™ã‚‹å&nbsp;´åˆã«ã¯, 
      é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ã¦å¤–å´çƒæ®»åŠå¾„ã‚’é¸ã¶ã®ãŒåˆ†ã‹ã‚Šã‚„ã™ã„. 
      ãã®å&nbsp;´åˆã«ã¯å„ã‚¹ã‚±ãƒ¼ãƒ«ã¯,
      %
      \begin{quote}
      \begin{tabular}{ll}
         é•·ã•ã‚¹ã‚±ãƒ¼ãƒ« &amp; å¤–å´çƒæ®»ã®åŠå¾„ $r_{o}$      \\
         é€Ÿåº¦ã‚¹ã‚±ãƒ¼ãƒ« &amp; åˆæœŸå€¤ã®å…¸åž‹çš„ãªé€Ÿåº¦ $U$, ã‚ã‚‹ã„ã¯å¤–åŠ› $\sqrt{D\mathcal{F}}$ \\
         æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ« &amp; $r_{o}/U$ ã‚ã‚‹ã„ã¯ $\sqrt{r_{o}/\mathcal{F}}$\\
         åœ§åŠ›ã®ã‚¹ã‚±ãƒ¼ãƒ« &amp; $\rho U^2$ ã‚ã‚‹ã„ã¯ $\rho r_{o}\mathcal{F}$ 
      \end{tabular}
      \end{quote}
       %
      ãã®çµæžœ, æ–¹ç¨‹å¼ç³»ã¯
      %
      \begin{eqnarray}
       &amp;&amp; 
        \Ddiv\Dvect{u} = 0, \\
       &amp;&amp;
        \DP{\Dvect{u}}{t} + (\Dvect{u}\cdot\Dgrad)\Dvect{u} 
          + \frac{1}{R_o^*}\Dvect{k}\times\Dvect{u}
        = - \Dgrad \pi  + \frac{1}{R_e^*}\Dlapla \Dvect{u} + \Dvect{\mathcal{F}}
      \end{eqnarray}
      %
      ã¨ãªã‚‹.
      ã“ã“ã§ $\Dvect{u},\pi,\Dvect{\mathcal{F}}$ ã¯å…ˆã¨åŒæ§˜, 
      å„ã€…ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸé€Ÿåº¦, åœ§åŠ›, å¤–åŠ›ã¨ã—ã¦ã‚ã‚‰ãŸã‚ã¦
      å®šç¾©ã—ãªãŠã—ãŸã‚‚ã®ã§ã‚ã‚‹.

      ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°ã«ã‚ˆã£ã¦ç¾ã‚ŒãŸç„¡æ¬¡å…ƒæ•°ã¯
      é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã®ç•°ãªã‚‹ãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•°ã¨ãƒ­ã‚¹ãƒ“ãƒ¼æ•°, 
      \begin{eqnarray}
         R_e^* &amp;\equiv&amp; \frac{\nu}{Ur_o} \quad 
                \mbox{or} \quad \frac{\nu}{\mathcal{F}},\\
         R_o^* &amp;\equiv&amp; \frac{U}{2\Omega r_o}\quad 
                \mbox{or} \quad \frac{\mathcal{F}}{2\Omega},
      \end{eqnarray}
      ã§ã‚ã‚‹. 
      ã“ã‚Œã«åŠ&nbsp;ãˆã¦ç³»ã«ç¾ã‚Œã‚‹ãƒ‘ãƒ©ãƒ¡ãƒ¼ã‚¿ã¨ã—ã¦
      å†…å¾„å¤–å¾„æ¯” $\zeta \equiv r_i/r_o$ ãŒã‚„ã¯ã‚Šå­˜åœ¨ã™ã‚‹.

      ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸå¢ƒç•Œæ¡ä»¶ã¯, 
      é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒ
      %
      \begin{equation}
        \Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒé‹å‹•å¢ƒç•Œæ¡ä»¶}
        u_r = 0, 
        \quad \mbox{at} \quad r=\zeta,1.
      \end{equation}
      %
      å¿œåŠ›ãªã—æ¡ä»¶ã¯
      %
      \begin{equation}
        \Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}
         \DP{}{r}\left(\frac{u_\lambda}{r}\right) 
         = \DP{}{r}\left(\frac{u_\varphi}{r}\right) = 0, 
           \quad \mbox{at} \quad r=\zeta,1.
      \end{equation}

      ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã¯
      %
      \begin{eqnarray}
        \Deqlab{å›žè»¢ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
         &amp; &amp; u_\lambda =0, \quad u_\varphi =0, 
              \quad \mbox{at} \quad r=\zeta, \\
         &amp; &amp; u_\lambda = \Omega_i \left(1\right)
                     [ -\cos\varphi_o\sin\varphi\cos(\lambda_o-\lambda)
                    + \sin\varphi_o\cos\varphi ], \\
         &amp; &amp; u_\varphi = \Omega_i \left(1\right)
                 [ \cos\varphi_o\sin(\lambda-\lambda_o)],
              \quad \mbox{at} \quad r=1, \\
      \end{eqnarray}
      %
      ã“ã“ã§ $\Omega_i$ ã¯æ™‚é–“ã‚¹ã‚±ãƒ¼ãƒ«åˆ†ã® 1: $U/r_o$ 
      ã‚ã‚‹ã„ã¯ $\mathcal{F}/r_o$ ã§ç„¡æ¬¡å…ƒåŒ–ã•ã‚ŒãŸã‚‚ã®ã§ã‚ã‚‹ã“ã¨ã«
      æ³¨æ„ã•ã‚ŒãŸã„. 

    \subsection{å„ã‚¹ã‚±ãƒ¼ãƒªãƒ³ã‚°æ–¹ç¨‹å¼ç³»ã®å¯¾å¿œé–¢ä¿‚}

      çƒæ®»ã®åŽšã•ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ã¦é¸ã‚“ã&nbsp;å&nbsp;´åˆã¨ã®æ–¹ç¨‹å¼ç³»ã¨æ¯”è¼ƒã™ã‚‹ã¨, 
      ç„¡æ¬¡å…ƒãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•°ã®å®šç¾©ã¨, å¢ƒç•Œã§ã®å‹•å¾„åº§æ¨™ãŒç•°ãªã‚‹ã&nbsp;ã‘ã§ã‚ã‚‹. 
      ãã“ã§ä»¥ä¸‹ã§ã¯çƒæ®»ã®åŽšã•ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ã¦é¸ã‚“ã&nbsp;å&nbsp;´åˆã®
      å®šå¼åŒ–ã®ã¿è¨˜ã™. 
      å¤–å´å¢ƒç•Œã®åŠå¾„ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ãŸå&nbsp;´åˆã®çµæžœã¯, ä»¥ä¸‹ã§ç™»å&nbsp;´ã™ã‚‹
      ãƒ¬ã‚¤ãƒŽãƒ«ã‚ºæ•° $R_e$ ã‚’ $R_e^*$ ã«, 
      ãƒ­ã‚¹ãƒ“ãƒ¼æ•° $R_o$ ã‚’ $R_o^*$ ã«, 
      å¢ƒç•Œã§ã®å‹•å¾„åº§æ¨™
      $r=\zeta/1-\zeta,1/1-\zeta$ ã‚’ $r=\zeta,1$ ã«èª­ã¿ã‹ãˆã‚Œã°è‰¯ã„. 

      å„ç‰©ç†é‡ã®å€¤ã‚’å¤‰æ›ã—ãŸã„å&nbsp;´åˆã«ã¯
      ä¸Šä»˜ã * å°ã‚’å¤–å´å¢ƒç•Œã®åŠå¾„ã‚’é•·ã•ã‚¹ã‚±ãƒ¼ãƒ«ã¨ã—ãŸå&nbsp;´åˆã®å¤‰æ•°ã¨ã—ã¦
      %
      \begin{eqnarray*}
        t^*      &amp;=&amp; (D/r_o)t = (1-\zeta)t, \\
        \Omega^* &amp;=&amp; (r_o/D)\Omega = (1/1-\zeta)\Omega, \\
        R_e^*    &amp;=&amp; (D/r_o)R_e = (1-\zeta)R_e, \\
        R_o^*    &amp;=&amp; (D/r_o)R_o = (1-\zeta)R_o.
      \end{eqnarray*}
      %
      ã¨è¨ˆç®—ã™ã‚Œã°è‰¯ã„. 

  \subsection{ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã«ã‚ˆã‚‹æ–¹ç¨‹å¼ç³»ã®è¡¨ç¾} 

    æ…£æ€§ç³»ã®å&nbsp;´åˆã¨åŒæ§˜, é€Ÿåº¦å&nbsp;´ã‚’ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒ»ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’
    ç”¨ã„ã¦è¡¨ç¾ã—, æ”¯é…æ–¹ç¨‹å¼ã¨å¢ƒç•Œæ¡ä»¶ã‚’æ§‹æˆã™ã‚‹. 

    \subsubsection{æ”¯é…æ–¹ç¨‹å¼}

       é€£ç¶šã®å¼ã®ã‚½ãƒ¬ãƒŽã‚¤ãƒ€ãƒ«æ¡ä»¶\Deqref{å›žè»¢ç³»ç„¡æ¬¡å…ƒé€£ç¶šã®å¼}ã‚ˆã‚Š, 
       é€Ÿåº¦å&nbsp;´ã‚’ãƒˆãƒ­ã‚¤ãƒ€ãƒ«ãƒ»ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‚’ç”¨ã„ã¦
       è¡¨ã™ã“ã¨ãŒã§ãã‚‹:
       %
       \begin{eqnarray}
        \Dvect{u} &amp;=&amp; \Drot(\psi \Dvect{r}) + \Drot\Drot(\phi\Dvect{r}).
       \end{eqnarray}
       %
       ã“ã‚Œã‚‰ã®ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ« $\psi, \phi $ ã‚’ç”¨ã„ã¦æ–¹ç¨‹å¼ç³»ã‚’è¡¨ã™. 
       é‹å‹•æ–¹ç¨‹å¼ \Deqref{å›žè»¢ç³»ç„¡æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼}ã«
       $\Dvect{r}\cdot\Drot$, $\Dvect{r}\cdot\Drot\Drot$ ã‚’ä½œç”¨ã™ã‚‹ã“ã¨ã§,
       \footnotemark[2]
       \begin{eqnarray}
        \Deqlab{å›žè»¢ç³»ãƒˆãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}
            \DP{}{t} (L_2 \psi)
         &amp;=&amp;   \frac{1}{R_o} \DP{\psi}{\lambda}
             - \frac{1}{R_o} Q \phi  
             + \frac{1}{R_e} L_2\Dlapla \psi \nonumber\\
         &amp; &amp; - \Dvect{r}\cdot\Drot [(\Dvect{u}\cdot\Dgrad)\Dvect{u}]
             + \Dvect{r}\cdot\Drot\Dvect{\mathcal{F}}, \\
        \Deqlab{å›žè»¢ç³»ãƒãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•}
           \DP{}{t} (L_2 \Dlapla \phi)
         &amp;=&amp;  \frac{1}{R_o}\DP{}{\lambda}(\Dlapla\phi)
             + \frac{1}{R_o} Q \psi  
             + \frac{1}{R_e} L_2\Dlapla\Dlapla\phi \nonumber\\
         &amp; &amp; + \Dvect{r}\cdot\Drot\Drot [(\Dvect{u}\cdot\Dgrad)\Dvect{u}]
             - \Dvect{r}\cdot\Drot\Drot\Dvect{\mathcal{F}}, 
       \end{eqnarray}
       %
       ãŒå¾—ã‚‰ã‚Œã‚‹. ã‚ã‚‹ã„ã¯, æ’ç­‰å¼
       $ \Dvect{u}\cdot\Dgrad\Dvect{u} 
        = \frac{1}{2}\Dgrad |\Dvect{u}|^2 + (\Drot\Dvect{u})\times\Dvect{u}
       $ ã‚’ç”¨ã„ã¦å¤‰å½¢ã—ãŸå¼, 
       %
       \begin{eqnarray}
        \Deqlab{å›žè»¢ç³»ãƒˆãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•ZxV}
            \DP{}{t} (L_2 \psi)
         &amp;=&amp;   \frac{1}{R_o} \DP{\psi}{\lambda}
             - \frac{1}{R_o} Q \phi  
             + \frac{1}{R_e} L_2\Dlapla \psi \nonumber\\
         &amp; &amp; - \Dvect{r}\cdot\Drot [(\Drot\Dvect{u})\times\Dvect{u}]
             + \Dvect{r}\cdot\Drot\Dvect{\mathcal{F}}, \\
        \Deqlab{å›žè»¢ç³»ãƒãƒ­ã‚¤ãƒ€ãƒ«é‹å‹•ZxV}
           \DP{}{t} (L_2 \Dlapla \phi)
         &amp;=&amp;  \frac{1}{R_o}\DP{}{\lambda}(\Dlapla\phi)
             + \frac{1}{R_o} Q \psi  
             + \frac{1}{R_e} L_2\Dlapla\Dlapla\phi \nonumber\\
         &amp; &amp; + \Dvect{r}\cdot\Drot\Drot [(\Drot\Dvect{u})\times\Dvect{u}]
             - \Dvect{r}\cdot\Drot\Drot\Dvect{\mathcal{F}}, 
       \end{eqnarray}
       %
       ãŒå¾—ã‚‰ã‚Œã‚‹. 
       ã“ã“ã§å¼ä¸­ã®æ¼”ç®—å­ $L_2$ ã¯
       åŠå¾„ 1 ã®çƒé¢ä¸Šã® 2 æ¬¡å…ƒãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³ã®é€†ç¬¦å·ã®ã‚‚ã®ã§ã‚ã‚Š, 
       æ…£æ€§ç³»ã§å®šç¾©ã—ãŸã‚‚ã®ã¨åŒã˜ã§ã‚ã‚‹. 
       æ¼”ç®—å­ $Q$ ã¯ 
       \begin{equation}
         Q  \equiv \Dvect{k}\cdot\Dgrad 
          - \Dinv{2}(    L_2 \Dvect{k}\cdot\Dgrad 
          + \Dvect{k}\cdot\Dgrad L_2   )
       \end{equation}
       ã§å®šç¾©ã•ã‚Œã‚‹æ¼”ç®—å­ã§ã‚ã‚Š, å…·ä½“çš„ã«ã¯, 
       %
       \begin{eqnarray}
         \Dvect{k}\cdot\Dgrad
             &amp;=&amp; \sin\varphi\DP{}{r}+\frac{\cos\varphi}{r}\DP{}{\varphi},\\
         Q   &amp;=&amp; \sin\varphi\DP{}{r}+\frac{\cos\varphi}{r}\DP{}{\varphi}\\
             &amp; &amp; - \Dinv{2}
                     \left[ 
                         L_2 \left(  \sin\varphi\DP{}{r}
                                    +\frac{\cos\varphi}{r}\DP{}{\varphi}\right)
                       + \left(  \sin\varphi\DP{}{r}
                                +\frac{\cos\varphi}{r}\DP{}{\varphi}\right)L_2
                   \right].
       \end{eqnarray}

    \subsubsection{é€Ÿåº¦å&nbsp;´ã¨ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã¨ã®é–¢ä¿‚å¼}

      ãƒˆãƒ­ã‚¤ãƒ€ãƒ«, ãƒãƒ­ã‚¤ãƒ€ãƒ«ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã‹ã‚‰
      é€Ÿåº¦å&nbsp;´ã‚’æ±‚ã‚ã‚‹å¼, ãŠã‚ˆã³ãã®é€†ã®å¼ã¯æ…£æ€§ç³»ã®ã‚‚ã®ã¨åŒã˜ã§ã‚ã‚‹:
      \begin{eqnarray}
         u_\lambda  &amp; = &amp; \DP[]{\psi}{\varphi} 
               + \Dinv{r\cos\varphi}
                    \DP[]{}{r}\left(r\DP[]{\phi}{\lambda}\right),\\
         u_\varphi &amp; = &amp;  - \Dinv{\cos\varphi}\DP[]{\psi}{\lambda}
               + \Dinv{r}
                    \DP[]{}{r}\left(r\DP[]{\phi}{\varphi} \right),\\
         u_r  &amp; = &amp; \frac{L_2 ~\phi}{r}. 
      \end{eqnarray}
      %
      \begin{equation}
            \phi = L_2^{-1}(ru_r), \quad
            \psi = L_2^{-1}[\Dvect{r}\cdot(\Drot\Dvect{u})]. 
      \end{equation}

    \subsubsection{å¢ƒç•Œæ¡ä»¶}

       é‹å‹•å­¦çš„å¢ƒç•Œæ¡ä»¶\Deqref{å›žè»¢ç³»ç„¡æ¬¡å…ƒé‹å‹•å¢ƒç•Œæ¡ä»¶}ã‚’
       ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã™ã‚Œã°,
       \begin{equation}
         \phi = 0, 
           \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},\frac{1}{1-\zeta}. 
       \end{equation}

       åŠ›å­¦çš„å¢ƒç•Œæ¡ä»¶ãŒå¿œåŠ›ãªã—ã®å&nbsp;´åˆã¯\Deqref{å›žè»¢ç³»ç„¡æ¬¡å…ƒå¿œåŠ›ãªã—æ¡ä»¶}ã‚’
       ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã—ã¦
       \begin{equation}
         \DP[2]{\phi}{r} = \DP{}{r}\left(\frac{\psi}{r}\right) = 0, 
         \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \frac{1}{1-\zeta}. 
       \end{equation}

       ç²˜ç€æ¡ä»¶ã®å&nbsp;´åˆã«ã¯, 
       \Deqref{å›žè»¢ç³»ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}ã‚’ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ã§è¡¨ç¾ã—ã¦
       %
       \begin{eqnarray*}
         \Deqlab{å›žè»¢ç³»ãƒãƒ†ãƒ³ã‚·ãƒ£ãƒ«ç„¡æ¬¡å…ƒç²˜ç€æ¡ä»¶}
            &amp; &amp;\DP{\phi}{r} = 0, \quad
               \psi = (\Dvect{r}\cdot\Dvect{\Omega_i})
               \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \\
            &amp; &amp; \DP{\phi}{r} = 0, \quad
               \psi = 0
               \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta},
       \end{eqnarray*}
       %
       æˆåˆ†ã§è¡¨ã›ã°
       %
       \begin{eqnarray}
         &amp; &amp; \DP{\phi}{r} = 0, \\
         &amp; &amp; \psi = \Omega_i \left(\frac{\zeta}{1-\zeta}\right)
                     [ \cos\varphi_i\cos\varphi\cos(\lambda_i-\lambda)
                      + \sin\varphi_i\sin\varphi ]
             \quad \mbox{at} \quad r=\frac{\zeta}{1-\zeta}, \\
         &amp; &amp; \DP{\phi}{r} = 0, \\
         &amp; &amp; \psi = 0
             \quad \mbox{at} \quad \frac{1}{1-\zeta}. 
       \end{eqnarray}
       %
       \footnotetext[4]{
         æ…£æ€§ç³»ã®å&nbsp;´åˆã¨åŒæ§˜, 
         $\psi(r=\zeta/(1-\zeta)$ ã‚’çƒé¢èª¿å’Œå‡½æ•°ã®å’Œã§
         \begin{displaymath}
           \psi(\lambda,\varphi,r=\zeta/(1-\zeta))=
            \sum_{n=1}^\infty\left[
                 a_n^0P_n^0(\sin\varphi) + 
                 \sum_{m=1}^n
                   P_n^m(\sin\varphi)[a_n^m\sqrt{2}\cos(m\lambda)
                                      -b_n^m\sqrt{2}\sin(m\lambda)]
              \right]
         \end{displaymath}
         %
         ã¨å±•é–‹ã™ã‚‹ã¨, $P_1^0(\sin\varphi)=\sin\varphi$,
         $P_1^1(\sin\varphi)=\cos\varphi$ ã§ã‚ã‚‹ã‹ã‚‰
         %
         \begin{eqnarray*}
           a_1^0 &amp;=&amp; \Omega_i\left(\frac{\zeta}{1-\zeta}\right)
                     \sin\varphi_i,\\
           a_1^1 &amp;=&amp; \Omega_i\left(\frac{\zeta}{1-\zeta}\right)
                       \frac{1}{\sqrt{2}}\cos\varphi_i\cos\lambda_i,\\
           b_1^1 &amp;=&amp; -\Omega_i\left(\frac{\zeta}{1-\zeta}\right)
                       \frac{1}{\sqrt{2}}\cos\varphi_i\sin\lambda_i
         \end{eqnarray*}
         %
         ã¨å±•é–‹ä¿‚æ•°ãŒä¸Žãˆã‚‰ã‚Œã‚‹. 
         }

  \subsection{è§£æžçš„ãªè§£ã®ä¾‹}

    \subsubsection{å¿œåŠ›ãªã—æ¡ä»¶ä¸‹ã§ã®å‰›ä½“å›žè»¢è§£}



%\include{horizon}
%\include{vertical}
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% å‚è€ƒæ–‡çŒ®
\markright{å‚è€ƒæ–‡çŒ®}
\thispagestyle{Dheadings}

\section*{å‚è€ƒæ–‡çŒ®}
\addcontentsline{toc}{section}{å‚è€ƒæ–‡çŒ®}
\begin{description}
%
\item Chandrasekhar,S.,1961: 
            {\em Hydrodynamic and Hydromagnetic Stability}. 
            Oxford University Press.
%
\item Zhang,K. and Busse,F.H.,1987: 
            On the onset of convection in rotating spherical shells. 
            {\em Geophys. Astrophys. Fluid Dyn.},  {\bf 39}, 119--147.
\item Landau, L. D., Lifshitz, E. M., 1987 : 
         Fluid Mechanics (Volume 6 of Course of theoretical physics), 
         2nd edition. Pergamon press, 539pp.

\end{description}
\end{document}

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