%#! platex master_thesis-M2.tex
\chapter{$B<!852r@O$K$h$k%(%M%k%.!<;60oGH?t$H(B\\$B%(%s%9%H%m%U%#!<;60oGH?t$NF3(B
$B=P(B} \label{chap:dim_analysis}

$B;~4V$H?eJ?J}8~$ND9$5$NE57?E*$J%9%1!<%k$r$=$l$>$l(B $T,L$ $B$H$9$k$H(B, $BG4@-$K(B
$B$h$k%(%M%k%.!<;60oN((B $\varepsilon$ $B$H%(%s%9%H%m%U%#!<;60oN((B $\eta$ $B$O$=(B
$B$l$>$l(B 
\[
 \varepsilon \sim \frac{L^2}{T^3}, \, \eta \sim \frac{1}{T^3}
\]
$B$H%9%1!<%k$5$l$k(B. $B$^$?(B, $BG4@-78?t$O12EYJ}Dx<0(B $\partial q / \partial
t=\cdots-\nu(\bnabla^2)^4 q$ $B$h$j(B
\[
 \nu \sim \frac{L^8}{T}
\]
$B$H%9%1!<%k$5$l$k(B. $B$3$3$G(B, $\varepsilon$ $B$H(B $\nu$ $B$rMQ$$$F(B $L$ $B$rI=$9$H(B,
\[
 \frac{\nu}{\varepsilon^{1/3}}=L^{8-\frac{2}{3}}=L^{\frac{22}{3}}
 \, \Leftrightarrow \, L=\left(\frac{\nu^{3}}{\varepsilon}\right)^{1/22}
\]
$B$H$J$k(B. $B%(%M%k%.!<;60oGH?t(B $k_{\mathrm{diss}}^{\mathcal{E}}$ $B$N<!85$O(B,
$k_{\mathrm{diss}}^{\mathcal{E}}\sim 1/L$ $B$G$"$k$+$i(B,
\begin{align}
 k_{\mathrm{diss}}^{\mathcal{E}} \sim \left(\frac{\varepsilon}{\nu^3}\right)^{1/22}
\end{align}
$B$H$J$k(B. $BF1MM$K(B, $\eta$ $B$H(B $\nu$ $B$G(B $L$ $B$rI=$9$H(B,
\[
 \frac{\nu}{\eta^{1/3}}=L^{8} \, \Leftrightarrow \,
 L=\left(\frac{\nu}{\eta^{1/3}}\right)^{1/8} 
\]
$B$H$J$k(B. $B$h$C$F(B, $B%(%s%9%H%m%U%#!<;60oGH?t(B
$k_{\mathrm{diss}}^{\mathcal{Q}}$ $B$O(B, 
\begin{align}
 k_{\mathrm{diss}}^{\mathcal{Q}} \sim
 \left(\frac{\eta^{1/3}}{\nu}\right)^{1/8} = \left(\frac{\eta}{\nu^3}\right)^{1/24}
\end{align}
$B$H$J$k(B.

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