はじめに

境界層内における伝熱が気になった．そこで基礎的な話から復習する．熱的な話に入る前に，まずは流速について考えてみる．高速流に発展させていくため圧縮性流体として議論を進める．

例題：境界層における速度場

基礎方程式

下準備として，基礎方程式並びに必要な関係式を列記する．圧縮性流体であることに注意．

連続の式: $$ \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} = 0 $$

ナビエストークス方程式(境界層方程式): $$ \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y} = - \frac{\rm{d} \it{p_e}}{\rm{d} \it{x}} + \frac{\partial }{\partial y} \left( \mu \frac{\partial u}{\partial y}\right) $$ $$ \frac{\partial p}{\partial y} ＝ 0 $$

座標変換の準備

速度場とエンタルピー場に関して，相似形のプロファイルを見出したいというモチベーションがある．これはそもそも， 「相似形のプロファイルが出現するには相似的な法則があると推測でき，そこには何らかの物理現象物理法則が存在する(そして物理学がさらに発展し人類に貢献できる)．」という物理学的な性質？によるものだと解釈している． そこで，以下の座標変換(イリングワース変換：Illingworth transformation)を適応する．

座標系の変換: $$ \xi = \int^x_0{\rho_e u_e \mu_e} \rm{d}\it{x} $$ $$ \eta = \frac{u_e}{\sqrt{2\xi}}\int^y_0{\rho} \rm{d}\it{y} $$

次に，チェーンルールに基づいた微分におけるの変形の下準備をしておくと，

微分の準備: $$ \frac{\partial \xi}{\partial x} = \rho_e u_e \mu_e $$ $$ \frac{\partial \xi}{\partial y} = 0 $$ $$ \frac{\partial \eta}{\partial x} = ??? (計算過程でキャンセルアウトされる) $$ $$ \frac{\partial \eta}{\partial y} = \frac{u_e \rho}{\sqrt{2\xi}} $$

$x$-方向: $$ \frac{\partial}{\partial x} = \left(\frac{\partial}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial x}\right) + \left(\frac{\partial}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial x}\right) = \left(\frac{\partial}{\partial \xi}\right) \rho_e u_e \mu_e + \left(\frac{\partial}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial x}\right) $$ $$ \therefore \frac{\partial}{\partial x} = \rho_e u_e \mu_e \frac{\partial}{\partial \xi} + \left(\frac{\partial \eta}{\partial x}\right) \frac{\partial}{\partial \eta} $$

$y$-方向: $$ \frac{\partial}{\partial y} = \left(\frac{\partial}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial y}\right) + \left(\frac{\partial}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial y}\right) = \left(\frac{\partial}{\partial \xi}\right) 0 + \left(\frac{\partial}{\partial \eta}\right) \frac{u_e \rho}{\sqrt{2\xi}} $$ $$ \therefore \frac{\partial}{\partial y} = \frac{u_e \rho}{\sqrt{2\xi}} \frac{\partial}{\partial \eta} $$ となる．

流れ関数による表記

流れ関数: $$ \frac{\partial \varPsi}{\partial y} = \rho u $$ $$ \frac{\partial \varPsi}{\partial x} = - \rho v $$

境界層方程式： $$ \frac{\partial \varPsi}{\partial y} \frac{\partial u}{\partial x} - \frac{\partial \varPsi}{\partial x} \frac{\partial u}{\partial y} = - \frac{\rm{d} \it{p_e}}{\rm{d} \it{x}} + \frac{\partial }{\partial y} \left( \mu \frac{\partial u}{\partial y}\right) $$

ここで$\xi - \eta$ 座標系に置き換えて， $$ \left[ \frac{u_e \rho}{\sqrt{2\xi}} \frac{\partial \varPsi}{\partial \eta} \right] \left[ \rho_e u_e \mu_e \frac{\partial u}{\partial \xi} + \left(\frac{\partial \eta}{\partial x} \right) \frac{\partial u}{\partial \eta} \right] + \left[ \rho_e u_e \mu_e \frac{\partial \varPsi}{\partial \xi} + \left(\frac{\partial \eta}{\partial x} \right) \frac{\partial \varPsi}{\partial \eta} \right] \left[ \frac{u_e \rho}{\sqrt{2 \xi}} \frac{\partial u}{\partial \eta} \right] = - \rho_e u_e \mu_e \frac{\rm{d} \it{p_e}}{\rm{d} \it{\xi}} + \frac{u_e \rho}{\sqrt{2 \xi}} \frac{\partial }{\partial \eta} \left( \frac{u_e \rho \mu}{\sqrt{2 \xi}} \frac{\partial u}{\partial \eta}\right) $$

$\sqrt{2 \xi} / u_e \rho$ を辺々にかけて， $$ \frac{\partial \varPsi}{\partial \eta} \left[ \rho_e u_e \mu_e \frac{\partial u}{\partial \xi} + \left(\frac{\partial \eta}{\partial x} \right) \frac{\partial u}{\partial \eta} \right] + \left[ \rho_e u_e \mu_e \frac{\partial \varPsi}{\partial \xi} + \left(\frac{\partial \eta}{\partial x} \right) \frac{\partial \varPsi}{\partial \eta} \right] \frac{\partial u}{\partial \eta} = - \sqrt{2 \xi} \frac{\rho_e}{\rho} \mu_e \frac{\rm{d} \it{p_e}}{\rm{d} \it{\xi}} + \frac{\partial }{\partial \eta} \left( \frac{u_e \rho \mu}{\sqrt{2 \xi}} \frac{\partial u}{\partial \eta}\right) $$ 以上より，$\xi - \eta$ 座標系における境界層方程式が得られた(多少残った$x-y$座標の項は後述で処理)．

プロファイル$f$の導入

$u$のプロファイルを $$ \frac{u}{u_e} = \frac{\partial f}{\partial \eta} = f' $$ と定義する．注意：微分記号 $'$ は $\eta$ の微分である．

$\xi - \eta$ 座標での微分を行うと， $$ \frac{\partial u}{\partial \xi} = \frac{\partial (f' u_e)}{\partial \xi} = f' \frac{\partial u_e}{\partial \xi} + u_e \frac{\partial (f')}{\partial \xi} = f' \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + u_e \frac{\partial (f')}{\partial \xi} (\because u_e = u_e(\xi)) $$ $$ \frac{\partial u}{\partial \eta} = \frac{\partial }{\partial \eta} \left( u_e \frac{\partial f}{\partial \eta} \right) = u_e \frac{\partial^2 f}{\partial {\eta}^2} = u_e f'' $$

$$ \left( \frac{\partial \varPsi}{\partial y} = \right) \frac{u_e \rho}{\sqrt{2\xi}} \frac{\partial \varPsi}{\partial \eta} = \rho u = \rho f' u_e $$ $$ \therefore \frac{\partial \varPsi}{\partial \eta} = \sqrt{2\xi} f' $$

$\eta$で積分して， $$ \varPsi = \sqrt{2\xi} f + F(\xi) $$ ただし，$F(\xi)$任意関数であるが，流れ関数の一般的な特性(壁面で流体の流出入なし）から，$\varPsi$に関して，$\varPsi(\xi,\eta=0) = 0$となる．上記の方程式において，$\eta=0$の任意の点において，$\varPsi=0$を満たすには， $f=0$かつ$F(\xi)=0$となる必要がある．ゆえに， $F(\xi)=0$として，元の方程式は， $$ \varPsi = \sqrt{2\xi} f $$

$$ \frac{\partial \varPsi}{\partial \xi} = \frac{\partial (\sqrt{2\xi} f)}{\partial \xi} = \sqrt{2\xi} \frac{\partial f}{\partial \xi} + \frac{1}{\sqrt{2\xi}} f $$ となる．

プロファイル$f$による表記

境界層方程式を再掲． $$ \color{red}{ \frac{\partial \varPsi}{\partial \eta} } \left[ \rho_e u_e \mu_e \color{red}{ \frac{\partial u}{\partial \xi} } + \left(\frac{\partial \eta}{\partial x} \right) \color{red}{ \frac{\partial u}{\partial \eta} } \right] + \left[ \rho_e u_e \mu_e \color{red}{ \frac{\partial \varPsi}{\partial \xi} } + \left(\frac{\partial \eta}{\partial x} \right) \color{red}{ \frac{\partial \varPsi}{\partial \eta} } \right] \color{red}{ \frac{\partial u}{\partial \eta} } = - \sqrt{2 \xi} \frac{\rho_e}{\rho} \mu_e \frac{\rm{d} \it{p_e}}{\rm{d} \it{\xi}} + \frac{\partial }{\partial \eta} \left( \frac{u_e \rho \mu}{\sqrt{2 \xi}} \color{red}{ \frac{\partial u}{\partial \eta} } \right) $$

上記の赤字に代入して， $$ \sqrt{2\xi} f' \left[ \rho_e u_e \mu_e \left( f' \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + u_e \frac{\partial (f')}{\partial \xi} \right) + \left(\frac{\partial \eta}{\partial x} \right) u_e f'' \right] - \left[ \rho_e u_e \mu_e \left( \sqrt{2\xi} \frac{\partial f}{\partial \xi} + \frac{1}{\sqrt{2\xi}} f \right) + \left(\frac{\partial \eta}{\partial x} \right) \sqrt{2\xi} f' \right] u_e f'' \\ = - \sqrt{2 \xi} \frac{\rho_e}{\rho} \mu_e \frac{\rm{d} \it{p_e}}{\rm{d} \it{\xi}} + \frac{\partial }{\partial \eta} \left( \frac{u_e^2 \rho \mu}{\sqrt{2 \xi}} f'' \right) $$

ここで，境界層縁での非粘性流に関してオイラー方程式より， $$ \rm{d} \it{p_e} = - \rho_e u_e \rm{d} \it{u_e} $$ であり，これを代入して，さらに辺々 $\sqrt{2 \xi} \rho_e u_e^2 \mu_e$ で割ると， $$ \frac{1}{u_e} f' \left[ \left( f' \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + u_e \frac{\partial (f')}{\partial \xi} \right) + \cancel{ \left(\frac{\partial \eta}{\partial x} \right) f'' } \right] - \left[ \left( \frac{\partial f}{\partial \xi} + \frac{1}{2\xi} f \right) + \cancel{ \frac{1}{u_e} \left(\frac{\partial \eta}{\partial x} \right) f' } \right] f'' = - \frac{\rho_e}{\rho} \frac{1}{u_e} \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + \frac{\partial }{\partial \eta} \left( \frac{1}{2 \xi} \frac{\rho \mu}{\rho_e \mu_e} f'' \right) $$

整理して， $$ \frac{1}{u_e} \left( f' \right)^2 \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + f' \frac{\partial (f')}{\partial \xi} - \frac{\partial f}{\partial \xi} f'' + \frac{1}{2\xi} f f'' = - \frac{\rho_e}{\rho} \frac{1}{u_e} \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + \frac{\partial }{\partial \eta} \left( \frac{1}{2 \xi} \frac{\rho \mu}{\rho_e \mu_e} f'' \right) $$

下記のような色分けで項を区別する． $$ \color{green}{\frac{1}{u_e} \left( f' \right)^2 \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}}} + \color{pink}{f' \frac{\partial (f')}{\partial \xi}} - \color{pink}{\frac{\partial f}{\partial \xi} f''} + \color{blue}{\frac{1}{2\xi} f f''} = - \color{green}{\frac{\rho_e}{\rho} \frac{1}{u_e} \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}}} + \color{red}{\frac{\partial }{\partial \eta} \left( \frac{1}{2 \xi} \frac{\rho \mu}{\rho_e \mu_e} f'' \right)} $$

移行して， $$ \frac{\partial }{\partial \eta} \left( \frac{1}{2 \xi} \frac{\rho \mu}{\rho_e \mu_e} f'' \right) + \frac{1}{2\xi} f f'' = \left[ \frac{1}{u_e} \left( f' \right)^2 - \frac{\rho_e}{\rho} \frac{1}{u_e} \right]\frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + f' \frac{\partial (f')}{\partial \xi} - \frac{\partial f}{\partial \xi} f'' $$

辺々に，$2\xi$ を掛けて，$C \equiv \frac{\rho \mu}{\rho_e \mu_e}$とすると， $$ \frac{\partial }{\partial \eta} \left( C f'' \right) + f f'' = \frac{2\xi}{u_e} \left[ \left( f' \right)^2 - \frac{\rho_e}{\rho} \right]\frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + 2\xi \left( f' \frac{\partial (f')}{\partial \xi} - \frac{\partial f}{\partial \xi} f'' \right) $$ ゆえに， $$ \left( C f'' \right)' + f f'' = \frac{2\xi}{u_e} \left[ \left( f' \right)^2 - \frac{\rho_e}{\rho} \right]\frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} + 2\xi \left( f' \frac{\partial (f')}{\partial \xi} - \frac{\partial f}{\partial \xi} f'' \right) $$ 以上より，速度$u$のプロファイル$f$を導入した境界層方程式が得られた．

本題：温度場とエンタルピー

基礎方程式

エネルギー保存則: $$ \rho u \frac{\partial h}{\partial x} + \rho v \frac{\partial h}{\partial y} = \frac{\partial}{\partial y} \left( k \frac{\partial T}{\partial y} \right) + u \frac{\rm{d} \it{p_e}}{\rm{d} \it{x}} + \mu \left(\frac{\partial u}{\partial y}\right)^2 $$

理想気体: $$ p = \rho R T $$

エンタルピー: $$ h = c_p T $$

プロファイル$g$の導入

$h$のプロファイルを $$ \frac{h}{h_e} = g(\xi.\eta)= g $$ と定義する．

微分の準備: $$ \frac{\partial \xi}{\partial x} = \rho_e u_e \mu_e $$ $$ \frac{\partial \xi}{\partial y} = 0 $$ $$ \frac{\partial \eta}{\partial x} = ??? (計算過程でキャンセルアウトされる) $$ $$ \frac{\partial \eta}{\partial y} = \frac{u_e \rho}{\sqrt{2\xi}} $$

$\xi - \eta$ 座標での微分を行うと， $$ \frac{\partial h}{\partial \xi} = \frac{\partial (g' h_e)}{\partial \xi} = g \frac{\partial h_e}{\partial \xi} + h_e \frac{\partial g}{\partial \xi} = g \frac{\rm{d} \it{h_e}}{\rm{d} \it{\xi}} + h_e \frac{\partial g}{\partial \xi} (\because h_e = h_e(\xi)) $$ $$ \frac{\partial h}{\partial \eta} = \frac{\partial }{\partial \eta} \left( h_e g \right) = h_e \frac{\partial g}{\partial \eta} = h_e g' $$

$x$-方向: $$ \frac{\partial h}{\partial x} \\ = \left(\frac{\partial h}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial x}\right) + \left(\frac{\partial h}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial x}\right) \\ = \left(\frac{\partial h}{\partial \xi}\right) \rho_e u_e \mu_e + \left(\frac{\partial h}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial x}\right) \\ = \left(g \frac{\rm{d} \it{h_e}}{\rm{d} \it{\xi}} + h_e \frac{\partial g}{\partial \xi} \right) \rho_e u_e \mu_e + \left( h_e g' \right) \left( \frac{\partial \eta}{\partial x} \right) $$

$y$-方向: $$ \frac{\partial h}{\partial y} \\ = \left(\frac{\partial h}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial y}\right) + \left(\frac{\partial h}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial y}\right) \\ = \left(\frac{\partial h}{\partial \xi}\right) \left( 0 \right) + \left(\frac{\partial h}{\partial \eta}\right) \left( \frac{u_e \rho}{\sqrt{2\xi}} \right) \\ = \left( h_e g' \right) \frac{u_e \rho}{\sqrt{2\xi}} $$ となる．

流れ関数: $$ \rho u = \frac{\partial \varPsi}{\partial y} \\ = \left(\frac{\partial \varPsi}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial y}\right) + \left(\frac{\partial \varPsi}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial y}\right) \\ = \left( \sqrt{2\xi} f' \right) \left( 0 \right) + \left( \sqrt{2\xi} \frac{\partial f}{\partial \xi} + \frac{1}{\sqrt{2\xi}} f \right) \left( \frac{u_e \rho}{\sqrt{2\xi}} \right) \\ = \left( u_e \rho \frac{\partial f}{\partial \xi} + \frac{u_e \rho}{2\xi} f \right) $$ $$ \rho v = - \frac{\partial \varPsi}{\partial x} \\ = - \left(\frac{\partial \varPsi}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial x}\right) - \left(\frac{\partial \varPsi}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial x}\right) \\ = - \left( \sqrt{2\xi} f' \right) \left( \rho_e u_e \mu_e \right) - \left( \sqrt{2\xi} \frac{\partial f}{\partial \xi} + \frac{1}{\sqrt{2\xi}} f \right) \left(\frac{\partial \eta}{\partial x}\right) \\ = - \sqrt{2\xi} \left( \rho_e u_e \mu_e f' + \frac{\partial f}{\partial \xi} \frac{\partial \eta}{\partial x} + \frac{1}{2\xi} f \frac{\partial \eta}{\partial x} \right) $$

オイラー方程式より， $$ \rm{d} \it{p_e} = - \rho_e u_e \rm{d} \it{u_e} $$

$y$-方向: $$ \frac{\partial u}{\partial y} \\ = \left(\frac{\partial u}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial y}\right) + \left(\frac{\partial u}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial y}\right) \\ = \left(\frac{\partial u}{\partial \xi}\right) \left( 0 \right) + \left(\frac{\partial u}{\partial \eta}\right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = \frac{u_e \rho}{\sqrt{2\xi}} \frac{\partial u_e f'}{\partial \eta} \\ = \frac{u_e \rho}{\sqrt{2\xi}} u_e f'' $$ となる

熱伝導の候に関して: $$ \frac{\partial T}{\partial y} \\ = \left(\frac{\partial T}{\partial \xi}\right) \left(\frac{\partial \xi}{\partial y}\right) + \left(\frac{\partial T}{\partial \eta}\right) \left(\frac{\partial \eta}{\partial y}\right) \\ = \left(\frac{\partial T}{\partial \xi}\right) \left( 0 \right) + \left(\frac{\partial T}{\partial \eta}\right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = \left(\frac{\partial T}{\partial \eta}\right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = \left(\frac{\partial (h/c_p)}{\partial \eta}\right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = \left(\frac{\partial h}{\partial \eta}\right) \frac{u_e \rho}{c_p \sqrt{2\xi}} \\ = \left(\frac{\partial (h_e g)}{\partial \eta}\right) \frac{u_e \rho}{c_p \sqrt{2\xi}} \\ = \left( \left(\frac{\partial h_e}{\partial \eta}\right) g + h_e \left(\frac{\partial g}{\partial \eta}\right) \right) \frac{u_e \rho}{c_p \sqrt{2\xi}} \\ = \left( \frac{\partial h_e}{\partial \eta} g + h_e g' \right) \frac{u_e \rho}{c_p \sqrt{2\xi}} $$ $$ \frac{\partial }{\partial y} \left( \frac{\partial T}{\partial y} \right) \\ = \frac{\partial}{\partial \eta} \left( \frac{\partial T}{\partial y} \right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = \frac{\partial}{\partial \eta} \left( \left( \frac{\partial h_e}{\partial \eta} g + h_e g' \right) \frac{u_e \rho}{c_p \sqrt{2\xi}} \right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = \frac{\partial}{\partial \eta} \left( \left( \frac{\partial h_e}{\partial \eta} g + h_e g' \right) \frac{u_e \rho}{c_p \sqrt{2\xi}} \right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = \left( \frac{\partial^2 h_e}{\partial \eta^2} g + \frac{\partial h_e}{\partial \eta} \frac{\partial g}{\partial \eta} + \frac{\partial h_e}{\partial \eta} \frac{\partial g}{\partial \eta} + \frac{\partial g'}{\partial \eta} \right) \frac{u_e^2 \rho^2}{c_p 2\xi} \\ = \left( \frac{\partial^2 h_e}{\partial \eta^2} g + 2 \frac{\partial h_e}{\partial \eta} g' + g'' \right) \frac{u_e^2 \rho^2}{c_p 2\xi} $$

エネルギー保存則: $$ \rho u \frac{\partial h}{\partial x} + \rho v \frac{\partial h}{\partial y} = \frac{\partial}{\partial y} \left( k \frac{\partial T}{\partial y} \right) + u \frac{\rm{d} \it{p_e}}{\rm{d} \it{x}} + \mu \left(\frac{\partial u}{\partial y}\right)^2 $$ 上述の変数を代入して， $$ \left[ \left( u_e \rho \frac{\partial f}{\partial \xi} + \frac{u_e \rho}{2\xi} f \right) \right] \left[ \left(g \frac{\rm{d} \it{h_e}}{\rm{d} \it{\xi}} + h_e \frac{\partial g}{\partial \xi} \right) \rho_e u_e \mu_e + \left( h_e g' \right) \left( \frac{\partial \eta}{\partial x} \right) \right] + \left[ - \sqrt{2\xi} \left( \rho_e u_e \mu_e f' + \frac{\partial f}{\partial \xi} \frac{\partial \eta}{\partial x} + \frac{1}{2\xi} f \frac{\partial \eta}{\partial x} \right) \right] \left[ \left( h_e g' \right) \frac{u_e \rho}{\sqrt{2\xi}} \right] \\ = \left[ k \left( \frac{\partial^2 h_e}{\partial \eta^2} g + 2 \frac{\partial h_e}{\partial \eta} g' + g'' \right) \frac{u_e^2 \rho^2}{c_p 2\xi} \right] + \left[ u_e f' \left( - \rho_e u_e \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} \right) \right] + \left[ \mu \left( \frac{u_e \rho}{\sqrt{2\xi}} u_e f'' \right)^2 \right] $$ ここでじっくり見ると$\frac{\partial \eta}{\partial x}$の項はキャンセルアウトされる． $$ \left[ \left( u_e \rho \frac{\partial f}{\partial \xi} + \frac{u_e \rho}{2\xi} f \right) \right] \left[ \left(g \frac{\rm{d} \it{h_e}}{\rm{d} \it{\xi}} + h_e \frac{\partial g}{\partial \xi} \right) \rho_e u_e \mu_e + \cancel{ \left( h_e g' \right) \left( \frac{\partial \eta}{\partial x} \right) } \right] + \left[ - \sqrt{2\xi} \left( \rho_e u_e \mu_e f' + \cancel{ \frac{\partial f}{\partial \xi} \frac{\partial \eta}{\partial x} + \frac{1}{2\xi} f \frac{\partial \eta}{\partial x} } \right) \right] \left[ \left( h_e g' \right) \frac{u_e \rho}{\sqrt{2\xi}} \right] \\ = \left[ k \left( \frac{\partial^2 h_e}{\partial \eta^2} g + 2 \frac{\partial h_e}{\partial \eta} g' + g'' \right) \frac{u_e^2 \rho^2}{c_p 2\xi} \right] + \left[ u_e f' \left( - \rho_e u_e \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} \right) \right] + \left[ \mu \left( \frac{u_e \rho}{\sqrt{2\xi}} u_e f'' \right)^2 \right] $$ ゆえに整理して， $$ \left( u_e \rho \frac{\partial f}{\partial \xi} + \frac{u_e \rho}{2\xi} f \right) \left(g \frac{\rm{d} \it{h_e}}{\rm{d} \it{\xi}} + h_e \frac{\partial g}{\partial \xi} \right) \rho_e u_e \mu_e - \sqrt{2\xi} \left( \rho_e u_e \mu_e f' \right) \left( h_e g' \right) \frac{u_e \rho}{\sqrt{2\xi}} \\ = k \left( \frac{\partial^2 h_e}{\partial \eta^2} g + 2 \frac{\partial h_e}{\partial \eta} g' + g'' \right) \frac{u_e^2 \rho^2}{c_p 2\xi} + u_e f' \left( - \rho_e u_e \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} \right) + \mu \left( \frac{u_e \rho}{\sqrt{2\xi}} u_e f'' \right)^2 $$ $C \equiv \rho \mu / \rho_e \mu_e$であり，さらに辺々 ${2 \xi} / {\rho \rho_e u_e^2 \mu_e h_e}$ を掛けると， $$ \left[ \left( u_e \rho \frac{\partial f}{\partial \xi} + \frac{u_e \rho}{2\xi} f \right) \right] \left[ \left(g \frac{\rm{d} \it{h_e}}{\rm{d} \it{\xi}} + h_e \frac{\partial g}{\partial \xi} \right) \rho_e u_e \mu_e + \left( h_e g' \right) \left( \frac{\partial \eta}{\partial x} \right) \right] + \left[ - \sqrt{2\xi} \left( \rho_e u_e \mu_e f' + \frac{\partial f}{\partial \xi} \frac{\partial \eta}{\partial x} + \frac{1}{2\xi} f \frac{\partial \eta}{\partial x} \right) \right] \left[ \left( h_e g' \right) \frac{u_e \rho}{\sqrt{2\xi}} \right] \\ = \left[ k \left( \frac{\partial^2 h_e}{\partial \eta^2} g + 2 \frac{\partial h_e}{\partial \eta} g' + g'' \right) \frac{u_e^2 \rho^2}{c_p 2\xi} \right] + \left[ u_e f' \left( - \rho_e u_e \frac{\rm{d} \it{u_e}}{\rm{d} \it{\xi}} \right) \right] + \left[ C \frac{u_e^2}{h_e} \left(f'' \right)^2 \right] $$

参考図書

John D. Anderson Jr., "Hypersonic and high-temperature gas dynamics", AIAA, 3rd Ed.(2019)
神元五郎，"機械工学大系10 高速流動"，コロナ社，1976
近藤次郎，"高速空気力学"，コロナ社，1977
日野幹雄，"理工学基礎講座16 流体力学"，朝倉書店，1974
日野幹雄，"流体力学"，朝倉書店，1992
巽友正，"新物理学シリーズ21 流体力学"，培風館，1982