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Reduction of Order

Starting from the FSC ODEs:

\[(Cf'')' + ff'' + \beta_H(\tau - f'^2) = 0\]
\[(Cg')' + fg' = 0\]
\[\begin{aligned} &\left(\frac{C}{\mathrm{Pr}}\tau'\right)' + (S-1)\left(C(f'^2)'\right)' + (K-1)S\left(C(g^2)'\right)' \\ &\quad + f\!\left[\tau' + (S-1)(f'^2)' + (K-1)S(g^2)'\right] = 0 \end{aligned}\]

where \(C = \rho\mu/(\rho_e\mu_e)\), \(f' = u/u_e\), \(g = w/w_e\), \(\tau = T/T_e\), and all primes denote differentiation with respect to \(\eta\).

Chapman–Rubesin factor

For a perfect gas \(\rho/\rho_e = 1/\tau\), so the Chapman–Rubesin factor simplifies to:

\[C = \frac{\rho\mu}{\rho_e\mu_e} = \frac{\mu}{\mu_e\tau}\]

Differentiating with respect to \(\eta\) via the quotient rule:

\[C' = \frac{d}{d\eta}\!\left(\frac{\mu}{\mu_e\tau}\right) = \frac{1}{\mu_e}\frac{\mu'\tau - \mu\tau'}{\tau^2}\]

and therefore:

\[\frac{C'}{C} = \frac{\mu'}{\mu} - \frac{\tau'}{\tau}\]

Expanding the compound derivatives

Applying the product rule to each term:

\[(Cf'')' = C'f'' + Cf'''\]
\[(Cg')' = C'g' + Cg''\]
\[\left(\frac{C}{\mathrm{Pr}}\tau'\right)' = \frac{C'}{\mathrm{Pr}}\tau' + \frac{C}{\mathrm{Pr}}\tau''\]
\[\left(C(f'^2)'\right)' = \left(2Cf'f''\right)' = 2\!\left(C'f'f'' + Cf''^2 + Cf'f'''\right)\]
\[\left(C(g^2)'\right)' = \left(2Cgg'\right)' = 2\!\left(C'gg' + Cg'^2 + Cgg''\right)\]

Substituting into the three ODEs:

x-momentum:

\[C'f'' + Cf''' + ff'' + \beta_H(\tau - f'^2) = 0\]

z-momentum:

\[C'g' + Cg'' + fg' = 0\]

Energy:

\[\frac{C'}{\mathrm{Pr}}\tau' + \frac{C}{\mathrm{Pr}}\tau'' + 2(S-1)\!\left(C'f'f'' + Cf''^2 + Cf'f'''\right) + 2(K-1)S\!\left(C'gg' + Cg'^2 + Cgg''\right) + f\!\left[\tau' + 2(S-1)f'f'' + 2(K-1)Sgg'\right] = 0\]

Isolating the highest derivatives

x-momentum → \(f'''\)

Divide by \(C\) and rearrange, using \(C'/C = \mu'/\mu - \tau'/\tau\):

\[\boxed{f''' = \frac{\tau'}{\tau}\,f'' - \frac{\mu'}{\mu}\,f'' - \frac{ff''}{C} - \frac{\beta_H(\tau - f'^2)}{C}}\]

z-momentum → \(g''\)

Divide by \(C\) and rearrange, using \(C'/C = \mu'/\mu - \tau'/\tau\):

\[\boxed{g'' = \frac{\tau'}{\tau}\,g' - \frac{\mu'}{\mu}\,g' - \frac{fg'}{C}}\]

Energy → \(\tau''\)

Solve for \(\tau''\), multiply by \(\mathrm{Pr}/C\), and substitute \(C'/C = \mu'/\mu - \tau'/\tau\):

\[\boxed{\begin{aligned} \tau'' &= \frac{\tau'^2}{\tau} - \frac{\mu'\tau'}{\mu} - \frac{\mathrm{Pr}\,f\tau'}{C} \\ &\quad - 2\mathrm{Pr}(S-1)\!\left[\left(\frac{\mu'}{\mu} - \frac{\tau'}{\tau}\right)f'f'' + f''^2 + f'f'''\right] \\ &\quad - 2\mathrm{Pr}(K-1)S\!\left[\left(\frac{\mu'}{\mu} - \frac{\tau'}{\tau}\right)gg' + g'^2 + gg''\right] \\ &\quad - \frac{2\mathrm{Pr}\,f}{C}\!\left[(S-1)f'f'' + (K-1)Sgg'\right] \end{aligned}}\]

Viscosity derivatives

The \(\mu'/\mu\) term requires the derivative of viscosity with respect to \(\eta\). By the chain rule:

\[\mu' = \frac{d\mu}{d\eta} = \frac{d\mu}{dT}\frac{dT}{d\eta}\]

Since \(T = T_e\tau(\eta)\):

\[\frac{dT}{d\eta} = T_e\frac{d\tau}{d\eta} = T_e\tau'\]

Therefore:

\[\mu' = \frac{d\mu}{dT}\,T_e\tau'\]

Substituting \(\mu'/\mu = (T_e/\mu)(d\mu/dT)\,\tau'\) into the intermediate expressions gives the final forms:

\[f''' = \frac{\tau'}{\tau}\,f'' - \frac{T_e}{\mu}\frac{d\mu}{dT}\,\tau'f'' - \frac{ff''}{C} - \frac{\beta_H(\tau - f'^2)}{C}\]
\[g'' = \frac{\tau'}{\tau}\,g' - \frac{T_e}{\mu}\frac{d\mu}{dT}\,\tau'g' - \frac{fg'}{C}\]
\[\begin{aligned} \tau'' &= \frac{\tau'^2}{\tau} - \frac{T_e}{\mu}\frac{d\mu}{dT}\,\tau'^2 - \frac{\mathrm{Pr}\,f\tau'}{C} \\ &\quad - 2\mathrm{Pr}(S-1)\!\left[\left(\frac{T_e}{\mu}\frac{d\mu}{dT} - \frac{1}{\tau}\right)\tau'f'f'' + f''^2 + f'f'''\right] \\ &\quad - 2\mathrm{Pr}(K-1)S\!\left[\left(\frac{T_e}{\mu}\frac{d\mu}{dT} - \frac{1}{\tau}\right)\tau'gg' + g'^2 + gg''\right] \\ &\quad - \frac{2\mathrm{Pr}\,f}{C}\!\left[(S-1)f'f'' + (K-1)Sgg'\right] \end{aligned}\]

where \(\mu = \mu(T_e\tau)\) and \(C = \mu/(\mu_e\tau)\) are evaluated locally at each \(\eta\), and \(f'''\) and \(g''\) are computed first.

State variables

The third-order equation in \(f\), the second-order equation in \(g\), and the second-order equation in \(\tau\) yield a seventh-order system. Define:

\[y_1 = f, \quad y_2 = f', \quad y_3 = f'', \quad y_4 = \tau, \quad y_5 = \tau', \quad y_6 = g, \quad y_7 = g'\]

First-order ODE system

\[y_1' = y_2\]
\[y_2' = y_3\]
\[y_3' = \frac{y_5}{y_4}\,y_3 - \frac{T_e}{\mu}\frac{d\mu}{dT}\,y_5 y_3 - \frac{y_1 y_3 + \beta_H(y_4 - y_2^2)}{C}\]
\[y_4' = y_5\]
\[\begin{aligned} y_5' &= \frac{y_5^2}{y_4} - \frac{T_e}{\mu}\frac{d\mu}{dT}\,y_5^2 - \frac{\mathrm{Pr}\,y_1 y_5}{C} \\ &\quad - 2\mathrm{Pr}(S-1)\!\left[\left(\frac{T_e}{\mu}\frac{d\mu}{dT} - \frac{1}{y_4}\right)y_5 y_2 y_3 + y_3^2 + y_2 y_3'\right] \\ &\quad - 2\mathrm{Pr}(K-1)S\!\left[\left(\frac{T_e}{\mu}\frac{d\mu}{dT} - \frac{1}{y_4}\right)y_5 y_6 y_7 + y_7^2 + y_6 y_7'\right] \\ &\quad - \frac{2\mathrm{Pr}\,y_1}{C}\!\left[(S-1)y_2 y_3 + (K-1)S\,y_6 y_7\right] \end{aligned}\]
\[y_6' = y_7\]
\[y_7' = \frac{y_5}{y_4}\,y_7 - \frac{T_e}{\mu}\frac{d\mu}{dT}\,y_5 y_7 - \frac{y_1 y_7}{C}\]

where \(C = \mu(T_e y_4)/(\mu_e y_4)\), \(T = T_e y_4\), and \(y_3'\), \(y_7'\) are computed before \(y_5'\).

Boundary conditions

Wall (\(\eta = 0\)):

\[y_1 = 0, \qquad y_2 = 0, \qquad y_6 = 0\]
  • Isothermal: \(y_4 = T_w/T_e\)
  • Adiabatic: \(y_5 = 0\)

Edge (\(\eta \to \infty\)):

\[y_2 = 1, \qquad y_4 = 1, \qquad y_6 = 1\]

The unknown wall values \(y_3(0) = f''(0)\), \(y_7(0) = g'(0)\), and (for isothermal walls) \(y_5(0) = \tau'(0)\) are determined by shooting to match the edge conditions.