Derivation (Illingworth-Stewartson)
Note
This derivation uses the Illingworth-Stewartson transformation following Derivation (Levy-Lees) .
Starting from the quasi-3D compressible BL equations :
\[
\frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} = 0
\]
\[
\rho\!\left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}\right)
= -\frac{\partial p}{\partial x} + \frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}\right)
\]
\[
\rho\!\left(u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y}\right)
= \frac{\partial}{\partial y}\!\left(\mu\frac{\partial w}{\partial y}\right)
\]
\[
\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= u\frac{\partial p}{\partial x}
+ \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \mu\!\left[\left(\frac{\partial u}{\partial y}\right)^{\!2}
+ \left(\frac{\partial w}{\partial y}\right)^{\!2}\right]
\]
The z-momentum equation carries no pressure-gradient term (dropped under the quasi-3D
assumption \(\partial p/\partial z = 0\) ), and the energy equation carries an extra
crossflow dissipation term \(\mu(\partial w/\partial y)^2\) relative to the 2D case.
Similarity Ansatz
Edge velocity power law (chordwise component):
\[u_e(x) = C x^m\]
The spanwise edge velocity \(w_e\) is constant along the chord (Cooke's independence
principle: the spanwise momentum equation decouples and \(w_e\) is a prescribed constant).
Dimensionless stream function and normalized dependent variables
\[\psi = \sqrt{2\xi}\,f(\eta), \qquad
f'(\eta) = \frac{u}{u_e}, \qquad
g(\eta) = \frac{w}{w_e}, \qquad
\tau(\eta) = \frac{T}{T_e}\]
Outer flow
The outer inviscid flow satisfies the Euler x-momentum equation (see
quasi-3D BL equations ):
\[-\frac{dp}{dx} = \rho_e u_e \frac{du_e}{dx}\]
For the power-law edge velocity \(u_e = Cx^m\) :
\[
\frac{du_e}{dx} = \frac{d}{dx}\left(Cx^{m}\right)
= m C x^{m-1} = \frac{m}{x} \underbrace{C x^m}_{u_e} = \frac{m u_e}{x}
\]
so the pressure gradient becomes:
\[-\frac{dp}{dx} = \rho_e u_e^2\frac{m}{x}\]
Definitions
Transformed coordinates (IS physical-to-transformed map). The IS transformation references freestream constants \(\rho_\infty\) , \(\mu_\infty\) , so \(\xi\) requires no assumption on the streamwise variation of edge quantities:
\[\bar{x} = \int_0^x \frac{\rho_e \mu_e}{\rho_\infty \mu_\infty}\,dx', \qquad
\bar{y} = \frac{1}{\rho_\infty}\int_0^y \rho\,dy'\]
In the transformed space \((\bar{x}, \bar{y})\) , the continuity equation takes the
incompressible form (unit density). The stream function \(\psi\) is therefore defined
without a density weighting:
\[\frac{\partial\psi}{\partial \bar{y}} = u, \qquad
\frac{\partial\psi}{\partial \bar{x}} = -\bar{v}\]
The similarity coordinates follow from \(\bar{x}\) and \(\bar{y}\) :
\[\xi = \int_0^{\bar{x}} u_e\,d\bar{x}' = \int_0^x \frac{\rho_e \mu_e u_e}{\rho_\infty\mu_\infty}\,dx',
\qquad
\eta = \frac{u_e}{\sqrt{2\xi}}\,\bar{y} = \frac{u_e}{\rho_\infty\sqrt{2\xi}}\int_0^y\rho\,dy'\]
Hartree parameter :
\[\beta_H = \frac{2m}{m+1}\]
Chapman-Rubesin factor (IS, referenced to freestream):
\[C = \frac{\rho\mu}{\rho_\infty\mu_\infty}\]
For a uniform freestream (\(\rho_e = \rho_\infty\) , \(\mu_e = \mu_\infty\) ) this coincides with
the edge-referenced form \(C = \rho\mu/(\rho_e\mu_e)\) used in the
ODE system and reduction of order .
Temperature ratio :
\[\tau = \frac{T}{T_e}\]
Ideal gas law :
From the ideal gas law \(p = \rho R T\) , the density is \(\rho = p/(RT)\) , and at the boundary
layer edge \(\rho_e = p_e/(RT_e)\) . From the
y-momentum boundary layer equation
\[
\frac{\partial p}{\partial y} = 0 \rightarrow p = p_e
\]
Therefore:
\[
\frac{\rho}{\rho_e}
= \frac{p/(RT)}{p_e/(RT_e)}
= \frac{T_e}{T}
= \frac{1}{\tau}
\]
Sweep decomposition and edge Mach number :
The total edge velocity \(Q_e\) splits into chordwise and spanwise components through the
local swept angle \(\Lambda\) :
\[u_e = Q_e\cos\Lambda, \qquad w_e = Q_e\sin\Lambda, \qquad
M_e = \frac{Q_e}{\sqrt{\gamma R T_e}}\]
With \(c_p = \gamma R/(\gamma-1)\) it follows that
\[
\frac{u_e^2}{c_p T_e} = (\gamma-1)M_e^2\cos^2\!\Lambda,
\qquad
\frac{w_e^2}{c_p T_e} = (\gamma-1)M_e^2\sin^2\!\Lambda
\]
Compressibility parameters :
\[
K = \frac{1 + \dfrac{\gamma-1}{2}M_e^2}{1 + \dfrac{\gamma-1}{2}M_e^2\cos^2\!\Lambda},
\qquad
S = 1 + \frac{\gamma-1}{2}M_e^2\cos^2\!\Lambda
\]
where \(\Lambda\) is the local swept angle and \(M_e\) the local edge Mach number. These
combine with the relations above into the compact identities used throughout the energy
assembly:
\[
2(S-1) = (\gamma-1)M_e^2\cos^2\!\Lambda = \frac{u_e^2}{c_p T_e},
\qquad
2(K-1)S = (\gamma-1)M_e^2\sin^2\!\Lambda = \frac{w_e^2}{c_p T_e}
\]
Deviation from Liu (2021)
Liu \(K\) and \(S\) in terms of a reference Mach number \(Ma_{e,\mathrm{ref}}\) and a
streamwise parameter \(\chi = (\tilde{\xi}/\tilde{\xi}_\mathrm{ref})^m\) that tracks the variation
of edge conditions along the surface. Here we adopt a locally self-similar formulation, which
sets \(\chi = 1\) and consequently \(Ma_{e,\mathrm{ref}} = M_e\) . The parameters \(\beta_H\) , \(K\) , \(S\) ,
and \(T_e\) are then known inputs evaluated from the local edge conditions at each station.
Prandtl number :
\[\mathrm{Pr} = \frac{\mu c_p}{k}\]
so that \(k = \mu c_p/\mathrm{Pr}\) .
Partial derivatives of the similarity coordinates
By the Leibniz integral rule:
\[\frac{\partial\xi}{\partial x} = \frac{\rho_e\mu_e u_e}{\rho_\infty\mu_\infty}, \qquad
\frac{\partial\eta}{\partial y} = \frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\]
For the power-law \(u_e = Cx^m\) with \(\rho_e\mu_e = \text{const}\) (uniform freestream, \(\rho_e = \rho_\infty\) , \(\mu_e = \mu_\infty\) ):
\[
\xi = \frac{\rho_\infty\mu_\infty}{\rho_\infty\mu_\infty}\int_0^x Cx'^m\,dx'
= \frac{\overbrace{Cx^m}^{u_e}\,x}{m+1}
= \frac{\rho_\infty\mu_\infty u_e x}{m+1}
\]
so that \(m/x = \beta_H\rho_\infty\mu_\infty u_e/(2\xi)\) . The \(\partial\eta/\partial x\) term cancels in the convective and energy terms.
\[\frac{\partial F}{\partial y}\bigg|_x
= \frac{\partial F}{\partial \xi}\bigg|_\eta \underbrace{\frac{\partial \xi}{\partial y}\bigg|_x}_{=\,0}
+ \frac{\partial F}{\partial \eta}\bigg|_\xi \frac{\partial \eta}{\partial y}\bigg|_x
= \frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\frac{\partial F}{\partial \eta}\bigg|_\xi\]
\[\frac{\partial F}{\partial x}\bigg|_y
= \frac{\partial F}{\partial \xi}\bigg|_\eta \frac{\partial \xi}{\partial x}\bigg|_y
+ \frac{\partial F}{\partial \eta}\bigg|_\xi \frac{\partial \eta}{\partial x}\bigg|_y\]
Streamwise velocity
From \(\partial\psi/\partial\bar{y} = u\) it follows that \(u = u_e f'(\eta)\) . The wall-normal derivative:
\[
\frac{\partial \overbrace{u}^{u_e f'(\eta)}}{\partial y}
= \frac{\partial \left(u_e f'(\eta)\right)}{\partial\eta} \overbrace{\frac{\partial\eta}{\partial y}}^{\frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}}
= \frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\frac{\partial(u_e f'(\eta))}{\partial\eta}
= \frac{\rho u_e^2}{\rho_\infty\sqrt{2\xi}}\,f''(\eta)
\]
The streamwise derivative:
\[
\frac{\partial \overbrace{u}^{u_e f'(\eta)}}{\partial x}\bigg|_y
= \overbrace{\frac{du_e}{dx}}^{mu_e/x} f'(\eta)
+ u_e\,f''(\eta)\,\frac{\partial\eta}{\partial x}\bigg|_y
= \frac{m u_e}{x}\,f'(\eta) + u_e\,f''(\eta)\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Continuity
Satisfied identically by construction of \(\psi\) .
x-momentum
Using the expressions derived above:
\(\rho u\frac{\partial u}{\partial x}\) term.
\[\rho u\frac{\partial u}{\partial x} = \rho u_e f' \left(\frac{m u_e}{x}f' + u_e f'' \frac{\partial\eta}{\partial x}\right)
= \frac{\rho u_e^2}{x}\!\left(m f'^2 + x f'' f'\frac{\partial\eta}{\partial x}\right)\]
\(\rho v \frac{\partial u}{\partial y}\) term.\(\partial\psi/\partial\bar{x} = -\bar{v}\) , applying the product rule to \(\psi = \sqrt{2\xi}\,f(\eta)\) with \(\rho_e = \rho_\infty\) , \(\mu_e = \mu_\infty\) :
\[
\rho v = -\frac{\rho_\infty\mu_\infty u_e}{\sqrt{2\xi}}\,f - \sqrt{2\xi}\,\rho_\infty\,f'\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Multiplying by \(\partial u/\partial y = \rho u_e^2 f''/(\rho_\infty\sqrt{2\xi})\) :
\[
\rho v\,\frac{\partial u}{\partial y}
= \left(-\frac{\rho_\infty\mu_\infty u_e}{\sqrt{2\xi}}\,f - \sqrt{2\xi}\,\rho_\infty\,f'\,\frac{\partial\eta}{\partial x}\bigg|_y\right)
\frac{\rho u_e^2 f''}{\rho_\infty\sqrt{2\xi}}
= -\frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,ff''
- \rho u_e^2\,f'f''\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Combined convective term. Adding both terms, the \(\partial\eta/\partial x\) pieces cancel:
\[
\rho u\,\frac{\partial u}{\partial x} + \rho v\,\frac{\partial u}{\partial y}
= \frac{m\rho u_e^2}{x}\,f'^2
+ \cancel{\rho u_e^2\,f'f''\,\frac{\partial\eta}{\partial x}}
- \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,ff''
- \cancel{\rho u_e^2\,f'f''\,\frac{\partial\eta}{\partial x}}
= \frac{m\rho u_e^2}{x}\,f'^2 - \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,ff''
\]
Recast \(\frac{m}{x}\) using \(\xi = \rho_\infty\mu_\infty u_e x/(m+1)\) :
\[
\frac{m}{\underbrace{x}_{\frac{\xi(m+1)}{\rho_\infty\mu_\infty u_e}}}
= \frac{m\rho_\infty\mu_\infty u_e}{\xi(m+1)}
= \overbrace{\frac{m}{(m+1)}}^{\beta_H/2} \frac{\rho_\infty\mu_\infty u_e}{\xi}
= \frac{\beta_H\rho_\infty\mu_\infty u_e}{2\xi}
\]
Substituting:
\[
\rho u\,\frac{\partial u}{\partial x} + \rho v\,\frac{\partial u}{\partial y}
= \overbrace{\frac{m}{x}}^{\frac{\beta_H\rho_\infty\mu_\infty u_e}{2\xi}} \rho u_e^2 f'^2
- \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,ff''
= \frac{\beta_H\rho\rho_\infty\mu_\infty u_e^3}{2\xi} f'^2 - \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,ff''
\]
\[\boxed{
\rho u\,\frac{\partial u}{\partial x} + \rho v\,\frac{\partial u}{\partial y}
= \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\!\left(\beta_H f'^2 - ff''\right)
}\]
Pressure term.
\[
-\frac{dp}{dx}
= \rho_e u_e^2\frac{m}{x}
= \rho_e u_e^2 \frac{\beta_H \rho_\infty\mu_\infty u_e}{2\xi}
= \overbrace{\rho_e}^{\rho\tau} \frac{\rho_\infty\mu_\infty u_e^3}{2\xi}\beta_H
= \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,\tau\beta_H
\]
\[\boxed{-\frac{dp}{dx} = \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,\tau\beta_H}\]
Viscous term.
\[
\mu\frac{\partial \overbrace{u}^{u_e f'}}{\partial y}
= \overbrace{\mu}^{C\rho_\infty\mu_\infty/\rho} \cdot \overbrace{\frac{\partial u}{\partial y}}^{\rho u_e^2 f''/(\rho_\infty\sqrt{2\xi})}
= \frac{C\mu_\infty u_e^2}{\sqrt{2\xi}}\,f''
\]
Applying the outer \(\partial/\partial y = (\rho u_e/(\rho_\infty\sqrt{2\xi}))\,\partial/\partial\eta\) :
\[
\frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}\right)
= \overbrace{\frac{\partial}{\partial y}}^{\frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\partial/\partial\eta}
\left(\overbrace{\mu\frac{\partial u}{\partial y}}^{C\mu_\infty u_e^2 f''/\sqrt{2\xi}}\right)
= \frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\cdot\frac{\rho_\infty\mu_\infty u_e^2}{\sqrt{2\xi}}\,(Cf'')'
\]
\[\boxed{\frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y}\right)
= \frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}\,(Cf'')'}\]
Assembly.
The x-momentum equation becomes:
\[
\cancel{\frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}}\!\left(\beta_H f'^2 - ff''\right)
= \cancel{\frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}}\beta_H\tau
+ \cancel{\frac{\rho\rho_\infty\mu_\infty u_e^3}{2\xi}}\,(Cf'')'
\]
Dividing through by \(\rho\rho_\infty\mu_\infty u_e^3/(2\xi)\) and rearranging:
\[(Cf'')' + ff'' + \beta_H(\tau - f'^2) = 0\]
Compare to Falkner-Skan
The x-momentum equation is identical to the
Falkner-Skan x-momentum —
the crossflow \(w\) does not appear.
z-momentum
The crossflow momentum equation has no pressure-gradient term:
\[
\rho\!\left(u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y}\right)
= \frac{\partial}{\partial y}\!\left(\mu\frac{\partial w}{\partial y}\right)
\]
With \(w = w_e\,g(\eta)\) and \(w_e = \text{const}\) , the crossflow is a function of \(\eta\) only,
so there is no \(m u_e/x\) contribution from the streamwise derivative:
\[
\frac{\partial \overbrace{w}^{w_e g(\eta)}}{\partial y}
= w_e\,g'(\eta)\,\overbrace{\frac{\partial\eta}{\partial y}}^{\frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}}
= \frac{\rho u_e w_e}{\rho_\infty\sqrt{2\xi}}\,g'(\eta),
\qquad
\frac{\partial w}{\partial x}\bigg|_y
= w_e\,g'(\eta)\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
\(\rho u\frac{\partial w}{\partial x}\) term.
\[
\rho u\frac{\partial w}{\partial x}
= \rho u_e f'\cdot w_e\,g'\,\frac{\partial\eta}{\partial x}
= \rho u_e w_e\,f'g'\,\frac{\partial\eta}{\partial x}
\]
\(\rho v\frac{\partial w}{\partial y}\) term.\(\rho v\) from the x-momentum section and multiplying by \(\partial w/\partial y = \rho u_e w_e g'/(\rho_\infty\sqrt{2\xi})\) :
\[
\rho v\,\frac{\partial w}{\partial y}
= \left(-\frac{\rho_\infty\mu_\infty u_e}{\sqrt{2\xi}}\,f - \sqrt{2\xi}\,\rho_\infty\,f'\,\frac{\partial\eta}{\partial x}\bigg|_y\right)
\frac{\rho u_e w_e\,g'}{\rho_\infty\sqrt{2\xi}}
= -\frac{\rho\rho_\infty\mu_\infty u_e^2 w_e}{2\xi}\,fg'
- \rho u_e w_e\,f'g'\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Combined convective term. Adding both terms, the \(\partial\eta/\partial x\) pieces cancel exactly as in the x-momentum case:
\[
\rho u\,\frac{\partial w}{\partial x} + \rho v\,\frac{\partial w}{\partial y}
= \cancel{\rho u_e w_e\,f'g'\,\frac{\partial\eta}{\partial x}}
- \frac{\rho\rho_\infty\mu_\infty u_e^2 w_e}{2\xi}\,fg'
- \cancel{\rho u_e w_e\,f'g'\,\frac{\partial\eta}{\partial x}}
\]
\[\boxed{
\rho u\,\frac{\partial w}{\partial x} + \rho v\,\frac{\partial w}{\partial y}
= -\frac{\rho\rho_\infty\mu_\infty u_e^2 w_e}{2\xi}\,fg'
}\]
Viscous term.
\[
\mu\frac{\partial \overbrace{w}^{w_e g}}{\partial y}
= \overbrace{\mu}^{C\rho_\infty\mu_\infty/\rho} \cdot \overbrace{\frac{\partial w}{\partial y}}^{\rho u_e w_e g'/(\rho_\infty\sqrt{2\xi})}
= \frac{C\mu_\infty u_e w_e}{\sqrt{2\xi}}\,g'
\]
Applying the outer \(\partial/\partial y = (\rho u_e/(\rho_\infty\sqrt{2\xi}))\,\partial/\partial\eta\) :
\[
\frac{\partial}{\partial y}\!\left(\mu\frac{\partial w}{\partial y}\right)
= \overbrace{\frac{\partial}{\partial y}}^{\frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\partial/\partial\eta}
\left(\overbrace{\mu\frac{\partial w}{\partial y}}^{C\mu_\infty u_e w_e g'/\sqrt{2\xi}}\right)
= \frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\cdot\frac{\rho_\infty\mu_\infty u_e w_e}{\sqrt{2\xi}}\,(Cg')'
\]
\[\boxed{\frac{\partial}{\partial y}\!\left(\mu\frac{\partial w}{\partial y}\right)
= \frac{\rho\rho_\infty\mu_\infty u_e^2 w_e}{2\xi}\,(Cg')'}\]
Assembly. Every term carries the common factor \(\rho\rho_\infty\mu_\infty u_e^2 w_e/(2\xi)\) :
\[
-\cancel{\frac{\rho\rho_\infty\mu_\infty u_e^2 w_e}{2\xi}}\,fg'
= \cancel{\frac{\rho\rho_\infty\mu_\infty u_e^2 w_e}{2\xi}}\,(Cg')'
\]
Dividing through by \(\rho\rho_\infty\mu_\infty u_e^2 w_e/(2\xi)\) and rearranging:
Energy
The energy equation carries an extra crossflow dissipation term:
\[
\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= u\frac{\partial p}{\partial x}
+ \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \mu\!\left[\left(\frac{\partial u}{\partial y}\right)^{\!2}
+ \left(\frac{\partial w}{\partial y}\right)^{\!2}\right]
\]
Note \(T = T_e\,\tau(\eta) \rightarrow\) function of \(\eta\) only
\[
\frac{\partial \overbrace{T}^{T_e\,\tau(\eta)}}{\partial y}
= T_e\,\overbrace{\frac{\partial\tau}{\partial\eta}}^{\tau'}\,
\overbrace{\frac{\partial\eta}{\partial y}}^{\rho u_e/(\rho_\infty\sqrt{2\xi})}
= \frac{\rho u_e T_e}{\rho_\infty\sqrt{2\xi}}\,\tau'
\]
\[
\frac{\partial \overbrace{T}^{T_e\,\tau(\eta)}}{\partial x}\bigg|_y
= T_e\,\overbrace{\frac{\partial\tau}{\partial\eta}}^{\tau'}\,\frac{\partial\eta}{\partial x}\bigg|_y
= T_e\,\tau'\,\frac{\partial\eta}{\partial x}\bigg|_y
\]
Convective term. Streamwise and wall-normal contributions combined (the \(\partial\eta/\partial x\) cross terms cancel):
\[
\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= \cancel{\rho c_p u_e T_e f'\tau'\frac{\partial\eta}{\partial x}}
- \frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\,f\tau'
- \cancel{\rho c_p u_e T_e f'\tau'\frac{\partial\eta}{\partial x}}
\]
\[\boxed{\rho c_p\!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y}\right)
= -\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\,f\tau'}\]
Diffusion term.
\[
k\frac{\partial T}{\partial y}
= \overbrace{k}^{C\rho_\infty\mu_\infty c_p/(\rho\,\mathrm{Pr})} \cdot
\overbrace{\frac{\partial T}{\partial y}}^{\rho u_e T_e\tau'/(\rho_\infty\sqrt{2\xi})}
= \frac{C\mu_\infty u_e c_p T_e}{\mathrm{Pr}\sqrt{2\xi}}\,\tau'
\]
Applying the outer \(\partial/\partial y = (\rho u_e/(\rho_\infty\sqrt{2\xi}))\,\partial/\partial\eta\) :
\[
\frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
= \overbrace{\frac{\partial}{\partial y}}^{\frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\partial/\partial\eta}
\left(\overbrace{k\frac{\partial T}{\partial y}}^{C\mu_\infty u_e c_p T_e\tau'/(\mathrm{Pr}\sqrt{2\xi})}\right)
= \frac{\rho u_e}{\rho_\infty\sqrt{2\xi}}\cdot\frac{\rho_\infty\mu_\infty u_e c_p T_e}{\mathrm{Pr}\sqrt{2\xi}}\,\left(\frac{C}{\mathrm{Pr}}\tau'\right)'
\]
\[\boxed{\frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
= \frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\left(\frac{C}{\mathrm{Pr}}\tau'\right)'}\]
Pressure work term. Using \(u_e^2/(c_p T_e) = 2(S-1)\) :
\[
u\frac{dp}{dx}
= \overbrace{u_e f'}^{u}\cdot\left(-\rho_e u_e^2\overbrace{\frac{m}{x}}^{\beta_H\rho_\infty\mu_\infty u_e/(2\xi)}\right)
= -\frac{\beta_H\rho_e\rho_\infty\mu_\infty u_e^4 f'}{2\xi}
= -\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\cdot
\overbrace{\frac{\beta_H\tau u_e^2 f'}{c_p T_e}}^{2(S-1)\beta_H\tau f'}
\]
\[\boxed{u\frac{dp}{dx}
= -\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\,2(S-1)\,\beta_H\tau f'}\]
Streamwise dissipation term. Using \(u_e^2/(c_p T_e) = 2(S-1)\) :
\[
\mu\!\left(\frac{\partial u}{\partial y}\right)^{\!2}
= \overbrace{\frac{C\rho_\infty\mu_\infty}{\rho}}^{\mu}
\left(\overbrace{\frac{\rho u_e^2 f''}{\rho_\infty\sqrt{2\xi}}}^{\partial u/\partial y}\right)^{\!2}
= \frac{C\rho\mu_\infty u_e^4 f''^2}{\rho_\infty\cdot 2\xi}
= \frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\cdot
\overbrace{\frac{C u_e^2 f''^2}{c_p T_e}}^{2(S-1)\,Cf''^2}
\]
\[\boxed{\mu\!\left(\frac{\partial u}{\partial y}\right)^{\!2}
= \frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\,2(S-1)\,Cf''^2}\]
Crossflow dissipation term. Using \(w_e^2/(c_p T_e) = 2(K-1)S\) :
\[
\mu\!\left(\frac{\partial w}{\partial y}\right)^{\!2}
= \overbrace{\frac{C\rho_\infty\mu_\infty}{\rho}}^{\mu}
\left(\overbrace{\frac{\rho u_e w_e g'}{\rho_\infty\sqrt{2\xi}}}^{\partial w/\partial y}\right)^{\!2}
= \frac{C\rho\mu_\infty u_e^2 w_e^2 g'^2}{\rho_\infty\cdot 2\xi}
= \frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\cdot
\overbrace{\frac{C w_e^2 g'^2}{c_p T_e}}^{2(K-1)S\,Cg'^2}
\]
\[\boxed{\mu\!\left(\frac{\partial w}{\partial y}\right)^{\!2}
= \frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}\,2(K-1)S\,Cg'^2}\]
Assembly. Every term carries the common factor \(\rho\rho_\infty\mu_\infty u_e^2 c_p T_e/(2\xi)\) :
\[
\begin{aligned}
-\cancel{\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}}\,f\tau'
&= -\cancel{\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}}\,2(S-1)\beta_H\tau f'
+ \cancel{\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}}\left(\frac{C}{\mathrm{Pr}}\tau'\right)' \\
&\quad + \cancel{\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}}\,2(S-1)Cf''^2
+ \cancel{\frac{\rho\rho_\infty\mu_\infty u_e^2 c_p T_e}{2\xi}}\,2(K-1)S\,Cg'^2
\end{aligned}
\]
Dividing through and rearranging gives the directly-derived static-temperature energy equation:
\[\left(\frac{C}{\mathrm{Pr}}\tau'\right)' + f\tau'
+ 2(S-1)\!\left[Cf''^2 - \beta_H\tau f'\right]
+ 2(K-1)S\,Cg'^2 = 0\]
This is the FSC analog of the
Falkner-Skan energy equation : setting \(w_e = 0\) and
\(\Lambda = 0\) makes \(2(S-1) \to (\gamma-1)M_e^2\) and drops the crossflow term, recovering
\((C/\mathrm{Pr}\,\tau')' + f\tau' + (\gamma-1)M_e^2[Cf''^2 - \beta_H\tau f'] = 0\) .
Boundary Conditions
Wall (\(\eta = 0\) ):
\[f = 0, \qquad f' = 0, \qquad g = 0\]
Isothermal : \(\tau(0) = T_w/T_e\) (prescribed)Adiabatic : \(\tau'(0) = 0\)
Edge (\(\eta \to \infty\) ):
\[f' = 1, \qquad g = 1, \qquad \tau = 1\]