3d equations
Assumptions
- Steady, three-dimensional, laminar flow
- Cartesian coordinates: \(x\) streamwise, \(z\) crossflow (spanwise), \(y\) wall-normal
- Calorically perfect gas (constant \(\gamma\), \(c_p\), \(\mathrm{Pr}\))
- Temperature-dependent viscosity (Sutherland or power law)
- Thin boundary layer (\(\delta \ll L\))
Governing Equations
The three-dimensional compressible laminar boundary layer equations are (see 1):
Continuity
\[
\frac{\partial (\rho u)}{\partial x}
+ \frac{\partial (\rho v)}{\partial y}
+ \frac{\partial (\rho w)}{\partial z} = 0
\]
x-momentum
\[
\rho \left( u \frac{\partial u}{\partial x}
+ v \frac{\partial u}{\partial y}
+ w \frac{\partial u}{\partial z} \right)
= -\frac{\partial p}{\partial x}
+ \frac{\partial}{\partial y}\!\left( \mu \frac{\partial u}{\partial y} \right)
\]
z-momentum
\[
\rho \left( u \frac{\partial w}{\partial x}
+ v \frac{\partial w}{\partial y}
+ w \frac{\partial w}{\partial z} \right)
= -\frac{\partial p}{\partial z}
+ \frac{\partial}{\partial y}\!\left( \mu \frac{\partial w}{\partial y} \right)
\]
y-momentum
\[
\frac{\partial p}{\partial y} = 0
\]
Energy
\[
\rho c_p \left( u \frac{\partial T}{\partial x}
+ v \frac{\partial T}{\partial y}
+ w \frac{\partial T}{\partial z} \right)
= \left( u \frac{\partial p}{\partial x} + w \frac{\partial p}{\partial z} \right)
+ \frac{\partial}{\partial y}\!\left( k \frac{\partial T}{\partial y} \right)
+ \mu \left[ \left(\frac{\partial u}{\partial y}\right)^{\!2}
+ \left(\frac{\partial w}{\partial y}\right)^{\!2} \right]
\]
Perfect gas state equation
\[
p = \rho R T
\]
From the y-momentum equation it follows that pressure is uniform across the layer and imposed entirely by the outer inviscid flow (see derivation below).
Outer Inviscid Flow
The boundary layer edge conditions are imposed by the inviscid outer flow:
\[
-\frac{\partial p}{\partial x} = \rho_e u_e \frac{\partial u_e}{\partial x}
+ \rho_e w_e \frac{\partial u_e}{\partial z}
\]
\[
-\frac{\partial p}{\partial z} = \rho_e u_e \frac{\partial w_e}{\partial x}
+ \rho_e w_e \frac{\partial w_e}{\partial z}
\]
Derivation from the Navier-Stokes equations
Compressible Navier-Stokes Equations (3D)
Start with the compressible NS equations for a Newtonian fluid with Stokes' hypothesis (\(\lambda = -2\mu/3\)).
Continuity
\[
\frac{\partial \rho}{\partial t}
+ \frac{\partial (\rho u)}{\partial x}
+ \frac{\partial (\rho v)}{\partial y}
+ \frac{\partial (\rho w)}{\partial z} = 0
\]
x-momentum
\[
\begin{aligned}
\rho \!\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right)
&= -\frac{\partial p}{\partial x}
+ \frac{\partial}{\partial x}\!\left(\frac{4\mu}{3}\frac{\partial u}{\partial x}
- \frac{2\mu}{3}\!\left(\frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right)\right) \\
&+ \frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y} + \mu\frac{\partial v}{\partial x}\right)
+ \frac{\partial}{\partial z}\!\left(\mu\frac{\partial u}{\partial z} + \mu\frac{\partial w}{\partial x}\right)
\end{aligned}
\]
z-momentum
\[
\begin{aligned}
\rho \!\left(\frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}\right)
&= -\frac{\partial p}{\partial z}
+ \frac{\partial}{\partial x}\!\left(\mu\frac{\partial w}{\partial x} + \mu\frac{\partial u}{\partial z}\right)
+ \frac{\partial}{\partial y}\!\left(\mu\frac{\partial w}{\partial y} + \mu\frac{\partial v}{\partial z}\right) \\
&+ \frac{\partial}{\partial z}\!\left(\frac{4\mu}{3}\frac{\partial w}{\partial z}
- \frac{2\mu}{3}\!\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)\right)
\end{aligned}
\]
y-momentum
\[
\begin{aligned}
\rho \!\left(\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}\right)
&= -\frac{\partial p}{\partial y}
+ \frac{\partial}{\partial x}\!\left(\mu\frac{\partial v}{\partial x} + \mu\frac{\partial u}{\partial y}\right) \\
&+ \frac{\partial}{\partial y}\!\left(\frac{4\mu}{3}\frac{\partial v}{\partial y}
- \frac{2\mu}{3}\!\left(\frac{\partial u}{\partial x} + \frac{\partial w}{\partial z}\right)\right)
+ \frac{\partial}{\partial z}\!\left(\mu\frac{\partial v}{\partial z} + \mu\frac{\partial w}{\partial y}\right)
\end{aligned}
\]
Energy (in terms of temperature)
\[
\begin{aligned}
\rho c_p \!\left(\frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + w\frac{\partial T}{\partial z}\right)
&= \left(\frac{\partial p}{\partial t} + u\frac{\partial p}{\partial x} + v\frac{\partial p}{\partial y} + w\frac{\partial p}{\partial z}\right) \\
&+ \frac{\partial}{\partial x}\!\left(k\frac{\partial T}{\partial x}\right)
+ \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \frac{\partial}{\partial z}\!\left(k\frac{\partial T}{\partial z}\right)
+ \Phi
\end{aligned}
\]
where
\[
\begin{aligned}
\Phi = \mu\Bigl[
&2\!\left(\frac{\partial u}{\partial x}\right)^2
+ 2\!\left(\frac{\partial v}{\partial y}\right)^2
+ 2\!\left(\frac{\partial w}{\partial z}\right)^2 \\
&+ \!\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)^2
+ \!\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right)^2
+ \!\left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}\right)^2 \\
&- \frac{2}{3}\!\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right)^2
\Bigr]
\end{aligned}
\]
Steady Flow
For steady flow, all \(\partial/\partial t\) terms vanish. The continuity, momentum, and energy equations reduce to:
\[
\frac{\partial (\rho u)}{\partial x}
+ \frac{\partial (\rho v)}{\partial y}
+ \frac{\partial (\rho w)}{\partial z} = 0
\]
\[
\begin{aligned}
\rho \!\left(u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right)
&= -\frac{\partial p}{\partial x}
+ \frac{\partial}{\partial x}\!\left(\frac{4\mu}{3}\frac{\partial u}{\partial x}
- \frac{2\mu}{3}\!\left(\frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right)\right) \\
&+ \frac{\partial}{\partial y}\!\left(\mu\frac{\partial u}{\partial y} + \mu\frac{\partial v}{\partial x}\right)
+ \frac{\partial}{\partial z}\!\left(\mu\frac{\partial u}{\partial z} + \mu\frac{\partial w}{\partial x}\right)
\end{aligned}
\]
\[
\begin{aligned}
\rho \!\left(u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}\right)
&= -\frac{\partial p}{\partial z}
+ \frac{\partial}{\partial x}\!\left(\mu\frac{\partial w}{\partial x} + \mu\frac{\partial u}{\partial z}\right)
+ \frac{\partial}{\partial y}\!\left(\mu\frac{\partial w}{\partial y} + \mu\frac{\partial v}{\partial z}\right) \\
&+ \frac{\partial}{\partial z}\!\left(\frac{4\mu}{3}\frac{\partial w}{\partial z}
- \frac{2\mu}{3}\!\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right)\right)
\end{aligned}
\]
\[
\begin{aligned}
\rho \!\left(u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}\right)
&= -\frac{\partial p}{\partial y}
+ \frac{\partial}{\partial x}\!\left(\mu\frac{\partial v}{\partial x} + \mu\frac{\partial u}{\partial y}\right) \\
&+ \frac{\partial}{\partial y}\!\left(\frac{4\mu}{3}\frac{\partial v}{\partial y}
- \frac{2\mu}{3}\!\left(\frac{\partial u}{\partial x} + \frac{\partial w}{\partial z}\right)\right)
+ \frac{\partial}{\partial z}\!\left(\mu\frac{\partial v}{\partial z} + \mu\frac{\partial w}{\partial y}\right)
\end{aligned}
\]
\[
\begin{aligned}
\rho c_p \!\left(u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} + w\frac{\partial T}{\partial z}\right)
&= u\frac{\partial p}{\partial x} + v\frac{\partial p}{\partial y} + w\frac{\partial p}{\partial z} \\
&+ \frac{\partial}{\partial x}\!\left(k\frac{\partial T}{\partial x}\right)
+ \frac{\partial}{\partial y}\!\left(k\frac{\partial T}{\partial y}\right)
+ \frac{\partial}{\partial z}\!\left(k\frac{\partial T}{\partial z}\right)
+ \Phi
\end{aligned}
\]
Boundary Layer Scaling
The 3D scaling follows the same argument as the 2D case. Both \(x\) and \(z\) are streamwise-like coordinates (\(\sim L\)), so \(z\)-gradients scale identically to \(x\)-gradients.
Step 1: get \(V\) scaling from continuity.
All three terms must balance:
\[
\frac{U_\infty}{L} \sim \frac{V}{\delta} \sim \frac{U_\infty}{L}
\qquad\Longrightarrow\qquad
V \sim \frac{\delta}{L}\,U_\infty
\]
Step 2: get \(\delta\) by balancing convection with wall-normal viscosity in x-momentum.
\[
\underbrace{\rho\,\frac{U_\infty^2}{L}}_{\text{inertia}} \sim \underbrace{\frac{\mu\,U_\infty}{\delta^2}}_{\text{viscosity}}
\qquad\Longrightarrow\qquad
\delta \sim \frac{L}{\sqrt{Re_L}}, \qquad Re_L = \frac{\rho U_\infty L}{\mu}
\]
Step 3: Define dimensionless variables
\[
\begin{aligned}
x^* &= \frac{x}{L}, &\quad
y^* &= \frac{y}{\delta}, &\quad
z^* &= \frac{z}{L}, \\[6pt]
u^* &= \frac{u}{U_\infty}, &\quad
v^* &= \frac{v}{\varepsilon U_\infty}, &\quad
w^* &= \frac{w}{U_\infty}, \\[6pt]
p^* &= \frac{p}{\rho_\infty U_\infty^2}, &\quad
\rho^* &= \frac{\rho}{\rho_\infty}, &\quad
\mu^* &= \frac{\mu}{\mu_\infty}, \\[6pt]
T^* &= \frac{T}{T_\infty}, &\quad
k^* &= \frac{k}{k_\infty}
\end{aligned}
\]
where \(\varepsilon = \delta/L = Re_L^{-1/2} \ll 1\).
Step 4: Substitute
Using \(\partial/\partial x = (1/L)\,\partial/\partial x^*\), \(\partial/\partial z = (1/L)\,\partial/\partial z^*\), and \(\partial/\partial y = (1/\delta)\,\partial/\partial y^*\), substitute into each equation and divide out the common dimensional factor. Terms small by \(\varepsilon\) or \(\varepsilon^2\) appear with an explicit prefactor.
Continuity
\[
\cancel{\frac{\rho_\infty U_\infty}{L}}\,\frac{\partial(\rho^* u^*)}{\partial x^*}
+ \cancel{\frac{\rho_\infty U_\infty}{L}}\,\frac{\partial(\rho^* v^*)}{\partial y^*}
+ \cancel{\frac{\rho_\infty U_\infty}{L}}\,\frac{\partial(\rho^* w^*)}{\partial z^*} = 0
\]
\[
\frac{\partial(\rho^* u^*)}{\partial x^*}
+ \frac{\partial(\rho^* v^*)}{\partial y^*}
+ \frac{\partial(\rho^* w^*)}{\partial z^*} = 0
\]
All three terms are \(\mathcal{O}(1)\) — nothing to drop.
x-momentum
\[
\begin{aligned}
&\cancel{\frac{\rho_\infty U_\infty^2}{L}}\rho^*\!\left(u^*\frac{\partial u^*}{\partial x^*}
+ v^*\frac{\partial u^*}{\partial y^*} + w^*\frac{\partial u^*}{\partial z^*}\right)
= -\cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{\partial p^*}{\partial x^*}
+ \cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial u^*}{\partial y^*}\right) \\[6pt]
&\quad+ \varepsilon^2\cancel{\frac{\rho_\infty U_\infty^2}{L}}\!\left[
\frac{\partial}{\partial x^*}\!\left(\frac{4\mu^*}{3}\frac{\partial u^*}{\partial x^*}
- \frac{2\mu^*}{3}\!\left(\frac{\partial v^*}{\partial y^*} + \frac{\partial w^*}{\partial z^*}\right)\right)
+ \frac{\partial}{\partial z^*}\!\left(\mu^*\frac{\partial u^*}{\partial z^*} + \mu^*\frac{\partial w^*}{\partial x^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*}\right)
\right]
\end{aligned}
\]
\[
\begin{aligned}
\rho^*\!\left(u^*\frac{\partial u^*}{\partial x^*} + v^*\frac{\partial u^*}{\partial y^*} + w^*\frac{\partial u^*}{\partial z^*}\right)
&= -\frac{\partial p^*}{\partial x^*}
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial u^*}{\partial y^*}\right) \\[6pt]
&+ \varepsilon^2\!\left[
\frac{\partial}{\partial x^*}\!\left(\frac{4\mu^*}{3}\frac{\partial u^*}{\partial x^*}
- \frac{2\mu^*}{3}\!\left(\frac{\partial v^*}{\partial y^*} + \frac{\partial w^*}{\partial z^*}\right)\right)
+ \frac{\partial}{\partial z^*}\!\left(\mu^*\frac{\partial u^*}{\partial z^*} + \mu^*\frac{\partial w^*}{\partial x^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*}\right)
\right]
\end{aligned}
\]
z-momentum
\[
\begin{aligned}
&\cancel{\frac{\rho_\infty U_\infty^2}{L}}\rho^*\!\left(u^*\frac{\partial w^*}{\partial x^*}
+ v^*\frac{\partial w^*}{\partial y^*} + w^*\frac{\partial w^*}{\partial z^*}\right)
= -\cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{\partial p^*}{\partial z^*}
+ \cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial w^*}{\partial y^*}\right) \\[6pt]
&\quad+ \varepsilon^2\cancel{\frac{\rho_\infty U_\infty^2}{L}}\!\left[
\frac{\partial}{\partial x^*}\!\left(\mu^*\frac{\partial w^*}{\partial x^*} + \mu^*\frac{\partial u^*}{\partial z^*}\right)
+ \frac{\partial}{\partial z^*}\!\left(\frac{4\mu^*}{3}\frac{\partial w^*}{\partial z^*}
- \frac{2\mu^*}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*}\right)\right)
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial v^*}{\partial z^*}\right)
\right]
\end{aligned}
\]
\[
\begin{aligned}
\rho^*\!\left(u^*\frac{\partial w^*}{\partial x^*} + v^*\frac{\partial w^*}{\partial y^*} + w^*\frac{\partial w^*}{\partial z^*}\right)
&= -\frac{\partial p^*}{\partial z^*}
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial w^*}{\partial y^*}\right) \\[6pt]
&+ \varepsilon^2\!\left[
\frac{\partial}{\partial x^*}\!\left(\mu^*\frac{\partial w^*}{\partial x^*} + \mu^*\frac{\partial u^*}{\partial z^*}\right)
+ \frac{\partial}{\partial z^*}\!\left(\frac{4\mu^*}{3}\frac{\partial w^*}{\partial z^*}
- \frac{2\mu^*}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*}\right)\right)
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial v^*}{\partial z^*}\right)
\right]
\end{aligned}
\]
y-momentum, common factor \(\varepsilon\,\rho_\infty U_\infty^2/L\):
\[
\begin{gathered}
\varepsilon\cancel{\frac{\rho_\infty U_\infty^2}{L}}\rho^*\!\left(u^*\frac{\partial v^*}{\partial x^*}
+ v^*\frac{\partial v^*}{\partial y^*} + w^*\frac{\partial v^*}{\partial z^*}\right)
= -\cancel{\frac{\rho_\infty U_\infty^2}{L}}\frac{1}{\varepsilon}\frac{\partial p^*}{\partial y^*} \\[6pt]
+ \varepsilon\cancel{\frac{\rho_\infty U_\infty^2}{L}}\!\left[
\frac{\partial}{\partial x^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*} + \mu^*\frac{\partial u^*}{\partial y^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\frac{4\mu^*}{3}\frac{\partial v^*}{\partial y^*}
- \frac{2\mu^*}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial w^*}{\partial z^*}\right)\right)
+ \frac{\partial}{\partial z^*}\!\left(\mu^*\frac{\partial v^*}{\partial z^*} + \mu^*\frac{\partial w^*}{\partial y^*}\right)
\right]
\end{gathered}
\]
Multiplying through by \(\varepsilon\):
\[
\begin{aligned}
\varepsilon^2\,\rho^*\!\left(u^*\frac{\partial v^*}{\partial x^*} + v^*\frac{\partial v^*}{\partial y^*} + w^*\frac{\partial v^*}{\partial z^*}\right)
&= -\frac{\partial p^*}{\partial y^*} \\[6pt]
&+ \varepsilon^2\!\left[
\frac{\partial}{\partial x^*}\!\left(\mu^*\frac{\partial v^*}{\partial x^*} + \mu^*\frac{\partial u^*}{\partial y^*}\right)
+ \frac{\partial}{\partial y^*}\!\left(\frac{4\mu^*}{3}\frac{\partial v^*}{\partial y^*}
- \frac{2\mu^*}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial w^*}{\partial z^*}\right)\right)
+ \frac{\partial}{\partial z^*}\!\left(\mu^*\frac{\partial v^*}{\partial z^*} + \mu^*\frac{\partial w^*}{\partial y^*}\right)
\right]
\end{aligned}
\]
At leading order \(\partial p^*/\partial y^* = 0\), so pressure is a function of \(x^*\) and \(z^*\) only.
Energy
\[
\begin{aligned}
&\cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\rho^*\!\left(u^*\frac{\partial T^*}{\partial x^*}
+ v^*\frac{\partial T^*}{\partial y^*} + w^*\frac{\partial T^*}{\partial z^*}\right) \\[6pt]
&\quad= \cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\mathrm{Ec}\!\left(u^*\frac{\partial p^*}{\partial x^*}
+ w^*\frac{\partial p^*}{\partial z^*}\right)
+ \cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{1}{\mathrm{Pr}}\frac{\partial}{\partial y^*}\!\left(k^*\frac{\partial T^*}{\partial y^*}\right) \\[6pt]
&\quad+ \cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left[
\!\left(\frac{\partial u^*}{\partial y^*}\right)^{\!2}
+ \!\left(\frac{\partial w^*}{\partial y^*}\right)^{\!2}\right] \\[6pt]
&\quad+ \varepsilon^2\cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{1}{\mathrm{Pr}}\!\left[
\frac{\partial}{\partial x^*}\!\left(k^*\frac{\partial T^*}{\partial x^*}\right)
+ \frac{\partial}{\partial z^*}\!\left(k^*\frac{\partial T^*}{\partial z^*}\right)\right] \\[6pt]
&\quad+ \varepsilon^2\cancel{\frac{\rho_\infty c_p T_\infty U_\infty}{L}}\frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left[
2\!\left(\frac{\partial u^*}{\partial x^*}\right)^{\!2}
+ 2\!\left(\frac{\partial v^*}{\partial y^*}\right)^{\!2}
+ 2\!\left(\frac{\partial w^*}{\partial z^*}\right)^{\!2}
+ \!\left(\frac{\partial u^*}{\partial z^*} + \frac{\partial w^*}{\partial x^*}\right)^{\!2}
- \frac{2}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*} + \frac{\partial w^*}{\partial z^*}\right)^{\!2}
\right]
\end{aligned}
\]
where \(\mathrm{Pr} = \mu_\infty c_p / k_\infty\) and \(\mathrm{Ec} = U_\infty^2/(c_p T_\infty)\).
\[
\begin{aligned}
\rho^*\!\left(u^*\frac{\partial T^*}{\partial x^*} + v^*\frac{\partial T^*}{\partial y^*} + w^*\frac{\partial T^*}{\partial z^*}\right)
&= \mathrm{Ec}\!\left(u^*\frac{\partial p^*}{\partial x^*} + w^*\frac{\partial p^*}{\partial z^*}\right)
+ \frac{1}{\mathrm{Pr}}\frac{\partial}{\partial y^*}\!\left(k^*\frac{\partial T^*}{\partial y^*}\right)
+ \frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left[\!\left(\frac{\partial u^*}{\partial y^*}\right)^{\!2}
+ \!\left(\frac{\partial w^*}{\partial y^*}\right)^{\!2}\right] \\[6pt]
&+ \varepsilon^2\frac{1}{\mathrm{Pr}}\!\left[
\frac{\partial}{\partial x^*}\!\left(k^*\frac{\partial T^*}{\partial x^*}\right)
+ \frac{\partial}{\partial z^*}\!\left(k^*\frac{\partial T^*}{\partial z^*}\right)\right] \\[6pt]
&+ \varepsilon^2\frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left[
2\!\left(\frac{\partial u^*}{\partial x^*}\right)^{\!2}
+ 2\!\left(\frac{\partial v^*}{\partial y^*}\right)^{\!2}
+ 2\!\left(\frac{\partial w^*}{\partial z^*}\right)^{\!2}
+ \!\left(\frac{\partial u^*}{\partial z^*} + \frac{\partial w^*}{\partial x^*}\right)^{\!2}
- \frac{2}{3}\!\left(\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*} + \frac{\partial w^*}{\partial z^*}\right)^{\!2}
\right]
\end{aligned}
\]
Step 5: drop \(\mathcal{O}(\varepsilon^2)\) terms.
Setting \(\varepsilon \to 0\) in each equation above:
\[
\frac{\partial(\rho^* u^*)}{\partial x^*}
+ \frac{\partial(\rho^* v^*)}{\partial y^*}
+ \frac{\partial(\rho^* w^*)}{\partial z^*} = 0
\]
\[
\rho^*\!\left(u^*\frac{\partial u^*}{\partial x^*} + v^*\frac{\partial u^*}{\partial y^*} + w^*\frac{\partial u^*}{\partial z^*}\right)
= -\frac{\partial p^*}{\partial x^*}
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial u^*}{\partial y^*}\right)
\]
\[
\rho^*\!\left(u^*\frac{\partial w^*}{\partial x^*} + v^*\frac{\partial w^*}{\partial y^*} + w^*\frac{\partial w^*}{\partial z^*}\right)
= -\frac{\partial p^*}{\partial z^*}
+ \frac{\partial}{\partial y^*}\!\left(\mu^*\frac{\partial w^*}{\partial y^*}\right)
\]
\[\frac{\partial p^*}{\partial y^*} = 0\]
\[
\rho^*\!\left(u^*\frac{\partial T^*}{\partial x^*} + v^*\frac{\partial T^*}{\partial y^*} + w^*\frac{\partial T^*}{\partial z^*}\right)
= \mathrm{Ec}\!\left(u^*\frac{\partial p^*}{\partial x^*} + w^*\frac{\partial p^*}{\partial z^*}\right)
+ \frac{1}{\mathrm{Pr}}\frac{\partial}{\partial y^*}\!\left(k^*\frac{\partial T^*}{\partial y^*}\right)
+ \frac{\mathrm{Ec}}{\mathrm{Pr}}\,\mu^*\!\left[\!\left(\frac{\partial u^*}{\partial y^*}\right)^{\!2}
+ \!\left(\frac{\partial w^*}{\partial y^*}\right)^{\!2}\right]
\]
Step 6: re-dimensionalize.
Reversing the substitutions and applying \(\partial p^*/\partial y^* = 0 \Rightarrow \partial p/\partial y = 0\) recovers the dimensional BL equations shown at the top of this page.
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Liu, Z. (2021). Compressible Falkner–Skan–Cooke boundary layer on a flat plate. Physics of Fluids, 33(12). DOI: 10.1063/5.0075233 ↩