\(\because \bar\delta \ll L \text{ when } \mathrm{Re}\gg 1\)
\(\therefore \bar\delta \sim \sqrt{\frac{\nu L}{U_\infty}}
\;\Rightarrow\;
\frac{\bar\delta}{L}\sim \sqrt{\frac{\nu}{U_\infty L}}
=\left(\frac{U_\infty L}{\nu}\right)^{-1/2}
=\mathrm{Re}^{-1/2}\ll1
\;\Rightarrow\; \bar\delta\ll L\)
\(\because \text{Continuity equation } \boxed{\partial u/\partial x+\partial v/\partial y=0} \text{ gives the scale for $v$}\)
\(\therefore \frac{\partial}{\partial x}\sim\frac{1}{L},\ \frac{\partial}{\partial y}\sim\frac{1}{\bar\delta},\ u\sim U_\infty \Rightarrow\quad \frac{U_\infty}{L}\sim \frac{v}{\bar\delta}
\;\Rightarrow\;
v \sim \frac{\bar\delta}{L}U_\infty
= U_\infty\,\mathrm{Re}^{-1/2}
\)
\(\because \text{Pressure scale at high } \mathrm{Re} \text{ (outer flow nearly irrotational)} \Rightarrow \text{the pressure variations scale with the fluid inertia}
\)
\(\therefore \rho(\mathbf{u}\cdot\nabla)\mathbf{u}\sim\nabla p
\;\Rightarrow\;
\frac{\rho U_\infty^2}{L}\sim \frac{p-p_\infty}{L}
\;\Rightarrow\;
p-p_\infty \sim \rho U_\infty^2
\)
\(\therefore \text{Choose nondimensional variables } x^*=\frac{x}{L},\quad
y^*=\frac{y}{\bar\delta},\quad
u^*=\frac{u}{U_\infty},\quad
v^*=\frac{v}{U_\infty}\mathrm{Re}^{1/2},\quad
p^*=\frac{p-p_\infty}{\rho U_\infty^2}
\)
Confirming these scalings make the continuity equation order-one
\(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}
=\frac{U_\infty}{L}\!\left(\frac{\partial u^*}{\partial x^*}+\frac{\partial v^*}{\partial y^*}\right)=0
\;\Rightarrow\;
\frac{\partial u^*}{\partial x^*}+\frac{\partial v^*}{\partial y^*}=0\)
$x$-momentum: \(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}
=-\frac{1}{\rho}\frac{\partial p}{\partial x}
+\nu\!\left(\frac{\partial^2 u}{\partial x^2}
+\frac{\partial^2 u}{\partial y^2}\right) \Rightarrow \frac{U_\infty^2}{L}
\!\left(
u^*\frac{\partial u^*}{\partial x^*}
+v^*\frac{\partial u^*}{\partial y^*}\right)
=\frac{U_\infty^2}{L}
\!\left(
-\frac{\partial p^*}{\partial x^*}
+\frac{1}{\mathrm{Re}}\frac{\partial^2 u^*}{\partial x^{*2}}
+\frac{\partial^2 u^*}{\partial y^{*2}}\right)\)
\(\Rightarrow u^*\frac{\partial u^*}{\partial x^*}
+v^*\frac{\partial u^*}{\partial y^*}
=-\frac{\partial p^*}{\partial x^*}
+\frac{1}{\mathrm{Re}}\frac{\partial^2 u^*}{\partial x^{*2}}
+\frac{\partial^2 u^*}{\partial y^{*2}}\)
$y$-momentum: \(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}
=-\frac{1}{\rho}\frac{\partial p}{\partial y}
+\nu\!\left(\frac{\partial^2 v}{\partial x^2}
+\frac{\partial^2 v}{\partial y^2}\right) \Rightarrow \frac{U_\infty^2}{L\,\mathrm{Re}^{1/2}}
\!\left(
u^*\frac{\partial v^*}{\partial x^*}
+v^*\frac{\partial v^*}{\partial y^*}\right)
=\frac{U_\infty^2}{L}
\!\left(
-\mathrm{Re}^{1/2}\frac{\partial p^*}{\partial y^*}
+\frac{1}{\mathrm{Re}^{3/2}}\frac{\partial^2 v^*}{\partial x^{*2}}
+\frac{1}{\mathrm{Re}^{1/2}}\frac{\partial^2 v^*}{\partial y^{*2}}\right)\)
\(\Rightarrow \frac{1}{\mathrm{Re}}\!\left(
u^*\frac{\partial v^*}{\partial x^*}
+v^*\frac{\partial v^*}{\partial y^*}\right)
=-\frac{\partial p^*}{\partial y^*}
+\frac{1}{\mathrm{Re}^2}\frac{\partial^2 v^*}{\partial x^{*2}}
+\frac{1}{\mathrm{Re}}\frac{\partial^2 v^*}{\partial y^{*2}}\)