Boundary Layer Theory

$$ \delta \sim \sqrt{\nu L / U_\infty} \quad \text{or} \quad \delta/L \sim 1/\sqrt{\text{Re}} $$ This estimate of $\delta$ can also be obtained by noting that viscous effects diffuse to a distance of order $(\nu t)^{1/2}$ in time $t$ and that the time-of-flight for a fluid element along a body of length $L$ is of order $L/U_\infty$. Substituting $L/U_\infty$ for $t$ in $(\nu t)^{1/2}$ suggests the viscous layer's diffusive thickness at $x = L$ is of order $(\nu L/U_\infty)^{1/2}$

$\ \delta \sim (\nu t)^{1/2},\quad t\sim \frac{L}{U_\infty}\ \Rightarrow\ \delta \sim \sqrt{\frac{\nu L}{U_\infty}} \ $
$\ \frac{\delta}{L} \sim \frac{1}{L}\sqrt{\frac{\nu L}{U_\infty}} = \frac{\sqrt{\nu}}{\sqrt{U_\infty}}\cdot\frac{\sqrt{L}}{L} = \frac{\sqrt{\nu}}{\sqrt{U_\infty}}\cdot \frac{\cancel{\sqrt{L}}}{\cancel{\sqrt{L}}\ \sqrt{L}} \ $
$\ \frac{\delta}{L} \sim \frac{\sqrt{\nu}}{\sqrt{U_\infty}}\cdot\frac{1}{\sqrt{L}} = \sqrt{\frac{\nu}{U_\infty L}} = \left(\frac{U_\infty L}{\nu}\right)^{-1/2} = \mathrm{Re}_L^{-1/2} \ $

A formal simplification of the equations of motion within the boundary layer can now be performed. The basic idea is that variations across the boundary layer occur over a much shorter length scale than variations along the layer $$ \frac{\partial}{\partial x}\sim \frac{1}{L},\qquad \frac{\partial}{\partial y}\sim \frac{1}{\bar\delta} $$ where $\boxed{\mathrm{Re}\gg 1} \ \Rightarrow\ \frac{\bar\delta}{L}\sim \mathrm{Re}^{-1/2}\ll 1 \ \Rightarrow\ \boxed{\bar\delta \ll L}$

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