Example $13\mathrm{.}1.$ The velocity distribution in the boundary layer is given by: $\frac{u}{U}=\frac{y}{},$ where $u$ is the velocity at a distance $y$ from the plate and $u=U$ at $y=,$ being boundary layer thickness. Find:

$($i$)$ The displacement thickness, $($ii$)$ The momentum thickness,

$($iii$)$ The energy thickness, and $($iv$)$ The value of $\frac{{}^{**}}{}.$

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Solution. Velocity distribution: $\frac{u}{x}=\frac{y}{x}$

$...($Given$)$