Consider in $y0$ the $2-$D flow

$u=x{f}^{\text{'}}(),v=-(v{)}^{\frac{1}{2}}f()$

where

$=(\frac{}{v}{)}^{\frac{1}{2}}y$

Show that it is an exact solution of the Navier$-$Stokes equations which

$($i$)$ satisfies the boundary conditions at the stationary rigid boundary

$y=0$ and $($ii$)$ takes the asymptotic form $ux,v-y$ far from the

Elementary viscous flow

Fig. $2\mathrm{.}17.$ The velocity profile in the boundary layer near a $2-$D

stagnation point.

boundary $($see Fig. $2\mathrm{.}13)$ if

with

$f^{\text{'}\text{'}\text{'}}+f{f}^{\text{'}\text{'}}+1-f^{\text{'}2}=0$

$f(0)=f^{\text{'}}(0)=0,f^{\text{'}}()=1$

$[$The differential equation for $f()$ is solved numerically, and $f^{\text{'}}()$ is

shown in Fig. $2\mathrm{.}17.$ Notably, $f^{\text{'}}(3)=0\mathrm{.}998,$ so beyond a distance of

$3(\frac{v}{}{)}^{\frac{1}{2}}$ from the boundary the flow is effectively inviscid and

irrotational, with $ux$ and $v-y]$