Consider a uniform stream of inviscid incompressible fluid incident onto a twodimensional body (such as a circular cylinder). Sketch the flow in the region close to the stagnation point, $S$, at the front of the body.

Let the fluid now have a small but non-zero viscosity. Using local co-ordinates $x$ along the boundary and $y$ normal to it, with the stagnation point as origin and $y>0$ in the fluid, explain why the local outer, inviscid flow is approximately of the form

$$\mathbf{u}=(E x,-E y)$$

for some positive constant $E$.

Use scaling arguments to find the thickness $\delta$ of the boundary layer on the body near $S$. Hence show that there is a solution of the boundary layer equations of the form

$$u(x, y)=E x f^{\prime}(\eta)$$

where $\eta$ is a suitable similarity variable and $f$ satisfies

$$f^{\prime \prime \prime}+f f^{\prime \prime}-f^{\prime^{2}}=-1 .$$

What are the appropriate boundary conditions for $(*)$ and why? Explain briefly how you would obtain a numerical solution to $(*)$ subject to the appropriate boundary conditions.

Explain why it is neither possible nor appropriate to perform a similar analysis near the rear stagnation point of the inviscid flow.