Write down the Navier-Stokes equations for an incompressible fluid.

Explain the concepts of the Euler and Prandtl limits applied to the Navier-Stokes equations near a rigid boundary.

A steady two-dimensional flow given by $(U, 0)$ far upstream flows past a semi-infinite flat plate, held at $y=0, x>0$. Derive the boundary layer equation

$$\frac{\partial \psi}{\partial y} \frac{\partial^{2} \psi}{\partial x \partial y}-\frac{\partial \psi}{\partial x} \frac{\partial^{2} \psi}{\partial y^{2}}=\nu \frac{\partial^{3} \psi}{\partial y^{3}}$$

for the stream-function $\psi(x, y)$ near the plate, explaining any approximations made.

Show that the appropriate solution must be of the form

$$\psi(x, y)=(\nu U x)^{1 / 2} f(\eta),$$

and determine the dimensionless variable $\eta$.

Derive the equation and boundary conditions satisfied by $f(\eta)$. [You need not solve them.]

Suppose now that the plate has a finite length $L$ in the direction of the flow. Show that the force $F$ on the plate (per unit width perpendicular to the flow) is given by

$$F=\frac{4 \rho U^{2} L}{(U L / \nu)^{1 / 2}} \frac{f^{\prime \prime}(0)}{\left[f^{\prime}(\infty)\right]^{2}} .$$