Fluid of density $\rho$ and dynamic viscosity $\mu$ occupies the region $y>0$ in Cartesian coordinates $(x, y, z)$. A semi-infinite, dense array of cilia occupy the half plane $y=0$, $x>0$ and apply a stress in the $x$-direction on the adjacent fluid, working at a constant and uniform rate $\rho P$ per unit area, which causes the fluid to move with steady velocity $\mathbf{u}=(u(x, y), v(x, y), 0)$. Give a careful physical explanation of the boundary condition

$$\left.u \frac{\partial u}{\partial y}\right|_{y=0}=-\frac{P}{\nu} \quad \text { for } \quad x>0$$

paying particular attention to signs, where $\nu$ is the kinematic viscosity of the fluid. Why would you expect the fluid motion to be confined to a thin region near $y=0$ for sufficiently large values of $x$ ?

Write down the viscous-boundary-layer equations governing the thin region of fluid motion. Show that the flow can be approximated by a stream function

$$\psi(x, y)=U(x) \delta(x) f(\eta), \quad \text { where } \quad \eta=\frac{y}{\delta(x)}$$

Determine the functions $U(x)$ and $\delta(x)$. Show that the dimensionless function $f(\eta)$ satisfies

$$f^{\prime \prime \prime}=\frac{1}{5} f^{\prime 2}-\frac{3}{5} f f^{\prime \prime}$$

What boundary conditions must be satisfied by $f(\eta)$ ? By considering how the volume flux varies with downstream location $x$, or otherwise, determine (with justification) the sign of the transverse flow $v$.