(i) Viscous, incompressible fluid of viscosity $\mu$ flows steadily in the $x$-direction in a uniform channel $0<y<h$. The plane $y=0$ is fixed and the plane $y=h$ has constant $x$-velocity $U$. Neglecting gravity, derive from first principles the equations of motion of the fluid and show that the $x$-component of the fluid velocity is $u(y)$ and satisfies

$$0=-P_{x}+\mu u_{y y},$$

where $P(x)$ is the pressure in the fluid. Write down the boundary conditions on $u$. Hence show that the volume flow rate $Q=\int_{0}^{h} u d y$ is given by

$$Q=\frac{U h}{2}-\frac{P_{x} h^{3}}{12 \mu}$$

(ii) A heavy rectangular body of width $L$ and infinite length (in the $z$-direction) is pivoted about one edge at $(x, y)=(0,0)$ above a fixed rigid horizontal plane $y=0$. The body has weight $W$ per unit length in the $z$-direction, its centre of mass is distance $L / 2$ from the pivot, and it is falling under gravity towards the fixed plane through a viscous, incompressible fluid. Let $\alpha(t) \ll 1$ be the angle between the body and the plane. Explain the approximations of lubrication theory which permit equations (1) and (2) of Part (i) to apply to the flow in the gap between the two surfaces.

Deduce that, in the gap,

$$P_{x}=\frac{6 \mu \dot{\alpha}}{x \alpha^{3}},$$

where $\dot{\alpha}=d \alpha / d t$. By taking moments about $(x, y)=(0,0)$, deduce that $\alpha(t)$ is given by

$$\frac{1}{\alpha^{2}}-\frac{1}{\alpha_{0}^{2}}=\frac{2 W t}{3 \mu L}$$

where $\alpha(0)=\alpha_{0}$.