(i) Incompressible fluid of kinematic viscosity $\nu$ occupies a parallel-sided channel $0 \leqslant y \leqslant h_{0},-\infty<x<\infty$. The wall $y=0$ is moving parallel to itself, in the $x$ direction, with velocity $\operatorname{Re}\left\{U e^{i \omega t}\right\}$, where $t$ is time and $U, \omega$ are real constants. The fluid velocity $u(y, t)$ satisfies the equation

$$u_{t}=\nu u_{y y}$$

write down the boundary conditions satisfied by $u$.

Assuming that

$$u=\operatorname{Re}\left\{a \sinh [b(1-\eta)] e^{i \omega t}\right\}$$

where $\eta=y / h_{0}$, find the complex constants $a, b$. Calculate the velocity (in real, not complex, form) in the limit $h_{0}(\omega / \nu)^{1 / 2} \rightarrow 0$.

(ii) Incompressible fluid of viscosity $\mu$ fills the narrow gap between the rigid plane $y=0$, which moves parallel to itself in the $x$-direction with constant speed $U$, and the rigid wavy wall $y=h(x)$, which is at rest. The length-scale, $L$, over which $h$ varies is much larger than a typical value, $h_{0}$, of $h$.

Assume that inertia is negligible, and therefore that the governing equations for the velocity field $(u, v)$ and the pressure $p$ are

$$u_{x}+v_{y}=0, p_{x}=\mu\left(u_{x x}+u_{y y}\right), p_{y}=\mu\left(v_{x x}+v_{y y}\right)$$

Use scaling arguments to show that these equations reduce approximately to

$$p_{x}=\mu u_{y y}, \quad p_{y}=0$$

Hence calculate the velocity $u(x, y)$, the flow rate

$$Q=\int_{0}^{h} u d y$$

and the viscous shear stress exerted by the fluid on the plane wall,

$$\tau=-\left.\mu u_{y}\right|_{y=0}$$

in terms of $p_{x}, h, U$ and $\mu$.

Now assume that $h=h_{0}(1+\epsilon \sin k x)$, where $\epsilon \ll 1$ and $k h_{0} \ll 1$, and that $p$ is periodic in $x$ with wavelength $2 \pi / k$. Show that

$$Q=\frac{h_{0} U}{2}\left(1-\frac{3}{2} \epsilon^{2}+O\left(\epsilon^{4}\right)\right)$$

and calculate $\tau$ correct to $O\left(\epsilon^{2}\right)$. Does increasing the amplitude $\epsilon$ of the corrugation cause an increase or a decrease in the force required to move the plane $y=0$ at the chosen speed $U ?$