A viscous fluid flows along a slowly varying thin channel between no-slip surfaces at $y=0$ and $y=h(x, t)$ under the action of a pressure gradient $d p / d x$. After explaining the approximations and assumptions of lubrication theory, including a comment on the reduced Reynolds number, derive the expression for the volume flux

$$q=\int_{0}^{h} u d y=-\frac{h^{3}}{12 \mu} \frac{d p}{d x}$$

as well as the equation

$$\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=0$$

In peristaltic pumping, the surface $h(x, t)$ has a periodic form in space which propagates at a constant speed $c$, i.e. $h(x-c t)$, and no net pressure gradient is applied, i.e. the pressure gradient averaged over a period vanishes. Show that the average flux along the channel is given by

$$\langle q\rangle=c\left(\langle h\rangle-\frac{\left\langle h^{-2}\right\rangle}{\left\langle h^{-3}\right\rangle}\right)$$

where $\langle\cdot\rangle$ denotes an average over one period.