What is lubrication theory? Explain the assumptions that go into the theory.

Viscous fluid with dynamic viscosity $\mu$ and density $\rho$ is contained between two flat plates, which approach each other at uniform speed $V$. The first is fixed at $y=0,-L<x<L$. The second has its ends at $\left(-L, h_{0}-\Delta h-V t\right),\left(L, h_{0}+\Delta h-V t\right)$, where $\Delta h \sim h_{0} \ll L$. There is no flow in the $z$ direction, and all variation in $z$ may be neglected. There is no applied pressure gradient in the $x$ direction.

Assuming that $V$ is so small that lubrication theory applies, derive an expression for the horizontal volume flux $Q(x)$ at $t=0$, in terms of the pressure gradient. Show that mass conservation implies that $d Q / d x=V$, so that $Q(L)-Q(-L)=2 V L$. Derive another relation between $Q(L)$ and $Q(-L)$ by setting the pressures at $x=\pm L$ to be equal, and hence show that

$$Q(\pm L)=V L\left(\frac{\Delta h}{h_{0}} \pm 1\right)$$

Show that lubrication theory applies if $V \ll \mu / h_{0} \rho$.