Explain the assumptions of lubrication theory and its use in determining the flow in thin films.

A cylindrical roller of radius a rotates at angular velocity $\Omega$ below the free surface at $y=0$ of a fluid of density $\rho$ and dynamic viscosity $\mu$. The gravitational acceleration is $g$, and the pressure above the free surface is $p_{0}$. The minimum distance of the roller below the fluid surface is $b$, where $b \ll a$. The depth of the roller $d(x)$ below the free surface may be approximated by $d(x) \approx b+x^{2} / 2 a$, where $x$ is the horizontal distance.

(i) State the conditions for lubrication theory to be applicable to this problem. On the further assumption that the free surface may be taken to be flat, find the flow above the roller and calculate the horizontal volume flux $Q$ (per unit length in the third dimension) and the horizontal pressure gradient.

(ii) Use the pressure gradient you have found to estimate the order of magnitude of the departure $h(x)$ of the free surface from $y=0$, and give conditions on the parameters that ensure that $|h| \ll b$, as required for part (i).

[Hint: Integrals of the form

$$I_{n}=\int_{-\infty}^{\infty}\left(1+t^{2}\right)^{-n} d t$$

satisfy $I_{1}=\pi$ and

$$I_{n+1}=\left(\frac{2 n-1}{2 n}\right) I_{n}$$

for $n \geqslant 1 .]$