Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 329 / 3 / b

This is a squeezing motion, not the rotational motion of the previous part. The gap changes at rate $h_t=V\sin \theta$. The Reynolds lubrication equation therefore givesThe outer wall also has tangential speed $V\cos \theta$, but its local Couette flow flux is $O(V\Delta )$ compared with the squeezing flux $O(Va)$ and does not affect the leading force. Thus $p_{\theta }=-12\mu aq/h^3$ at leading order. Periodic pressure gives $C=0$, because the integral of $\cos \theta /h^3$ vanishes over a full circle. Balancing pressure traction on the fixed inner cylinder and using integration by parts,The supplied integral gives the squeeze resistance of an eccentric journal bearing:The fluid tends to pull the inner cylinder upwards with the moving outer wall, so a downward holding force is required when $V>0$. At concentricity the translation resistance is $R_0=12\pi \mu a^3/{\Delta }^3$. The omitted wall-tangential flux and shear traction give smaller thin-gap contributions.