Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 2 / c / Solution

Integrate the profile over the thickness:This is a volume flux per unit axial length. In a steady layer with no evaporation, sources or axial variation, mass conservation gives ${\partial }_{\theta }Q=0$ to leading lubrication order. Therefore $Q$ is constant around the cylinder even though the local speed and thickness vary.

Assume $U,Q>0$ for the wall-entrained branch, and substitute $h=(Q/U)H$ into the flux formula. After division by $Q$,ConsequentlyThe cubic correction measures gravitational drainage relative to transport by the rotating wall. The nondimensionalization would need modification for zero or oppositely directed mean flux.