On lubrication with comparable viscous and inertia forces

In this paper we re-examine the flow of fluid in a slender channel with nonuniform cross-section driven by the relative motion of the boundaries in which viscous and inertia forces are comparable. We describe the first-order-accurate asymptotic solutions in the limits of large and small Reynolds number as well as two approximate solutions. These analytical solutions are compared with numerically calculated solutions of the problem which improve those previously available. In particular, we clarify the confusion in the literature over the behaviour of the solution near the inlet of the channel.