Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 36A / b / Solution

Integrating the velocity profiles over their respective half-channels gives volume flux per unit widthWith $r={\mu }_{+}/{\mu }_{-}$, the requested ratio isAt $r=1$, both fluxes equal $G{a}^3/(3{\mu }_{-})$ and the ratio is one. For $r\to \infty$, $Q_{-}\to G{a}^3/(12{\mu }_{-})$, $Q_{+}\sim 7G{a}^3/(12{\mu }_{+})$ and the ratio is asymptotic to $7/r$, tending to zero. The lower fluid flows between the wall and an interface which becomes effectively stationary; the upper fluid carries vanishing flux despite finite limiting stress.