Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 1 / b / ii / Solution

Let $b=h-\delta$ be the half-width of the cell-rich core. In fully developed pressure-driven flow the shear stress is fixed by momentum balance, independently of the local viscosity:With no slip at $y=h$,Interchanging the order of integration gives the total fluxBy definition, the homogeneous effective fluid hasWriting $\epsilon =\delta /h$ therefore givesor the reciprocal of the right-hand side.