Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 352 / 1 / b / Solution

Set $G=\Delta p/L>0$, so $dp/dz=-G$. For the unidirectional velocity $\mathbf{u}=u(r){\mathbf{e}}_z$, the axial inertialess Cauchy momentum equation isThe free surface is shear-free, ${\tau }_{rz}(b)=0$, and integration givesThis result follows from force balance alone, so it is independent of the constitutive equation and of ${\sigma }_y$.

The axial shear force exerted by the liquid on the cylindrical fibre is thereforeIts direction is downstream; the reaction on the fluid is upstream. In particular, the yield stress changes the velocity field but not the total force required by the imposed pressure drop.