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

The local motion is simple shear with flow direction $z$, gradient direction $r$, and vorticity direction $\theta$. Writing $q=du/dr$, a second-order fluid has first and second normal-stress differencesUp to an isotropic contribution absorbed into pressure, a convenient representation isThus both the flow-direction and gradient-direction normal stresses can be nonzero, while the radial momentum equation becomesThe first normal-stress coefficient affects ${\tau }_{zz}$ but cancels from radial balance.

In the yielded layer,Moreover,Consequently $dp/dr$ has the sign opposite to ${\Psi }_2$ and is independent of ${\Psi }_1$. For the common case ${\Psi }_2<0$, pressure increases radially outward toward the free surface.