Put $G=\Delta p/L>0$, with flow in the positive $z$ direction, so $dp/dz=-G$. Balancing pressure forces and wall shear on a cylindrical fluid volume of radius $R$ and length $L$ gives
$$
\Delta p\,\pi R^2=2\pi RL\,\tau_R,
\qquad
\boxed{\tau_R=\frac{\Delta p\,R}{2L}=\frac{GR}{2}}.
$$
Thus the wall stress is fixed before any <constitutive equation> is specified.

For fully developed <pipe flow>, the axial Cauchy equation reduces to
$$
-G=\frac1r\frac d{dr}(r\tau_{rz}).
$$
Regularity at the axis removes the $1/r$ integration constant, so
$$
\boxed{\tau_{rz}(r)=-\frac{Gr}{2}
=-\tau_R\frac rR}.
$$
Its magnitude rises linearly from zero at the axis to $\tau_R$ at the wall.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 352 / 1 / c / Solution

{wiki=Pipe_flow}

Pipe flow is fluid motion through a closed cylindrical conduit. In fully developed axisymmetric flow, the axial velocity depends only on radius and the axial pressure gradient sets a shear stress linear in radius.


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