 Pressure-driven annular Bingham flow with a free surface

For a Bingham layer on a stationary cylinder of radius $a$ with a shear-free cylindrical surface at $b$, an axial pressure gradient $-G$ produces ${\tau }_{rz}=G(b^2/r-r)/2$. Flow starts when $G(b^2-a^2)/(2a)$ exceeds the yield stress.