 Annular return flow around a slumping column

In a closed-base cylindrical container, mass conservation makes the annular volume flux opposite to the core flux. With $h=R-a\ll a$ and annular viscosity $\lambda \mu$, Couette-Poiseuille flow in a thin gap gives $q_g=wh/2-h^3{P}_z/(12\lambda \mu )\simeq -aw/2$, where $P$ removes the annular hydrostatic pressure. Hence $P_z=6\lambda \mu aw/h^3[1+O(h/a)]$. The resulting column equation is $(a^2{w}_z{)}_z/a^2=\Delta \rho g/(3\mu )+2\lambda aw/h^3$. Curvature and the Couette flux change only the neglected relative order $h/a$.