= Annular return flow around a slumping column
{title2=$P_z\sim6\lambda\mu aw/h^3$}
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 $q_g=wh/2-h^3P_z/(12\lambda\mu)\simeq-aw/2$, 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 $(a^2w_z)_z/a^2=\Delta\rho g/(3\mu)+2\lambda aw/h^3$. Curvature and the Couette flux change only the neglected relative order $h/a$.
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