= Axisymmetric viscous-sheet stretching equations
For thickness $h(r,t)$, radial velocity $u(r,t)$ and external pressure $p_{\rm ext}$, the leading <Newtonian fluid stress tensor> and <incompressibility> give
$$
\sigma_{rr}=-p_{\rm ext}+2\mu(2u_r+u/r),\qquad
\sigma_{\theta\theta}=-p_{\rm ext}+2\mu(u_r+2u/r).
$$
Radial <force balance> on an annular sector includes the inward projection of hoop <traction> and external pressure on the sloping broad faces. Together with <conservation of mass> it yields
$$
2\mu\left[\partial_r(2rhu_r+hu)-h(2u/r+u_r)\right]=rh\,\partial_rp_{\rm ext},\qquad
h_t+r^{-1}\partial_r(rhu)=0.
$$
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