= Capillary closure of a falling hollow thread
For steady equal-pressure flow, <mass conservation> gives $Aw=A_0w_0$ and the <kinematic boundary condition> gives $(D\sqrt w)_z=\gamma/(2\mu\sqrt w)$, where $D=R_e-R_i$. Therefore finite-height closure occurs exactly when
$$
\frac\gamma{2\mu\sqrt{w_0}}\int_0^{z_*}\frac{dz}{\sqrt{w(z)}}=\sqrt{A_0}-D_0.
$$
For positive <velocity> the <integral> is increasing. If it diverges, closure is finite; if its limiting value lies below the right-hand threshold, there is no finite closure. Quadratic freely falling profiles have logarithmically divergent <integrals>, while sufficiently strongly drawn exponentially accelerating profiles can have finite <integrals>. Thus nozzle speed alone is insufficient to guarantee closure: the axial tension or downstream condition matters.
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