 Capillary closure of a falling hollow thread

For steady equal-pressure flow, mass conservation gives $Aw=A_0{w}_0$ 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 whenFor 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.