Universal Free-Fall Law for Liquid Jets under Fully Developed Injection Conditions

We show that vertical slender jets of liquid injected in air with a fully-developed outlet velocity profile have a universal shape in the common case in which the viscous force is much smaller than the gravitational force. The theory of ideal flows with vorticity provides an analytical solution that, under negligible surface tension forces, predicts $R_j(Z)=[(1+Z\mathrm{/}4{)}^{1\mathrm{/}2}-(Z\mathrm{/}4{)}^{1\mathrm{/}2}{]}^{1\mathrm{/}2}$, where $R_j$ is the jet radius scaled with the injector radius and $Z$ is the vertical distance scaled with the gravitational length, $l_g=u_o^2\mathrm{/}2g$, where $u_o$ is the mean velocity at the injector outlet and $g$ is the gravitational acceleration. In contrast with Mariotte’s law, $R_j=(1+Z{)}^{-1\mathrm{/}4}$, previously reported experiments employing long injectors collapse almost perfectly onto the new solution.