Scaling law of contraction in the non-self-similar region of a pendant drop under finite gravity

Pendant-drop neck contraction under external stretching and moderate finite gravity left parenthesis 0.1 less than upper B o less than or equivalent to 0.5 right parenthesis ( 0.1 < B o ≲ 0.5 ) $(0.1\lt Bo\lesssim 0.5)$ is investigated via theoretical modelling and high-speed photography, focusing on the dynamic surface-tension-balancing mechanism and its transition to the inertial–capillary self-similar regime. A slender-jet model is developed to account for the coupled effects of inertia, viscosity, surface tension and external stretching. From the stretching dynamics and mass conservation, a viscosity-independent scaling h tilde tau Superscript 1 divided by 2 h ∼ τ 1 / 2 $h\sim \tau ^{1/2}$ for the neck radius is derived for italic We tilde 1 We ∼ 1 $ \textit{We}\sim 1$ . Experiments reveal a three-stage contraction: an early balancing regime with h tilde tau Superscript 1 divided by 2 h ∼ τ 1 / 2 $h\sim \tau ^{1/2}$ , followed by an inertial–capillary equilibrium stage and finally an inertial–capillary self-similar regime with h tilde tau Superscript 2 divided by 3 h ∼ τ 2 / 3 $h\sim \tau ^{2/3}$ . A smaller italic We We $ \textit{We}$ prolongs the tau Superscript 1 divided by 2