Fig. 4.

Role of the turbulence in kink formation. (A) Ensemble average of the normalized average neck diameter $⟨\overline{d}/d_{\text{n}}⟩$ over all experiments for various needles and levels of turbulence, with the time until pinch-off normalized by the turbulent timescale at the needle length scale, $t_{\text{turb}}={\left(d_{\text{n}}^2/ϵ\right)}^{1/3}$. The shaded area indicates the standard deviation, and the black line indicates the $\alpha =1/2$ inertial power law scaling. Inset shows the distribution of the exponent $\alpha$ obtained by fitting $\overline{d}\propto {\left(t_0-t\right)}^{\alpha }$ for each experiment (Materials and Methods). The distribution is centered around 0.5. (B) The portion of cases with a resolved kink at each experimental condition as a function of $d_{\text{n}}/d_{\text{H}}$, with kinks rarely appearing at low $d_{\text{n}}/d_{\text{H}}$ but almost always being resolved as the ratio approaches 1. (C) The neck size when the kink appears, $d_k$, as a function of $d_{\text{n}}/d_{\text{H}}$, which exhibits a power law relation $\left({\overline{d}}_{\text{k}}/d_{\text{H}}\right)\sim {\left(d_{\text{n}}/d_{\text{H}}\right)}^{\beta }$, with $\beta =2.1$ obtained by a fit (which is shown in green). (D) Snapshots of the bubble at $t_0-t=1\hspace{0.17em}\text{ms}$ (Upper) and 2/3 of the way between kink formation and pinch-off (Lower) for cases at 3 different $d_{\text{n}}/d_{\text{H}}$, with the background shading mapped to the color bar in B. When $d_{\text{n}}/d_{\text{H}}$ increases, larger initial deformations lead to earlier and larger kink formation. (Scale bar: 1 mm.)