Liquid Helix: How Capillary Jets Adhere to Vertical Cylinders

From everyday experience, we all know that a solid edge can deflect a liquid flowing over it significantly, up to the point where the liquid completely sticks to the solid. Although important in pouring, printing, and extrusion processes, there is no predictive model of this so-called “teapot effect.” By grazing vertical cylinders with inclined capillary liquid jets, here we use the teapot effect to attach the jet to the solid and form a new structure: the liquid helix. Using mass and momentum conservation along the liquid stream, we first quantitatively predict the shape of the helix and then provide a parameter-free inertial-capillary adhesion model for the jet deflection and critical velocity for helix formation.

Figure 1

(a) Schematic of the experiment indicating the initial inclination angle ${\psi }_0$ and helical pitch ${\lambda }_0$. (b) Side-view pictures of a sequence of experiments showing the deviation of a 0.5 mm water jet grazing a 3.0 mm glass cylinder. The flow rate $Q$ is decreased until the penultimate image and then increased again to illustrate the hysteresis in the sticking transition.

Figure 2

(a) Sketch of the (unwrapped) rivulet model. The impacting jet turns into a rivulet whose centerline (drawn as a red dashed line) is parametrized by the arc length $s$ and the angle with respect to the vertical axis $\psi (s)$. Inset: Initial helix pitch ${\lambda }_0/{\mathcal{R}}_h$ as a function of the jet inclination angle ${\psi }_0$ for varying $D_j$, $D_c$, and initial velocity $U_{0,X}$ (color coded in m/s). Dashed line is the inertial prediction ${\lambda }_0=2\pi {\mathcal{R}}_h/\tan {\psi }_0$. (b) Theoretical helix shape including the effect of gravity with (orange curve) and without (blue curve) viscous effects for typical experimental parameters ($D_c=10\mathrm{mm}$, $D_j=0.8\mathrm{mm}$, ${\psi }_0=64°$, $U_0=2.1\mathrm{m}/\mathrm{s}$, and $C=11$).

Figure 3

Comparison between experiments and theory for a range of experimental parameters. From left to right: $D_c=1$, 5, 5, 10 mm; $D_j=1$, 0.3, 0.5, 0.3 mm; ${\psi }_0=26.3$, 40.3, 47.8, 68.6 deg; $U_0=1.0$, 4.9, 3.0, $5.4\mathrm{m}/\mathrm{s}$; $C=15$, 11.5, 7, 9.5.

Figure 4

(a) Top view of a typical experiment ($D_j=0.5\mathrm{mm}$, glass cylinder $D_c=10\mathrm{mm}$, $Q=0.77\mathrm{mL}/\mathrm{s}$, ${\psi }_0=91.5°$). (b) Sketch of the 2D model defining the geometric parameters. The control volume for which we consider the momentum balance is indicated by the red dashed line. (c) Jet deviation angle $\alpha$ as a function of the Weber number We for different inclination angles ${\psi }_0$ and dimensionless cylinder radii $\tilde{R}=D_c/(2D_j)$ (glass, $\theta =32\pm 15°$). (d) Same data plotted as a function of the parallel Weber number ${\mathrm{We}}_{\parallel }=\mathrm{We}{\sin }^2({\psi }_0)$. Dashed lines are results from our (parameter-free) theory [Eqs. (S9) and (S10) in the Supplemental Material [18] ]. (e) Critical Weber number ${\mathrm{We}}_{\parallel c}$ as a function of $\tilde{R}(1+\cos \theta )$. All experimental parameters ($D_j$, $D_c$, $\theta$, and ${\psi }_0$) are varied. The purple solid curve is the full numerical solution [Eqs. (S9) and (S10) in the Supplemental Material [18] ] (the gray area represents the variations as $\theta$ is varied from 0° to 179°). The black dashed line is the asymptotic expansion in the large $\tilde{R}$ limit.