Dripping of a Liquid from a Tube in the Absence of Gravity

In zero gravity, liquid does not drip from a tube but hangs from it for all time as a pendant drop. It is shown that a drop can be made to fall off a tube by heating the contact line (CL), which causes a temperature gradient between the drop’s CL and its tip. Simulations show that whether a drop detaches from or remains attached to the tube is set by the competition between the Marangoni stress that causes flow away from its CL, where the surface tension $\sigma$ is low, toward its tip, where $\sigma$ is high, and the capillary pressure that causes flow from the tip back toward the CL. The method can be used to make microscopic (nanoscopic) drops by dripping in ordinary gravity, which had heretofore been unattainable.

Figure 1

Evolution in time $t$ of shapes of drops and flow and temperature fields when $\alpha =0.8$ and $\mathrm{Bi}=0.001$ for $\mathrm{Pe}=1$. The initial state of the drop, whose shape is a spherical section with its center located by the symbol ×, is depicted at $t=0$ before heating. At each subsequent time, streamlines are shown to the right of $r=0$ and temperature fields, contour values of which are indicated in the legend, are shown to its left. In this and subsequent figures, $\mathrm{Re}=100$, $\mathrm{Br}=0.001$, ${\mu }_p=0.3$, and ${\beta }_p=0.7$. The dashed and solid bold arrows at $t=1.01$ indicate the directions of the Marangoni stress $\mathrm{Re}{\nabla }_s\sigma$ and the negative of the capillary pressure gradient $-\nabla p_c$.

Figure 2

Same as Fig. 1 except for $\mathrm{Pe}=100$.

Figure 3

Evolution in time $t$ of the variation along the drop surface of temperature $\theta$ and Marangoni stress $\mathrm{Re}{\nabla }_s\sigma$ when $\alpha =0.8$ and $\mathrm{Bi}=0.001$ for (a) $\mathrm{Pe}=1$ and (b) $\mathrm{Pe}=100$.

Figure 4

Evolution in time of the drop length $L(t)$ when $\alpha =0.8$ and $\mathrm{Bi}=0.001$ for $\mathrm{Pe}=1$ (solid line) and $\mathrm{Pe}=100$ (dashed line). The time at which drop breakup occurs is indicated by the symbol × on each curve. The drops oscillate when $t<2$, which is shown more clearly by the inset.

Figure 5

Evolution in time of the drop length $L(t)$ when $\alpha =0.5$ and $\mathrm{Pe}=1$ for $\mathrm{Bi}=0.001$. The drop exhibits a stable damped oscillatory response without breakup.

Figure 6

Variation of breakup time $t_b$ with drop size parameter $\alpha$ when $\mathrm{Pe}=10$ for $\mathrm{Bi}=0.001$ and $\mathrm{Bi}=1$. The shape insets show drop profiles and temperature fields within the drops at the incipience of breakup.