Thermocapillary effect on the dynamics of viscous beads on vertical fiber

The gravity-driven flow of a thin liquid film down a uniformly heated vertical fiber is considered. This is an unstable open flow that exhibits rich dynamics including the formation of droplets, or beads, driven by a Rayleigh-Plateau mechanism modified by the presence of gravity as well as the variation of surface tension induced by temperature disturbance at the interface. A linear stability analysis and a nonlinear simulation are performed to investigate the dynamic of axisymmetric disturbances. The results showed that the Marangoni instability and the Rayleigh-Plateau instability reinforce each other. With the increase of the thermocapillary effect, the fiber flow has a tendency to break up into smaller droplets.

Figure 1

Sketch of the geometry of a fiber flow.

Figure 2

The dispersion relation of the real growth rate vs the wave number. (a) For various Marangoni numbers at $\text{Bi}=1.0$, (b) for various Biot number at $\text{Ma}=1.0$. Other parameters are $\epsilon =0.2,a=0.5$.

Figure 3

Effect of $\text{Bi}$ on the real growth rate and the wave number of the most unstable mode for various $a$. (a) The curves of ${\lambda }_m$ vs $\text{Bi}$, (b) the curves of $k_m$ vs $\text{Bi}$. The other parameters are $\epsilon =0.2,\text{Ma}=1.0$.

Figure 4

Effect of $\text{Ma}$ on the profiles of the interface via numerical simulations for the most unstable disturbances. (a), $a=0.2551,\epsilon =0.2915,k=2.47,2.78,3.09$ for $\text{Ma}=0$, 1, 2; (b), $a=0.2856,\epsilon =0.233,k=3.09,3.47,3.73$ for $\text{Ma}=0,1,2;$ (c), $a=0.3262,\epsilon =0.178,k=4.045,4.3820,4.8876$ for $\text{Ma}=0,1,2.$

Figure 5

Time snapshots showing the evolution of the most unstable mode in a periodic domain of $2\pi /k_m$ from $t=0$ to 20 for $a=0.2551,\epsilon =0.2915$. Interfacial profiles $S$ are shown at time intervals $\Delta t=0.2$. (a) $k=2.47,\text{Ma}=0$; (b) $k=3.09,\text{Ma}=2$.

Figure 6

The profiles of the interface via transient numerical simulations for various Marangoni numbers. Other parameters are $l=15,a=0.2551,\epsilon =0.2915$. $\text{Ma}=0,1,2$ for (a), (b), and (c).

Figure 7

The profiles of the interface via transient numerical simulations for various Marangoni numbers. Other parameters are $l=15,a=0.2856,\epsilon =0.233$. $\text{Ma}=0,1,2$ for (a), (b), and (c).

Figure 8

The profiles of the interface via transient numerical simulations for various Marangoni numbers. Other parameters are $l=15,a=0.3262,\epsilon =0.178$. $\text{Ma}=0,1,2$ for (a), (b), and (c).

Figure 9

Traveling wave solution: spacings $l$ and propagating speeds $c$ for various $a$ and $\text{Ma}$ at $\epsilon =0.2$.

Figure 10

The profiles of the interface and the isostreamlines for $a=0.25$. Other parameters are (a) $\text{Ma}=0,l=2$, (b) $\text{Ma}=0.8,l=2$, (c) $\text{Ma}=0,l=20$, (d) $\text{Ma}=0.8,l=20$. Negative and positive values of the stream function are denoted by dashed and solid lines.

Figure 11

The profiles of the interface and the isostreamlines for $a=0.5$. Other parameters are (a) $\text{Ma}=0,l=5$, (b) $\text{Ma}=0.8,l=5$, (c) $\text{Ma}=0,l=20$, (d) $\text{Ma}=0.8,l=20$. Negative values are denoted by dashed lines.