One-dimensional reduction of viscous jets. II. Applications

In a companion paper [Phys. Rev. E 97, 043115 (2018)], a formalism allowing to describe viscous fibers as one-dimensional objects was developed. We apply it to the special case of a viscous fluid torus. This allows to highlight the differences with the basic viscous string model and with its viscous rod model extension. In particular, an elliptic deformation of the torus section appears because of surface tension effects, and this cannot be described by viscous string nor viscous rod models. Furthermore, we study the Rayleigh-Plateau instability for periodic deformations around the perfect torus, and we show that the instability is not sufficient to lead to the torus breakup in several droplets before it collapses to a single spherical drop. Conversely, a rotating torus is dynamically attracted toward a stationary solution, around which the instability can develop freely and split the torus in multiple droplets.

Figure 1

Notation in the plane of constant $\theta$.

Figure 2

The initial torus shape is ${\epsilon }_i=\frac{1}{3}$ in both cases. (a) A low-viscosity case ($\mu =0.01{\mu }_0$) and (b) a high-viscosity case ($\mu =100{\mu }_0$). The continuous lines correspond to our model, the dashed lines to the basic rod model which is equivalent to the viscous string model in this case. The central curves correspond to the central line, the top curves to the outer intersection of the section with the torus plane ($z=0$), and the lower curves to the inner intersection. In (a), the inviscid solution [Eq. (3.15)] is depicted in a thin and continuous gray line.

Figure 3

The initial aspect ratio is ${\epsilon }_i=0.1$. (a) The dots are the adimensionalized growth rate $\alpha R_i^{3/2}{\nu }^{-1/2}$ as a function of the mode integer $n$, for an initial shape characterized by ${\epsilon }_i=0.1$. The continuous line corresponds to the result obtained for the Rayleigh-Plateau instability on an infinite straight line. From top to bottom we show the cases $\mu =0.1\sqrt{\nu R_i}$, $\mu =\sqrt{\nu R_i}$, and $\mu =10\sqrt{\nu R_i}$. (b) Growth rate relative difference with respect to the one obtained for the Rayleigh-Plateau instability of an infinite straight line. Our model (circles) and the improved rod model (crosses) is evaluated for large symbols with $\mu =0.1\sqrt{\nu R_i}$ (low viscosity) and for small symbols with $\mu ={10}^3\sqrt{\nu R_i}$ (high viscosity). In all cases, the dipolar mode ($n=1$) growth rate is identically vanishing as required by center of mass conservation.

Figure 4

(a), (b) $\delta R/\delta R_i$. (c), (d) $\delta r/\delta R_i$. The initial shape is characterized by ${\epsilon }_i=0.1$. The viscosity is low ($\mu =0.1{\mu }_0$) for (a) and (c) and high ($\mu =10{\mu }_0$) for (b) and (d). The instability modes are plotted from $n=1$ to 10 for (a) and (c), or from $n=2$ to 10 for (b) and (d), starting from the thinnest line to the thickest line.

Figure 5

For both figures ${\overline{\mathcal{V}}}_i=0.8\sqrt{\nu /R_i}$, $\mu ={\mu }_0$. (a) ${\epsilon }_i=\frac{1}{3}$. The continuous lines correspond to our model and the dashed lines correspond to the viscous string model with circular sections. The thick dotted-dashed line is $-\mathcal{E}{R}^2$ whereas the thin dotted-dashed line is $R^2/\left(4r^2\right)$. (b) ${\epsilon }_i=0.1$ and the lines (from thin to thick) correspond to the modes from $n=1$ to 8 of $\left|\delta R\right|/\delta R_i$. The dipolar mode ($n=1$) does not grow because of center of mass conservation.