Revisiting the Taylor-Culick approximation: Retraction of an axisymmetric filament

We numerically study the retraction of an axisymmetric viscous filament in a passive surrounding fluid. The analysis focuses on the evolution of the tip velocity, from the early stage of the filament retraction until it reaches its final equilibrium spherical shape. The problem is governed by two control parameters: the Ohnesorge number, Oh, which measures the relative importance of viscous and surface tension effects, and the initial aspect ratio of the filament, $\mathcal{A}$. We investigate the influence of Oh over a wide range of aspect ratios. The small-Oh regime is characterized by the occurrence of a spherical blob at the extremity of the filament. This feature has a key impact on the tip dynamics, which moves with an oscillating velocity whose mean value is close to the Taylor-Culick prediction. The oscillatory behavior of the tip velocity is explained through a simple mass-spring model. This regime is also characterized by the presence of capillary waves, with a phase velocity slightly larger than the Taylor-Culick velocity. Surface oscillations are also observed when the filament reaches its final spherical shape; the corresponding period agrees well with predictions of the linear theory. At intermediate Oh and large $\mathcal{A}$, the tip velocity reaches a value close to the Taylor-Culick prediction. However, for smaller aspect ratios, the maximum tip velocity is much smaller than this prediction, and does not exhibit any oscillation. The recoil dynamics is qualitatively and quantitatively different at high Oh. In this case, the radius of the filament grows uniformly over time and no blob forms, making the tip velocity decrease after a short transient. A self-similar solution is found to closely match the numerical results in this regime.

Figure 1

Sketch of the physical configuration, with the definition of the geometrical and physical parameters used.

Figure 2

Sketch of the computational domain for $\mathcal{A}=5$. Each gray region corresponds to a specific level of grid refinement. The cell size varies by a factor of two between two successive levels.

Figure 3

Successive shapes of a retracting filament with $\text{Oh}=0.1$ and $\mathcal{A}=10$ (assuming ${\mathcal{R}}_0=1$). The time step between two successive snapshots is $\mathrm{\Delta }t=2t_i$.

Figure 4

Tip velocity vs time during the early stages of the retraction process for $\text{Oh}=0.1$ and different initial aspect ratios. $–+–$ : $\mathcal{A}=5$, $–\times –$ : $\mathcal{A}=10$, $–*–$ : $\mathcal{A}=20$, $–\square –$ : $\mathcal{A}=40$, $\_$ : prediction (10), $\_\_$ : guide for the eye with slope $3U_i/\left(2t_i\right)$, $■$: numerical results from Ref. [10] for $\mathcal{A}=15$.

Figure 5

Tip velocity and neck radius vs time for $\text{Oh}=0.1$ and different aspect ratios $–+–$: $\mathcal{A}=5$, $–\times –$: $\mathcal{A}=10$, $–*–$: $\mathcal{A}=20$, $–\square –$ : $\mathcal{A}=40$. $ꖴ$ : neck radius for $\mathcal{A}=40$.

Figure 6

Interface and vorticity contours for $\text{Oh}=0.1$ and $\mathcal{A}=20$ at (a) time $t/t_i=10$ and (b) time $t/t_i=12$; the azimuthal vorticity ${\omega }_{\theta }$ is normalized with $U_i/{\mathcal{R}}_0$.

Figure 7

Tip velocity vs time for $\text{Oh}=0.1$ and different initial aspect ratios. $–*–$ : $\mathcal{A}=20$, $–\square –$ : $\mathcal{A}=40$, $-$ : $U_i\left[1-{(6t_i/t)}^{2/3}\right]$.

Figure 8

Late evolution of the filament for $\text{Oh}=0.1$ and different aspect ratios. (a) Tip velocity. $–+–$: $\mathcal{A}=5$, $–\times –$ : $\mathcal{A}=10$, $–*–$ : $\mathcal{A}=20$, $–\square –$ : $\mathcal{A}=40$. (b)–(e) Distribution of the normalized pressure $p/\left(\rho U_i^2\right)$ within the filament for $\mathcal{A}=5$. Snapshots in the bottom row are taken at time instants (increasing from left to right) identified with a vertical dashed line in the top figure; the time step between two successive images is $0.6t_i$.

Figure 9

Characteristics of capillary oscillations during the final stage for $\text{Oh}=0.1$. Left: Period of the oscillations vs the initial aspect ratio, $\mathcal{A}$; $+$: Simulation results, $-$: linear prediction (16) for mode $l=2$. Right: final decay of the oscillations vs time; $–\times –$: $\mathcal{A}=10$, $–\square –$: $\mathcal{A}=40$, dotted line: linear prediction (16) for mode $l=2$.

Figure 10

Tip velocity vs time for a filament with $\text{Oh}=1$ and different initial aspect ratios. Left: Early stage of recoil; right: intermediate and late stages. $–+–$: $\mathcal{A}=5$, $–\times –$: $\mathcal{A}=10$, $–*–$: $\mathcal{A}=20$, $–\square –$: $\mathcal{A}=40$; $\_$: short-time prediction (10), $■$: numerical results from Ref. [10] for $\mathcal{A}=15$.

Figure 11

Tip velocity vs time for $\text{Oh}=10$ and different initial aspect ratios during the initial transient. $–+–$: $\mathcal{A}=5$, $–\times –$: $\mathcal{A}=10$, $–*–$: $\mathcal{A}=20$, $–\square –$: $\mathcal{A}=40$, $\_\_$: prediction (22).

Figure 12

Tip velocity vs time for $\text{Oh}=10$ and different initial aspect ratios. $–+–$: $\mathcal{A}=5$, $–\times –$: $\mathcal{A}=10$, $–*–$: $\mathcal{A}=20$, $–\square –$: $\mathcal{A}=40$.

Figure 13

Successive shapes of a retracting filament with $\text{Oh}=10$ and $\mathcal{A}=10$ (assuming ${\mathcal{R}}_0=1$). The time step between two successive snapshots is $\mathrm{\Delta }t=1.2t_v$.

Figure 14

Evolution of the filament radius in the midplane $z=0$ for $\text{Oh}=10$. $–+–$: $\mathcal{A}=5$, $–\times –$: $\mathcal{A}=10$, $–*–$: $\mathcal{A}=20$, $–\square –$: $\mathcal{A}=40$, $\_\_$: theoretical prediction (24).

Figure 15

Evolution of the tip velocity for $\text{Oh}=10$. $–+–$: $\mathcal{A}=5$, $–\times –$: $\mathcal{A}=10$, $–*–$: $\mathcal{A}=20$, $–\square –$: $\mathcal{A}=40$; dotted lines: theoretical prediction (25).

Figure 16

Tip velocity versus time for $\text{Oh}=0.1$ and $\mathcal{A}=20$. (left) Early stage stage of the recoil process; (right): intermediate stage. $--$: prolate spheroidal end; $–*–$: spherical end; $-$: prediction (10); $■$: numerical results from Ref. [10] for $\mathcal{A}=15$.

Figure 17

Same as Fig. 16 for $\text{Oh}=0.05$.