Diffuse-interface approach to competition between viscous flow and diffusion in pinch-off dynamics

The pinch-off dynamics of a liquid thread has been studied through numerical simulations and theoretical analysis. Occurring at small length scales, the pinch-off dynamics admits similarity solutions that can be classified into the Stokes regime and the diffusion-dominated regime, with the latter being recently experimentally observed in aqueous two-phase systems [Phys. Rev. Lett. 123, 134501 (2019)]. Derived by applying Onsager's variational principle, the Cahn–Hilliard–Navier–Stokes model is employed as a minimal model capable of describing the interfacial motion driven by not only advection but also diffusion. By analyzing the free energy dissipation mechanisms in the model, a characteristic length scale is introduced to measure the competition between diffusion and viscous flow in interfacial motion. This length scale is typically of nanometer scale for systems far from the critical point, but can approach micrometer scale for aqueous two-phase systems close to the critical point. The Cahn–Hilliard–Navier–Stokes model is solved by using an accurate and efficient spectral method in a cylindrical domain with axisymmetry. Ample numerical examples are presented to show the pinch-off processes in the Stokes regime and the diffusion-dominated regime respectively. In particular, the crossover between these two regimes is investigated numerically and analytically to reveal how the scaling behaviors of similarity solutions are to be qualitatively changed as the characteristic length scale is inevitably accessed by the pinching neck of the interface. Discussions are also provided for numerical examples that are performed for the breakup of long liquid filaments and show qualitatively different phenomena in different scaling regimes.

Figure 1

(a) The dimensionless neck radius ${\overline{r}}_n$ plotted as a function of the dimensionless time to pinch-off ${\overline{t}}^{*}={\overline{t}}_s-\overline{t}$. Using $\overline{H}=1.5$ and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.4037$ and ${\overline{r}}_0(\pm \overline{H})=0.5463$, numerical results are obtained for three different combinations of $\overline{\xi }$ and $B$. The dash-dotted lines are used to indicate the linear dependence, from which the slopes are found to be close to $0.0316\frac{\overline{\xi }}{B}$. (b–g) Six snapshots taken at $\overline{t}=0.02$, 12.78, 13.00, 13.10, 13.12, and 14.00 for $\overline{\xi }=0.005$ and $B=0.0005$ in (a).

Figure 2

Formation of mother drops and satellite drops in the fragmentation of long liquid filaments. Using $\overline{H}=12$ and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.28$ and ${\overline{r}}_0(\pm \overline{H})=0.32$, numerical results are obtained for $\overline{\xi }=0.02$ and $B=0.0005$. The eight snapshots are taken at $\overline{t}=0.008$, 1.64, 1.8, 1.92, 2, 2.072, 2.112, and 3.2. Note that the initial interfacial profile involves only one period of undulation.

Figure 3

(a) The dimensionless neck radius ${\overline{r}}_n$ plotted as a function of the dimensionless time to pinch-off ${\overline{t}}^{*}={\overline{t}}_s-\overline{t}$. Using $\overline{H}=2$ and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.25$ and ${\overline{r}}_0(\pm \overline{H})=0.35$, numerical results are obtained for $\overline{\xi }=0.005$ and $B=0.0005$. The dash-dotted line is used to indicate the linear dependence, from which the slope is found to be close to $0.0316\frac{\overline{\xi }}{B}$. (b)–(g) Six snapshots taken at $\overline{t}=0.02$, 1.80, 2.06, 2.08, 2.12, and 2.13. Note that the initial interfacial profile involves only one period of undulation.

Figure 4

Two sequences of multiple breakup events, one in the diffusion-dominated regime (a)–(h) and the other in the Stokes regime (i)–(p): (a)–(h) for $\overline{H}=8$, $\overline{\xi }=0.02$, $B=10$ and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.28$ and ${\overline{r}}_0(\pm \overline{H})=0.32$, and the eight snapshots are taken at $\overline{t}=0.8$, 36, 52, 64.8, 76, 112, 120.8, and 160; (i)–(p) for $\overline{H}=8$, $\overline{\xi }=0.02$, $B=0.0005$ and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.28$ and ${\overline{r}}_0(\pm \overline{H})=0.32$, and the eight snapshots are taken at $\overline{t}=0.008$, 1.92, 1.96, 2.16, 2.28, 2.68, 2.88, and 4.00.

Figure 5

The dimensionless neck radius ${\overline{r}}_n$ plotted as a function of ${\left({\overline{t}}^{*}\right)}^{1/3}$, with ${\overline{t}}^{*}=\overline{t_s}-\overline{t}$ being the dimensionless time to pinch-off. The data are obtained by using $B=0.1$ and $B=10$ and by solving the CH equation with $\overline{\mathbf{v}}=0$ (corresponding to $B=\infty$): (a) for $\overline{H}=2$, $\overline{\xi }=0.005$, and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.15$ and ${\overline{r}}_0(\pm \overline{H})=0.5$, and (b) is the same as (a), but with $\overline{\xi }=0.0025$.

Figure 6

The slope determined from $\frac{d{\overline{r}}_n^3}{d{\overline{t}}^{*}}$ vs $\overline{\xi }$ for $\overline{\xi }\le 0.006$. The data are obtained by solving the CH equation with $\overline{\mathbf{v}}=0$ (corresponding to $B=\infty$) for $\overline{H}=2$ and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.15$ and ${\overline{r}}_0(\pm \overline{H})=0.5$.

Figure 7

Time dependence of ${\overline{r}}_n$ showing two different scaling behaviors in the initial and final stages respectively, with the crossover occurring around $r_n\approx 2l_c$: (a) for $\overline{H}=2$, $\overline{\xi }=0.005$, $B=0.0015$, and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.15$ and ${\overline{r}}_0(\pm \overline{H})=0.8$. Panel (b) is the same as (a), but with $B=0.002$. The insets shows the Stokes regime (upper right) and the diffusion-dominated regime (lower left). The point of $r_n=2l_c$ (i.e., ${\overline{r}}_n=\sqrt{2B}$) is marked to indicate the occurrence of crossover. Each inset uses its own time of pinch-off $t_s$ to define ${\overline{t}}^{*}$(i.e., ${\overline{t}}_1^{*}$ or ${\overline{t}}_2^{*}$) for a linear fitting.

Figure 8

Contributions of advection and diffusion to the evolution of interface, obtained for the case in Fig. 7 with $B=0.002$ (i.e., $\frac{l_c}{L}=0.0316$): (a)–(c) are taken at ${\overline{r}}_n=0.128$ in the Stokes regime, showing the spatial distributions of $-\overline{\mathbf{v}}·\overline{\mathbf{\nabla }}\overline{\varphi }$ in (a), $\frac{1}{2}{\overline{\nabla }}^2\overline{\mu }$ in (b), and $\mathbf{v}$ (with the interface indicated by the color field) in (c), and (d)–(f) are taken at ${\overline{r}}_n=0.031$ in the diffusion-dominated regime, showing those corresponding to (a), (b), and (c).

Figure 9

$\mathcal{Y}$ vs $\mathcal{X}$ plotted in two different ways, where $\mathcal{Y}$ measures the ratio of diffusion to advection at the neck as defined in the main text, and $\mathcal{X}$ is defined by $\mathcal{X}=\frac{l_c}{r_n}=\frac{1}{{\overline{r}}_n}\sqrt{\frac{B}{2}}$. Here $\overline{H}=1.5$, $\overline{\xi }=0.005$, and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.125$ and ${\overline{r}}_0(\pm \overline{H})=0.85$. Different values of $B$ lead to different ranges of $\mathcal{X}$ measured from the evolving interfaces, and data collected from different evolution processes all follow $\mathcal{Y}=4.455{\mathcal{X}}^2$ approximately, showing semiquantitative agreement with Eq. (13).

Figure 10

Self-similar profiles in the pinching neck obtained at different time instants with different neck radius: (a) in the Stokes regime, computed for $\overline{H}=2$, $\overline{\xi }=0.005$, $B=0.0005$, and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.25$ and ${\overline{r}}_0(\pm \overline{H})=0.35$, in correspondence with Fig. 3, and (b) in the diffusion-dominated regime, computed for $\overline{H}=1.5$, $\overline{\xi }=0.0025$, $B=10$, and the initial interfacial profile with ${\overline{r}}_0\left(0\right)=0.15$ and ${\overline{r}}_0(\pm \overline{H})=0.85$.