Dynamics of retracting surfactant-laden ligaments at intermediate Ohnesorge number

The dynamics of ligaments retracting under the action of surface tension occurs in a multitude of natural and industrial applications; these include inkjet printing and atomization. We perform direct, fully three-dimensional, two-phase numerical simulations of the retracting process over a range of system parameters that account for surfactant solubility, sorption kinetics, and Marangoni stresses. Our results indicate that the presence of surfactant inhibits the “end-pinching” mechanism and promotes neck reopening through Marangoni-flow; this is induced by the formation of surfactant concentration gradients that drive flow-reversal toward the neck. The vortical structures associated with this flow are also analyzed in detail. We also show that these Marangoni stresses lead to interfacial rigidification, observed through a reduction of the retraction velocity and ligament kinetic energy.

Figure 1

Schematic representation of the initial shape of the ligament (a) highlighting the fluid quantities and nondimensional parameters of the system, and (b) initial shape of the ligament with the computational domain of size $8R\times 8R\times 32R$ in a three-dimensional Cartesian domain $\mathbf{x}=(x,y,z)$ and divided into $6\times 6\times 12$ subdomains; the Cartesian resolution is set to $32\times 32\times 64$ per subdomain, and the global resolution is $192\times 192\times 768$. Typical interfacial shape for a surfactant-free case at $t=7$ for $L_o=15$ and $\text{Oh}={10}^{-2}$ (c) with definitions of particular locations and features whose dynamics will be discussed in the present work. (d) Typical interfacial shape for a surfactant-laden case at $t=7$ with $L_o=15$, $\text{Oh}={10}^{-2}$, $\text{Bi}=1$, ${\text{Pe}}_s=10$, and $\chi =0.9$ with color bars indicating the magnitude of the surfactant interfacial and bulk concentrations in the left and right halves of the ligament, respectively.

Figure 2

Spatiotemporal evolution of a retracting ligament for $L_o=15$ and $\text{Oh}={10}^{-2}$. The solid lines correspond to the results of the present study and the diamonds are collected from Ref. [8].

Figure 3

Surfactant-free ligament retraction for $L_o=15$ and $\text{Oh}={10}^{-2}$: (a) Temporal evolution of the location of both ligament tips, and a three-dimensional representation of the interface for the dimensionless times shown in the panel in which the color bar depicts the velocity magnitude; (b) temporal evolution of the total, kinetic and surface energies, $E_T$, $E_K$, and $E_S$, respectively, and the energy dissipated, $E_D$; (c) scaling of the minimum radius with respect to the pinchoff time $t_o=11.115$, which agrees with that predicted by inertio-viscous scaling theory of Ref. [17]; (d)–(f) time-space plots of the interface, $p$, and $u_z$, respectively, with snapshots shown between $t=013.5$, with time intervals of 1 between $t=011$ and 0.5 between $t=1113.5$; here, the red profiles are associated with $t=11.5$ and $t=12$, the instances at which pinchoff occurs.

Figure 4

Ligament retraction with an insoluble surfactant for $L_o=15$, $\text{Oh}={10}^{-2}$, ${\text{Pe}}_s=10$, ${\beta }_s=0.3,$ and ${\mathrm{\Gamma }}_o={\mathrm{\Gamma }}_{\infty }/2$: (a) temporal evolution of the north tip location for the surfactant-free and surfactant-laden cases, and a three-dimensional representation of the interface for the dimensionless times shown in the panel in which the color bar depicts the magnitude of the surfactant interfacial concentration, $\mathrm{\Gamma }$; (b) temporal evolution of the Kinetic energy $E_K$ for the surfactant-free and surfactant-laden cases; (c) temporal evolution of the neck radius highlighting the two escapes of pinchoff; (d)–(g) time-space plots of the interface, $\mathrm{\Gamma }$, $p$, $u_z$ with snapshots shown between $t=016$ at equal time intervals; here, the red profiles are associated with $t=10$.

Figure 5

Spatial variation of the interfacial shape, $\mathrm{\Gamma }$, and $\tau$ (a), the tangential interfacial velocity, $u_t$, for the surfactant-free and surfactant-laden cases (b), and the streamline structure within the retracting ligament, (c); the parameter values are the same as in Fig. 4 with $t=10$.

Figure 6

Dynamics of a retracting, surfactant-covered ligament with isolating the Marangoni effect: (a) temporal evolution of the north tip location for the surfactant-free and surfactant-laden cases, and a three-dimensional representation of the interface for the dimensionless times shown in the panel in which the color bar depicts the magnitude of the surfactant interfacial concentration, $\mathrm{\Gamma }$; (b) temporal evolution of the kinetic energy $E_K$ for the surfactant-free and surfactant-laden cases.

Figure 7

Illustration of the surfactant-mediated mechanism underlying the first escape from pinchoff in Fig. 4 depicting the vorticity ${\omega }_{\theta }$ and streamlines for the surfactant-free case at $t=10.6$, 10.8, and 11 (a), and the surfactant laden base case (b) at $t=12.0$, 12.3, and 12.4, with the same parameters as in Fig. 4. The framed regions of the ligament show a magnified view of the flow direction at every snapshot.

Figure 8

Spatiotemporal evolution of the interfacial shape, $u_z$, and $p$ for the surfactant-free case (a), and the surfactant-laden case (b), for the same parameters as those used to generate Fig. 7. In (b), we have also plotted the evolution of $\mathrm{\Gamma }$ and $\tau$, and labeled the the direction of the Marangoni-induced and capillary-driven flows with red and blue arrows, respectively. The dimensionless times are shown in each panel.

Figure 9

Illustration of the surfactant-mediated mechanism underlying the second escape from pinchoff in Fig. 4 depicting the vorticity ${\omega }_{\theta }$ and streamlines (a) at $t=13.8$, 14.4, and 14.5. The framed regions of the ligament show a magnified view of the flow direction at each time. In (b), we have plotted the spatiotemporal evolution of the interfacial shape, $\mathrm{\Gamma }$, $u_z$, and $\tau$ and labeled the direction of the Marangoni-induced and capillary-driven flows with red and blue arrows, respectively.

Figure 10

Effect of ${\beta }_s$ on the retraction dynamics for $L_o=15$, $\text{Oh}={10}^{-2}$, ${\text{Pe}}_s=10$, and ${\mathrm{\Gamma }}_o=\chi {\mathrm{\Gamma }}_{\infty }/2$: (a, b) temporal evolution of the north-tip location, and the kinetic energy, $E_k$, respectively; panels (a) and (b) also show three-dimensional representations of the interface for ${\beta }_s=0.5$ and 0.1, respectively, and for the dimensionless times shown in the panels in which the color bar depicts the magnitude of the surfactant interfacial concentration, $\mathrm{\Gamma }$. Panels (c), (e) and (d), (f) show the spatial variation of the interfacial shape, $\mathrm{\Gamma }$, and $\tau$, and the tangential interfacial velocity, $u_t$, for the surfactant-free and surfactant-laden cases, respectively, at $t=10$. In panels (c), (e) and (d), (f), ${\beta }_s=0.1$ and 0.5, respectively. Note: in (c) interface location is superimposed, however, axis is not shown.

Figure 11

Illustration of the no-escape from pinchoff for ${\beta }_s=0.1$ at $t=13$ and 13.2. The rest of the parameters are the same as in Fig. 10.

Figure 12

Effect of ${\text{Pe}}_s$ on the retraction dynamics for $L_o=15$, $\text{Oh}={10}^{-2}$, ${\beta }_s=0.3$, $\chi =0.9$ and ${\mathrm{\Gamma }}_o=\chi {\mathrm{\Gamma }}_{\infty }/2$: (a) temporal evolution of the north-tip location and three-dimensional representations of the interface for ${\text{Pe}}_s=0.1$ and for the dimensionless times shown in the panels in which the color bar depicts the magnitude of the surfactant interfacial concentration, $\mathrm{\Gamma }$; (b) temporal evolution of the kinetic energy, $E_k$. respectively; panels (a) and (b) also show ${\beta }_s=0.5$ and 0.1, panels (c) and (d) show the spatial variation of the interfacial shape, $\mathrm{\Gamma }$, and $\tau$, and the tangential interfacial velocity, $u_t$, for the surfactant-free and surfactant-laden cases, respectively, for ${\text{Pe}}_s=0.1$ and at $t=10$. Note: in panels (c) and (e) interface location is superimposed; however, axis is not shown.

Figure 13

Effect of Bi on the retraction dynamics for $L_o=15$, $\text{Oh}={10}^{-2}$, ${\beta }_s=0.3$, ${\text{Pe}}_s=10$, ${\text{Pe}}_b=10,$ and ${\mathrm{\Gamma }}_o=\chi {\mathrm{\Gamma }}_{\infty }/2$: (a) and (b) temporal evolution of the north-tip location, and the kinetic energy, $E_k$, respectively; panels (a) and (b) also show three-dimensional representations of the interface for $\text{Bi}=1$ and ${10}^{-3}$, respectively, and for the dimensionless times shown in the panels in which the color bar depicts the magnitude of the surfactant interfacial concentration, $\mathrm{\Gamma }$. Panels (c) and (e), and panels (d) and (f) show the spatial variation of the interfacial shape, $\mathrm{\Gamma }$, and $\tau$, and the tangential interfacial velocity, $u_t$, for the surfactant-free and surfactant-laden cases, respectively, at $t=10$. In panels (c) and (e), and panels (d) and (f), $\text{Bi}={10}^{-3}$ and 1, respectively.

Figure 14

Effect of $\chi$ on the retraction dynamics for $L_o=15$, $\text{Oh}={10}^{-2}$, ${\beta }_s=0.3$, ${\text{Pe}}_s=10$, ${\text{Pe}}_b=10,$ and $\text{Bi}=0.1$: (a) and (b) temporal evolution of the north-tip location, and the kinetic energy, $E_k$, respectively; panels (a) and (b) also show three-dimensional representations of the interface for $\chi =0.1$ and 0.99, respectively, and for the dimensionless times shown in the panels in which the color bar depicts the magnitude of the surfactant interfacial concentration, $\mathrm{\Gamma }$. Panels (c) and (e), and panels (d) and (f) show the spatial variation of the interfacial shape, $\mathrm{\Gamma }$, and $\tau$, and the tangential interfacial velocity, $u_t$, for the surfactant-free and surfactant-laden cases, respectively, at $t=10$. In panels (c) and (e), and panels (d) and (f), $\chi =0.1$ and 0.99, respectively. Note: in panels (c) and (e) interface location is superimposed; however, axis is not shown.

Figure 15

Mesh study for “surfactant-free” (panels (a), (c), and (e) on the left) when $L_o=15$ and $\text{Oh}={10}^{-2}$, and “surfactant-laden base case” (panels (b), (d), and (f) on the right) when $L_o=15$, $\text{Oh}={10}^{-2}$, ${\beta }_s=0.3,$ and ${\text{Pe}}_s=10$. The panels highlight the temporal evolution of the north and south tips location (a), (b), the kinetic energy $E_k$ (c), (d), and the relative variation of the liquid volume (e), (f).