Simulation of surfactant-mediated tipstreaming in a flow-focusing geometry

Simulations are performed of a surfactant-laden drop that is stretched by an imposed uniaxial extension flow at infinity with flow focusing provided by two transverse, coaxial, annular baffles placed symmetrically to either side of the drop. The geometry is axisymmetric, with additional symmetry in the transverse plane that contains the drop center. Under suitable conditions, the drop can enter a mode of drop breakup referred to as tipstreaming, in which a thin elongated filament or thread is emitted from the drop ends and which subsequently breaks up into small droplets via capillary instability. The influence that flow focusing has on the conditions required for tipstreaming and on quantities such as the thread radius are investigated by study of sample simulations and the extent of flow focusing is varied by changing the inner or aperture radius of the annular baffles. The surfactant is soluble and bulk-interface surfactant exchange is in the mixed-kinetic or finite-Biot-number regime. The boundary-integral method is used for the underlying two-phase Stokes flow solver, combined with a finite-difference scheme for evolution of adsorbed surfactant on the interface. The dynamics of dissolved bulk phase surfactant is resolved by a large-bulk-Péclet-number asymptotic approach. Results on the conditions for tipstreaming in the simulations are compared to separate experimental results on conditions for tipstreaming in a microfluidic flow-focusing device.

Figure 1

(a) A drop is situated in a uniaxial extension flow facing a narrow opening in a baffle on either side. The geometry is axisymmetric. (b) Half of the drop in $x>0$ showing a conical tip, in a 2D profile. (c) The 3D view.

Figure 2

(a) Schematic diagram (left) indicating flow of the continuous and dispersed phases, and an image (right) of the aperture region with interface not tipstreaming. The dimensions shown are $W_{\text{up}}=280\mu \mathrm{m}$, $a=90\mu \mathrm{m}$, $\mathrm{\Delta }Z=180\mu \mathrm{m}$, and $W_{\text{or}}=34\mu \mathrm{m}$. (b) Image showing streaklines in the dispersed phase during formation of a tipstreaming thread (vertical length bar $50\mu \mathrm{m}$). (Figures have been reprinted with permission from Ref. [10].)

Figure 3

(a) Velocity components $u_x$ and $u_r$ versus $r$, evaluated at the midpoint between each of the $N$ computational mesh points on the baffle $S_b$ in the absence of the drop, with values of $N$ from 16 to 128 as indicated in each panel. The baffle occupies the region $r\in [1,10]$ and $\text{Ca}=1$. (b) Plot of the $L^2$ error of the velocity components versus $N$ on log-log scales. The error is of $O\left(h\right)$.

Figure 4

Drop shape and bulk surfactant concentration with capillary number $\text{Ca}=0.04$ and initial equilibrium surfactant concentration ${\mathrm{\Gamma }}_0=1/3$, shown at times (a) $t=50$, (b) $t=200$, and (c) $t=500$. At the latest time, the surface and transition layer bulk surfactant concentration profiles have reached a steady state. For other parameter values see the text.

Figure 5

(a) Surfactant-laden drop, which tipstreams at values of the Biot number indicated when $\text{Bi}\lesssim 1.0$. The main parameter values are $\lambda =0.05$, $\text{Ca}=0.1$, and the initial surface coverage ${\mathrm{\Gamma }}_0=0.6$; for other parameters see the text. Tipstreaming profiles are shown just before pinch-off: in the insoluble limit when $\text{Bi}=0.0$ at time $t=52.0$ and with surfactant solubility when $\text{Bi}=0.01$ at time $t=52.0$, when $\text{Bi}=0.1$ at time $t=64.5$, and when $\text{Bi}=1.0$ at time $t=88.0$. (b) Bulk surfactant concentration for the simulation of (a) when $\text{Bi}=0.01$. (c) Surface surfactant concentration $\mathrm{\Gamma }$ (left-hand scale) with tangential component $u_s$ and normal component $u_n$ of the interfacial fluid velocity (right-hand scale) versus $x$.

Figure 6

Insoluble surfactant ($\text{Bi}=0$) with aperture radius $r_0=0.75$ and $\text{Ca}=0.74$. (a) and (e) Surfactant-free interface (${\mathrm{\Gamma }}_0=0$) shown at time $t=86.3$, (a) in profile and (e) in perspective. In the remaining panels the interface is surfactant laden, shown in profile just before pinch-off on the left and with the distribution of $\mathrm{\Gamma }$ versus $x$ at the same time on the right, with (b) and (f) ${\mathrm{\Gamma }}_0=0.025$ at time $t=13.2$, (c) and (g) ${\mathrm{\Gamma }}_0=0.05$ at time $t=12.1$, and (d) and (h) ${\mathrm{\Gamma }}_0=0.10$ at time $t=11.5$.

Figure 7

(a)–(d) Influence of Biot number Bi on drop profiles just before pinch-off and (e)–(h) the corresponding surface surfactant distribution $\mathrm{\Gamma }$, with $r_0=0.75$, $\text{Ca}=0.74$, and ${\mathrm{\Gamma }}_0=0.10$. The other parameters are (a) and (e) $\text{Bi}=0$, the insoluble limit, at $t=11.5$, per Figs. 6 and 6; (b) and (f) $\text{Bi}=0.10$ at $t=11.3$; (c) and (g) $\text{Bi}=0.25$ at $t=11.1$; and (d) and (h) $\text{Bi}=0.50$ at $t=10.9$.

Figure 8

Bulk surfactant concentration $C$ with the same parameter values and time as in Figs. 7 and 7.

Figure 9

Insoluble surfactant ($\text{Bi}=0$) with aperture radius $r_0=0.50$ and $\text{Ca}=1.82$. (a) and (e) Surfactant-free interface (${\mathrm{\Gamma }}_0=0$) shown at time $t=50$, (a) in profile and (e) in perspective. In the remaining panels the interface is surfactant laden, shown in profile just before pinch-off on the left and with the distribution of $\mathrm{\Gamma }$ and interface velocity components $(u_s,u_n)$ versus $x$ at the same time on the right, with (b) and (f) ${\mathrm{\Gamma }}_0=0.025$ at time $t=7.6$, (c) and (g) ${\mathrm{\Gamma }}_0=0.05$ at $t=9.0$, and (d) and (h) ${\mathrm{\Gamma }}_0=0.10$ at $t=9.5$.

Figure 10

(a)–(d) Influence of Biot number Bi on drop profiles just before pinch-off and (e)–(h) the corresponding surface surfactant distribution $\mathrm{\Gamma }$ and interface velocity components $(u_s,u_n)$ with $r_0=0.50$, $\text{Ca}=1.82$, and ${\mathrm{\Gamma }}_0=0.10$. The other parameters are (a) and (e) $\text{Bi}=0$, the insoluble limit, at $t=9.5$, per Figs. 9 and 9; (b) and (f) $\text{Bi}=0.10$ at $t=9.2$; (c) and (g) $\text{Bi}=0.25$ at $t=8.4$; and (d) and (h) $\text{Bi}=0.50$ at $t=7.8$.

Figure 11

Bulk surfactant concentration $C$ with the same parameter values and time as in Figs. 10 and 10.

Figure 12

Thread evolution with aperture radius $r_0=0.50$, $\text{Ca}=1.82$, and ${\mathrm{\Gamma }}_0=0.10$. Between each frame a tip drop pinches off; it is then removed from the computation and is not shown in the figure. (a) $\text{Bi}=0.25$ and the time interval between frames is $\mathrm{\Delta }t=0.3$ with the first (top) frame at $t=8.4$. (b) $\text{Bi}=0.50$ and the frame time interval is $\mathrm{\Delta }t=0.3$ with the first (top) frame at $t=7.8$. (c) $\text{Bi}=0.50$ and the frame time interval is $\mathrm{\Delta }t=0.2$ with the first (top) frame at $t=9.2$.

Figure 13

Steady drop with a conical tip in the absence of surfactant, with the aperture radius $r_0=0.25$ and capillary number $\text{Ca}=12.90$ shown at time $t=38.6$, in (a) profile view and (b) perspective view.

Figure 14

The addition of insoluble surfactant at initial concentration ${\mathrm{\Gamma }}_0=0.10$ but with all other parameters as in Fig. 13 induces tipstreaming. Data are shown just before pinch-off, at time $t=9.0$. (a) Interface profile. (b) Surface surfactant concentration $\mathrm{\Gamma }$ with tangential component $u_s$ and normal component $u_n$ of the interfacial fluid velocity versus $x$.

Figure 15

The inclusion of surfactant solubility with $\text{Bi}=0.1$ but with all other conditions as in Fig. 14 leads to a shorter and slightly narrower thread just before pinch-off, at time $t=7.6$, with aperture size $r_0=0.25$, capillary number $\text{Ca}=12.90$, and ${\mathrm{\Gamma }}_0=0.10$. (a) Interface profile. (b) Surface surfactant concentration $\mathrm{\Gamma }$ and interfacial fluid velocity components $(u_s,u_n)$ versus $x$. (c) Bulk surfactant concentration $C$.

Figure 16

Data in the $(\overline{Q},\overline{C})$ state-space plane. (a) Conditions for tipstreaming observed experimentally by Moyle et al. [11] are denoted by squares. Lines denote boundaries outside which tipstreaming is not expected to occur based on a model of the setup and due to physical limits as indicated in the plot. [Panel (a) has been reproduced with permission from Ref. [11].] (b) Conditions for tipstreaming predicted by the flow-focusing simulations of this study, with symbol and capillary number Ca as indicated. Close agreement is found for the range of the dimensionless bulk surfactant concentration data $\overline{C}$. The difference in the range of values for the dimensionless flow rate $\overline{Q}$ is attributed mostly to the difference in the aperture radius $r_0$.