Emulsification with rectangular tubes

The flow of two immiscible liquids or fluids in bounded systems where confinement geometry varies can lead to drop or bubble formation. This phenomenon has been reported in the context of oil recovery and named snap-off, or exploited for making emulsions, and then foams, by using microfluidic systems, namely, microchannel emulsification or step emulsification. We report a comprehensive experimental investigation of such an emulsification process occurring at the end of a glass rectangular tube filled with oil and immersed in a water bath. This allows us to clearly visualize the breakup event of the dispersed phase liquid finger at the capillary's end. Below a critical flow rate, the drop size varies slowly with the flow rate and it is linked to the pinching time of the dispersed phase. A semiempirical law that gives the resulting drop size as a function of fluid and geometrical properties is proposed. However, this feature is altered for an aspect ratio of the rectangular tube below 2.5 where the forming drop hinders the counterflow of the continuous phase leading to larger drops. Then, above a critical flow rate, or capillary number that weakly depends on the viscosity ratio of the two liquids, the neck adopts a quasistatic shape well accounted for by a model based on a Hele-Shaw flow. In that case, drop formation is driven by gravity and a transition from a dripping regime to a jetting regime is observed at higher flow rates. Monodisperse foam can also be formed by injecting air. While the overall dynamics of bubble formation shares similarities with an incompressible fluid, the bubble size and the critical capillary number do not follow the same scaling laws.

Figure 1

Front view of a rectangular glass capillary having a height $h$ of $96\mu \mathrm{m}$ and a width $w$ of $1050\mu \mathrm{m}$.

Figure 2

(a) Time sequence showing the drop formation from a rectangular glass capillary observed from both sides ($w=1000\mu \mathrm{m}$ and $h=100\mu \mathrm{m}$). Time is indicated in milliseconds and $t=0$ matches the time when the neck breaks up. Gravity is pointing downward, in the $x$ direction. (b) Spatiotemporal diagram in the $x$ direction in the midline of the channel from which characteristic timescales of drop formation are evaluated. (c) Spatiotemporal diagram in the $y$ direction at the location of the neck breakup. (d) Time evolution of the radii of curvature $r_{xy}$ and $r_{xz}$ evaluated during a period of drop formation, the reference time being taken at breakup, i.e., at $t_b$. The time period when the meniscus is stable is indicated by closed symbols and the unstable phase by open ones.

Figure 3

(a) Evolution of the drop diameter $d$ ($•$) and the necking time ${\tau }_n$ ($\circ$) as a function of the flow rate. The height of the capillary is $50\mu \mathrm{m}$, the width is $500\mu \mathrm{m}$, and the dispersed phase is the silicone oil SO5. The solid line corresponds to Eq. (1). (b) Variation of the drop size $d_0$ with the height $h$ of the capillary. The dispersed phase is the silicone oil SO5. The solid line is a purely linear fit with a slope of 4.4. The inset shows the drop size normalized by the capillary height as a function of the viscosity ratio ${\eta }_i/{\eta }_o$ for silicone oils with capillaries having an aspect ratio $w/h$ equal to 2 ($\square$), 11 ($•$), and 20 ($\circ$) and fluorocarbon oils with $w/h=11$ ($▴$).

Figure 4

Characteristic timescale of drop formation ${\tau }_0$ as a function of (a) the inner viscosity ${\eta }_i$ for two aspect ratios $w/h$ equal to 11 ($•$) and 20 ($\circ$) with ${\eta }_o=1 \mathrm{mPa}\mathrm{s}$ and $h=50\mu \mathrm{m}$, (b) the outer viscosity with ${\eta }_i=4.65 \mathrm{mPa}\mathrm{s}$ and $h=50\mu \mathrm{m}$, (c) the interfacial tension $\gamma$ for silicon oil ($•$) and fluorocarbon oil ($▴$) with $h=50\mu \mathrm{m}$, and (d) the height $h$ with ${\eta }_i=4.65 \mathrm{mPa}\mathrm{s}$ and ${\eta }_o=1 \mathrm{mPa}\mathrm{s}$. The solid lines represent linear functions or are proportional to $1/\gamma$ in (c). (e) Characteristic timescale ${\tau }_0$ versus the empirical scaling law where $a=375, \alpha =4.7\times {10}^{-2}$, and $\beta =0.33$. The symbols represent data sets with the same fluid properties and tube geometry as in Fig. 3. The solid line is a purely linear function with a prefactor of 1. The inset shows correlation between the average necking time ${\tau }_n$ and ${\tau }_0$. The solid line is a purely linear fit with a prefactor of 1.36.

Figure 5

(a) Evolution of the necking time ${\tau }_n$ as a function of the aspect ratio $h/w$ for SO20 along with two snapshots prior to breakup. The scale bar is 1 mm. (b) Growth of the normalized drop size $d/d_0$ as a function of the flow rate ratio $q$ divided by the critical flow rate ratio $q_c$ at the transition for two aspect ratios, 2 (squares) and 10 (circles), and various formulation and capillary geometries. The width and the height of the rectangular capillaries are indicated in micrometers in the legend. The lines correspond to Eq. (5) for $b$ equal to 0.45 (solid line) and 4.8 (dashed line).

Figure 6

(a) Correlation between the critical flow rate $q_c$ above which the meniscus is stabilized by the flow and the flow rate created by the characteristic timescale ${\tau }_0$ and drop size $d_0$. The solid line is a linear fit with a coefficient of proportionality equal to 0.45. The inset shows the critical capillary number Ca times the aspect ratio of the rectangular tube $w/h$ as a function of the viscosity ratio ${\eta }_i/{\eta }_o$. The oblique line corresponds to a power law of 0.2. The symbols represent data sets with the same fluid properties and tube geometry as in Fig. 3. (b) Streamlines in the continuous phase revealed by micrometer-size particles when the tip is approaching the tube's end and when the neck is dynamically stabilized. The images are a superposition of 100 snapshots taken every millisecond. The width of the capillary is $1000\mu \mathrm{m}$.

Figure 7

Quasistatic shapes of the meniscus for increasing flow rates, and thus capillary number Ca, and various viscosities ${\eta }_i$. The value of ${10}^3\text{Ca}$ is indicated on each picture and ${\eta }_i$ is equal to (a) 4.65 $\mathrm{mPa}\mathrm{s}$, (b) 48 $\mathrm{mPa}\mathrm{s}$, and (c) 106 $\mathrm{mPa}\mathrm{s}$. The height and the width of the glass capillaries are about 100 $\mu \mathrm{m}$ and $1000\mu \mathrm{m}$, respectively. The white solid lines are the equilibrium shapes predicted by the integration of Eq. (11).

Figure 8

(a) Time sequence showing the bubble formation from a rectangular glass capillary ($w=1000\mu \mathrm{m}$ and $h=100\mu \mathrm{m}$). Time is indicated in milliseconds and $t=0$ matches the time when the neck breaks up. Gravity is pointing downward. The injection system is case i sketched in (c). (b) Evolution of the bubble's diameter $d$ divided by the height $h$ as a function of the flow rate $q$ for different injection conditions. The inset shows the corresponding evolution of the necking time ${\tau }_n$ as a function of $q$. (c) Schematics of the different injection systems (see the text for more details).