Symmetry breaking of a parallel two-phase flow in a finite length channel

Parallel two-phase flows are omnipresent in technological applications that require contact between two immiscible fluids for a finite amount of time. Precise control over the flow and separation of the fluids once they have been in contact are therefore the key challenges in these applications. Here, using experiments and numerical simulations, we show that the interface between two immiscible fluids flowing at the same flow rate in a symmetric channel can become unstable locally near the exit junction, where the two fluids are separated. This instability leads to the shedding of the droplets of one phase into the other, preventing a complete separation. We characterize this instability and show that the period of drop shedding is inversely proportional to the flow rate. We derive a stability criterion based on the balance between the Laplace pressure across the liquid-liquid interface and viscous pressure drop along each flow stream. The stability criterion and our experimental results are used to highlight the extreme sensitivity of this flow system to the parameters involved such as viscosity difference and exit geometry, which introduces gravitational effects and characteristics of the exit tubing.

Figure 1

The experimental setup and the instability. (a) Schematic showing the flow system: we inject the two immiscible liquids via the two inlets at equal flow rates $Q$, and the outlets drain into respective fluid reservoirs where the exit pressure is $p_0$ (atmospheric pressure). PTFE tubes of length $L_t$ and radius $R_t$ carry the liquids from the syringes to the inlet and from the exit to their respective reservoirs. (b) 3D schematic of the channel showing the various geometric parameters. (c) Schematics showing the shapes of the liquid-liquid interface on the modified surfaces of the channel. The advancing (a) and receding (r) contact angles of oil on the oil side (schematic on the left) and the water on the water side (schematic on the right) are shown based on the values given in Table 1. (d) When the system is unstable, the interface between the two fluids periodically sheds droplets near the exit junction where the two fluids are separated. The flow is from left to right, and the flow rate is $Q$ = 0.1 ml/min in this case. The light gray fluid is perfluorodecalin (oil), and the dark gray fluid is dyed water-glycerol mixture (water). The geometric parameters of the channel are $H$ = $W$ = 0.75 mm, $L_m$ = 12.5 mm, $L_e$ = 30 mm, and $\alpha$ = ${45}^{\circ }$. (e) The snapshots here show the formation of interfacial bumps in the channels with different lengths: (i) $L_m=12.7\text{mm}$, (ii) $L_m=21.4\text{mm}$, and (iii) $L_m=50.9\text{mm}$. The flow rate in all cases is $Q$ = 0.1 ml/min. All scale bars represent 2 mm.

Figure 2

The spatiotemporal evolution of the unstable interface. (a) A bump, characterized by the length scale $y_{\max }$, develops near the exit junction where the two fluids are being separated. The red dot represents the maximum point on the interface, and the dashed black line in the inset represents the mean interface location $y_{\mathrm{mean}}$. A space-time diagram shows the progression of the interface profile during a drop shedding event. The evolution of the magnitude of $y_{\max }$ in time and the horizontal location along the channel where it occurs is tracked in panels (b) and (c), respectively. Different colors represent different drop shedding events within the same experiment at a constant flow rate $Q$ = 0.1 ml/min using fluid pair B. The bump approximately follows an exponential growth in this case as shown in (b).

Figure 3

The drop shedding caused by the instability is periodic in time. (a) The red, green, and blue data points represent the temporal evolution of the height of three different points along the interface near the bump. The bump grows in time until it touches the wall on the opposite side of the channel. A drop is then shed and the interface relaxes, leading to a sudden decrease in the interface location. The interface growth is then repeated periodically in time. (b) The period of the drop shedding decreases as the flow rate increases, showing a scaling $\tau \sim 1/U$. Varying the geometric features of the device affects the period. In the legend the reference geometry corresponds to $H$ = $W$ = 0.75 mm, $L_m$ = 12.5 mm, $L_e$ = 30 mm, and $\alpha$ = 45${}^{\circ }$. All other cases shown in the legend are variations of the reference geometry, where one of the geometric features is changed. Fluid pair A was used for all of the closed data points, and $h_r$ was such that the flow was unstable for all of the flow speeds studied. The open data points correspond to situations where the critical flow speed below which the flow is stable, indicated by the dashed line of the same color, was within the window of flow speeds studied. The blue open squares correspond to fluid pair B with $h_r=0$ mm, the red open triangles correspond to fluid pair C with $h_r=0$ mm, and the black open circles correspond to fluid pair D with $h_r=0$ mm.

Figure 4

Results from our numerical study with two fluids of viscosity ratio ${\mu }_2/{\mu }_1=0.92$. Fluid 1 is below the interface, and fluid 2 is above the interface. Fluid properties used for this simulation are as follows: ${\mu }_1=5.6\mathrm{mPa}\mathrm{s},{\rho }_1=1124\mathrm{kg}/{\mathrm{m}}^3,{\rho }_2=1930\mathrm{kg}/{\mathrm{m}}^3$, and $\gamma =37.0\mathrm{mN}/\mathrm{m}$. (a) Base (steady) state interface profiles for three different flow rates. Only the largest flow rate, $Q=0.3$ ml/min, is unstable. (b) Spatiotemporal evolution of the interface for the unstable case $Q=0.3$ ml/min showing the interface deforming more from its base state over time. The local growth of the interface near the exit junction is similar to that seen in experiments. (c) Evolution of the pressure field in the channel as the interface deforms and becomes unstable for the case with $Q=0.3$ ml/min. The solid black line is the interface. Note that most of the changes occur locally near the exit junction. (d) The growth rate ${\omega }_i$ of the linear instability as a function of flow rate. The flow is globally unstable when ${\omega }_i>0$.

Figure 5

Instability phase diagram for the parallel two-phase flow. The closed data points are from our experiments with all four fluid pairs, and the open data points are from the literature [17]. The invading fluid (fluid 1) is always water in our experiments, and it is oil in the data from [17]. The solid line represents the stability criterion [Eq. (5)]. Here unstable (red squares) corresponds to imperfect separation of the two fluids, and stable (green circles) corresponds to perfect separation.