4.1. Description of stratified coflow in a microchannel

The instabilities generated due to the viscosity stratification in coflowing fluids strongly depend on the interface location and local velocity variation (Govindarajan & Sahu Reference Govindarajan and Sahu2014). In order to describe the unstable regimes, we first analyse the primary flow (figure 1a) using the theoretical formulations given in § 3.1. This allows us to identify the reference interface location and determine the characteristic velocities for phases P1 and P2.

Initially, we analytically determine the interface location by calculating the low-viscosity (P1) and high-viscosity (P2) stream width ($W_1$ and $W_2$), as detailed in Appendix C.1. The variation of width ratio $( {W_{2}}/{W})$ depends on the flow rate ratio ($Q_{r}$) and the viscosity ratio ($m$). We plot the variation of $W_2/W$ with $(Q_r m)^{-1}$ at different values of $m$ as shown in figure 2(a), where $(Q_r m)^{-1}= (Q_2 \mu _2)/(Q_1 \mu _1)$. Here, the plots for different $m$ values collapse onto a single curve given by a theoretical fit: $W_2/W \approx (1+1.67 (Q_r m)^{1/3})^{-1}$. We use this fit to determine the interfacial location (or fluid stream width ratio) and the local velocity variations. We observe that, with the increase in both flow rate and viscosity of phase P2 (while keeping the phase P1 flow rate and viscosity constant), the width of second phase or $W_2$ increases (Hu & Cubaud Reference Hu and Cubaud2018). Owing to the thin layer effect (Renardy & Joseph Reference Renardy and Joseph1985), at very high $(Q_r m)^{-1}$ values, the phase 1 shrinks to a small stream width $W_1$ which can be obtained as $W_1=W-W_2$.

Figure 2.

Theoretical study of width ratios, velocity profiles and shear stress distribution. (a) Evolution of fluid stream width ratio $(W_2/W)$ against $(Q_r m)^{-1}=(Q_2/Q_1)(\mu _2/\mu _1)$ for various viscosity ratios ($m$), which is represented by a theoretical fit of the form $W_2/W \approx (1+1.67 (Q_r m)^{1/3})^{-1}$. (b) Velocity profile of coflow combinations at various $m$ values for $Q_r=4$ are shown, where phase 1, phase 2 and the interface location are indicated. (c) Interfacial shear stress ($\bar {\sigma }_{xy}|_{y=0}$) variation with $Q_2/Q_1$ for various $m$ values. The viscosity ratios $(m=\mu _1/\mu _2)$ are considered as $m=1/100$, $1/200$, $1/250$ and $1/500$.

We obtain the flow velocity profiles for various coflow combination with different viscosity ratios at a fixed flow rate ratio ($Q_{r} = 4.0$) from (3.35)–(3.37), as presented in figure 2(b). It is found that, with decreasing $m$, velocity profiles become more skewed, which can give rise to interfacial instability (Hu & Cubaud Reference Hu and Cubaud2018). As expected theoretically (Hazra et al. Reference Hazra, Mitra and Sen2022b), the maximum velocity occurs in P1 phase having a lower viscosity. Therefore, at a fixed flow rate of P2, with decreasing $m$, the maximum velocity in P1 phase increases (Hazra et al. Reference Hazra, Mitra and Sen2022b). The variation of interfacial shear stress ($\bar {\sigma }_{xy}$) with varying flow rate ratio ($Q_{r}$) is depicted in figure 2(c). The interfacial shear stress at $y=0$ is calculated as $\bar {\sigma }_{xy}|_{y=0} \approx ({1}/{R_2}) ({\partial U_2}/{\partial y})=({r}/{R_1}) ({\partial U_1}/{\partial y})$. The results demonstrate that, for a fixed viscosity ratio ($m$), the interfacial shear stress ($\bar {\sigma }_{xy}|_{y=0}$) increases with a decrease in the flow rate ratio ($Q_{r}$). Additionally, for a fixed $Q_{r}$, a decrease in the viscosity ratio ($m$) leads to an increase in $\bar {\sigma }_{xy}|_{y=0}$. This relationship between $\bar {\sigma }_{xy}$ and $Q_{r}$ is crucial for understanding the transition between different flow regimes.

4.2. Description of flow regimes

Viscous and elastic stratification give rise to different flow regimes in microfluidic confinements. However, understanding the individual effect of viscosity and elasticity on the total instability is challenging due to the coupled variation of viscosity and elasticity. We conduct experiments and use the formulations given in § 3.3 to explore the decoupled effects of viscous and elastic stratification. In this section, we describe the experimentally observed flow regimes and use the variation of complex wave speed $(c_i)$ to explain the transition between stable and unstable regimes. The earlier study (Yih Reference Yih1967) for the case of Poiseuille flow of coflowing Newtonian fluids reported only an unstable regime for a fluid density ratio $r=1$ and stream width ratio $n=1$. However, we observe different flow morphologies: (i) a stable regime (S) with a straight interface, (ii) an unstable regime (U) with wavy interface or droplets (dripping) from low-viscosity fluid; for the case of coflowing Newtonian and viscoelastic fluids. The experimental images of the different regimes observed from a large set of experiments are shown in figure 3(a). The images are captured at a fixed axial location, $\tilde {x} \approx 3$ mm with various coflow combinations (at a fixed $m$) by varying the flow rate ($Q_{2}$) of the high-viscosity fluid, keeping the flow rate ($Q_{1}$) of the low-viscosity fluid fixed.

Figure 3.

Description of experimentally observed distinct flow regimes via theoretical study depicting the variation of $J(=J_1+J_2), \ J_1$ (viscous stratification component), and $J_2$ (elastic stratification component) and a comprehensive regime plot. (a) Experimental images (pertaining to data points marked with $\bigstar$ in c) different regimes: stable (grey), unstable waviness (red) and unstable droplet (blue) are shown in a tabular form for varying flow rate ratio ($Q_{r}$) and viscosity ratio ($m$). Scale measures 300 $\mathrm {\mu }$m. (b) Theoretical variation of $J$, $J_1$, and $J_2$ corresponding to (a) for different fluid combinations with fixed $Q_1$: (i) FC40-SiO 1000, $Q_1 = 45$ $\mathrm {\mu }$l min$^{-1}$, (ii) FC40-PEO 3 %, $Q_1 = 50$ $\mathrm {\mu }$l min$^{-1}$, (iii) FC40-PEO 4 %, $Q_1 = 45$ $\mathrm {\mu }$l min$^{-1}$ and (iv) FC40-PEO 5 %, $Q_1 = 40$ $\mathrm {\mu }$l min$^{-1}$. The colour bands represent the experimental stable (S), unstable-waviness (U-W) and unstable-droplet (U-D) regimes for a range of $Q_2$. From theory, the critical values of $Q_2$ and $J$ for the transition from the stable (S) to the unstable (U) regime are indicated by $Q_2^{c1}$ (black vertical dashed line) and $J^{c1}$ (black horizontal dashed line). The transition from the unstable-waviness to the unstable-droplet regime is marked by $Q_2^{c2}$ (green vertical dashed line) and $J^{c2}$ (green horizontal dashed line). (c) Regime plot showing stable coflow (grey), unstable-waviness (red) and unstable-droplet (blue) regimes for FC40-PEO 3 % (triangle symbol, double compound line), FC40-PEO 4 % (square symbol, dash-dot line), FC40-PEO 5 % (diamond symbol, dash-dot-dot line) and FC40-SiO 1000 (circle symbol, dash line) coflow combinations over a range of capillary numbers, ${Ca}_{1}$ and ${Ca}_{2}$. The subscript 1 and 2 are used for low-viscosity fluid (FC40) and the high-viscosity fluid, respectively. Different versions and colours of the corresponding symbols are used to indicate flow rates ($Q_{1}$, $Q_{2})$ over which the coflowing fluid combinations undergo various regime transitions, as shown in the legend. A more detailed version of the regime plot is provided in Appendix D.

In general, our experiments (shown in figure 3a) show that instabilities are generated due to the viscosity and elasticity difference between the liquids in a coflow. In this situation, an increase in the flow rate of the high-viscosity fluid causes a transition from stable to convective and eventually to absolute instability (Guillot et al. Reference Guillot, Colin, Utada and Ajdari2007; Utada et al. Reference Utada, Fernandez-Nieves, Gordillo and Weitz2008; Hemachandran et al. Reference Hemachandran, Hoque, Laurell and Sen2021). Transition between the different regimes (stable and unstable) in microfluidic coflows can be understood further using the expression for the complex wave speed ($c_{i}$) obtained from the temporal stability analysis given in § 3.3, where $c_i=k R_2 J$ and $J=J_1+J_2$. Generally in a temporal stability analysis, the stability of a system is determined by observing the time evolution of perturbations (Yih Reference Yih1967; Govindarajan & Sahu Reference Govindarajan and Sahu2014). By assuming the wavenumber to be real ($k = k_r$), temporal instability is distinguished from spatial instability (Utada et al. Reference Utada, Fernandez-Nieves, Gordillo and Weitz2008). A zero value of the complex wave speed of the perturbations, $c_i$, indicates a neutrally stable case. However, if $c_i$ is negative ($c_i<0$), the disturbances diminish over time, indicating a flat interface in the stable regime. Conversely, if $c_i$ is positive ($c_i >0$), the disturbances increase exponentially, indicating that the system is unstable. Since our theoretical analysis assumes the wavenumber to be real, we use the above mentioned $c_i$ conditions to predict the regimes. The nature of the instability is found to be either convective or absolute. In order to differentiate between these two, a spatio-temporal analysis with the Briggs–Bers criterion is used in the literature (Guillot et al. Reference Guillot, Colin, Utada and Ajdari2007; Utada et al. Reference Utada, Fernandez-Nieves, Gordillo and Weitz2008). The complex wave speed, $c_i$, and growth rate, $\omega _i$, are related to the wavenumber, $k$, by the equation $\omega _i = k c_i$, where the values of $\omega _0$ and $k_0$ are obtained from $D(\omega _0, k_0)=0$ and $v_{group}=\partial \omega / \partial k |_{\omega =\omega _0, k=k_0}$ (Huerre & Monkewitz Reference Huerre and Monkewitz1990; Sahu & Govindarajan Reference Sahu and Govindarajan2014). Finally, the regimes obey the following conditions with regards to the instability: (i) convectively unstable, $\omega _i>0, v_{group}>0$; (ii) absolutely unstable, $\omega _i>0, v_{group}<0$.

First, we investigate the effect of viscous stratification by considering the case of N–N coflows: FC40-SiO 1000 ($m = 1/250$). As mentioned in the earlier § 3.3, the formulations for $c_i$ reduces to (3.97) for the case of a FC40-SiO 1000 coflow, where the elastic stratification (Weissenberg number, $Wi\approx 0$) effects become negligible ($J_2=0$, $J=J_1$) and the instabilities are generated only due to the variation in the viscous stratification ($J_1$). Through (3.94)–(3.97), we obtain the variations in $J_1$, $J_2$, $J$ and $c_i$ for a wide range of values of $r$ and $n$, and represent for the case of a FC40-SiO 1000 coflow having $m=1/250$ and $r=1.89$ via figure 3(b-i). As $Q_2$ increases, the viscosity stratification causes $J_1$ to increase – flipping from a negative to a positive value. Further, $J_1$ becomes maximum at a substantially high flow rate ($Q_2$) as shown in figure 3(b-i). At the same time, the fluid stream width ratio ($W_2/W$) and shear stress follow the trend as explained in the previous section and shown in figure 2. Since at lower $Q_2$, $J_1 < 0$, therefore $c_i < 0$, where $c_i = R_2 k J_1 = R_2 k J$; which indicates a stable regime. Here, due to the high viscosity difference, the width of the high-viscosity phase (P2) is greater than that of the low-viscosity phase (P1) even at a smaller $Q_2$ and the shear stresses at interface is less (refer figure 2), which stabilizes the flow (Charru & Hinch Reference Charru and Hinch2000; Govindarajan & Sahu Reference Govindarajan and Sahu2014). In order to characterize the transition between stable and unstable regimes, we define the critical parameters: $Q_r^{c1}$ and $J^{c1}$, where $Q_r^{c1}$ is the critical flow rate ratio, $Q_r^{c1}=Q_1/Q_2^{c1}$ and $J^{c1}$ corresponds to the neutrally stable condition. Beyond $Q_2 > 17$ $\mathrm {\mu }$l min$^{-1}$ (in figure 3b-i with the critical flow ratio for S–U regime transition being $Q_{r}^{c1} \approx 2.65$), the system becomes unstable with $J_1 > 0$, $J > 0$ and $c_i > 0$. Here, the normal stress difference starts to exceed the Laplace pressure, causing the interface to deform. The stable (S), unstable-waviness (U-W) and unstable-droplet (U-D) regimes experimentally observed at different ranges of the flow rates are depicted using colour bands in figures 3(b-i) to 3(b-iv). The variation of $J$ aligns with our experimental observations, as shown in figures 3(a-i) and 3(b-i). Low positive values of $J$ indicate an unstable regime with waviness, where the shear stress is optimal. At extreme values of $J$ (or at higher P2 phase velocity), FC40 cannot withstand the higher shear stress at the interface and breaks into droplets, which occurs when $c_i > 0$ and $v_{group} < 0$, leading to absolute instability (Huerre & Monkewitz Reference Huerre and Monkewitz1990; Guillot et al. Reference Guillot, Colin, Utada and Ajdari2007; Utada et al. Reference Utada, Fernandez-Nieves, Gordillo and Weitz2008). In a similar way, we experimentally define the critical parameters for the transition between waviness and the droplet realm within the unstable regime: $Q_r^{c2}$ and $J^{c2}$, where $Q_r^{c2}$ is the critical flow rate, $Q_r^{c2}=Q_1/Q_2^{c2}$. By directly comparing the experiments with the theory as shown in figure 3(b-i), we find $Q_r^{c2} \approx 1.125$ and $J^{c2} \approx 3.8$.

So far, we have discussed the sole effect of viscous stratification in the case of a N–N coflow inside a microchannel. Further, we extend this discussion to the combined effect of viscous and elastic stratification in the case of a N–VE coflow, as shown in figures 3(a-ii–iv) and 3(b-ii–iv). Here, the competition between viscous stratification ($J_1$) and elastic stratification ($J_2$) governs the dynamics and hence determines the flow regimes. In the case of FC40-PEO 3 % coflow (N–VE, $m = 1/200$, $\tau = 0.009$ s), an increase in $Q_2$ leads to a transition from the stable (S) regime to the unstable (U) regime, similar to the FC40-SiO 1000 coflow case (see figures 3a-i and 3b-i). However, the transition from stable to unstable occurs at a lower $Q_2$ (or a higher $Q_{r}^{c1} \approx 4.55$) compared with FC40-SiO 1000 coflow. This difference is mainly due to the elastic stratification effect and the higher viscosity ratio ($m = 1/200$) in the FC40-PEO 3 % system. In this case, the variation of $J_1$ is minimal at lower $Q_2$ and more at higher $Q_2$, as shown in figure 3(b-ii). Elastic stratification causes a non-monotonic variation in $J_2$ with increasing $Q_2$. Initially, $J_1 \approx 0$, $J_2 < 0$ and $|J_2| > |J_1|$, resulting in $J < 0$ and $c_i < 0$, indicating a stable regime, as seen in experiments. With further increases in $Q_2$ (or decrease in $Q_r$), $J_2$ becomes positive ($J_2 > 0$) and $J_2 > J_1$, which give rise to $J > 0$ and $c_i > 0$, indicating an unstable regime, as shown in figure 3(b-ii). We further experimentally distinguish the transition between unstable waviness and unstable droplet (in figure 3a). The extra normal stress from the elasticity of phase 2 ($J_2 > 0$) helps to break the FC40 phase into droplets at lower interfacial shear stress as compared with the N–N coflow case, occurring when $c_i > 0$ and $v_{group} < 0$, leading to absolute instability (Huerre & Monkewitz Reference Huerre and Monkewitz1990; Guillot et al. Reference Guillot, Colin, Utada and Ajdari2007; Utada et al. Reference Utada, Fernandez-Nieves, Gordillo and Weitz2008). An earlier study (Azaiez & Homsy Reference Azaiez and Homsy1994) reported that an Oldroyd-B fluid is significantly more stable than a Newtonian fluid, with elasticity effects stabilizing the flow. However, in our study, we observed a contrasting phenomenon where elastic effects can destabilize the flow. Specifically, in the case of FC40-PEO 3 %, the elastic effects of the viscoelastic fluid stabilize the coflow at higher flow rate ratios but destabilize the flow at lower flow rate ratios. Moreover, viscous stratification also destabilizes the flow at lower flow rate ratios. The combined destabilization effects of viscous and elastic stratification shift the critical flow rate ratio for regime transition and convective to absolute instability transition to a lower $Q_2$ (or to a higher flow rate ratio) value as compared with the viscous case without elastic effects. In our study, we obtained $Q_{r}^{c1} \approx 4.55$ (with $J^{c1} \approx 0$) and $Q_{r}^{c2} \approx 3.85$ (with $J^{c2} \approx 2.13$). Thus our findings show that, in contrast to the previous study (Azaiez & Homsy Reference Azaiez and Homsy1994), elasticity can both stabilize and destabilize coflows depending on the flow rate ratio, thus demonstrating that the role of elasticity in flow instability is more complex and varies with fluid and flow parameters.

To explore further variations in the viscous and elastic properties of the fluid in the unstable regimes, we replace phase 2 (the high-viscosity phase) with PEO 4 % and PEO 5 %, where the viscosity and relaxation time are further increased compared with PEO 3 % (refer to table 1). The FC40-PEO 4 % ($m=1/750$, $\tau =0.03$ s) and FC40-PEO 5 % ($m= 1/2250$, $\tau =0.06$ s) cases show a stark contrast to the previous cases. Interestingly, we observe that the instability of the interface is suppressed with increasing phase 2 flow rate ($Q_2$), leading to transitions from unstable (U) to stable (S) regimes, as shown in figure 3(a-iii,iv). In both cases, the very high viscosity contrast causes an initial increase in $J_1$ with $Q_2$, followed by a decrease at high $Q_2$ due to the thin layer effect, which is more evident in the case of FC40-PEO 4 %, depicted in figure 3(b-iii,iv). However, the increase in relaxation time or the Weissenberg number ($Wi$) compared with PEO 3 % changes the effects of elastic stratification. With an increase in $Q_2$, $J_2$ becomes more and more negative with a magnitude far higher than $J_1$. At lower $Q_2$ (or higher flow rate ratio), $J_1 > 0$ and $J_2 < 0$, where viscous stratification tends to destabilize the flow and elastic stratification tends to stabilize it. However, since $|J_1| > |J_2|$, viscous stratification suppresses elastic stratification, resulting in $J > 0$ and $c_i > 0$, indicating an unstable regime, as seen in experiments (figure 3b-iii,iv). In contrast, at higher $Q_2$ (or low flow rate ratio), $J_1 > 0$, $J_2 < 0$ and $|J_2|> |J_1|$, which shows that the elastic effects generated in the flow due to elastic stratification suppresses the viscous stratification undulations, resulting in $J < 0$ and $c_i < 0$, indicating a stable regime. It is important to mention that the unstable regime for both combinations of fluids shows only waviness in the interface. As evident from figures 3(b-iii,iv) and 2(a), with increasing $Q_2$, the elastic effects dominate and the P1 phase width ($W_{1}$) becomes thinner (‘thin layer effect’), which prevents the interface breaking into droplets (Renardy & Joseph Reference Renardy and Joseph1985). Overall, in the case of a fluid with very high viscosity (or very small $m$) and relaxation time (or high $Wi$), the elastic effects of the viscoelastic fluid stabilize the flow at lower flow rate ratios (Azaiez & Homsy Reference Azaiez and Homsy1994), while viscous stratification destabilizes it at higher flow rate ratios. The critical flow rate for the unstable to stable transition is found to be $Q_{r}^{c1} \approx 9$ for the FC40-PEO 4 % combination and $Q_{r}^{c1} \approx 13.33$ for the FC40-PEO 5 % combination.

The regimes described above are obtained by varying the flow rate of phase 2 (high-viscosity fluid), $Q_2$ at fixed phase 1 (low-viscosity fluid) flow rate, $Q_1$. To generalize the flow regimes for various $Q_1$ and $Q_2$ combinations, we describe the flow regimes in terms of the capillary numbers of the two coflowing phases, $Ca_1$ and $Ca_2$ (Anna Reference Anna2016), presented in figure 3(c) (see Appendix D for a more detailed regime plot). In the case of FC40-SiO 1000 coflow (N–N, $m = 1/250$), we observe a transition from the stable (S) to the unstable-droplet (U-droplet) regime for $Ca_1 < 0.01$ and a transition from stable to unstable-waviness (U-waviness) to U-droplet regime for $Ca_1 > 0.01$ as shown in figure 3(a-i,b-i,c). This transitional behaviour of stable to U-waviness to U-droplet continues even at higher $Ca_1$ (see Appendix D). In the case of FC40-PEO 3 % coflow (N–VE, $m = 1/200$), for a small $Ca_1$ (i.e. $Ca_1 < 0.005$), an increase in $Ca_2$ leads to a transition from the stable regime to the U-droplet regime. For higher $Ca_1$ (i.e. $Ca_1 > 0.005$), we observe that the stable coflow (S) regime first changes to the U-waviness regime and eventually the P1 phase breaks into droplets, giving rise to the U-droplet regime, as shown in figure 3(a-ii,b-ii,c). At even higher $Ca_1$ (i.e. $Ca_1 > 0.01$), we only observe a stable regime, which can be attributed to the ‘thin layer effect’ (Renardy & Joseph Reference Renardy and Joseph1985) that dominates at a higher $Ca_2$. However, in the case of FC40-PEO 4 % and FC40-PEO 5 % coflows, we observe only unstable to stable transitions with increasing $Ca_2$ at all values of $Ca_1$ as shown in figure 3(a-iii,iv,b-iii,iv,c). Previous studies reported that, at high $Ca$ values, the growth of instabilities is suppressed, the flow becomes stable and droplets are not formed (Azaiez & Homsy Reference Azaiez and Homsy1994; Ambravaneswaran et al. Reference Ambravaneswaran, Subramani, Phillips and Basaran2004; Hemachandran et al. Reference Hemachandran, Hoque, Laurell and Sen2021). In contrast, we observe droplet generation even at high $Ca$ in a variety of Newtonian and viscoelastic coflowing fluid combinations, and report that using N–VE combination with moderate relaxation times (PEO $\lesssim$3 %) allows for control of the critical capillary number, which demarcates the U-waviness and U-droplet regimes.

Based on the preceding analysis, we have identified distinct unstable-waviness and unstable-droplet regimes that significantly differ from the stable configuration. These findings are supported by both experimental observations and theoretical insights. In the following sections, we will experimentally explore and delineate the physical characteristics of these regimes.

4.3. Parametric study of unstable-waviness regime

The propagation of the instability waves are characterized by frequency $\tilde {f}$, wavelength $\tilde {\lambda }$, wave speed $\tilde {c}$ and amplitude $\tilde {A}$, where $\tilde {c}=\tilde {\omega }/\tilde {k}=\tilde {f} \tilde {\lambda }$. A dispersion relation in the instability analysis provides the information about the space–time variation of the perturbations. Therefore, we delve into the dispersion relations for both N–VE and N–N coflows using experiments and theory. In experiments, the flow rate ratio ($Q_{r}$) is gradually changed, and frequency ($\tilde {f}$) and wavelength ($\tilde {\lambda }$) are measured for each case, thus angular frequency $\tilde {\omega }_r= 2{\rm \pi} \tilde {f}$ and wavenumber ($\tilde {k} = {2{\rm \pi} }/\tilde {{\lambda }})$ are calculated. From experimental videos, for each $Q_{r}$, we calculate $\tilde {f}$ by counting the number of peaks/troughs formed at a fixed axial location per second and $\tilde {\lambda }$ is calculated as the distance between two adjacent troughs (figure 1a-ii). Subsequently, these values are used to calculate $\tilde {\omega }$ and $\tilde {k}$. It is observed that wavelength, $\tilde {\lambda }$ for the N–VE coflow ($\tilde {\lambda } \approx 1200$ $\mathrm {\mu }$m) is higher than that for the N–N coflow ($\tilde {\lambda } \approx 800$ $\mathrm {\mu }$m) (for $Q_{r}=6.25$ and 4.55, respectively), as shown in figure 4(a). When $Q_2/Q_1$ (or $Q_r^{-1}$) increases, frequency ($\,\tilde {f}$) increases but wavelength ($\tilde {\lambda }$) for both N–VE and N–N cases decreases significantly, as evident from figure 4. This suggests that both dimensionless wavenumber ${k}$ ($=\tilde {k} W_2$) and angular frequency ${\omega _r}$ ($\widetilde {\omega _{r}} W_2/U_0$) follow a similar trend of variation, as shown in figure 4(b). We find an expression for $\omega _r$ from (3.78), where $c_0=c'_0+b$, $c_0=\tilde {c}_0/U_0$. Therefore $\tilde {\omega }_r=\tilde {c}_0 \tilde {k}$, can be represented as $\tilde {\omega }_r=\tilde {k} U_0 c'_0+\tilde {k} U_0 b$. Here, the variation of dimensionless wavenumber, ${k}$ with angular frequency, ${\omega }$ for N–VE and N–N coflows obtained from experiments and theory show a good agreement, as shown in figure 4(b). The variation of wavenumber, ${k}$ with angular frequency, ${\omega }_{r}$ for the N–N coflow and N–VE coflow cases follow a similar trend but with a different slope. The percentage deviation of $k$ between theory and experiments is less than 10 %.

Figure 4.

Variation of wave wavelength and frequency, and the dispersion relation for N–VE and N–N coflowing combinations. (a) Experimental images showing variation of wavelength with flow rate ratio, $Q_{r}$ for N–VE and N–N coflow at $\tilde {x} \sim 3$ mm. (b) Experimental fit and theoretical variation of $k$ vs $\omega _{r}$ (dispersion relation) for N–VE and N–N coflow. (c) Variation of wave frequency, $\tilde {f}$ with $Q_{r}$. (d) Variation of wave wavelength, $\tilde {\lambda }$ with $Q_{r}$. Scale represents 300 $\mathrm {\mu }$m.

Now we look into the effect of flow conditions and fluid properties on the physical characteristics of the unstable-waviness nature of the instability, by characterizing in terms of the dimensionless wavelength $\lambda \ (=\tilde {\lambda }/W_2)$ and amplitude $A (=\tilde {A}/W_2)$. The axial evolution of instability for different coflow combinations is captured at various axial locations ($\tilde {x}$), which is depicted in figure 5. In most cases, we observe that instability is suppressed along the channel length downstream. It is also evident that the decay in $A$ is accompanied by an increase in $\lambda$, as shown in figure 5, suggesting a complex relationship between the two parameters. We investigate the variation of $A$ with $x$ for a range of flow rate ratios ($Q_r$) as well as for the corresponding capillary number ratios ($Ca_r=Ca_1/Ca_2$) in figure 6.

Figure 5.

Experimental images of axial evolution of instability of FC40-SiO 1000, FC40-PEO 3 %, FC40-PEO 4 % and FC 40-PEO 5 %, coflow.

Figure 6.

Experimental variation of amplitude, $A$ vs $x$ for (a) FC40-SiO 1000, (b) FC40-PEO 3 %, (c) FC40-PEO 4 % and (d) FC40-PEO 5 % coflow. The dashed lines indicate the experimental fit of the form $A \approx A_0 \,\textrm {e}^{-k_i x}$, where $k_i$ is the decay rate.

The axial variations in amplitude $A$ for different coflow combinations are depicted in figure 6. With an increase in $Q_r$, $A$ decreases across all coflow combinations, reflecting reduced fluid field perturbations due to viscous and elastic stratification. As discussed in figure 5, the amplitude $A$ of waviness decays along the majority of the channel length, modelled as $A \propto \textrm {e}^{-k_i x}$, where $k_i$ represents the complex wavenumber, indicating the decay rate with $x$. Experimental data in figure 6 give $k_i$ values: (i) FC40-SiO 1000, $k_i \approx 0.0125{-}0.0160$; (ii) FC40-PEO 3 %, $k_i \approx 0.0069{-}0.0089$; (iii) FC40-PEO 4 %, $k_i \approx 0.0118{-}0.0194$; and (iv) FC40-PEO 5 %, $k_i \approx 0.0112{-}0.0308$. These values indicate the maximum decay rate for FC40-PEO 5 % and the minimum for FC40-PEO 3 %, which follow the variation of viscosity $\mu _{{PEO} \ 3\,\%}<\mu _{{SiO}-1000}<\mu _{{PEO} \ 4\,\%} <\mu _{{PEO} \ 5\,\%}$ and relaxation time $\tau _{{PEO} \ 3\,\%}<\tau _{{PEO} \ 4\,\%} <\tau _{{PEO} \ 5\,\%}$. This indicates the effects of varying viscous and elastic stratification along the channel, which cause elongation of the wavelet, as indicated by blue arrows in figure 5. In addition to the decreasing trend of $A$, an increasing trend in amplitude is observed for FC40-SiO 1000 in figure 6(a) for $\tilde {x} > L/2$, where wavelets contract, leading to an increase in the amplitude. The evolution of amplitude $A$ along $x$ is intricately linked with wavelength $\lambda$, as shown in Appendix E. For FC40-PEO 3 % and FC40-SiO 1000 coflows, experimental findings yield $\tilde {A}^{m}\tilde {\lambda }^{n} = \tilde {A}\tilde {\lambda }^{1.7} = \textrm {constant} \sim 10^7$, where $\tilde {A}$ and $\tilde {\lambda }$ are in $\mu m$ (see Appendix E). Additionally, the visco-capillary length scale $\ell _c$ is calculated to determine whether wavelengths fall in the long- or short-wave regime: $Ca = {\mu _{2}{\dot {\gamma }}_{2}l_{c}}/{\gamma _{12}} \sim 1$ gives $l_{c} \approx {{\gamma _{12}}/{\mu _{2}{\dot {\gamma }}_{2}}} \sim 100$ $\mathrm {\mu }$m. Considering the channel width $W \approx 300$ $\mathrm {\mu }$m as a reference geometric length scale, it can be inferred that $\lambda > \ell _c$ and $\lambda > W$, suggesting a long-wave regime.

4.4. Droplet generation regime

Finally, we investigate the mechanism of droplet generation for FC40-PEO 1.7 %, FC40-PEO 3 % and FC40-SiO 1000 coflow combinations. In the above cases, time lapse images are captured at a location, $\tilde {x} \approx 5$ mm to observe the time evolution of the interfaces. In the case of FC40-PEO 1.7 % with $m \approx 1/20$ and FC40-PEO 3 % with $m \approx 1/200$, from § 4.2, we get $c_i>0$, $v_{group}<0$ leading to the absolute instability (Huerre & Monkewitz Reference Huerre and Monkewitz1990; Guillot et al. Reference Guillot, Colin, Utada and Ajdari2007; Utada et al. Reference Utada, Fernandez-Nieves, Gordillo and Weitz2008; Sahu & Govindarajan Reference Sahu and Govindarajan2014), wherein instability would increase over time, resulting in breaking of the FC40 phase into droplets. From experimental results shown in figure 7(a), we observe that the interface deforms significantly and assumes $\tilde {\lambda } \approx 400$ $\mathrm {\mu }$m before the FC40 phase breaks into drops. Variation of the local vertical distance of the marked point (initially $\tilde {y} = 80$ $\mathrm {\mu }$m at $\tilde {t} < 0$ s) on the interface from lower/P1 sidewall (figure 7b, FC40-PEO 1.7 % case) is measured with time and presented in a $\tilde {y}$ vs $\tilde {t}$ plot (figure 7b). Droplet pinching first initiates (at $\tilde {t} = 0$ s) at some distance downstream of the marked point but gradually the droplet pinching point recedes to the left until it reaches the marked point beyond which it stabilizes. Hence the amplitude of the interfacial disturbance increases for $0\ \textrm {s} < \tilde {t} < 1\ \textrm {s}$. Subsequently, we do not observe any further leftward movement of the pinching point (figure 7b) and the amplitude of interfacial disturbance remains fixed at ${\sim }80\ \mathrm {\mu } \textrm {m}$ for $1\ \textrm {s} < t < 2\ \textrm {s}$, implying the pinching event is periodic. Similar to FC40-PEO 3 %, in FC40-SiO 1000 coflow, increasing $Q_2$ leads to breaking of the FC40 phase into droplets when $c_i>0$ and $v_{group}<0$, shown in figure 3. The droplet generation phenomenon is studied by capturing high-speed videos for sufficient duration $({\sim }4\ \textrm {s})$. The flow rate ratio ($Q_{r}$) for FC40-PEO 1.7 %, FC40-PEO 3 % and FC40-SiO 1000 coflow combinations, is kept constant at $Q_{r} \sim 2.78$, 3.85 and 2.33, respectively.

Figure 7.

(a) Experimental time lapse images of droplet generation for FC40-PEO 1.7 %, FC40-PEO 3 % and FC40-SiO 1000 coflows. (b) Time evolution of interface distance from lower (P1) sidewall for FC40-PEO 1.7 % coflow. Droplet breakup point and $t_{pinch}$ is shown for zoomed-in view of an interval of $0.1$ s.

Generally, the droplet generation is associated with a pinch-off time $\tilde {t}_{pinch}$, which can be considered as the time between two successive droplet pinch offs, as described in the literature (Guerrero et al. Reference Guerrero, Chang, Fragkopoulos and Fernandez-Nieves2019). In the present case of stratified coflow, $\tilde {t}_{pinch}$ can be calculated from the time gap between the two consecutive droplet pinch offs for $1\ \textrm {s} < \tilde {t} < 2\ \textrm {s}$, as shown in figure 7(b), wherein the amplitude of instability is constant. Using this argument, a small interval of 0.1 s from a $\tilde {y}$ vs $\tilde {t}$ plot (figure 7b) is analysed and we get $\tilde {t}_{pinch} \sim 0.01$ s. The droplet pinch-off time for the FC40-SiO 1000 coflow combination can be similarly obtained as $0.07$ s.

In summary, here we investigated interfacial instabilities resulting from the coflow of N–N and N–VE fluids within a microfluidic confinement. Our theoretical formulations elucidate the experimentally observed regimes and their transition, where we discuss the decoupled influences of viscous and elastic stratification. Additionally, we analyse the dispersion relation, variations in wavelet amplitude and wavelength and the phenomena of droplet formation. Our findings are pertinent to a wide array of applications in microfluidics, including: chemical, biological and biochemical applications.