3. Flow regimes

We systematically examine microflow arrangements as a function of flow rates, Q ₁ and Q ₂, for various solvent concentrations Φ_S (figure 2(a)). While a wide range of flow morphologies are obtained from the beginning to the end of microchannels, three main types of regimes are identified in the field of view near the fluid contactor, ranging from x = 0 at the needle tip to approximately x/h ∼ 10, including (a) droplet flow patterns, which comprise the dripping regime, where droplets form from a growing meniscus of L1 at the tip of the microneedle, and the jetting regime, where droplets are emitted at the tip of a slender jet of L1 (figure 2(c)), (b) wetting and phase inversion flow regimes, where dynamic wetting leads to the encapsulation of L2 or the formation of complex stratifications at the walls (figure 2(b)) and (c) jet regimes, which are characterised by various core–annular flow patterns (figure 2(d)]). Such generic flow regimes are also observed for pure fluids at Φ_S = 0 with notably wetting regimes found at low Q ₂ and jet regimes obtained at large Q ₁. Initially continuous jets eventually breakup into droplets due to the Rayleigh–Plateau instability, however, due to the convective nature of the instability, at large injection velocities, jets remain locally stable and form continuous streams over distances x/h>10 that are of the order of the typical size of microfluidic systems.

Figure 2. (a) Maps of flow regimes based on injection rates Q ₁ and Q ₂, including dripping(), jetting(), phase inversion()and core–annular flows(), at various solvent concentrations (i) Φ_S = 0.0, (ii) 0.2, (iii) 0.4, (iv) 0.6 and (v) 0.8. (b, c and d) micrographs of typical flow regimes, flow rates in μl/min, from top to bottom. (b) Phase inversion/wetting regime: (i) phase inversion of large droplets (Φ_S, Q ₁, Q ₂) = (0.0, 2, 1) and (0.4, 1, 1), (ii) wall coalescence of small droplets (0.6, 0.6, 5) and (0.6, 0.9, 5) and (iii) wetting jets (0.6, 40, 10) and (0.8, 0.5, 2). (c) Dripping and jetting droplets regimes, (i) dripping droplets at low Φ_S (Φ_S, Q ₁, Q ₂) = (0.0, 2, 10) and (0.0, 6, 10), (ii) dripping droplets at moderate Φ_S (0.4, 3, 5) and (0.6, 4, 20) and (iii) jetting droplets (0.6, 1, 20) and (0.6, 3, 50). (d) Jet regimes, (i) quasi-straight jets (Φ_S, Q ₁, Q ₂) = (0.8, 20, 200) and (0.6, 50, 50), (ii) swelling jets (0.8, 0.5, 50) and (0.8, 0,2, 20) and (iii) varicose jets (0.4, 170, 50) and (0.6, 15, 50).

Overall, the main influence of solvent concentration Φ_S on flow maps is to reduce the operating range of droplet dispensation regimes, which progressively shift from nearly full range at Φ_S = 0 (figure 2(a)(i)) to null at large Φ_S = 0.8 (figure 2(a)(v)). While dripping flows are readily produced for purely immiscible fluids at Φ_S = 0 (figures 2(c)(i)), the oil–solvent continuous phase becomes progressively turbid due to the spontaneous emulsification of a diffusive fluid layer — referred to as conjugate or consolute fluid layer — around microfluidic droplets, which recirculate in segmented flows as Φ_S increases (figure 2(c)(ii)). At moderate Φ_S = 0.6, dripping flows consists of compact arrangements of droplets evolving in time in a sheath of conjugate fluids and significant shifts toward lower values of flow rates are observed for both the droplet/jet regime transition as well as for the dripping/jetting transition. Previous work on multiphase flow in microchannels have highlighted the importance of the capillary number Ca = ηJ/γ, where η is the dynamic viscosity and the velocity J = (Q ₁ + Q ₂)/h ², on transition between dispersed and separated flows (Hu & Cubaud, Reference Hu and Cubaud2020). Here, the dripping/jetting transition at low Q ₁ occurs around Q ₂ ∼ 10² μl/min^–1 for low Φ_S between 0 and 0.4 but decreases to 2 × 10¹ μl/min^–1 at moderate Φ_S = 0.6 and disappears at large Φ_S = 0.8, which suggests that the interfacial tension γ remains relatively constant at low Φ_S and sharply diminishes at moderate Φ_S before vanishing at large Φ_S. A similar behaviour is observed for the droplet/jetting transition at large Q ₁ with a steep decrease of approximately an order of magnitude in critical Q ₁ between Φ_S = 0.4 and 0.6 (figure 2(a)(iv)]). The morphological evolution of droplet flows based on solvent concentration is examined in the next section.

At low velocities, another regime of interest consists of the partial wetting of ethanol droplets and jets at the walls of microchannels made of borosilicate glass. Dynamic wetting properties typically induce the phase inversion of liquid–liquid dispersions with the generation of solvent-rich oil L2 droplet in a continuous phase of ethanol L1 (figure 2(b)(i)). As Φ_S increases, significant mass transfer across the interface results in the rupture of the intercalating film of L2 between walls and droplets with the formation of complex stratifications with spontaneous emulsification at the walls (figure 2(b)(ii)). This phenomenon is also observed during the formation of jets at high Φ_S (figure 2(b)(iii)).

Finally, continuous separated flows are generally obtained at large velocities with a variety of core–annular flow patterns resulting, including straight jets (figure 2(d)(i)), swelling jets (figure 2(d)(ii)) as well as varicose and distorted jets (figure 2(d)(iii)). Strongly diffusive jets are found to enlarge in sheaths of spatially developing layers of emulsifying conjugate fluids with the formation of periodic structures at high Φ_S. In the following, we study the evolution of the three main regimes, including droplet, phase inversion and jet flow patterns.

4. Droplets

We first examine the evolution of quantities associated with immiscible multiphase flows made of ethanol and pure oil, i.e. at null solvent concentration Φ_S = 0. In particular, the streamwise length of droplets d/h is measured in the dripping and jetting regimes at fixed side flow rate Q ₂ and varying central flow rate Q ₁ (figure 3(a)). For large droplets, the size is found to scale as φ = Q ₁/Q ₂ according to

(4.1) \begin{equation}d/h=k_{\rm D} \varphi,\end{equation}

where the constant k _D = 0.97, as expected from the typical dripping regime of immiscible flows at low χ « 1 (Hu & Cubaud, Reference Hu and Cubaud2020). By contrast, for small sizes d/h<1, the scaling of jetting flow is recovered, such as

(4.2) \begin{equation}d/h= k_j\varphi ^{1/3}, \end{equation}

with k _J = 1.15. The distinction between dripping and jetting regimes is not so sharp for very small droplet sizes, which follow a similar scaling, as can be seen in figure 3(b) for low φ. Overall, while small variations of d/h are observed based on flow velocity Q _T/h ², the droplet size remains largely independent of flow velocity for Φ_S = 0. In comparison with d/h, the spacing between droplets L/h is found to mainly depends on flow rate Q ₁ at large values and φ at lower values. In particular, data show that the spacing decreases according to L/h ∼ Q ₁^–1/2 (figure 3(c)). A useful parameter of segmented flows is the wavelength λ = d + L of droplet patterns, which reaches a minimum value at the transition between diluted and concentrated flows (figure 3(d)). The lowest value of λ also marks the transitions in scaling laws between d/h ∼ φ ^1/3 and d/h ∼ φ. For diluted flows, the wavelength of repeating units of segmented flows is strongly dominated by the droplet size d/h, whereas λ is controlled by L/h. Hence, the normalised wavelength scales as λ/h ∼ φ for large φ (figure 3(e)) and λ/h ∼ Q ^–1/3 for low φ (figure 3(f)). In both cases, various iso-Q ₂ branches of λ/h are observed in graphs with a reduction of the wavelength as the side flow rate Q ₂ increases. At larger Q ₂, the transition to the jetting morphology with low values of both d and L is found to follow

(4.3) \begin{equation}\lambda / h= k_{\mathrm {L}} \varphi ^{1/6}, \end{equation}

where k _L = 1.6 as well as the scaling relationship λ/h ∼ Q ₁^1/6. Overall, these relationships, obtained for pure fluid pairs in the absence of mass transfer across the interface provide a useful metric to characterise droplet formation at low χ in coaxial microchannels.

Figure 3. Reference segmented flow with pure fluids at Φ_S = 0. (a) Evolution of droplet size d/h with flow rate ratio φ. Solid line: Eq. (4.1). Dashed-line: Eq. (4.2). (b) Micrographs of ethanol droplet formation in pure oil at fixed Q ₂ = 50 μl/min, from bottom to top: Q ₁ = 1, 3, 15, 50, 130 μl/min. (c) Droplet spacing L/h as a function of inner phase flow rate Q ₁. Solid line: L/h = 2.7 Q ₁^–1/2. Dashed line: L/h = 0.25. (d) Variations of segmented flow wavelength λ/h with φ based on d/h and L/h at fixed Q ₂ = 50 μl/min and varying Q ₁. (e) Normalised wavelength λ/h versus flow rate ratio φ. Solid line: λ/h = 0.97φ. Dashed line: λ/h = 1.6φ ^1/6. (f) Evolution of λ/h with Q ₁. Solid line: λ/h = 3.2Q₁^–1/3. Dashed line: λ/h = 0.78Q₁^1/6.

We now turn our attention to the role of the solvent concentration Φ_S on droplet morphology and examine the evolution of d/h in the diluted regime at low φ. In general, the droplet size is found to slightly increase with Φ_S and follows Eq. (4.2) similar to that of the pure oil case, where k _J depends on both Φ_S and the side flow rate Q ₂ as shown in figures 4(a) to 4(c). Data for each iso-Q ₂ curve is fitted with the previous equation and values of the measured factor k _J are plotted for each Φ_S as a function Q ₂ (figure 4(d)). While a small dependence on Q ₂ is found according to k _J = k ₀Q ₂^–0.15, the dimensional prefactor k ₀ is well fit with a restricted equation of the form k ₀ = 1.8Φ_S^0.87 (figure 4(d)-inset). The overall increase in droplet size d/h at low velocities is expected from the reduction in capillary number albeit this is not significant at Φ_S = 0 over the range of parameters investigated in this work. Despite the complexity in the evolution of droplet flows in a solvent-rich oil phase, the change in initial droplet size remains relatively modest as Φ_S increases. At moderate concentration Φ_S = 0.4, the continuous phase becomes progressively cloudy due to the interdiffusion of solvent and ethanol across the droplet interface and observations of fine dispersions in the continuous phase at low velocities suggest the presence of a flux of ethanol from droplets into the external phase, which is driven by large solvent concentration near the interface (figure 4(e)). Over time, streams of miniature droplets coalesce into bigger droplets as can be seen in figure 4(e), downstream of the reference droplet at longer times. While the continuous phase morphology spatially evolves at modest Φ_S = 0.4, segmented flow characteristics, such d/h, L/h and λ/h, remain relatively constant along the flow direction. At larger concentration Φ_S = 0.6, however, significant droplet size d/h enlargement and spacing L/h reduction are uncovered along the flow direction. To illustrate this behaviour, segmented flow features are compared for identical values of flow rates in figures 4(f) and 4(g) for Φ_S = 0.4 and 0.6. At larger solvent concentration, it is found that, while d/h and L/h vary along the flow direction, the wavelength of segmented λ/h remains essentially constant (figure 4(h)). In this case, droplet growth indicates a flux of solvent from the continuous phase to the droplet, which is accompanied by a drastic modifications of the morphology of flow pattern unit cells, having a fixed spatial period.

Figure 4. Droplet formation at low φ. Flow rates in μl/min. (a) Evolution of d/h as a function of φ for Φ_S = 0.2. Solid line: d/h = 0.95φ ^1/3. Dashed- ine: d/h = 1.67φ ^1/3. (b) Variation of droplet size with flow rate ratio for Φ_S = 0.4. Solid line: d/h = 0.91φ ^1/3. Dashed line: d/h = 1.68φ ^1/3. (c) Droplet size d/h versus φ for Φ_S = 0.6. Solid line: d/h = 1.3φ ^1/3. Dashed line: d/h = 1.9φ ^1/3. (d) Role of flow rate Q ₂ on prefactor k_J for Φ_S = 0.2(), 0.4()and 0.6().Solid lines: k _J = k ₀Q ₂^–0.15. Inset: k ₀ versus Φ_S. Solid line: k ₀ = 1.8Φ_S^0.87. (e) Time series of micrographs in the droplet reference frame at Φ_S = 0.4 for (Q ₁, Q ₂) = (3, 5), Δt = 2 s. (f) Spatial evolution of d/h, L/h, and λ/h at Φ_S = 0.4 for (Q ₁, Q ₂) = (1, 10). (g) Flow spatial evolution at Φ_S = 0.6 for (Q ₁, Q ₂) = (1, 10). (h) Micrograph showing spatial evolution of droplet microflows at Φ_S = 0.6 for (Q1, Q2) = (1, 10), Δx/h ∼ 3.

As the wavelength of flow patterns remains spatially constant at low and large concentrations, the parameter λ/h provides a useful metric to examine multiphase flows. From small to moderate Φ_S = 0.2 and 0.4, the spatial period of regular patterns of dripping droplets follows a very similar behaviour as that of the pure fluid case at Φ_S = 0, with scaling laws such as λ/h ∼ φ at large φ and λ/h ∼ Q ₁^–1/3 at low velocities, as displayed with solid lines in figures 5(a) and (b). Likewise, Eq. (4.3) associated with the wavelength of jetting droplets is recovered, namely λ/h ∼ φ ^1/6 with similar prefactors k _L as for Φ_S = 0 and λ/h ∼ Q ₁^1/6. For the case of large concentrations, including Φ_S = 0.6, however, data for both dripping and jetting droplets collapse onto a curve defined as λ/h ∼ φ ^1/6 and the wavelength is seen to decrease with side flow rates Q ₂ in figure 5(c), which is contrast with observed behaviour at low Φ_S as illustrated in figure 5(d)(i), where λ/h remains constant at varying Q ₂ for Φ_S = 0.4, and in figure 5(d)(ii), where λ/h decreases with Q ₂ for Φ_S = 0.6. While iso-Q ₂ curves of wavelength λ/h plotted as a function of Q ₁ scale with exponent 1/6, the prefactors k _L in Eq. (4.3) are larger than the ones for jetting in the pure fluid case, which is indicated with a dashed line in figure 5(c) — bottom, and shown in figure 5(d)(iii). Therefore, segmented flows at large Φ_S display peculiar behaviours with the formation of apparent dripping droplets adopting the behaviour of jetting droplets as a result of the solvent–droplet exchange process. The significant mass transfer occurring during droplet growth and detachment from the microneedle, with a large accumulation of solvent in the aqueous phase and a relatively slow infusion of L1 into the area with high solvent concentration around the droplet, leads to the formation of dripping flows having small droplet spacing L/h and uniform size d/h ∼ 1. As previously shown from the spatial evolution of droplets, the spacing L/h is further reduced downstream due to the recirculation of the solvent-rich phase through forced convection rolls during transport between droplets. Hence, the overall effect of solvent is the reduction of droplet spacing L/h and as a result λ/h compared with the regular dripping case. Therefore, the morphology of flows observed at low φ ∼ O( –1) resembles those expected at large flow rate ratio φ ∼ O(0) where d ∼ L, as seen in figure 5(d)(ii).

Figure 5. Characteristics of segmented flows at various solvent concentration Φ_S. (a) Evolution of wavelength λ/h as a function φ and Q ₁ at Φ_S = 0.2. Top: solid line, λ/h = 0.97φ; dashed line, λ/h = 1.46φ ^1/6. Bottom: solid line, λ/h = 3.2Q ₁^–1/3; dashed line, λ/h = 0.68Q ₁^1/6. (b) Wavelength λ/h versus φ and Q ₁ at Φ_S = 0.4. Top: solid line, λ/h = 1.15φ; dashed line, λ/h = 1.3φ ^1/6. Bottom: solid line, λ/h = 4Q ₁^–1/3; dashed line, λ/h = 0.53Q ₁^1/6. (c) Variations of λ/h with φ and Q ₁ at Φ_S = 0.6. Top: dashed line, λ/h = 1.66φ ^1/6. Bottom: solid line: λ/h = 1.53Q ₁^1/6; dashed line: λ/h = 0.78Q ₁^1/6. (d) Experimental micrographs. Flow rates in μl/min. (i) Role of Q ₂ for Φ_S = 0.4, from bottom to top (Q ₁, Q ₂) = (4, 5), (4, 10) and (4, 20). (ii) Role of Q ₂ for Φ_S = 0.6, from bottom to top (Q ₁, Q ₂) = (1, 5), (1, 10) and (1, 20). (iii) Jetting morphology at φ ∼ 3 × 10^–3 , from bottom to top, (Φ_S, Q ₁, Q ₂) = (0.2 , 3, 100), (0.4 , 7, 200) and (0.6, 0.6, 20).

The reduction of initial droplet spacing L/h as the solvent concentration Φ_S increases is due in part to the combination of mass exchange between internal and external phases as well as the change of viscosity contrasts and local interfacial tension as fluids partially mix. In turn, modifications of the ratio between viscous and interfacial tension stresses during droplet formation lead to a change of flow morphology. As a result, a general trend consists in the sharp drop of the wavelength λ/h with Φ_S at low φ when d/h ∼ 1, as illustrated in figure 6(a) for identical flow rates Q ₁ = 1 and Q ₂ = 5 μl/min across all fluid pairs. Corresponding micrographs are displayed in figure 6(b) and show the main features of solvent-infused multiphase flows investigated in this work, including a nearly immiscible two-phase flow behaviours from low Φ_S = 0.2 to moderate Φ_S = 0.4 with the emergence of darker streams of droplets resulting from the spontaneous emulsification of the aqueous phase in the solvent-rich oil phase at Φ_S = 0.4. At larger concentrations, Φ_S = 0.5 and 0.6, droplets appear surrounded by an external envelope of consolute fluids undergoing complex rearrangement processes. In addition to the influence of φ, such intriguing microstructures are also strongly dependent on absolute flow velocity Q _T = Q ₁ + Q ₂, which sets convective time scales and residence times of droplets in the field of view. At large Φ_S = 0.6, droplet flows obtained at fixed φ but various Q _T show similar spatial evolution of d/h and L/h with constant λ/h (figure 6(c)) but varying temporal evolution of multiphase flows parameters (figure 6(d)). The strong dependence of the rate of deformation of d/h and L/h on absolute flow rate Q _T shows the advantage of microfluidic droplet flows to enhance mass transfer with recirculating flow between droplets with segmented flows mixing faster at large velocities.

Figure 6. Morphology of segmented flows. Flow rates in μl/min. (a) Role of solvent concentration Φ_S in cell characteristics, including d/h, L/h and λ/h, at fixed (Q ₁, Q ₂)= (1, 5). (b) Micrographs of droplet formation at (Q ₁, Q ₂)= (1, 5) at various Φ_S. (c) Spatial evolution of flow pattern characteristic at Φ_S = 0.6 and φ = 10^–1 (i) Dashed lines: (Q ₁, Q ₂)= (1, 10); (ii) solid lines: (Q ₁, Q ₂)= (2, 20). Inset: micrographs of (i) and (ii) at x/h ∼ 2. (d) Temporal evolution of d/h, L/h and λ/h at Φ_S = 0.6 for (i) (Q ₁, Q ₂)= (1, 10) and (ii) (Q ₁, Q ₂)= (2, 20). Inset: micrographs of (i) and (ii) at t ∼ 2 sec.

Overall, the role of solvent concentration on microfluidic multiphase flows is diverse and varied and each combination of flow rates and Φ_S display unique, time-evolving microstructures. To gain better insights into microflows in the presence of spontaneous emulsification, we also document flow patterns with high-resolution photographs as shown in figure 7. This approach reveals the local partitioning of solvent around the meniscus with the formation of a thin, smooth film of isopropanol at the meniscus at moderate Φ_S = 0.4 and the inception of complex filamentous structures at larger Φ_S = 0.5 (figure 7(a)). In both cases, the solvent appears to permeate the meniscus near the microneedle and re-emerge at the meniscus tip to form a continuous stream of conjugate fluids, which envelops droplets further downstream. The permeation of solvent through the interface also leads to complex breakup processes, as shown in figure 7(b) with the formation of spiky microstructures in the interfacial neck during pinch-off at large Φ_S. In the case of segmented flows with large droplet sizes d/h, figure 7(c) shows the progressive influence of solvent with significant turbidity observed at moderate Φ_S = 0.4. The central stream made of consolute fluid flows at the peak velocity of parabolic flows ∼ 2.1Q _T/h ² in square channels, which is larger than the speed of droplets, typically adopts the average flow velocity ∼ Q _T/h ². As a result, the central stream recirculates between droplets and affects the stability of the intercalating fluids between L1 and the walls. In general, very fine microstructures of conjugate fluids form around droplets and lead to intricate, yet regular flow morphologies around droplets, as shown in figures 7(d) to 7(e).

Figure 7. Photographs of microfluidic droplets in a solvent-rich oil phase. (a) Meniscus morphology at moderate Φ_S = 0.4 (top) and large Φ_S = 0.5 (bottom). (b) Role of solvent concentration on neck breakup, Φ_S = 0.2(top) and 0.4 (bottom) with formation of conjugate fluid spikes. (c) Influence of Φ_S on concentrated segmented flows. (d) Spontaneous emulsification streams around a large droplet, Φ_S = 0.4. (e) Trail of consolute fluids in the wake of a small droplet, Φ_S = 0.4. (f) Elongated droplets in a cloak of spiky conjugate fluids, Φ_S = 0.6.

5. Phase inversion and partially wetting regimes

Another regime of interest consists of the phase inversion of segmented flows, which results from the liquid–liquid dynamic wetting properties of ethanol and silicone oil on borosilicate glass. While fully lubricated droplet flows are readily formed at large velocities Q _T/h ² and low flow rate ratios φ, i.e. for small d/h<1, partially wetting flows are observed at low Q _T/h ² and large φ. Indeed, measurements of static contact angles between ethanol droplets deposited on borosilicate glass in a bath of silicone oil show values θ _E0 ∼ 30⁰. Hence, ethanol droplets are prone to wet the hydrophilic walls of the square microchannel made of borosilicate glass at rest. As the velocity increases, however, the advancing contact angle θ _E,A is expected to rise according to a Cox–Voinov relationship of the form $\theta^3_{E,A} = \theta^3_{E0} + wV$ , where w is a constant that depends on material properties (Voinov, Reference Voinov1976; Cox, Reference Cox1998). Therefore, a dynamic wetting flow transition is expected for θ _E,A ∼ π above a critical velocity. Here, dynamic wetting regimes are observed at large Q ₁ and low Q ₂, i.e. for large droplets at small velocities, as seen in flow maps (figure 2(a)). An example of a partially wetting flow regime of droplets in coaxial microchannels is shown in the form of a time series for Φ_S = 0 in figure 8(a) and the corresponding spatio-temporal diagram of the centreline is displayed in figure 8(b). This case study shows, in particular, the growth of a dewetting hole in the film of L2 between a long droplet made of L1 and the walls in figure 8(a)(ii). As L1 makes contact with the wall it encapsulates L2, which locally forms a pinching neck (figure 8(a)(iii)) that breaks to form of a partially non-wetting oil droplet having an advancing contact angle θ _OA = π – θ _ER, where θ _ER is the receding contact angle of ethanol, and an advancing contact angle θ _OR = π – θ _EA. Overall, this mechanism can lead to the formation of periodic flows of oil droplets (figure 8(b)). For the case of very long droplets d/h » 1, the dynamic dewetting of the intercalating film resembles that of a bag breakup of a liquid droplet, as seen in figure 8(c). A variety of intriguing partially non-wetting flow morphologies are also observed in the presence of solvent, which induces multiphase flow inversion with the formation of spontaneously emulsifying oil droplets and solutal Marangoni flows further downstream (figures 8(d) to 8(f)).

Figure 8. Dynamic wetting and phase inversion regime. Flow rates in μl/min. (a) Time series of phase inversion process due to dynamic wetting at Φ_S = 0 and (Q ₁, Q ₂) = (2, 1). (b) Spatio-temporal diagram associated with (a). (c) Time series of dewetting hole growth resembling bag break at Φ_S = 0.2, (Q ₁, Q ₂) = (9, 1), Δt = 3.3 × 10^–2s. (d) Oil droplet formation at Φ_S = 0.4 , (Q ₁, Q ₂) = (8,5), Δt = 3.3 × 10^–1 s . (e) Oil droplet formation at Φ_S = 0.4 , (Q ₁, Q ₂) = (6,1) Δt = 6.7 × 10^–1 s. (f) Evolution of oil droplet in the droplet reference frame at Φ_S = 0.2, (Q ₁, Q ₂) = (3, 2). Δt = 2.7 s.

6. Jets

Finally, a variety of aqueous jets are formed in a solvent-rich oil phase at very large concentration Φ_S = 0.8. Core–annular flows represent an important class of separated flow pattern complementary to dispersed flow patterns made of droplets. At such large solvent concentrations, direct formation of droplets is impeded by the large solubility of the conjugate fluid layer around L1 at the fluid contactor. Hence, continuous streams of L1 are surrounded by conjugate fluids and are emitted from the cylindrical microneedle into the square microchannels at relatively large Q _T. Such jets are seen to reach a minimal diameter ε _M/h near the fluid contactor before enlarging due to effective diffusion with the surrounding fluids further downstream. A simple model of jet diameter ε ₀/h is developed for core–annular flow based Stokes equations in circular pipes in the absence of mass transfer and interfacial tension (Cao et al. Reference Cao, Ventresca, Sreenivas and Prasad2003). In particular, the core diameter can be estimated according to

(6.1) \begin{equation} \frac{\varepsilon_0 }{h}=\left(\frac{1+\varphi -\sqrt{1+\varphi \chi ^{-1}}}{2+\varphi -\chi ^{-1}}\right)^{1/2}. \end{equation}

Since for a given viscosity contrast χ = η ₁/η ₂, the width of the central stream is expected to depend on the flow rate ratio φ, we conduct a series of experiments at fixed φ and varying Q _T to clarify the role of the absolute velocity of the flow morphology. We find, in particular, that for a given φ, ε _M/h is large at low Q _T and tends to ε ₀/h at high Q _T (figure 9(a)). Examples of flow patterns are displayed in figure 9(b) where a wide range of morphologies are observed for fixed flow rate ratio φ = Q ₁/Q ₂ = 2 × 10^–2 as Q _T = Q ₁ + Q ₂ varies over two orders of magnitudes from a thin jet at large Q _T to a thick core ensheathed in a layer of spontaneously emulsifying fluids at low Q _T, which indicates significant mass transfer.

Figure 9. Formation of jets at high solvent concentration Φ_S = 0.8. (a) Evolution of iso-φ curves of minimum jet diameter ε _M/h as a function of flow rate ratio φ. Solid line: Eq. (6.1). (b) Micrographs showing change in flow morphology as a function of Q _T for fixed φ = 2 × 10^–2. (c) Variations of iso-φ curves of ε _M/h with Q _T. Solid lines: Eq. 6.2. (d) Iso-φ curves of ε _M/h versus central flow rate Q ₁. Solid line: Eq. 6.2 with Q _T = Q ₁(1 + 1/φ) (e) Comparison of measured ε _M/h with predicted ε _M/h^*. Solid line: ε _M/h = ε _M/h^*. (f) Micrograph of jets with fixed side flow rate Q ₂ = 40 μl/min and varying Q ₁.

Plotting the diameter ε _M/h as function of Q _T reveals the strong influence of convective–diffusive phenomena with a jet diameter essentially controlled by Q _T at low velocities (figure 9(c)). Since the minimum diameter ε _M/h tends to unity in the highly diffusive regime at low velocities and tends to ε ₀/h in the convective regime at large velocities, similar to our previous work on the stability of core–annular flows made of miscible fluids (Cubaud, Reference Cubaud2020), we model the jet evolution as function of φ and Q _T using bounded functions, which find good agreement with data for the expression

(6.2) \begin{equation} \frac{\varepsilon _{M}}{h}=1-\left[\frac{1}{1-\varepsilon _{0}/h}+\left(\frac{Q_{T}}{Q_{c}}\right)^{-0.7}\right]^{-1} ,\end{equation}

where the critical flow rate Q _C = 13 μl/min depends on fluid properties, including Φ_S = 0.8, and the exponent –0.7 represents the steepness of curves displayed in figure 9(c). While data for ε _M/h collapse onto a single curve when plotted as function of Q _T a low velocities, the evolution of ε _M/h for each iso-φ curves remains ungrouped when plotted as function of Q ₁ as seen in figure 9(d) where fitting curves are developed based on Eq. (6.2) and the relation Q _T = Q ₁(1 + 1/φ). Overall, we compare measurements of ε _M/h with Eq. 6.2 and find remarkable agreement with our simple phenomenological model for the jet minimum diameter, which we refer to as ε _M^*/h (figure 9(e)). Detailed views of jet morphology at fixed Q ₂ = 40 μl/min and two widely different Q ₁ = 0.2 and 20 μl/min are displayed in figure 9(f) and show the presence of periodic structures of miniature droplets of conjugate fluids at the jet interface due to the solutal Marangoni effect. Convection–diffusion phenomena are usually treated with the Péclet number Pe = Jh/D ₁₂, where D ₁₂ is the diffusion coefficient of the liquid pair L1/L2. Previous work on the swelling of miscible threads in microchannels has shown the similarity in critical Pe_C ∼ Q_C/(hD ₁₂) ∼ 850 across various fluid pairs, which provides a mean for estimating D ₁₂ ∼ Q _C of swelling threads. Here, the value Q _C = 13 μl/min for the composite jet at Φ_S = 0.8 indicates a much larger value of effective D ₁₂ compared with that of the mixture of 100-cSt silicone/isopropanol where Q _C = 4.5 μl/min and D ₁₂ ∼ 3.5 × 10^–2 m²s^–1 (Cubaud et al. Reference Cubaud, Conry, Hu and Dinh2021). As a result, significant mass transfer is observed across the jet interface due to fast solvent shifting compared with miscible fluid diffusion. Material fluxes much larger than that due to pure molecular diffusion were also computationally observed in ternary liquid mixtures (Park et al. Reference Park, Mauri and Anderson2012).

At last, we investigate the role of solvent concentration on jet behaviour at moderate and large solvent concentrations Φ_S. In particular, we examine the morphology of jets at Φ_S = 0.6 at fixed values of Q ₂ and various Q ₁ (figure 10(a)). Similar to the previous case at Φ_S = 0.8, the minimum jet diameter ε _M/h is seen to tend toward ε ₀/h as the flow velocity increases. We find that iso-Q ₂ curves of ε _M/h at Φ_S = 0.6 are well fit with Eq. (6.2) using the simple identity Q _T = Q ₂(1 + φ) for each curve obtained at fixed Q ₂ on the graph of ε _M/h as a function of φ. In this case, the only fitting parameter in this family of curves corresponds to Q _C = 3.6 μl/min for Φ_S = 0.6. Hence, similitude arguments based on Péclet number suggest more than a threefold decrease in effective diffusion coefficient D ₁₂ between Φ_S = 0.8 and Φ_S = 0.6. Examples of jet morphologies at Φ_S = 0.6 are displayed in figure 10(b), where the surrounding layer of consolute fluid can be seen to develop curly structures at different jet sizes and velocities. Overall, we compare measurements of ε _M/h with our phenomenological model of minimum jet diameter ε _M^*/h based on bounded functions and find satisfactory agreement between data and theory (figure 10(c)). We also examine the spatial evolution of jet morphology as a function of moderate and large solvent concentrations Φ_S at φ = 1 for identical flow rates Q ₁ = Q ₂ = 100 μl/min (figure 10(d)). These micrographs show the spatial development of conjugate fluid layers ensheathing the jet at different concentrations with a nearly regular jet near the fluid contactor at moderate Φ_S = 0.4, which further develop into a sparce, low-density filamentous microstructure further downstream. At medium Φ_S = 0.5, the conjugate fluid layer appears to undergo a form a spinodal dewetting near the microneedle and develops into a compact, high-density filamentous structures wrapping up the jet further downstream. At large Φ_S = 0.6, the conjugate fluid layer appears thicker and continuous with the development of curly microstructures further downstream. Overall, the formation of periodic and aperiodic structures made of filaments and droplets at the jet interface in the presence of solvent constitutes an intriguing topic of investigations. More work is required to fully elucidate the dynamic morphogenesis of surrounding fluid layers in ternary fluid systems based on fluid and flow properties together with Marangoni flows.

Figure 10. Jets at moderate and large solvent concentrations Φ_S. (a) Evolution of iso-Q ₂ curves of minimum jet diameter ε _M/h as a function of flow rate ratio φ at Φ_S = 0.6. Solid lines: Eq. (6.2) with Q _T = Q ₁ + Q ₂. (b) Chart of jet micrographs near the fluid contactor at Φ_S = 0.6. (c) Comparison of measured ε _M/h with predicted ε _M/h^*. Solid line: ε _M/h = ε _M/h. (d) Micrograph showing the influence of solvent concentration on upstream and downstream jet morphology at φ = 1 with Q ₁ = Q ₂ = 100 μl/min for all fluids.