3.1. Flow modes and phase diagram

By varying the dispersed-phase flow rate (e.g. $0.5 \le Q_{\mathrm{d}} \le 90\,\mathrm{\unicode {x03BC} l}\,\mathrm{min}^{-1}$ ) and the rotational velocity (e.g. $0.003 \le \varOmega R \le 0.482\,\mathrm{m\,s}^{-1}$ ), four fundamental modes of droplet generation are observed in RFF: squeezing, dripping, jetting and tip-streaming modes. Representative images of each mode are shown in figure 2(a). In the squeezing and dripping modes, the dispersed phase undergoes periodic stretching and breakup at the needle tip, forming highly uniform microdroplets within the slit. Mechanically, these two modes are fundamentally similar, both resulting from the interaction between the shear forces of the two-phase fluid and interfacial tension, which leads to droplet formation. However, the morphological characteristics differ between them. In the squeezing mode, the dispersed phase entirely fills the rotor gap before breakup, producing droplets much larger than the rotor spacing. In contrast, in the dripping mode, the dispersed phase does not contact the rotor wall upon droplet formation, resulting in smaller droplets. Unlike FF or microchannels with fixed walls, the RFF in squeezing mode, characterised by a lower rotational velocity, restricts the dispersed phase from moving against the rotor wall during the initial droplet formation stage. As the dispersed phase fills the front of the gap, the rotor wall’s stretching action enables it to traverse the gap and form droplets. This dynamic rotor wall mechanism allows the RFF technique to efficiently generate larger droplets. In the jetting mode, the dispersed phase is ejected from the needle, forming a jet that breaks into droplets at its tip due to capillary instability. Unlike conventional jet flows, the slit shape constraints result in a longer conical structure in front of the needle, with the jet exhibiting a larger diameter at both ends and a narrower centre. The formation mechanism of this unique jet structure will be further analysed based on numerical simulations of the external flow field. In the tip-streaming mode, a stable cone of dispersed fluid forms at the needle tip, from which a microjet emerges and subsequently breaks into small droplets. These droplets are typically one to two orders of magnitude smaller than the needle diameter. Through these four flow modes, RFF demonstrates the ability to produce droplets spanning at least three orders of magnitude in size (micrometre to millimetre), combining the advantages of FF while exhibiting novel characteristics of droplet formation.

Figure 2. (a) Representative images of RFF in various flow modes. (b) Dimensionless phase diagram of RFF, with the transverse axis $Ca_{\mathrm{c}}$ representing rotational velocity ( $\varOmega R$ ) and the longitudinal axis $Re_{\mathrm{d}}$ representing the dispersed-phase flow rate ( $Q_{\mathrm{d}}$ ). The dashed line represents $Ca_{\mathrm{c}}+We_{\mathrm{d}}=1$ . (Geometric parameters: $L = 2\,\mathrm{mm}$ and $d = 0.3\,\mathrm{mm}$ .)

The dimensionless phase diagram for RFF is presented in figure 2(b). As $Ca_{\mathrm{c}}$ increases, the interfacial shear force also increases, leading to a transition from squeezing to dripping, and ultimately to jetting mode. For microchannel flow, jetting mode occurs when the sum of the viscous forces exerted by the continuous fluid and the dispersed fluid inertia overcomes the interfacial tension forces, which is expressed as $Ca_{\mathrm{c}}+We_{\mathrm{d}}\ge 1$ (Utada et al. Reference Utada, Fernandez-Nieves, Stone and Weitz2007), as indicated by the dashed line in figure 2(b). Given that the experimental parameters yield $We_{\mathrm{d}}=\rho _{\mathrm{d}}u_{\mathrm{d}}^{2}d_{\mathrm{i}}/\gamma \sim 10^{-2}$ , the dashed line can be approximated as $Ca_{\mathrm{c}} \approx 1$ . This result suggests that, under the experimental conditions, the dripping-to-jetting transition in RFF follows a similar criterion to that of microchannel flow. At low capillary numbers (e.g. $Ca_{\mathrm{c}}\lt 1$ ), the surface tension dominates and maintains the dispersed phase in the dripping mode; whereas at $Ca_{\mathrm{c}}\gt 1$ , the interfacial shear stress of the continuous phase exceeds the interfacial tension, initiating the transition to the jetting mode. In the tip-streaming mode, the transition from a conical tip to a fine jet requires that the interfacial shear force overcomes the constraint imposed by the interfacial tension at the characteristic scale of the jet diameter. Under such high-speed external flow conditions, the shear force is predominantly governed by the viscosity of the continuous phase, indicating that the capillary number of the continuous phase should satisfy $Ca_{\mathrm{c}} \geq 1$ . Additionally, the Reynolds number of the dispersed phase should remain sufficiently low (e.g. $Re_{\mathrm{d}} \ll 1$ ) to ensure that the viscous forces dominate, allowing the momentum transferred at the interface to be effectively diffused within the dispersed phase (Marín et al. Reference Marín, Campo-Cortés and Gordillo2009).

3.2. Quantitative analysis of squeezing and dripping modes

In the squeezing and dripping modes, the droplet size is comparable to the needle diameter, typically of the order of hundreds of micrometres. Figure 3(a) presents droplets generated under specific parameters, which exhibit good uniformity. During the experiment, droplets collected under different parameters are measured. Figure 3(b) shows the influence of flow control parameters – rotational velocity ( $\varOmega R$ ) and dispersed-phase flow rate ( $Q_{\mathrm{d}}$ ) – on the dimensionless droplet size ( $D_{\mathrm{d}}/d_{\mathrm{s}}$ ). These parameters correspond to the dimensionless numbers $Ca_{\mathrm{c}}$ and $Re_{\mathrm{d}}$ , respectively. The variable $d_{\mathrm{s}}$ represents the diameter of the contact line between the interface and the needle, as depicted in figure 3(c). When $Re_{\mathrm{d}}$ is held constant, an increase in $Ca_{\mathrm{c}}$ , which enhances the shear force between the interfaces, results in a reduction of droplet size. Additionally, a decrease in $Re_{\mathrm{d}}$ , representing a reduction in the dispersed-phase flow rate, also leads to smaller droplets.

Figure 3. Squeezing and dripping modes in RFF (geometric parameters: $L=2\,\mathrm{mm}$ and $d=0.3\,\mathrm{mm}$ ). (a) Images of resulting droplets, with a scale bar of $500\,\mathrm{\unicode {x03BC} m}$ ( $Ca_{\mathrm{c}}=0.548$ , $Re_{\mathrm{d}} = 0.406$ ). (b) Dimensionless droplet size, calculated from experiments and theoretical predictions, as a function of $Ca_{\mathrm{c}}$ . The dashed lines represent the results of (3.3). (c) Morphology of the dispersed-phase interface when the droplet is about to break ( $Ca_{\mathrm{c}}=0.455$ , $Re_{\mathrm{d}} = 0.406$ ).

To quantitatively explore the relationship between droplet diameter and experimental parameters, the detachment of the droplets at the needle tip is analysed. Figure 3(c) shows that droplets can be approximated as ellipsoidal during separation. The two minor axes of the ellipsoid are approximately equal and are denoted as $D_{\mathrm{short}}$ , while the long axis is denoted as $D_{\mathrm{long}}$ . The ratio of the long axis to the minor axis is defined as $K = D_{\mathrm{long}}/D_{\mathrm{short}}$ , a value primarily determined by $Re_{\mathrm{d}}$ .

During droplet detachment, the force preventing droplets from detaching from the needle tip is the interfacial tension, given by $F_{\gamma }=\pi \gamma d_{\mathrm{s}}$ . In contrast, the forces promoting droplet generation include the momentum transfer force $F_{I}=4\rho _{\mathrm{d}}Q_{\mathrm{d}}^{2}/(\pi d_{\mathrm{s}}^{2})$ and the viscous force $F_{\mu }=3\pi \mu _{\mathrm{c}}(u_{\mathrm{c}}-u_{\mathrm{droplet}})(D_{\mathrm{short}}-d_{\mathrm{s}})$ . The viscous force is derived from a modified Stokes formula, in which the characteristic length is defined as $(D_{\mathrm{short}}-d_{\mathrm{s}})$ . Here, $D_{\mathrm{short}}$ represents the droplet diameter in the flow direction, while $d_{\mathrm{s}}$ accounts for the influence of the needle on the force acting upon the droplet (Umbanhowar et al. Reference Umbanhowar, Prasad and Weitz1999). In the equation for viscous force, the droplet velocity $u_{\mathrm{droplet}}=4Q_{\mathrm{d}}/(\pi D_{\mathrm{short}}^{2})$ and the continuous-phase velocity is $u_{\mathrm{c}}= \varOmega R$ . Additionally, the droplets are subjected to buoyancy in the vertical direction, given by $F_{g}=\pi KD_{\mathrm{short}}^{3}g(\rho _{\mathrm{c}}-\rho _{\mathrm{d}})/6$ . Since the buoyancy and momentum transfer forces are three orders of magnitude smaller than the viscous forces and surface tension within the experimental parameter range, they are considered negligible. By balancing the viscous force and interfacial tension, the following equation is obtained:

(3.1) \begin{eqnarray} \pi \gamma d_{\mathrm{s}}=3\pi \mu _{\mathrm{c}}(u_{\mathrm{c}}-u_{\mathrm{droplet}})(D_{\mathrm{short}}-d_{\mathrm{s}}). \end{eqnarray}

After simplification and non-dimensionalisation, the following equation is derived:

(3.2) \begin{eqnarray} \bar {d}^{3}-\left(1+\frac {1}{3Ca_{\mathrm{c}}}\right)\bar {d}^{2}-\varphi \bar {d}+\varphi =0, \end{eqnarray}

where $\bar {d}=D_{\mathrm{short}}/d_{\mathrm{s}}$ and $ \varphi =4Q_{\mathrm{d}}/(\pi \varOmega Rd_{\mathrm{s}}^{2})$ . Elliptical droplets become spherical under the influence of surface tension after passing through the narrow slit. Let the diameter of the spherical droplet be $D_{\mathrm{d}}$ . Using the volume conservation principle, the volumes of spherical droplet and the elliptical droplet can be give as: $\frac {4}{3}\pi D_{\mathrm{d}}^{3}=\frac {4}{3}\pi KD_{\mathrm{short}}^{3}$ . This yields the following relation:

(3.3) \begin{eqnarray} D_{\mathrm{d}}=AK^{1/3}\bar {d}d_{\mathrm{s}}, \end{eqnarray}

where $A$ is a fitting constant, with a value of $A = 0.81$ , and $d_{\mathrm{s}} = 181\,\unicode {x03BC} \mathrm{m}$ in the experiment. The value of $K$ under different $Re_{\mathrm{d}}$ values is given as follows: $K \approx 1.61$ when $Re_{\mathrm{d}} = 0.135$ , $K \approx 2.44$ when $Re_{\mathrm{d}} = 0.406$ , $K \approx 3.42$ when $Re_{\mathrm{d}} = 0.677$ and $K \approx 4.39$ when $Re_{\mathrm{d}} = 0.948$ . Figure 3(b) shows the theoretical prediction curve spanning the squeezing and dripping modes, with the experimental agreement deteriorating at $Ca_{\mathrm{c}}$ extremes. At $Ca_{\mathrm{c}}\lt 0.2$ , the rotor wall confinement and flow non-uniformity limit the droplet growth despite theoretical predictions of size increase from reduced external flow velocity, yielding the experimental diameters smaller than the theoretical values. Conversely, near the dripping-to-jetting transition boundary ( $Ca_{\mathrm{c}}\gt 0.7$ ), satellite droplets exceeding 25 % of the main droplet diameter reduce effective main droplet volume, causing experimental sizes to undershoot predictions, and the flow instability broadens the droplet size distribution. Despite these deviations, a strong agreement persists in central squeezing and dripping modes.

3.3. Quantitative analysis of jetting mode

The jetting mode is one of the most significant modes for the preparation microdroplets in open-space microfluidics (Zhu et al. Reference Zhu, Chen, Huang, Wang, Zhu, Xu and Si2023). Figure 4(a) illustrates a typical jet in RFF, highlighting the key parameters of the jet: jet diameter ( $d_{\mathrm{j}}$ ), droplet size ( $D_{\mathrm{j}}$ ), unstable wavelength ( $\lambda$ ) and jet length ( $L_{\mathrm{j}}$ ). Variations in slit width significantly influence the downstream development of the jet. So $d_{\mathrm{j}}$ is defined as the width at the centre of the slit, while $L_{\mathrm{j}}$ extends from the conical end of the dispersed fluid to the jet breakup point, as shown in figure 4(a). Figures 4(b)–4(d) present the dimensionless droplet size, dimensionless jet length and dimensionless jet diameter for different flow control parameters, with the characteristic length being the inner diameter of the needle $d_{\mathrm{i}}$ . As $Ca_{\mathrm{c}}$ increases, both the jet diameter and droplet size decrease, while the jet length increases. Conversely, an increase in $Re_{\mathrm{d}}$ results in larger jet diameters, droplet sizes and jet lengths.

Figure 4. Jetting mode in RFF (geometric parameters: $L=2\,\mathrm{mm}$ and $d=0.3\,\mathrm{mm}$ ). (a) Typical images ( $Ca_{\mathrm{c}} = 1.38$ , $Re_{\mathrm{d}} = 0.406$ ). (b) Dimensionless jet diameter $d_{\mathrm{j}}/d_{\mathrm{i}}$ , calculated from experiments and theoretical derivations, as a function of $Ca_{\mathrm{c}}$ under varying $Re_{\mathrm{d}}$ values. (c) Dimensionless droplet diameter $D_{\mathrm{j}}/d_{\mathrm{i}}$ , calculated from experiments and theoretical derivations, as a function of $Ca_{\mathrm{c}}$ under varying $Re_{\mathrm{d}}$ values. (d) Dimensionless jet length $L_{\mathrm{j}}/d_{\mathrm{i}}$ as a function of $Ca_{\mathrm{c}}$ under varying $Re_{\mathrm{d}}$ values.

Figure 5. Numerical simulation of the rotational shear flow field (rotational velocity $\varOmega R = 0.3\,\mathrm{m\,s}^{-1}$ for a–c). (a) Velocity and pressure contour maps between narrow slits (origin at the centre of the slit throat). (b) Velocity distribution in the $x$ direction ( $u_{x}$ ) and pressure ( $p$ ) along the $x$ axis. (c) Velocity $u_{x}$ across the channel cross-section at different $x$ positions. (d) Velocity $u_{x}$ along the $x$ axis at varying rotational velocity.

To quantify the experimental observations and analyse the global evolution of the dispersed phase within narrow slits, the continuous-phase flow field between the rotors is numerically simulated. The geometric parameters and fluid properties are set to match the experimental conditions, with the rotational velocity specified as $\varOmega R = 0.3\,\mathrm{m\,s}^{-1}$ (detailed simulation methods are provided in Appendix A). Figure 5(a) displays the simulated velocity and pressure fields between two rotors (origin: the slit centre). The magnitude and direction of velocity contours and streamlines in the upper section are quantified. Flow contraction at the entrance accelerates the continuous fluid to maximum velocity at the throat, with a velocity decay downstream due to the flow expansion. Counter-flow vortices develop at both ends of the slit, opposing the primary flow. This flow topology aligns with prior double-rotor and four-rotor flow field studies (Dormy & Moffatt Reference Dormy and Moffatt2024; Jeffery Reference Jeffery1922; Taylor Reference Taylor1934). The lower-section pressure distribution in figure 5(a) shows the entrance peak and the symmetric exit minimum. Figure 5(b) quantifies the continuous-phase influence via centreline axial velocity ( $u_{x}$ ) and pressure ( $p$ ) profiles, directly explaining the biconical jet morphology in figure 4(a). The mass conservation dictates jet necking at the throat forming characteristic waist-like profiles. Concurrently, the frontal pressure peaks drive post-needle dispersed-phase accumulation, generating the conical structures.

The velocity profiles at different positions within the slit are presented in figure 5(c), where they closely approximate parabolic shapes. To quantify the external flow field characteristics, the dimensionless velocity ratio $\beta = u_{x}/(\varOmega R)$ is defined as the local $x$ -direction velocity normalised by the rotational velocity. At $x = \pm 0.79\,\mathrm{mm}$ , the $x$ -direction velocity is nearly equal to the rotor wall velocity, yielding $\beta \approx 1$ , whereas at the origin ( $x = 0, y=0$ ), $\beta$ reaches a maximum of approximately 1.47. Figure 5(d) presents the x-velocity profiles and $\beta$ -variation curves along the $x$ -axial direction under varying rotational velocities. The results reveal that increasing the rotational velocity proportionally increases the velocity magnitude throughout the flow field; the overall structure of the velocity profile and the position of the stationary point remain unchanged.

After determining the basic structure of the external flow field, the interfacial evolution of the dispersed phase can be analysed. Within the experimental parameter range, it can be assumed that the flow velocity of the dispersed phase $u_{\mathrm{d}}$ is equal to that of the continuous phase $u_{\mathrm{c}}$ . According to the conservation of mass:

(3.4) \begin{eqnarray} Q_{\mathrm{d}}=u_{\mathrm{d}}\frac {\pi d_{\mathrm{j}}^{2}}{4}=\beta \varOmega R\frac {\pi d_{\mathrm{j}}^{2}}{4}, \end{eqnarray}

and the dimensionless jet diameter can be related to $Ca_{\mathrm{c}}$ as follows:

(3.5) \begin{equation} d_{\mathrm{j}}/d_{\mathrm{i}}=\sqrt {\frac {4Q_{\mathrm{d}}\mu _{\mathrm{c}}}{\pi \beta \gamma d_{\mathrm{i}}^{2}}} Ca_{\mathrm{c}}^{-1/2}. \end{equation}

Since the jet diameter is measured at the narrowest point of the slit, the value of $\beta$ is 1.47. Figure 4(b) shows that the theoretical and experimental values of the jet diameter are in good agreement, which indirectly validates the numerical simulation.

As the jet develops downstream, it breaks up into a series of droplets due to capillary instability. The conservation of volume is satisfied during the jet breakup:

(3.6) \begin{eqnarray} \lambda _{\mathrm{max}}\frac {\pi d_{\mathrm{j}}^{2}}{4}=\frac {4}{3}\pi D_{\mathrm{j}}^{3}, \end{eqnarray}

where $\lambda _{\mathrm{max}}$ is the most unstable wavelength. By substituting the most unstable wavelength of the jet $\lambda _{\mathrm{max}}=\pi d_{\mathrm{j}}/k_{\mathrm{max}}$ and (3.5) into (3.6), the following can be obtained:

(3.7) \begin{eqnarray} \frac {D_{\mathrm{j}}}{d_{\mathrm{i}}} =\frac {1}{d_{\mathrm{i}}}\left(\frac {3}{2k_{\mathrm{max}}\sqrt {\pi }}\right)^{1/3}\left(\frac {\mu _{\mathrm{c}}Q_{\mathrm{d}}}{\gamma \beta }\right)^{1/2} Ca_{\mathrm{c}}^{-1/2}. \end{eqnarray}

In the analysis, two key dimensionless parameters are identified: the critical wavenumber $k_{\text{max}}$ and the velocity ratio $\beta$ . To facilitate the theoretical investigation of $k_{\text{max}}$ , the model is simplified. Based on axisymmetric linear stability analysis (Si et al. Reference Si, Li, Yin and Yin2009), it is found that the dimensionless critical wavenumber remains approximately constant at $k_{\text{max}} = 0.55$ under given experimental conditions (detailed calculations are provided in Appendix B). Regarding the velocity ratio $\beta$ , it is observed that as the jet length increases, the value of $\beta$ at the jet breakup point decreases from 1.17 to 0.42. Due to the rapid variation in the flow field near the jet breakup location, obtaining accurate $\beta$ values under specific experimental conditions is challenging. Moreover, the upstream flow field also influences the droplet formation process. Therefore, in the theoretical formulation of (3.7), a representative value of $\beta = 1$ is adopted for all calculations. The corresponding theoretical predictions, indicated by the dashed lines in figure 4(c), exhibit good agreement with the experimental results.

It is important to note that variations in flow parameters also influence the jet length $L_{\mathrm{j}}$ , which follows the scaling law $L_{\mathrm{j}} \sim u_{\mathrm{d}} t_{\mathrm{j}}$ . The jet velocity $u_{\mathrm{d}}$ , as established earlier, is approximated by the continuous-phase velocity $u_{\mathrm{c}}$ in the jetting mode. The jet breakup time $t_{\mathrm{j}}$ is governed by the Ohnesorge number (Lister & Stone Reference Lister and Stone1998), which is defined as $Oh_{\mathrm{d}}=\mu _{\mathrm{d}}/\sqrt {\rho _{\mathrm{d}}d_{\mathrm{j}}\gamma }$ . Specifically, when $Oh \ll 1$ , the breakup time is given by $t_{\mathrm{j}} = \sqrt {\rho _{\mathrm{d}}d_{\mathrm{j}}^{3}/\gamma }$ , while for $Oh \sim 1$ , it follows $t_{\mathrm{j}} = \mu _{\mathrm{d}}d_{\mathrm{j}}/\gamma$ . Under the experimental conditions, the Ohnesorge number of the dispersed phase is expressed as

(3.8) \begin{eqnarray} Oh_{\mathrm{d}}=\mu _{\mathrm{d}}\left(\frac {\pi \beta \varOmega R}{4Q_{\mathrm{d}}\rho _{\mathrm{d}}^{2}\gamma ^{2}}\right)^{1/4}\sim O(0.1). \end{eqnarray}

Accordingly, both time scales are considered separately, as presented in (3.9) and (3.10):

(3.9) \begin{align} L_{\mathrm{j}}\sim \left(\frac {4Q_{\mathrm{d}}}{\pi }\right)^{3/4}\left(\frac {\rho _{\mathrm{d}}^{2}\beta }{\mu _{\mathrm{c}}\gamma }\right)^{1/4} Ca_{\mathrm{c}}^{1/4}, \end{align}

(3.10) \begin{align} L_{\mathrm{j}}\sim \mu _{\mathrm{d}}\sqrt {\frac {4\beta Q_{\mathrm{d}}}{\pi \gamma \mu _{\mathrm{c}}}} Ca_{\mathrm{c}}^{1/2}. \end{align}

The actual jet length scale should lie between these two scaling laws. Figure 4(d) plots the two scaling curves. Experimental data indicate that when $L_{\mathrm{j}}/d_{\mathrm{i}} \lt 20$ , the relative jet length follows the expected scaling law. However, as $L_{\mathrm{j}}/d_{\mathrm{i}}\gt 20$ , the increase in relative jet length slows with rising $Ca_{\mathrm{c}}$ . This deviation is attributed to the fact that at $L_{\mathrm{j}}/d_{\mathrm{i}} = 20$ , the velocity ratio $\beta =1$ at the jet breakup point, while the axial velocity of the downstream continuous-phase flow rapidly decreases. In such a rapidly decelerating external flow field, the development of the dispersed-phase jet is hindered, leading to a departure from the theoretical scaling behaviour. Therefore, when employing the jetting mode for droplet generation, excessive rotational velocity and dispersed-phase flow rate should be avoided to ensure that $\beta \gt 1$ at the jet breakup position, thereby promoting the droplet formation in a relatively uniform flow field.

3.4. Experimental results on tip-streaming mode

In the tip-streaming mode, a conical fluid structure forms at the needle tip, where a slender jet emerges and subsequently breaks up into fine droplets. The droplet diameter can be one to two orders of magnitude smaller than the characteristic length scale of the device. However, the experimental observations indicate that stable tip-streaming becomes difficult to achieve when the rotor spacing $d$ is small and the needle-to-rotor-connection-midpoint distance $L$ is large. When geometric parameters are set to $d = 0.3\,\mathrm{mm}$ and $L = 2\,\mathrm{mm}$ , interfacial oscillations occur with intermittent tip-jet emergence, preventing uniform microdroplet formation. Conversely, adjusting parameters to $d = 0.8\,\mathrm{mm}$ and $L = 0\,\mathrm{mm}$ – effectively widening the slit and aligning the needle with the slit throat – establishes stable tip-streaming under identical flow conditions. Subsequent tip-streaming investigations maintain $d = 0.8\,\mathrm{mm}$ and $L = 0\,\mathrm{mm}$ , while § 3.5.1 details geometric parameter effects on interface evolution. As shown in figure 6(a), variations in $Ca_{\mathrm{c}}$ do not significantly alter the conical shape at the needle tip, although the droplet size changes substantially. To quantify the relationship between experimental parameters and droplet size, droplets are collected under a range of controlled conditions. The dimensionless droplet size as a function of $Ca_{\mathrm{c}}$ at different $Re_{\mathrm{d}}$ is presented in figure 6(b). The droplet diameter decreases monotonically with increasing $Ca_{\mathrm{c}}$ and decreasing $Re_{\mathrm{d}}$ . The droplet size is of the order of $10^{-5}$ m, while the production frequency can exceed the kilohertz range.

Figure 6. Tip-streaming mode in RFF (geometric parameters: $L=0\,\mathrm{mm}$ and $d=0.8\,\mathrm{mm}$ ). (a) Images of conical regions and droplets at different $Ca_{\mathrm{c}}$ ( $Re_{\mathrm{d}} = 0.0135$ ). (b) Dimensionless droplet diameter obtained from experiments as a function of $Ca_{\mathrm{c}}$ under varying $Re_{\mathrm{d}}$ .

3.5. Discussion

3.5.1. Geometrical parameters

In the preceding analysis, the geometric parameters of the RFF device are not considered as a primary focus. However, as previously discussed, stable tip-streaming mode becomes difficult to achieve when the rotor spacing $d$ is small and the needle-to-rotor-connection-midpoint distance $L$ is large, suggesting that geometric parameters play a critical role in the flow dynamics.

Figure 7. (a) Pressure distribution along the $x$ axis for different values of $d$ . (b) Velocity distribution along the $x$ axis in the $x$ direction for different values of $d$ .

Figure 8. (a) Influence of varying $L$ on the dripping mode under constant flow parameters ( $Re_{\mathrm{d}} = 0.406$ , $Ca_{\mathrm{c}} = 0.362$ ). (b) Influence of varying $L$ on the jetting mode under constant flow parameters ( $Re_{\mathrm{d}} = 0.677$ , $Ca_{\mathrm{c}} = 1.85$ ). (c) Influence of varying $L$ on the boundary of mode transition. (d) Influence of varying $d$ on the dripping mode under constant flow parameters ( $Re_{\mathrm{d}} = 0.406$ , $Ca_{\mathrm{c}} = 0.362$ ). (e) Influence of varying $d$ on the jetting mode under constant flow parameters ( $Re_{\mathrm{d}} = 0.677$ , $Ca_{\mathrm{c}} = 1.85$ ). (f) Influence of varying $d$ on the boundary of mode transition.

Figure 5(b) presents the pressure and velocity distributions along the x axis of the external flow field. A distinct pressure peak is observed near the front end of the slit, and its magnitude increases with higher rotational velocity. Under conditions of low dispersed-phase flow rate and high rotational velocity – characteristic of the tip-streaming mode – the dispersed phase accumulates and compresses upstream of the pressure peak, forming a conical interface. A fine jet is produced only after the tip of this cone crosses the pressure peak, resulting in the characteristic tip-streaming morphology. Once part of the accumulated fluid is ejected through the fine jet, the conical tip retracts upstream to the pressure peak, repeating the accumulation–jetting cycle. To establish a stable tip-streaming mode, $L$ must be sufficiently small so that the needle tip resides downstream of the pressure peak. Reduced $L$ also increases the velocity of the continuous phase at the needle tip, further promoting tip-streaming formation. Complementarily, figure 7 demonstrates how increased rotor spacing $d$ reduces the pressure peak while accelerating and homogenising flow at the slit entrance – conditions favouring tip-streaming generation.

The influence of geometric parameters on other flow modes and their transition boundaries is also investigated. Figure 8(a,b) illustrates the effects of variations in $L$ on the dripping and jetting modes, while figure 8(c) presents how changes in $L$ affect the mode transition boundaries. Increased $L$ reduces needle-tip velocity and enlarges the needle-to-pressure-peak distance. Consequently, droplet size expands in dripping mode, and the mode transition boundaries shift towards higher $Ca_{c}$ (i.e. higher rotational velocity). Notably, the dripping to tip-streaming boundary for $L=2.5\,\mathrm{mm}$ is absent in figure 8(c) due to insufficient continuous-phase velocity for tip-streaming within tested parameters. In the jetting mode, $L$ variations minimally affect jet breakup location and droplet diameter, as they solely alter the flow field at the needle outlet without affecting the downstream evolution process after jet formation. Figures 8(d) and 8(e) show the effects of varying rotor spacing on the dripping and jetting modes, respectively, and figure 8(f) illustrates the corresponding impact on the mode transition boundaries. Conversely, increased $d$ reduces the pressure peak while boosting dispersed-phase flux at the needle tip. These alterations yield smaller droplets in the dripping mode and facilitate transitions to jetting and tip-streaming modes. Larger $d$ also promotes uniform droplet formation in jetting mode by mitigating flow disturbances from rotor-shaft vibrations. Such vibrations – arising from inherent challenges in maintaining perfect concentricity between shaft and support – disrupt slit-proximal flow at high rotational velocity. Similarly, other geometric parameters are found to influence interfacial evolution. For instance, variations in rotor radius affect both the flow field structure and the pressure gradient within the slit, and an increased rotor radius promotes a more parallel flow field, facilitating stable droplet generation. Moreover, an increase in rotor height helps to suppress vertical three-dimensional effects, further contributing to the stability of interfacial evolution. These findings collectively indicate that, within an appropriate range, increasing $d$ and $R$ while decreasing $L$ promotes the stable production of uniform microdroplets.