3.1. Flow pattern transitions in oblique impinging jets

Figure 3 illustrates a series of flow patterns formed by the oblique impingement of two microjets at an impingement half-angle of $\alpha = 35^\circ$ . These patterns are captured synchronously from the views of the front (top images) and side (bottom images).

As $\textit{We}_j$ increases, distinct transitions in the flow morphology are observed, with two primary transitions between stable and unstable flow patterns. At low $\textit{We}_j$ values (figure 3 a), the jets merge to form a ‘liquid chain’ structure (Bush & Hasha Reference Bush and Hasha2004). Small oscillations in the upper region of the liquid chain are visible from the side view. With careful observation from the high-speed movies, we find that the stationary capillary waves (Lighthill Reference Lighthill2002) are generated at the interaction zone of the colliding jets. The interfacial deformation induced by the waves may lead to small oscillations. More details about the interaction between capillary waves and the upper region of the liquid chain are the focus of the present study. As $\textit{We}_j$ increases, the liquid chain enlarges and forms a stable liquid sheet with a rim along its edges (figure 3 b). The interface of the first liquid chain is stable and free of oscillations.

At intermediate $\textit{We}_j$ values (figure 3 c), the rim transits into a ‘wavy rim’ pattern. From the side view, periodic oscillations in the upper region are observed propagating downstream, inducing a sinuous motion in the rim. With further increases in $\textit{We}_j$ (as shown in figure 3 d), these oscillations intensify, leading to periodic atomisation along the rim, resulting in a symmetric ‘fishbone’ pattern (Bush & Hasha Reference Bush and Hasha2004), characterised by evenly spaced liquid beads that elongate into threads and subsequently break into droplets downstream. This observation reveals a strong correlation between the dynamics of upper region of the liquid sheet and rim instability, suggesting that oscillations in the upper region of the liquid sheet may play a critical role in developing fishbone patterns. These findings provide important insights into the origin of fishbone instabilities in impinging jets, which are further substantiated by subsequent numerical analysis.

As $\textit{We}_j$ continues to increase, the fishbone pattern expands laterally and eventually fragments into smaller droplets (figure 3 e–g). Despite the increasing complexity, the fishbone pattern maintains its periodicity. At higher $\textit{We}_j$ values, the liquid sheet transits back to a stable configuration (figure 3 h–i), forming a ‘closed rim’ pattern. In this regime, a stable rim surrounds the liquid sheet, and no significant atomisation occurs. At the highest $\textit{We}_j$ tested (figure 3 j), the rim becomes unstable again, resulting in an ‘open rim’ pattern. This instability is likely caused by infinitesimal disturbances along the rim that grow slowly and eventually lead to breakage. Notably, while the rim destabilises, the upper region remains tilted and stable. This rim instability mechanism differs from the fishbone pattern observed at lower $\textit{We}_j$ values. Note that the upper region of the liquid sheet, instead of maintaining a planar tilt, exhibits a pronounced leftward curvature at We $_j$ = 472.6. This behaviour is likely attributed to the sheet becoming sufficiently thin such that gravitational forces, which are typically negligible relative to inertial forces in thicker sheets, exert a non-negligible influence. The resulting imbalance causes the sheet to sag asymmetrically, leading to a curved morphology.

3.2. Evolution and dynamics of fishbone patterns

After identifying the role of the upper region of the liquid sheet in driving the flow dynamics, we focus on the evolution of fishbone patterns. Figure 4(a) shows the periodic oscillation of the upper region and its influence on the liquid rim and sheet interface for $\alpha = 35^\circ$ and $\textit{We}_j = 99.3$ . Four moments within one period $T$ are captured from the side and front views, with red arrows indicating the motion of the upper region. The lateral motion of the liquid sheet deforms the rim, generating disturbances that propagate along it to form the fishbone pattern. Two liquid beads are created per period, as marked by blue arrows at $T$ . As the beads move downstream, they coalesce, with their positions tracked. The coalescence of two beads is marked by numbers 1–4, enclosed in ellipses of different colours. The bead in the number 1 ellipse is tracked across different moments, demonstrating periodic behaviour. After one period, the pair of beads evolves into the same location and shape as the number 2 ellipse at the starting point.

At $\textit{We}_j = 122.1$ , the flow pattern exhibits an altering morphology, with period-doubling behaviour (figure 4 b), a characteristic phenomenon of nonlinear systems such as dripping faucets (Ambravaneswaran, Phillips & Basaran Reference Ambravaneswaran, Phillips and Basaran2000), submerged nozzles (Tuson Reference Tuson1955) and microfluidic bubble generators (Garstecki, Fuerstman & Whitesides Reference Garstecki, Fuerstman and Whitesides2005). In figure 4(b), two oscillation cycles of the upper region are explicitly marked by the period $T$ to illustrate the temporal regularity. During 2 $T$ , four liquid beads develop into repeating flow structures from the front view, highlighted by blue and green arrows at $t_2$ . The three liquid beads highlighted by the blue arrows merge with each other to form a flow structure, while the liquid bead highlighted by the green arrow develops into the other flow structure. Like figure 4(a), the flow structures marked by the number 1 ellipse are tracked over time and eventually evolve into the exact location and shape of the number 2 ellipse. Since the period of the fishbone pattern is twice that of the upper region, this flow pattern is identified as a period-2 fishbone. The corresponding wave on the downstream liquid sheet shows a darker wave that repeats once per period. At a higher $\textit{We}_j$ of 160.8, a period-4 fishbone morphology emerges, as shown in figure 4(c). Similar to the observation figure 4(b), the flow structures marked by the number 1 ellipse are tracked over time and eventually evolve into the exact location and shape as the number 2 ellipse, while eight liquid beads develop into repeating flow structures.

Figure 4.

Evolution of typical fishbone patterns within one period for different fishbone morphologies with $\alpha$ = 35 $^\circ$ . (a) Under $\textit{We}_j = 99.3$ with $T = 4.95\times 10^{-4}$ s; (b) under $\textit{We}_j = 122.1$ with $T = 4.20\times 10^{-4}$ s; (c) under $\textit{We}_j = 160.8$ with $T = 3.78\times 10^{-4}$ s.

Overall, figure 4 demonstrates how oscillations in the upper region of the liquid sheet drive successive period-doubling bifurcations, a hallmark of nonlinear systems. These bifurcations highlight the transition from a simple to an increasingly complex dynamics.

3.3. Interfacial and internal flow evolution in the upper region

In order to study the detailed flow dynamics of the upper region of the liquid sheet, direct numerical simulations are carried out for selected flow conditions. To validate the numerical model, the same physical and flow parameters as those in the experimental conditions are adopted for a direct comparison under $\alpha = 35^\circ$ and $\textit{We}_j$ = 122.1. The numerical simulation results, presented in figure 5(a), closely correspond to the experimental results shown in figure 5(b), confirming the model’s accuracy. Therefore, numerical simulations are a reliable tool for analysing the impact region’s complex interfacial evolution dynamics and flow field. However, it should be noted that these simulations are focused solely on the flow dynamics near the impact region of the liquid jets. The grid resolution is inadequate to resolve the flow field farther downstream, where the liquid sheet becomes thinner than the grid size, potentially leading to unphysical breakups.

Figure 5.

Comparison of typical flow patterns obtained from numerical simulation (a) and experiments (b) for impinging jets at $\alpha = 35^\circ$ and $\textit{We}_j$ = 122.1.

Figure 6(a) shows a stable liquid film at a $\textit{We}_j$ of 57.9, where the upper region remains undisturbed. As the velocity increases, small amplitude oscillations appear in the upper region. At a $\textit{We}_j$ of 63.5, figure 6(b) illustrates the liquid rim undergoing oscillations over one oscillation period with $T = 4.25\times 10^{-4}$ s. The upper region oscillates perpendicular to the impingement plane, generating initial disturbances propagating along the liquid rim. At a higher $\textit{We}_j$ of 122.1, figure 6(c) captures the complex three-dimensional variations of the liquid film with $T = 3.63\times 10^{-4}$ s. These results align with experimental observations and provide further insight into the underlying flow dynamics.

Figure 6.

Numerical results showing detailed insight into interfacial and internal flow dynamics. Panels (a), (b) and (c) depict the interfaces near the impact point of the liquid jets for three different $\textit{We}_j$ of 57.9, 63.5 and 122.1, respectively; (d), (e) and (f) illustrate the velocity vectors at the central cross-section for the three conditions.

The velocity vector field at the central cross-section of the liquid sheet is presented in figure 6(d–f), corresponding to figure 6(a–c), respectively. At lower velocities (figure 6 d), the upper region exhibits a roughly circular profile and remains stable, containing two counter-rotating vortices. These vortices are asymmetrical due to the pre-impingement jet length asymmetry. At higher velocities, figure 6(e) highlights the interaction between the vortices and surface tension, which drives the back-and-forth oscillation of the upper region of the liquid sheet. Figure 6(f) reveals a more complex dynamics at the highest velocity, showing a single distinct vortex instead of a vortex pair.

The dynamic processes depicted in figures 6(e) and 6(f) are discussed in figure 7. With $\textit{We}_j$ of 63.5, the upstream liquid sheet transitions to an unstable state, as shown in figure 7(a). At $t_1$ , the upper region shifts closer to the left jet, with the majority of the fluid rotating counterclockwise toward the left, while a smaller portion flows clockwise toward the right. As time progresses, at $t_4$ , the sheet briefly returns to a central position between the two jets, where the flow begins to divide asymmetrically, resulting in uneven fluid distribution to the left and right sides. By $t_5$ , the flow to the left diminishes while the flow to the right increases, causing the stagnation point to shift from the left side of the centreline to the right as the upstream liquid sheet moves closer to the right jet. Between $t_6$ and $t_{10}$ , the flow to the left gradually increases, while the flow to the right decreases, leading the upstream liquid sheet to shift back toward the left jet. During this process, the stagnation point returns to the left side of the centreline, completing one entire oscillation cycle. These findings highlight the dynamic interplay between fluid inertia and interfacial forces that govern the periodic oscillatory behaviour of the upstream liquid sheet. The stability and morphology of the liquid sheet are strongly influenced by $\textit{We}_j$ and the balance of forces driving the flow.

Figure 7.

Interface morphology and velocity field evolution under two $\textit{We}_j$ conditons for $\alpha$ = 35 $^\circ$ . (a) Under $\textit{We}_j$ = 63.5, while $T = 4.25\times 10^{-4}$ s; (b) under $\textit{We}_j$ = 122.1, while $T = 3.63\times 10^{-3}$ s.

With an increase in $\textit{We}_j$ to 122.1, the dynamics of the upper region of the liquid sheet becomes more pronounced, as shown in figure 7(b). From time $t_1$ to $t_3$ , the top of the liquid film and the left root of the upper region near the stagnation point gradually move closer to the centreline. Between $t_4$ and $t_5$ , the upper region begins to tilt toward the right jet, with the stagnation point crossing the centreline. During this phase, the fluid in the upper region exhibits a counterclockwise circular motion. As the fluid travels from the root to the top, its velocity progressively decreases until it approaches zero. By $t_6$ , the fluid in the upper region transitions to a clockwise circular motion. From $t_7$ to $t_9$ , the top of the liquid film and the stagnation point shift leftward across the centreline, while the left root of the upper region moves farther from the centreline. Finally, at $t_{10}$ , the internal velocity field of the upper region once again exhibits counterclockwise rotation, returning to the initial state observed at $t_1$ and thereby completing an entire oscillation cycle. This cyclical behaviour highlights the intricate interplay between inertial forces and interfacial dynamics within the upper region. The periodic shifts in velocity direction and stagnation point position underscore the sensitivity of the liquid film’s motion to changes in $\textit{We}_j$ and the complex internal flow dynamics driving the flow.

Figure 8.

Evolution of surface waves under $\alpha = 35^\circ$ and $\textit{We}_j = 122.1$ . (a) Temporal evolution of the liquid sheet interface over one oscillation period ( $T = 3.63 \times 10^{-4}$ s); (b) location of three selected transverse cross-sectional planes; (c) transverse cross-sectional views at different downstream positions, with the $x$ -axis coordinates magnified by a factor of 8 for clarity.

3.4. Propagation and origin of surface waves

Numerical simulations reveal two distinct types of waves on the liquid sheet and their interactions under typical conditions of $\alpha = 35^\circ$ and $\textit{We}_j = 122.1$ . As shown in figure 8(a), the first type is a short wave, marked by yellow arrows, which propagates circumferentially along the liquid rim with a wavelength comparable to $D_j$ . Yellow dots indicate the intersection points between the short waves and the liquid rim from $t_1$ to $t_6$ , highlighting the circumferential propagation of these disturbances. The second type is a long wave, marked by blue arrows, which propagates downward with a wavelength much larger than $D_j$ , expanding laterally as it moves downstream. The short waves are associated with undulations in the upper region of the liquid sheet, induced by oscillations of the rim, while the long waves originate from velocity oscillations within the downstream region of the sheet. Both wave types occur with a fundamental period $T$ , governed by the oscillatory behaviour of the upper region. The red-marked region at $t_1$ shows that the two wave modes remain disconnected, confirming that their formation mechanisms are independent. Furthermore, the long waves exhibit negligible influence on the rim dynamics, reinforcing the distinct roles played by each wave type in the overall flow behaviour.

In figure 8(b), three distances from the impact point, labelled as $d_1$ , $d_2$ and $d_3$ , are selected for transverse cross-sectional extraction. Transverse cross-sectional views at different axial positions are shown in figure 8(c) to illustrate wave propagation further. The spatial fluctuations of the short wave are evident (marked by elliptical shapes) and exhibit a relatively short wavelength, confirming that these fluctuations originate from the motion of the liquid rim. The figure also demonstrates minimal interaction between the two waves, supporting the conclusion that the rim of the sheet is not perturbed by shear instability, contrary to the hypothesis by Bremond & Villermaux (Reference Bremond and Villermaux2006). These findings align with the view initially proposed by Bush & Hasha (Reference Bush and Hasha2004), which suggests that fishbone instability is primarily driven by the rim rather than the sheet.

To investigate the generation mechanism of the long wave, figure 9(a) illustrates the velocity distribution across a cross-section of the liquid sheet, clearly revealing the presence of a long-wave structure. Figure 9(b) further presents the velocity vector field within the cross-sectional area, focusing on the region marked by the dashed box in figure 9(a) over one oscillation period $T$ . In this figure, blue dots indicate stagnation points in the velocity field, while red dots mark the jet impact points where the centrelines of the two impinging jets converge. The periodic motion of the blue stagnation points, alternating between the left and right sides of the centreline, demonstrates that oscillations in the upper region of the liquid sheet directly modulate the downstream flow field. This dynamic coupling highlights the pivotal role of upstream oscillations in driving the formation of long waves on the liquid sheet.

To further illustrate this influence, we analyse the lateral (x-direction) velocity component at the jet impact point, indicated by the red dot. This component is critical in generating and sustaining surface waves on the liquid film, which gradually attenuate as they propagate downstream. Figure 9(c) shows the time-dependent variation of the normalised x-direction velocity, $u_x/u_j$ , with velocity normalised by the jet velocity and time normalised by the oscillation period $T$ . During the first half of the oscillation cycle, the lateral velocity decreases as the liquid sheet shifts from left to right. In contrast, in the second half, the velocity increases as the sheet moves from right to left. The asymmetry between the maximum and minimum velocity magnitudes is attributed to the velocity difference between the two jets at the collision plane. This periodic modulation of the lateral velocity serves as the primary driver of long-wave formation on the liquid sheet, underscoring the essential influence of upstream oscillations on the downstream flow dynamics.

In summary, the simulations reveal the coexistence of two distinct wave modes on the liquid sheet: short waves, which propagate circumferentially along the rim and originate from rim-induced undulations; and long waves, which travel downstream and are generated by periodic modulation of the internal velocity field induced by upstream oscillations. These two wave types exhibit different spatial paths, arise from independent mechanisms and remain decoupled throughout their evolution. The minimal interaction observed confirms that the rim dynamics is not significantly affected by the long waves, reinforcing the view that the fishbone instability is primarily governed by rim oscillations rather than global shear modes in the sheet body.

Figure 9.

Characterisation of the downstream flow response to periodic upstream oscillations under $\alpha = 35^\circ$ , $\textit{We}_j = 122.1$ and $T = 3.63\times 10^{-4}$ s. (a) Cross-sectional velocity distribution showing the presence of a long wave; (b) velocity vector field illustrating stagnation point oscillations induced by upstream motion; (c) time variation of the normalised lateral velocity at the jet impact point.

Figure 10.

Different flow patterns resulting from varying $\alpha$ and $\textit{We}_j$ . Panels (a)–(f) are under $\alpha$ = 25 $^\circ$ ; (g)–(l) are under $\alpha$ = 45 $^\circ$ ; (m)–(r) are with varying $\alpha$ and similar $\textit{We}_j$ .

3.5. Flow patterns and frequency analysis at varying angles

Figure 10(a–f) illustrates the characteristic evolution of the liquid sheet under $\alpha = 25^\circ$ . The flow transitions from a liquid chain pattern to a fishbone pattern, then to a closed rim pattern, ultimately leading to the open rim pattern where the liquid rim breaks downstream. The sequence of flow transitions observed closely mirrors that reported in figure 3, indicating a consistent instability development pathway under $We_j$ conditions considered here. A similar trend is observed in figure 10(g–l) for an increased $\alpha$ of $45^\circ$ , where the liquid sheet initially remains stable but then develops rim perturbations, followed by the emergence of fishbone-like, closed rim and open rim structures. Notably, despite the change in $\alpha$ , the qualitative sequence of instability evolution – including rim destabilisation, periodic amplification and subsequent ligament breakup – remains largely unchanged. This observation suggests that the primary mechanisms driving the formation of fishbone patterns and rim instabilities are robust against variations in $\alpha$ , primarily governed by the intrinsic oscillatory behaviour of the upper region and the dynamic response of the rim region. Figure 10(m–r) further illustrates the evolution of flow patterns across a range of $\alpha$ values at approximately similar $\textit{We}_j$ . As $\alpha$ increases, the liquid sheet initially develops rim instabilities and fishbone-like structures. With a further increase in  $\alpha$ , the fishbone patterns become more complex, and eventually the flow stabilises progressively, leading to the formation of a closed rim. This morphological transition from a fishbone pattern to a closed rim closely mirrors the behaviour observed with increasing $\textit{We}_j$ under a fixed $\alpha$ . The earlier onset of rim undulations followed by their suppression at higher $\alpha$ values underscores the critical role of impingement geometry in modulating not only the instability dynamics but also the eventual steady-state morphology of the liquid sheet.

To further elucidate the oscillatory mechanisms underlying the observed instability evolution, a detailed POD analysis is conducted. Figure 11 systematically compares the dynamic characteristics of the first few POD modes (modes 1, 3 and 5) by jointly presenting their spatial-domain representations and PSD and under $\alpha = 35^\circ$ for three different $\textit{We}_j$ . The analyses are conducted from both side and front views. Figure 11(a) shows the POD results from the side view at $\textit{We}_j = 99.3$ . Mode 1 exhibits a distinct single-frequency peak at 2023.59 Hz, with a higher PSD than modes 3 and 5. This dominant frequency corresponds to the oscillation frequency of the upper region, denoted as $f_u$ . (Note that $f_u = 1/T$ .) The corresponding front-view results in figure 11(b) reveal the same dominant frequency of 2023.59 Hz in mode 1, which corresponds to the frequency of the fishbone pattern, $f_r$ . The equality between $f_u$ and $f_r$ confirms the single-period (marked as period-1) behaviour observed in figure 4(a). At a higher $\textit{We}_j$ of $122.1$ , figure 11(c) presents the side-view POD results, where mode 1 exhibits a dominant frequency of 2321.18 Hz ( $f_u$ ). In contrast, the front-view results in figure 11(d) reveal a dominant frequency of 1130.83 Hz ( $f_r$ ), approximately half of $f_u$ . The halving between $f_u$ and $f_r$ supports the period-doubling behaviour previously identified in figure 4(b). At the highest $\textit{We}_j$ considered, 160.8, figure 11(e) presents the side-view POD results, where mode 1 shows a dominant frequency of 2678.28 Hz ( $f_u$ ). In the corresponding front-view results shown in figure 11(f), the dominant frequency shifts to 654.69 Hz ( $f_r$ ), approximately one fourth of $f_u$ . This quartering of the frequency between $f_u$ and $f_r$ is consistent with the period-4 behaviour observed in figure 4(c). Overall, the results presented in figure 11 demonstrate that the oscillations in the upper region of the liquid sheet maintain a dominant frequency that increases with increasing $\textit{We}_j$ , while the fishbone frequency exhibits successive bifurcations from the frequency of the upper region. These findings confirm the presence of a hierarchical mode transition as $\textit{We}_j$ increases, highlighting the progressive complexity of the instability dynamics.

Figure 11.

Spatial and frequency-domain characteristics of the first few POD modes under $\alpha = 35^\circ$ at different $\textit{We}_j$ . Panels (a) and (b) are of the side and front views, respectively, for $\textit{We}_j = 99.3$ ; (c) and (d) are for $\textit{We}_j = 122.1$ ; (e) and (f) are for $\textit{We}_j = 160.8$ .

Figure 12 presents a special case at $\alpha = 45^\circ$ and $\textit{We}_j = 46.2$ . Figures 12(a) and 12(b) show that the dominant frequency of the upper region is 654.69 Hz ( $f_u$ ), while the frequency associated with the fishbone pattern is 1249.87 Hz ( $f_r$ ), approximately twice  $f_u$ . Examination of the fishbone evolution over one oscillation period of the upper region, as shown in figure 12(c), reveals that each liquid bead generates one thread during $T/2$ . This indicates that the flow structure has half the period of the upper region of the liquid sheet. However, since the liquid beads are generated by the oscillation of the upper region within one period, their positions in the side view do not repeat. As a result, this flow pattern is classified as a period-1 fishbone, even though two droplets are formed in each period.

Figure 12.

Spatial and frequency-domain characteristics of the first few POD modes under $\alpha = 45^\circ$ and $\textit{We}_j = 46.2$ , along with evolution of the corresponding fishbone pattern. Panels (a) and (b) are for POD modes from the side and front views; (c) evolution of the fishbone pattern.

Figure 13(a) illustrates $f_u$ and $f_r$ at different $\alpha$ and $\textit{We}_j$ values. In the figure, P-m denotes a period-m pattern of the fishbone structures, where m represents the number of cycles of oscillation observed in the upper region. The hydrodynamic resonance between the oscillations of the upper region and the rim dynamics governs the frequency behaviour. The diagram highlights the relationship between $f_u$ and $f_r$ . For $\alpha = 25^\circ$ , both $f_u$ and $f_r$ increase synchronously with $\textit{We}_j$ , indicating that the upper region oscillations directly influence the periodic formation of droplets, resulting in a stable and regular fishbone pattern along the liquid rim.

Figure 13.

Evolution of the upper region oscillation frequency ( $f_u$ ) and rim frequency ( $f_r$ ) as functions of $\textit{We}_j$ and $\alpha$ . (a) The main diagram showing synchronous frequency growth and the emergence of bifurcation behaviour with solid symbols representing $f_u$ and hollow symbols representing $f_r$ ; (b) special case of a P-3 mode observed under $\alpha$ = 45 $^\circ$ and $\textit{We}_j$ = 66.46; (c) special case of a P-7 mode observed under $\alpha$ = 30 $^\circ$ and $\textit{We}_j$ = 237.13.

As $\alpha$ increases, a divergence between $f_u$ and $f_r$ becomes evident, as shown in figure 13(a), indicating the onset of bifurcation behaviour. Bifurcations up to period-7 are observed, marking the emergence of complex interactions between the upper region oscillations and rim dynamics. In addition to the previously observed P-1, P-2 and P-4 modes, two special cases are identified: a P-3 mode at $\alpha = 45^\circ$ and a P-7 mode at $\alpha = 30^\circ$ , as shown in figures 13(b) and 13(c), respectively. Observing high-order bifurcations in the oscillation modes indicates that the system exhibits strong nonlinear characteristics. Such nonlinearity likely arises from multiple interacting mechanisms, including the upper region’s finite-amplitude oscillations, nonlinear coupling between the sheet and rim dynamics and competition between inertial and capillary forces. While these features suggest the presence of significant nonlinear interactions governing the flow evolution, a complete characterisation of the system as strongly nonlinear would require further dynamical analyses.

3.6. Regime transition and stability criteria

Figure 14 shows a regime diagram of the flow patterns observed across various $\alpha$ and $\textit{We}_j$ values. Stable–unstable–stable transitions are identified for all $\alpha$ values examined in this study. The lower boundary indicates the transition from the stable liquid chain pattern to the unstable wavy rim or fishbone pattern. In contrast, the upper boundary transitions from the unstable fishbone pattern to the stable closed rim pattern. Both critical $\textit{We}_j$ values decrease as $\alpha$ increases, reflecting the increasing susceptibility of the liquid sheet to instability at higher $\alpha$ .

Figure 14.

Regime diagram illustrating different flow patterns in the parameter spaces of $\alpha$ and $\textit{We}_j$ . The two lines indicate the boundaries of the fishbone patterns.

Our experimental and numerical studies demonstrate that the formation and stability of various flow patterns in microjet impingement are fundamentally influenced by the dynamics of the upper region of the liquid sheet and the presence or absence of a liquid rim. The destabilised motion of the vortex pair drives the transition between the liquid chain and the unstable patterns of the wavy rim or fishbone. However, predicting the critical condition for this destabilisation remains challenging due to the complex coupling between inertia, viscosity and surface tension.

To understand the transition between the unstable fishbone and stable closed rim patterns, we refer to the theoretical models of Hasson & Peck (Reference Hasson and Peck1964) and Bremond & Villermaux (Reference Bremond and Villermaux2006). Based on the method of Bremond & Villermaux (Reference Bremond and Villermaux2006), the upstream extent of the sheet, $L_{{e}}$ , is expressed as

(3.1) \begin{align}\nonumber\\[-24pt] L_{{e}} = d_j \textit{We}_j \frac {\sin ^3 \alpha }{8(1+\cos \alpha )^2}. \end{align}

For a cylindrical jet sectioned by a plane at an angle $\alpha$ , the half-major axis of the resulting ellipse is $d_j / (2 \sin \alpha )$ , with its focus offset by $d_j / (2 \tan \alpha )$ from the jet centre (Hasson & Peck Reference Hasson and Peck1964). For the liquid sheet to form, $L_{{e}} / d_j$ must exceed $(1 /\! \sin \alpha - 1 /\! \tan \alpha ) / 2$ , yielding the critical $\textit{We}_j$ (Bremond & Villermaux Reference Bremond and Villermaux2006)

(3.2) \begin{align} \nonumber\\[-18pt]\textit{We}_j^c = \frac {4(1+\cos \alpha )}{(\sin \alpha )^2}. \end{align}

While this theory predicts the trend that $\textit{We}_j^c$ decreases as $\alpha$ increases, it does not quantitatively match experimental results. This discrepancy arises because the theory does not account for the complex shape near the transition region from the liquid jet to the liquid rim at the upper region of the liquid sheet (as illustrated in figure 6 d–6 f). Additionally, liquid jet inertia must overcome viscosity and surface tension at the junction between the liquid jet and the liquid rim to form the thin liquid sheet assumed in these theoretical models.

The model proposed by Hasson and Peck (Hasson & Peck Reference Hasson and Peck1964) predicts that for a liquid sheet formed by impinging jets

(3.3) \begin{equation} h_{e} L_{{e}} = d_j^2 \frac {\sin ^3 \alpha }{4(1+\cos \alpha )^2}, \quad u_{e} = u_j, \end{equation}

where, $h_e$ is the thickness of the liquid sheet at the location where the upstream extent $L_e$ is achieved. The location is also the junction of the liquid sheet and the liquid rim at the upper region. Substituting (3.1) into (3.3), the Weber number of the liquid sheet at the rim junction, $\textit{We}_s^{e}$ , is derived as

(3.4) \begin{equation} \textit{We}_s^{{e}} = \frac {\rho u_{e}^2 h_{e}}{\sigma } = 2. \end{equation}

The value of $\textit{We}_s^{e}$ is consistent with the Taylor–Culick velocity (Taylor Reference Taylor1959; Culick Reference Culick1960), characterising the retraction speed of a liquid film driven by surface tension. By balancing the capillary retraction force, $2\sigma$ , with the incident momentum flux, $\rho u_e^2 h_e$ (Gordillo, Lhuissier & Villermaux Reference Gordillo, Lhuissier and Villermaux2014), the exact value of $\textit{We}_s^{e}$ can be derived. This relationship implies that as long as the stable liquid sheet exists, its minimum Weber number, $\textit{We}_s$ is 2, as the sheet is thinnest at its junction with the liquid rim at the upper region. According to (3.3), the condition $\textit{We}_s \geqslant 2$ is maintained throughout the upper region at the central cross-section.

Figure 15.

Cross-sectional view of the velocity inside the upper region of the liquid sheet for estimated the Weber number of the upper region of the liquid sheet $\textit{We}_s$ at $\alpha$ = 35 $^\circ$ and $\textit{We}_j$ = 122.1. Panels (a) and (b) correspond to $t_1$ and $t_{10}$ in figure 7(b).

To validate the above discussion, numerical simulations under conditions of $\alpha = 35^\circ$ and $\textit{We}_j = 122.1$ , representing a scenario in which a stable liquid sheet cannot form, are analysed. Figure 15 presents the cross-sectional velocity fields within the upper region of the sheet for two moments corresponding to $t_1$ and $t_{10}$ in figure 7(b), obtained by slicing a plane through the jet centreline. As the fluid travels from the base to the top of the liquid film, the velocity gradually decreases. Five characteristic positions are marked by coloured lines for each moment. At each position, the local sheet thickness $h_s$ and mean velocity $u_s$ are extracted to compute the local Weber number using $\textit{We}_s = {\rho h_s u_s^2}/{\sigma }$ . The analysis shows that at the red-marked location, $\textit{We}_s = 2$ ; above this point (blue lines), $\textit{We}_s \lt 2$ , indicating that inertial forces are insufficient to overcome surface tension, while below (yellow line), $\textit{We}_s \gt 2$ . The results indicate locations where $\textit{We}_s \lt 2$ in the upper region. Under these conditions, the rim fails to form, leading to the destabilisation of the liquid sheet. This observation confirms that the instability of the liquid sheet is directly associated with the violation of the $\textit{We}_s \geqslant 2$ condition, reinforcing the theoretical prediction. Interestingly, these findings align with the theoretical prediction by Lin, Lian & Creighton (Reference Lin, Lian and Creighton1990), which states that a liquid sheet becomes absolutely unstable when $\textit{We}_s \lt 2$ . In such instances, capillary effects dominate, leading to the rapid disintegration of the sheet.