2. Results and discussion

Figure 1(a) shows the generation process of an impinging liquid sheet: two identical cylindrical jets with a diameter of $R_j=25\ \mathrm{\unicode {x03BC} m}$ , ejected from capillaries, impinge at $2\theta =60^{\circ }$ and form a leaf-like sheet in the orthogonal direction. The high jet velocities ( $u_j \sim 20 \mathrm{\ m}\, \mathrm{s}^{-1}$ ) render the effects of gravity negligible. We measure the absolute thickness at specific points via white-light reflection spectrum, while the relative thickness distribution can be determined through the interference fringes of monochromatic, as illustrated in figure 1(b). The accuracy of this novel technique is 0.5 %, and the spatial resolution is $5\ \mathrm{\unicode {x03BC} m}$ (Peng et al. Reference Peng2024).

Figure 1.

(a) Schematic of the liquid-sheet generation. The ellipse with the red edge is a jet cross-section. (b) Interference fringes and the retrieved thickness distribution. (c) Phase diagram of kinematic viscosity and surface tension of the ternary liquid mixtures in the experiment. The pie charts show the mass composition of the mixtures.

To differentiate the effects of viscosity and surface tension, we blend three liquids – water, ethanol and glycerol – to create a mixed solution. Figure 1(c) shows the phase diagram of these mixtures, where the small pie charts depict the mass proportions of each constituent liquid. Numerous experiments indicate that the diagram can be divided into three regions. In the high-pressure region, highly viscous mixtures can cause excessive pressure that exceeds the fluid system’s capacity. In the unstable region, the mixtures have low viscosity or high surface tension. The unbalance between viscous damping and surface-tension restoration results in unstable sheets, consistent with previous findings (Bush & Hasha Reference Bush and Hasha2004; Sanjay & Das Reference Sanjay and Das2017). Consequently, most of the mixtures investigated in this work are in the stable region, where exceptionally stable liquid-sheet chains form. All data are measured at a temperature of 24.4 to $24.8\,^{\circ } \mathrm{C}$ .

Figure 2.

(a,b) Thickness distribution of two mixtures with $\nu =3.6\ \mathrm{mm}^2 \, \mathrm{s}^{-1}$ and $\nu =19.5\ \mathrm{mm}^2 \, \mathrm{s}^{-1}$ , respectively. (c) Thickness along the $z$ -axis for mixtures with similar surface tension but different kinematic viscosity. (d) Thickness a fixed point on the $z$ -axis as a function of Reynolds number and Weber number. The flow rate of a single jet is held constant at $2 \mathrm{\ mL}\, \mathrm{min}^{-1}$ for all data presented in figure 2.

Figures 2(a) and 2(b) shows the measured thickness distribution of two mixtures with kinematic viscosity of $\nu =3.6\ \mathrm{mm}^2\, \mathrm{s}^{-1}$ and $\nu =19.5\ \mathrm{mm}^2\, \mathrm{s}^{-1}$ , respectively. Their surface tensions are similar $ (\sigma =25\pm 1 \ \mathrm{mN} \, \mathrm{m}^{-1} )$ , and all other parameters are identical. The thickness decreases with distance $r$ but increases with viscosity at a given position, especially along the $z$ -axis. The separation point S is the origin (Hasson & Peck Reference Hasson and Peck1964), and the $z$ -axis is $0^{\circ }$ . The thickness discrepancy between the two is approximately $30 \,\%$ at $\phi =0^{\circ }$ azimuth, decreasing to less than $10 \,\%$ at $\phi =10^{\circ }$ azimuth. Such a trend was observed widely in the experiments. We plot the thickness $h$ along the $z$ -axis in figure 2(c) for four other mixtures with similar surface tension but different kinematic viscosity. One can see the thicknesses for all the mixtures are nearly identical at small $r$ , and become smaller with the increase of $r$ . The mixtures with higher viscosity are significantly thicker at given $r$ , which is beyond the prediction of the H&P model. We found that $h \propto r^{-0.86}$ for $\nu =11.0$ , and $h \propto r^{-0.97}$ for $\nu =2.3$ , while the H&P model (Hasson & Peck Reference Hasson and Peck1964) states that the thickness follows $h \propto r^{-1}$ (dashed line in figure 2 c):

(2.1) \begin{equation} h r=R_j^2 \sin ^3 \theta /(1-\cos \phi \cos \theta )^2. \end{equation}

While the viscosity profoundly affects the thickness of the sheets, the surface tension exhibits negligible influence. We use Reynolds number ( $\textit{Re} =2 R_j u_j / \nu$ ) and Weber number ( $We=2 \rho R_j u_j{ }^2 / \sigma$ ), $\rho$ is density of the liquid in figure 2(d) to illustrate the thickness at $r=2 \mathrm{\ mm}$ on the $z$ -axis. The gradient of the iso-thickness line is determined mainly by viscosity, represented by $R e$ , with surface tension, represented by $W e$ , having a relatively minor impact. Surface tension acts primarily on the rim of the liquid sheets, affecting their overall shape, but its effects on thickness will be disregarded in the subsequent discussion.

The liquid viscosity not only affects the variation of thickness with $r$ but also significantly influences its dependence on the azimuth angle $\phi$ . Figure 3(a) presents experimental results for two mixtures with $\nu =2.7\ \mathrm{mm}^2\, \mathrm{s}^{-1}$ and $\nu =13.2\ \mathrm{mm}^2\, \mathrm{s}^{-1}$ at different $\phi$ . Beyond the prediction of the H&P model, experimental results indicate a complex dependence of the thickness–distance product $hr$ on $\phi$ for mixtures with different viscosity. At $0^{\circ }$ , curves with greater thickness are positioned above curves with smaller thickness; however, this trend reverses approximately at $21^{\circ }$ . This indicates that the thickness decreases more rapidly than $r^{-1}$ , resulting in thinner sheets at large azimuth angles, a phenomenon not previously observed (Choo & Kang Reference Choo and Kang2001; Lu & Corvalan Reference Lu and Corvalan2014; Yang et al. Reference Yang, Zhao, Fu and Cui2014). With $r$ fixed at 1 mm , we show this azimuth-dependent viscosity effect in figure 3(b). Viscosity affects sheet thickness oppositely around $21^{\circ }$ , consistent with the trends shown in figure 3(a), resulting in a steeper gradient, with thicker sheets at small angles and thinner sheets at large angles. The thickness characteristics of these viscous liquid sheets may benefit applications such as laser ion acceleration (Chen et al. Reference Chen, Pukhov, Yu and Sheng2009).

Figure 3.

(a) The thickness–distance product $hr$ as a function of azimuth angle. The thickness of each curve is fixed. (b) Thickness at $r=1 \mathrm{\ mm}$ for different azimuth angles. The dashed line represents the predictions of our model presented in this work.

To further understand these angle-dependent viscous effects, we develop a theoretical model that describes the thickness distribution by considering momentum exchange and mass transport resulting from viscous interactions between adjacent radial regions. We assume a steady flow of the ultra-thin liquid sheet, neglecting gravity, evaporation, air friction and $x$ -direction velocity. The coordinate system and the elliptical cross-section of the impinging jet follows figure 1(a). After impingement, the flow forms a radial velocity field originating from the separation point S . For inviscid liquids, this radial field is maintained as the sheet evolves, allowing mass conservation to be $2 \sin \theta {\rm d} \phi \int _0^{q_j(\phi )} q u_j {\rm d} q=u_s h r {\rm d} \phi$ , where $u_j$ and $u_s$ are the velocities in the jet and the sheet, respectively. The distance between S and the jet boundary $q_j$ is

(2.2) \begin{align} q_j(\phi ) &= \Big\{b \cos \phi \sin ^2 \theta +\big[R_j^2\big(1-\cos ^2 \phi \cos ^2 \theta \big) -b^2 \sin ^2 \phi \sin ^2 \theta \big]^{1 / 2}\Big \} \Big/ \nonumber\\& \quad \big(1-\cos ^2 \phi \cos ^2 \theta \big). \end{align}

Here the parameter $b$ is the distance of S to the central point of the jet, as shown in figure 1(a). It determines the initial flow distribution from the jet to the sheet. For the subsequent analysis, a constant $u_j$ over the whole cross-section is assumed, yielding $h r=q_j^2 \sin \theta u_j / u_s$ . We experimentally ensure this assumption by setting a sufficiently long pre-impinging length (free jet length before impinging), which is also validated by OpenFOAM simulations (see supplementary material).

In the H&P model, it is assumed that $u_s$ equals $u_j$ everywhere, thus leading to $b=R_j \cot \theta$ and (2.1). However, $u_s$ is actually non-uniform (Choo & Kang Reference Choo and Kang2002), and deviates from a purely radial pattern for viscous liquids. With a perturbative treatment of viscous effects, $u_s$ is a sum of zeroth- and first-order terms: $\boldsymbol{u}_s= (u+u_r^{(1)} ) \boldsymbol{e}_r+u_\phi ^{(1)} \boldsymbol{e}_\phi$ . Under this assumption, it can be proved from the 2-D Navier–Stokes equations that $u=u(\phi )$ is independent of $r$ , and $u_\phi ^{(1)}(r, \phi )={2 \nu \partial u}/{u r \partial \phi } \ln ({r}/{R})$ increases monotonously with the azimuth angle from zero. The angle-dependent $u(\phi )$ signifies a direct viscosity-driven momentum exchange in adjacent radial regions, while the presence of angular velocity $u_\phi ^{(1)}$ implies mass transport across these regions. With this understanding of $u_s$ , we can add a perturbation term to $h r=q_j{ }^2 \sin \theta [1+\delta (r, \phi )]$ . The equation governing $\delta$ is derived by applying this with the flow conservation equation and the Navier–Stokes equations (see supplementary material). The results reveal a weak dependence of $\partial \delta / \partial r$ on $r: \partial \delta / \partial r \propto a(\phi )[1+o(r, \phi )]$ , validated experimentally as shown in figure 4(a). In the first-order approximation, we neglect the higher-order terms of $\delta$ with respect to $r$ and get an elegant thickness model:

(2.3) \begin{equation} h=\frac {q_j^2(\phi ) \sin \theta }{r}\left [1+a(\phi ) \frac {\nu r}{Q}\right ]. \end{equation}

Here $Q$ is the jet flow rate, and the dimensionless viscous coefficient $a(\phi )$ is a function of $u / u_j$ . This model exhibits a remarkably simple structure: the thickness varies as $1 / r$ plus a constant $a(\phi ) \nu / Q$ at any given azimuth. Consequently, with all other factors fixed, altering the liquid viscosity causes a proportional change in thickness at any position. In contrast to the H&P model, $h=q_j{ }^2 \sin \theta / r, a(\phi )$ describes the angular distribution of the viscous effects, while $\nu , r$ and $Q$ determine their strength. The physics of $\delta = a(\phi ) \nu r / Q$ is clear: larger $r$ indicate more complete momentum and mass exchange, leading to stronger viscous effects; larger $Q$ values, conversely, correspond to shorter interaction times and thus weaker viscous effects.

Figure 4.

(a) The proportional relationship between thickness deviation $\delta$ and $r$ . (b) Linear fitting of the slope of $\partial \delta / \partial r$ with respect to $\nu$ . The units of $Q$ are $\mathrm{mL} \mathrm{min}^{-1}$ . (c) Angular distribution of the viscous coefficient $a(\phi )$ . (d) Measured interference fringes of a mixture with $\nu =18.0$ . (e,f) Interference fringes predicted by our model and by the H&P model, respectively. (g) The experimental results compared with our model and the H&P model. (h) The length of liquid sheets $(\mathrm{mm})$ as a function of $R e$ and $W e$ .

Since $\delta$ is proportional to $r$ , figure 4(b) shows the slope of $\delta$ with $r$ at various flow rates and azimuth angles, using the form $h=h_0(1+k r)$ . With $\phi$ and $Q$ fixed, $k$ is proportional to the viscosity, and this proportionality is inversely proportional to the flow rate. These results support $\delta \propto \mathrm{\nu }r / Q$ described by our model in (2.3).

Another important aspect is the determination of $a(\phi )$ , which describes the angular distribution of momentum exchange and mass transport. It depends on the zeroth-order radial velocity $u(\phi )$ and the angular velocity $u_\phi ^{(1)}(r, \phi )$ . Indeed, $u_\phi ^{(1)}$ is also a function of $u(\phi )$ . Given the symmetry and periodicity of $u(\phi )$ , we can employ a cosine series expansion to investigate its form: $u=u_j \sum _k a_k \cos (k \phi )$ , ak refers to the expansion coefficients at each level of trigonometric expansion. $u(\varphi)$ depends on $q_j$ and the conservation equations of mass, momentum and energy from the jet to the sheet. Generally, $b$ can be expressed as $b=\beta R_j \cot \theta$ (Choo & Kang Reference Choo and Kang2007). Our experimental data yield a best-fit value of $\beta =0.69$ , and obtain

(2.4) \begin{equation} \begin{gathered} u / u_j=0.147+0.408 \cos \phi +0.879 \cos 2 \phi \\ \quad -0.767 \cos 3 \phi +0.186 \cos 4 \phi. \end{gathered} \end{equation}

Equation (2.4) demonstrates universal applicability at $\beta =0.69$ , i.e. the impinging angle $\theta =30^{\circ }$ . Figure 4(c) shows the theoretical value of $a(\phi )$ (dashed line) and experimental points. It changes from positive near the $z$ -axis to negative at larger azimuth angles, reaching zero at approximately $21^{\circ }$ , consistent with figure 3(b). The physics is explained as follows: at small azimuth angles, viscous momentum exchange is predominant, where liquid elements decelerate and transfer momentum to larger angles, thereby increasing thickness. The radial viscous force increases with the azimuth angle, thereby producing a torque that drives rotation from larger to smaller azimuth angles. Significant angular velocity $u_\phi$ at large azimuth angles drives mass transport from larger to smaller angles, resulting in a rapid decrease in thickness. Figure 4(c) shows the momentum exchange contribution to $a(\phi )$ as the solid line; the remaining portion represents angular mass transport.

Figure 4(d–f) compares the measured fringes with theoretical predictions from our thickness model and the H&P model for a mixture with $\nu =18.0$ . Our model closely matches experimental results; conversely, the H&P model exhibits significant discrepancies in both fringe orders and spatial distribution. Figure 4(g) illustrates the relative deviation of our model from the H&P model. While the experimental results show nearly perfect consistency with our model, they deviate from the H&P model by a factor of 0.4–1.6. The extent of viscous effects is determined by the maximum value that $r$ can take, and figure 4(h) shows the liquid sheet length $L$ , which can be fitted as $L / R_j \propto W e R e^{0.2}$ . Overall, regulating viscosity and surface tension offers precise in situ control over liquid-sheet size and thickness, overcoming current limitations that rely on mechanical regulation, particularly in microfluidic chips (Loh et al. Reference Loh2020; Treffert et al. Reference Treffert2022). This can be achieved by adjusting solution components and solute or, for magnetorheological fluids, by applying a magnetic field (Du et al. Reference Du, Zhao, Liu, Ma, Dong and Huang2024).

The viscous effects are significant when the Ohnesorge number in the sheet is comparable to 1: $O h=\nu(h \sigma / \rho )^{-1 / 2} \sim 1$ . In this work, $O h \sim 10{-}100$ , but for large-scale, low-speed liquid systems with $R_j\gt 1\ \mathrm{mm}$ , the viscous effects are weak as $Oh \ll 1$ . Considering $h \propto R_j^2,\ Q=\pi u_j R_j^2,\ u_j R_j \sim \textrm {const.}$ , we can estimate that $\delta \propto Q^{-1} \propto O h$ . For this reason, the H&P model, in which thickness is considered independent of $\nu$ and $Q$ , has been adequate for many years due to the relatively large flow rate. However, with the increasing importance of microscale, ultra-thin liquid sheets, viscous effects must now be taken into consideration. Moreover, the momentum exchange and mass transport are fundamental aspects for the further understanding of liquid sheets, including those formed by converging (Crissman et al. Reference Crissman2022) and gas-assisted nozzles, as the underlying physics is similar (Woo et al. Reference Woo2018).