2.1. Experimental set-up

The liquid jet used throughout this study is typically set into motion by an overpressure applied by a pressure controller (Elveflow OB1 Mk4) on a water volume contained in a glass bottle (Schott). This forcing drives the liquid through the tubing connected to a stainless steel nozzle (Nordson EFD Optimum). The inclination of the nozzle is adjusted to produce a jet impacting the water bath at an angle $\alpha\! \in [21,49]^\circ$ . By changing the nozzle, the radius of the jet $R$ is varied between 0.12 and 0.73 mm. The flow rate is measured using a graduated cylinder and a timer, allowing the determination of the mean speed of the jet $V\! \in [0.8; 5.4]\ \textrm {m s}^{-1}$ .

The bath is contained in a cubic quartz tank (6 cm side length). To ensure repeatability of the experiments, the water must be free of contaminants. For this reason, all tubing used in the set-up is made of fluoropolymer (Versilon FEP, Saint Gobain), the water is ultrapure (resistivity greater than 18.2 $\textrm {M}\Omega\,\textrm {cm}$ ) and the bath is changed at least every 30 minutes.

We observe the interface using a LED backlight (PHLOX) and a digital camera (Basler a2A1920-160umPRO) equipped with a lens (telecentric SilverTL 0.75x when the jet radius $R$ is less than 0.25 mm, and Ricoh Lens FL-CC5028-2M 50 mm with an 8 mm extender ring otherwise). To be able to image the interface without deflection by menisci on the glass, we only keep the images taken when the tank is filled to the brim with water so that the interface is nearly flat.

2.2. Observation of the cavity

As shown in figure 1(a), when the jet impacts the bath vertically, we observe an axisymmetric meniscus in all our experiments. This is consistent with observations of smooth millimetric jets in the literature (Hancock & Bush Reference Hancock and Bush2002; Patrascu & Balan Reference Patrascu and Balan2021; Gaichies et al. Reference Gaichies, Salonen, Antkowiak and Rio2024). In this case, the flow separates from the interface, and the outer part of the interface can be described by hydrostatic equations (Gaichies et al. Reference Gaichies, Salonen, Antkowiak and Rio2024).

Figure 1.

(a) Experimental images of the interface impacted by a jet of radius $R = 0.19$ mm with a speed $ V = 2.8\ \mathrm{m\,s^{-1}}$ , with various angles $\alpha$ between the jet and the bath. (b) Measurements of the width of the cavity $W$ for jets of varying speeds, radii and inclinations with respect to the bath.

However, when the jet inclination $\alpha$ reaches around $50 ^\circ$ , a cavity appears in front of the jet. As the jet approaches the horizontal, the cavity widens. To characterise this evolution, we define the typical width of the cavity $W$ , measured at a depth below the interface equal to 10 % of the distance between the interface and the deepest point of the cavity. Due to the waves travelling along the cavity, the typical uncertainty on $W$ can be up to 10 % for large cavities. The experimental measurements are displayed in figure 1(b) for various radii, speeds and inclinations. The cavity is wider for larger and faster jets. Crucially, the experimental data highlight that the jet inclination is the dominant parameter governing the appearance and geometry of the cavity. The cavity width is up to 10 times larger for very inclined jets.

We will see in § 4 that the flow separation from the interface is still present in this inclined configuration. We can then expect that, similarly to the case of the vertical jet, parts of the interface can be described by hydrostatic equations. It is thus relevant to study the static equivalent of our system, an inclined fibre in a liquid bath, to isolate the effect of inclination on the shape of a meniscus.

5.1. Model for the outer part of the interface

As the liquid velocity in the outer part, which is separated from the inner, fast-flowing region, is low, we propose to describe this part of the interface with hydrostatic equations. With the notations of figure 5, we write the projection along the $z$ axis of the equilibrium between the Laplace pressure at the interface $P_0+\gamma \kappa$ , with $\kappa$ the curvature and the hydrostatic pressure $P_0 -\rho g z$ , to obtain, at $z=h(r,\theta )$ ,

(5.1) \begin{equation} \rho gh = -\gamma \kappa . \end{equation}

Within the slender slope approximation we write $\kappa = -\Delta h(r,\theta )$ . We use $l_c=(\gamma /\rho g)^{1/2}$ as a characteristic length and define the non-dimensional height as $\bar h= h/l_c$ . Our equation can then be rewritten as the non-dimensional Poisson equation $\Delta \bar h=\bar h$ . Assuming azimuthal symmetry of the form $\bar h = H(r)\exp (im\theta )$ , we obtain a Bessel equation for $H(r)$ :

(5.2) \begin{equation} \bar {r}^2H'' + \bar r H' - (m^2 + \bar r^2)H = 0. \end{equation}

The solutions of these equations are the modified Bessel functions $I_m$ and $K_m$ . As the $I_m$ diverge for $\bar r \to \infty$ , this leaves $K_m$ as the only candidates fulfilling the boundary conditions. The symmetry of our system, which goes from a meniscus in the acute part ( $\theta = 0$ ) to a cavity in the obtuse part ( $\theta = \pi$ ), suggests $m=1$ . As a result, the final dimensionalised solution takes the form

(5.3) \begin{equation} h(r,\theta ) = A_1l_cK_1\left(\frac {r}{l_c}\right){\rm Re} ({\rm e}^{i\theta }). \end{equation}

To test this model, we fit it on the profile of the interfaces extracted from various experimental images, leaving $A_1$ as a fitting parameter. The origin ( $r=0$ ) is chosen manually, as the fitting procedure failed when it was left as a free parameter. Its position is, however, severely constrained as the fit is done on both the acute and obtuse parts simultaneously. The origin is always situated in the middle of the straight part of the cavity, as indicated by a cross in the images of figure 5. These fits are represented on the experimental images in figure 5. We observe that the function $h(r,\theta )$ describes the outer part of the meniscus and the cavity simultaneously, with a single fitting parameter, which validates the hydrostatic modelling of the interface in the outer zone.

Figure 5.

Images of the interfaces, with the adjusted (5.3) represented in red. The origin for the fit is marked with a red cross. The conditions for the jets are, from left to right: ( $R = 0.12\,\textrm {mm}$ , $V = 3.6\,\textrm {m s}^{-1}$ , $ \alpha = 21^\circ$ ), ( $R = 0.12\,\textrm {mm}$ , $V = 2.9\,\textrm {m s}^{-1}$ , $ \alpha = 41^\circ$ ), ( $R = 0.73\,\textrm {mm}$ , $V = 0.9\,\textrm {m s}^{-1}$ , $ \alpha = 26^\circ$ ), ( $R = 0.73\,\textrm {mm}$ , $V = 1.2\,\textrm {m s}^{-1}$ , $ \alpha = 42^\circ$ ).

Figure 6.

Pressure fields (minus the hydrostatic pressure) (on the left) and velocity fields (on the right) in the x–y plane for various heights $z$ , from the simulation of a jet impacting the bath with $\alpha = 31^\circ$ . The increase of the velocity after impact is highlighted by the colour code, where values superior to the injection velocity are shown in black. The experimental image is of a jet ( $R = 0.19\,\textrm {mm}$ , $V = 2.8\,\textrm {m s}^{-1}$ , $ \alpha = 33^\circ$ ) impacting a water bath. The white line is a sketch of the streamline used to write Bernoulli’s equation. Grey arrows indicate the approximate width of the fields plotted on the side.

5.2. Model of the width of the cavity

We propose that the driving force behind the cavity’s creation is the contraction of the velocity field, which lowers the pressure below the interface, creating a pulling force on the interface. To verify this claim, we return to the numerical simulations. We plot in figure 6 the velocity fields and the pressure deviation from hydrostatics, in the x–y plane. We can see that above the bath level (at point 1), the velocity of the liquid in the jet is equal to the speed of injection. The local overpressure is due to the jet’s curvature. Along the cavity, the velocity increases and is concentrated near the free surface. This is denoted by the black colour in the velocity maps, black corresponding to velocities higher than the injection velocity. Below the cavity (at point 2), we observe a depression that correlates partly with the region of high speed. The simulations thus give us the hint that the depression results from the liquid acceleration. We can also observe that near the impact point, there exists a strong pressure gradient. It arises due to differences in the curvature: in the acute part, which is connected to the meniscus, the pressure is lower than in the obtuse part, where the curvature is positive. This pressure gradient can reorient the jet, which participates in its contraction. As this gradient exists over the typical height of the asymmetry of the meniscus, we propose that the velocity under the cavity (at point 2) $V_2$ scales with inclination in the same manner as $\Delta H$ . We thus write the simplest possible scaling for the speed $V_2 \sim \beta f(\alpha ) V$ , with $\beta$ a proportionality coefficient to be fitted and

(5.4) \begin{equation} f(\alpha ) = \ln \left ( \frac {1+\cos (\alpha )}{1-\cos (\alpha )}\right )\!, \end{equation}

the geometric factor obtained in (3.2).

If we follow the streamline represented in figure 6, we can write at point 1 Bernoulli’s equation $C = P_0+ \rho V^2/2$ , with $P_0$ the reference pressure at $z=0$ . We neglect the curvature of the jet, which leads to an overpressure $\gamma /R \sim 0.072/ 0.0002 \sim 360$ Pa which is negligible in front of the typical dynamic pressure $\rho V^2/2 \sim 2$ kPa.

At point 2, we have $C = P_j + \rho (Vf(\alpha ))^2 /2+ \rho gz$ , where $P_j$ is the local pressure. The deviation from hydrostatic pressure under the cavity is then $P_D = P_0 -\rho g z - P_j = ({1}/{2})$ $\rho V^2(\beta ^2 f(\alpha )^2 - 1)$ . The suction force at the origin of the cavity creation then scales as $ R^2 P_D$ . The opposing forces are the weight of the displaced water $F_g$ and the force exerted by surface tension $F_\gamma$ . We propose to write $F_g \sim \rho g W^3$ , $W$ being the typical length scale characteristic of the cavity. To have an estimate of $F_\gamma$ , we adapt the work of Raufaste et al. (Reference Raufaste, Kirstetter, Celestini and Cox2012), where they find, through simulations, that a soap film exerts a purely vertical force on a tilted cylinder (of radius $R_c$ ): $ F_\gamma =2\gamma R_c\pi /\sin (\alpha ) \sim \gamma R /\sin (\alpha )$ . Finally, by doing a one-dimensional vertical balance of the driving and opposing forces, as sketched in figure 7(a), we obtain a prediction for $W$ :

(5.5) \begin{equation} W = \left ( \frac {(\textit{RV})^2 (\beta ^2f(\alpha )^2 -1)}{2g} - l_c^2R/\sin (\alpha ) \right )^{1/3}\!. \end{equation}

Figure 7.

(a) Sketch of the force balance on the stationary cavity. (b) Comparison of the measured cavity’s width against the prediction of (5.5).

A comparison of the measured $W$ with the model is shown in figure 7(b), with $\beta = 0.68$ for all experiments. We can see that this simple model gives a reasonable estimate of the cavity size. However, the model overestimates the size of small cavities, and underestimate the size of cavities with $W \gt 4$ mm. This estimation would be improved with more delicate scalings of the cavity’s volume and surface, taking into account the three-dimensional nature of the cavity.