Effective water/water contact angle at the base of an impinging jet

The base of a jet impinging on an ultrapure water bath is studied experimentally. At the impact point, a train of capillary waves develops along the jet. By performing particle tracking velocity measurements, we show that there is a boundary layer separation between the jet and the meniscus. We thus describe the shape of this meniscus with a hydrostatic model. A striking observation is the existence of an effective nonzero water/water contact angle between the jet and the meniscus. The rationalization of this finite contact angle requires a full description of the shape of the interface. By doing an analytical matching between the meniscus and the jet, we show that the capillary waves can be considered as reflected waves present to ensure pressure continuity. It is finally shown that the value of the apparent contact angle is fixed by energy minimization, with an excellent agreement between prediction and experiment for small jets.

Figure 1

(a) (ii) Experimental image of a water jet of radius $R=100\mu \mathrm{m}$ and speed $V=2.1\mathrm{m}{\mathrm{s}}^{-1}$ impacting a bath of water. (iii) Zoom on the capillary waves developing at the jet surface fitted by a function of the form $r\left(z\right)=R+acos(kz+\phi )exp(-\frac{z}{z_{\mathrm{att}}})$. (i) Zoom on the connection between the meniscus and the jet, where the nonzero contact angle appears. (d) Comparison between the values of $z_{\mathrm{att}}$ and $\lambda$ extracted from the experimental fit and the ones calculated with Eq. (6) in Ref. [13] for a jet of radius $R=160\mu \mathrm{m}$.

Figure 2

(a) Fit of the meniscus, on the right and on the left, by Eq. (1) for a jet of radius $R=280\mu \mathrm{m}$ and speed $V=2.65\mathrm{m}{\mathrm{s}}^{-1}$ and for (b) a glass fiber of $R=250\mu \mathrm{m}$. (c) Results of the PTV experiments under the water surface for a jet of radius $R=280\mu \mathrm{m}$ and speed $V=2\mathrm{m}{\mathrm{s}}^{-1}$ and a jet of radius $R=80\mu \mathrm{m}$ and speed $V=2.8\mathrm{m}{\mathrm{s}}^{-1}$.

Figure 3

(a) Sketch of the jet impacting a bath. In the first virtual case, a straight jet connects to the meniscus, and exposes the impossibility of a pressure continuity between the two regions. In the second case, the capillary waves allow for pressure continuity. (b) Fitted interfaces of jets of various radii and speeds. The red line is the fitted meniscus, the blue line the wave field with the calculated amplitude, and the violet cross the calculated matching point $(r_0,z_0)$.

Figure 4

(a) Energy excess of the meniscus ($\mathrm{\Delta }{E}_{\mathrm{jet}}$), of the jet ($\mathrm{\Delta }{E}_{\mathrm{meniscus}}$), and their sum ($\mathrm{\Delta }{E}_{\mathrm{tot}}$) plotted against the speed for a jet of radius $R=190\mu \mathrm{m}$. The inserted sketches show the difference between the shape of the interface for a low and high contact angle. (b) Apparent contact angle fitted and predicted by our model for jets of various radii, plotted against the speed of the jet. The two experimental images show two reasons for the deviation from our model, the first one being instationarity (at the top) and the second one (at the bottom) the loss of axisymmetry, even when the stationarity is restored.