Effect of ambient gas on cavity formation for sphere impacts on liquids

Formation of a splash crown and a cavity following the impact of a sphere on a body of liquid is a classical problem. In the related problem of a droplet splashing on a flat surface, it has been established that the properties of the surrounding gas can influence the splashing threshold. At lower impact speeds, this is due mainly to the influence of gas kinetic effects, since the height of the gas lubrication film which is displaced during dynamic wetting is often comparable to the mean free path of the gas. At higher Weber and Reynolds numbers, on the other hand, inertial effects dominate and the density of the gas becomes important in determining whether a splash occurs. In this article, sphere impacts on a liquid body are investigated in a rarefied atmosphere using high-speed photography. It is found that the threshold entry speed for cavity formation is influenced by the density of the surrounding gas, whereas changing the mean free path of the gas has no effect. We attribute this phenomenon to the gas slowing the sealing of the thin crown sheet behind the sphere. This assertion is supported with experimental measurements of the liquid sheet thickness. In the range of parameters considered, the splash crown influences the movement of the contact line.

Figure 1

Chronophotography of a 15 mm sphere plunging into 1.74 cP silicone oil at $V=2$ m ${\mathrm{s}}^{-1}$ at times 1.6, 3.2, 4.8, and 6.4 ms after touchdown. The pressures are (a) 1 bar, (b) 0.53 bar, and (c) 0.5 bar, respectively. Videos no. 1, 2, and 3 in the Supplemental Material [32].

Figure 2

Diagram of cavity/no-cavity regimes at varying gas density ratio $\rho /{\rho }_{\text{atm}}$ and capillary number Ca. Three parameters are varied as indicated in the legend: sphere diameter, molecular mass of the gas, and dynamic viscosity of the liquid. Cavity/no-cavity pairs with the same sphere size, oil viscosity, and gas type are connected by a line to indicate uncertainty of the cavity/no-cavity threshold (red, blue, and green lines are used for the 15, 8, and 2 mm balls, respectively). Black diamonds indicate regimes shown in Fig. 1. The red triangle and the black dashed line correspond to the experimental data and the theoretical prediction from [8].

Figure 3

Diagram of cavity/no-cavity regimes similar to that in Fig. 2 with the inverse mean free path ${\ell }_{\text{atm}}/\ell$ at the horizontal axis. Note that, although a trend related to the change in gas density is present, a number of experiments with He prevents us from plotting an unequivocal divider between splash and no-splash regimes.

Figure 4

Testing the influence of gas molecular mass (and thus mean free path). Comparison of the impact in air at $p=140$ mbar (left panels of each image) and in helium at $p=950$ mbar (right panels), videos no. 4 and 5 in the Supplemental Material [32]. Both cases correspond to the same gas density of 0.14 atmospheric. The 15 mm ball falls at 2.03 m ${\mathrm{s}}^{-1}$. Time separation between frames is 1 ms.

Figure 5

Testing the influence of gas density. Comparison of the impact in air at the density of 0.525 atmospheric (left panels) and in air at the density of 0.5 atmospheric (right panels), videos no. 2 and 3 in the Supplemental Material [32]. The 15 mm ball falls at 2.06 m ${\mathrm{s}}^{-1}$. Time separation between frames is 1 ms. Parameters are chosen close to the threshold so that the increase in the gas density by 5% leads to cavity formation.

Figure 6

Growth rate for wetted part of sphere against time. The dashed black line shows the radius of the cross section of the sphere at the level of the liquid. Red markers correspond to the experiment where cavity was formed (air at 1000 mbar), blue markers to the experiment with no cavity (air at 150 mbar). The impact speed is $V\approx 2.3$ m ${\mathrm{s}}^{-1}$ in both cases.

Figure 7

Spatiotemporal image of the curtain propagating towards the pole in the experiments shown in Fig. 5 (left panel is air at $p=525$ mbar and right panel is air at $p=500$ mbar). The dashed line indicates position of the unperturbed liquid surface. At lower gas density (right panel), the crown deceleration is not sufficient to prevent it from collapsing at the pole. At higher gas density (left panel), deceleration is higher and the liquid surface overtakes the crown, forming a cavity.

Figure 8

Oil vapor and laser sheet visualization of the gas flow behind the falling sphere (left) and of the vortex formed behind the crown. $V\approx 1.6$ m ${\mathrm{s}}^{-1}$, air at atmospheric pressure. Video no. 8 in the Supplemental Material [32]. The bottom of the image coincides with the unperturbed liquid level.

Figure 9

Asymmetric splash at low gas density at higher impact speed: Formation of cavities on one side while the most of circumference is cavity-free. $V\approx 2.8$ m ${\mathrm{s}}^{-1}$, helium, $p=138$ mbar left, side view and $p=200$ mbar right, top view. Videos no. 6 and 7 in the Supplemental Material [32].

Figure 10

Sheet thickness measured by the blue-colored oil: Contour levels of the liquid sheet thickness and the corresponding optical images. Atmospheric pressure, 13 cm drop height. Levels are plotted with the twofold increments in the liquid thickness. Observe the thickness corresponding to the major part of the curtain corresponds to the interval 80–$160\mu \text{m}$ so that, accounting for the two parts of the curtain captured, the thickness of the curtain is half this value, approximately 50–100 $\mu \mathrm{m}$.

Figure 11

Schematic in the frame of reference moving with the root of the crown. In the absence of the ambient gas, propagation of the front is defined by the balance between capillary pressure under the concave liquid surface and viscous forces. When the ambient gas density is high enough, it slows the crown (dashed line), which changes the shape of the curved interface and slows down propagation of the liquid front.