Observation of the pressure effect in simulations of droplets splashing on a dry surface

At atmospheric pressure, a drop of ethanol impacting on a solid surface produces a splash. Reducing the ambient pressure below its atmospheric value suppresses this splash. The origin of this so-called pressure effect is not well understood, and this study presents an in-depth comparison between various theoretical models that aim to predict splashing and simulations. In this paper, the pressure effect is explored numerically by resolving the Navier-Stokes equations at a 3-nm resolution. In addition to reproducing numerous experimental observations, it is found that different models all provide elements of what is observed in the simulations. The skating droplet model correctly predicts the existence and scaling of a gas film under the droplet, the lamella formation theory is able to correctly predict the scaling of the lamella ejection velocity as a function of the impact velocity for liquids with different viscosity, and lastly, the dewetting theory's hypothesis of a lift force acting on the liquid sheet after ejection is consistent with our results.

Figure 1

Time series of the impact of a droplet on a dry surface. The top frames (a)–(c) show simulation results for atmospheric pressure at different times. The bottom frames show (d) liquid-sheet breakup at atmospheric pressure, (e) no breakup at reduced pressure, and (f) the central air bubble.

Figure 2

Time series of the droplet interface showing the evolution of the contact line at atmospheric pressure. Light blue represents the liquid phase. The vertical axis shows the distance normal to the surface and the horizontal axis shows the distance parallel to the surface relative to the center of the droplet. Both axes are in $\mu \mathrm{m}$.

Figure 3

Gas film profiles of different droplets approaching the wall. The vertical axis shows the height above the wall and the horizontal axis the radial distance away from the center. Both axes are in $\mu \mathrm{m}$. The figures in the top row (a)–(d) show an impact velocity of $2.5\mathrm{m}{\mathrm{s}}^{-1}$ and the time difference between successive lines is $1.0\mu \mathrm{s}$. The figures in the bottom row (e)–(h) show an impact velocity of $10.0\mathrm{m}{\mathrm{s}}^{-1}$ with a time difference of $0.1\mu \mathrm{s}$. (a) and (e) show the results for ethanol at atmospheric ambient pressure, (b) and (f) show the results for the high-viscosity liquid (i.e., a viscosity ten times higher than the viscosity of ethanol), (c) and (g) show the results for the liquid with a high surface tension (i.e., a surface tension 20 times higher than the surface tension of ethanol in air), and (d) and (h) show the results for ethanol at reduced ambient pressure (i.e., an ambient pressure $1/100$ of atmospheric pressure).

Figure 4

Height of the gas film under the drop at the moment of lamella formation for various velocities, viscosities, and surface tensions. The line shows the theoretical predictions by Mandre and Brenner [11].

Figure 5

The minimum thickness of the gas film as function of time comparing two simulations: one simulation with a slip length of $\lambda =1\mathrm{nm}$, and one simulation with a slip length of $\lambda =100\mathrm{nm}$. Both simulations are performed at reduced ambient pressure. The inset shows a magnified view of the droplets approaching the wall.

Figure 6

Time series of the droplet interface with the vertical axis plotted on a logarithmic scale. The axis are in $\mu \mathrm{m}$ and the vertical axis is plotted on a logarithmic scale. Since the vertical axis cannot go to zero, it is cut off at ${10}^{-3}\mu \mathrm{m}$.

Figure 7

Time series of the droplet interface, velocity magnitude, and streamlines, showing the evolution of the contact line at atmospheric pressure. This time series is magnified 15 times compared to Fig. 2. The velocity magnitude is in $\mathrm{m}{\mathrm{s}}^{-1}$, and the axes are in $\mu \mathrm{m}$. The small arrows on the streamlines indicate the direction of the flow.

Figure 8

Time series of the droplet interface and vorticity $\omega$ showing the evolution of the contact line at atmospheric pressure. This time series is magnified 15 times compared to Fig. 2. The axes are in $\mu \mathrm{m}$ and the contours in ${\mathrm{s}}^{-1}$.

Figure 9

Lamella profiles of different droplets approaching the wall. The vertical axis shows the height above the wall and the horizontal axis the radial distance away from the center. Both axes are in $\mu \mathrm{m}$. The figures in the top row (a)–(d) show an impact velocity of $2.5\mathrm{m}{\mathrm{s}}^{-1}$ and the time difference between successive lines is $2.0\mu \mathrm{s}$. The figures in the bottom row (e)–(h) show an impact velocity of $10.0\mathrm{m}{\mathrm{s}}^{-1}$ with a time difference of $0.4\mu \mathrm{s}$. (a) and (e) show the results for ethanol at atmospheric ambient pressure, (b) and (f) show the results for the high-viscosity liquid (i.e., ten times higher than the viscosity of ethanol), (c) and (g) show the results for the liquid with a high surface tension (i.e., 20 times higher than the surface tension of ethanol in air), and (d) and (h) show the results for ethanol at reduced ambient pressure (i.e., $1/100$ of atmospheric ambient pressure).

Figure 10

Various lamella properties as a function of its ejection time. In all three plots the results are shown for an ethanol droplet in a gas at atmospheric ambient pressure ($\circ$), and ethanol drop in a gas at reduced ambient pressure ($\diamond$), a droplet with a high surface tension in a gas at atmospheric ambient pressure ($▵$), and a droplet with a high viscosity ($\square$). (a) Lamella height $z_e$ as a function of ejection time $t_e$. The three data points per symbol correspond to impact velocities of 2.5, 5.0, and $10.0\mathrm{m}{\mathrm{s}}^{-1}$ with $10.0\mathrm{m}{\mathrm{s}}^{-1}$ resulting in the smallest ejection time. (b) Lamella ejection velocity $v_e$ as a function of ejection time $t_e$. The ejection velocity can be nondimensionalized using the impact velocity $v_0$ and droplet diameter $D$, the result of which can be seen in (c). The data follow a scaling of $v_e/v_0\propto t^{-1/2}$.

Figure 11

The radial and vertical velocity of the liquid sheets ejected from the droplet as a function of time. $t_e$ is defined as the moment of sheet ejection.

Figure 12

Lamella ejection velocity as function of the impact velocity of a droplet for various velocities, viscosities, and surface tensions. Ethanol refers to the simulations of ethanol in air, “high viscosity” refers to simulations of a fluid with ten times the viscosity of ethanol in air, and “high surface tension” refers to simulations of a fluid with 20 times the surface tension of ethanol in air. The lines show theoretical predictions by Mandre and Brenner [11] ($v_e\propto v_0^{4/3}$), Riboux and Gordillo [17] ($v_e\propto v_0^{1.3}$ in the current regime), and Mongruel et al. [18] ($v_e\propto v_0^{3/2}$). The error bars in this figure are smaller than the symbols.

Figure 13

(a) Interface evolution of droplet impact at $2.5\mathrm{m}{\mathrm{s}}^{-1}$ with a liquid viscosity ten times that of ethanol. A maximum in the horizontal velocity ($\circ$) can be observed on the interface early after impact and can be traced to the time of lamella formation. At the moment of lamella formation there is a bifurcation for points following the cusp ($\square$) and the lamella ($▵$). (b) The radial position plotted as a function of time for the same data points as in (a). The $t^{1/2}$ scaling suggests that the flow is dominated by inertia. (c) The vertical position of the velocity maximum, lamella, and cusp as a function of time.

Figure 14

(a) Nondimensional vertical position of the velocity maximum as a function of nondimensional time for $2.5\mathrm{m}{\mathrm{s}}^{-1}$ ($▵$), $5.0\mathrm{m}{\mathrm{s}}^{-1}$ ($\square$), and $10.0\mathrm{m}{\mathrm{s}}^{-1}$ ($\circ$). The surface tension is that of ethanol and the viscosity is ten times that of ethanol. The scaling suggests that at early times, lamella formation is dominated by viscosity. (b) Nondimensional vertical position of the velocity maximum as a function of nondimensional time for $2.5\mathrm{m}{\mathrm{s}}^{-1}$ ($▵$), $5.0\mathrm{m}{\mathrm{s}}^{-1}$ ($\square$), and $10.0\mathrm{m}{\mathrm{s}}^{-1}$ ($\circ$). The viscosity is the viscosity of ethanol and the surface tension is 20 times that of ethanol. The scaling suggests that at late times, the scaling for these droplets is dominated by surface tension.