Boundary-layer effects in droplet splashing

A drop falling onto a solid substrate will disintegrate into smaller parts when its impact velocity $V$ exceeds the so-called critical velocity for splashing, i.e., when $V>V^{*}$. Under these circumstances, the very thin liquid sheet, which is ejected tangentially to the solid after the drop touches the substrate, lifts off as a consequence of the aerodynamic forces exerted on it. Subsequently, the growth of capillary instabilities breaks the toroidal rim bordering the ejecta into smaller droplets, violently ejected radially outward, provoking the splash [G. Riboux and J. M. Gordillo, Phys. Rev. Lett. 113, 024507 (2014)]. In this contribution, the effect of the growth of the boundary layer is included in the splash model presented in Phys. Rev. Lett. 113, 024507 (2014), obtaining very good agreement between the measured and the predicted values of $V^{*}$ for wide ranges of liquid and gas material properties, atmospheric pressures, and substrate wettabilities. Our description also modifies the way at when the liquid sheet is first ejected, which can now be determined in a much more straightforward manner than that proposed in Phys. Rev. Lett. 113, 024507 (2014).

Figure 1

Sketch of the lamella for $T>T_e$ for $V>V^{*}$, i.e., for impact velocities above which the lamella dewets the substrate. The regions in which the pressure is larger or smaller than the reference atmospheric pressure are indicated with either a plus or a minus sign. The lift force responsible for droplet splashing results from the integration of the pressure distribution along the edge of the lamella. This figure also illustrates the definitions of the different variables needed to describe the position of the rim.

Figure 2

(a) Computed shapes of a drop using the potential flow numerical code described in [37] for a value of the Weber number $\mathrm{We}=100$. The inset indicates the position of the stagnation point existing in the flow in a relative frame of reference translating with a velocity $\dot{a}$ for different instants of time. (b) Comparison between the computed values of the radial velocity $v_r(r,z=0)$ corresponding to the different drop shapes depicted in part (a) (continuous lines) and the radial velocity field assumed ad hoc in [38] (dashed lines). The origin $r=0$ corresponds to the impact point. The vertical lines indicate the radial position of the root of the lamella, $r=a=\sqrt{3t}$.

Figure 3

(a) Time evolution of the values of the radial velocity profiles depicted in Fig. 2 represented in a frame of reference translating at a velocity $\dot{a}=1/2\sqrt{3/t}$. The values of the radial velocity are normalized by $\dot{a}$ and are represented as a function of the distance to the root of the lamella, $r-a$. Notice that the relative radial velocity is $\dot{a}$ at $r=a$ and it is zero at $r=r_s$, with $r_s$ indicating the radial position of the stagnation point in the relative frame of reference, marked using a colored dot. (b) The radial velocity $v_r$ varies linearly between $r=r_s$ and $r=a$, namely the spatial region located between the stagnation point of the flow in the relative frame of reference and the root of the lamella. Indeed, notice that distances are normalized here by $r_s-a=c_n{h}_a\left(t\right)$, with $c_n=1.5$ a constant (see the inset). The relevant region for the development of the boundary-layer flow entering into the lamella is $0<\left(r-r_s\right)/\left(c_n{h}_a\right)<1$, whereas the region $\left(r-r_s\right)/\left(c_n{h}_a\right)>1$ corresponds to that of the ejected liquid sheet, which can be described using a ballistic approximation [37, 42].

Figure 4

Log-log plot of the radial velocity as a function of the distance to the root of the lamella. In agreement with Eq. (9), the radial velocity decays as ${\left(a-r\right)}^{-1/2}$ for $r\gtrsim r_s$.

Figure 5

Sketch of the flow developing between the stagnation point in the relative frame of reference and the root of the lamella.

Figure 6

Solution of Eq. (17) subjected to the boundary conditions given in Eq. (18).

Figure 7

Sketch of the forces decelerating the advancing front of the lamella.

Figure 8

Continuous lines represent the values of $t_e$ calculated either as $t_e=1.05{\mathrm{We}}^{-2/3}$ for ${\mathrm{Re}}^{1/6}{\mathrm{Oh}}^{2/3}<0.25$ or as $t_e=0.6{\mathrm{Re}}^{-1/3}$ for ${\mathrm{Re}}^{1/6}{\mathrm{Oh}}^{2/3}>0.25$. Dashed lines represent the values of $t_e$ obtained solving Eq. (1). The numerical values associated with each symbol represent $1000\times \mathrm{Oh}$. The value of the Ohnesorge number is $\mathrm{Oh}=2.3\times {10}^{-3}$ for the case of the experiments reported in [9].

Figure 9

(a) Comparison between the critical velocity $V^{*}$ measured experimentally for the case ${\mathrm{Re}}^{1/6}{\mathrm{Oh}}^{2/3}<0.25$ and the corresponding velocities predicted by Eq. (30). The material properties of the different liquids used, the type of solid substrate, the radii of the impacting drops, and the corresponding values of the Ohnesorge number are summarized in Table 2. The inset represents the comparison between predicted and measured values of $V^{*}$ when ${\mathrm{Re}}^{1/6}{\mathrm{Oh}}^{2/3}>0.25$. In part (a), the surrounding gas is air at normal atmospheric conditions (see Table 1). (b) Comparison between the predicted and measured values of the critical splash velocity for the case of the experiments reported in [4]. In this case, ${\mathrm{Re}}^{1/6}{\mathrm{Oh}}^{2/3}<0.25$ and the material properties of the gases and liquids used are provided in Tables 1 and 3, respectively. Continuous lines represent the predicted value of $V^{*}$ for $K_h=2$, while dashed lines represent the corresponding values of $V^{*}$ for $K_h=2.5$.