Dipping into a new pool: The interface dynamics of drops impacting onto a different liquid

When a drop impacts onto a pool of another liquid, the common interface will move down at a well-defined speed for the first few milliseconds. While simple mechanistic models and experiments with the same fluid used for the drop and pool have predicted this speed to be half the impacting drop speed, this is only one small part in a rich and intricate behavior landscape. Factors such as viscosity and density ratios greatly affect the penetration speed. By using a combination of high-speed photography, high-resolution numerical simulations, and physical modeling, we disentangle the different roles that physical fluid properties play in determining the true value of the postimpact interfacial velocity.

Figure 1

(a) Diagram of the experimental setup. Two cameras observe the drop impact above and below the surface of the pool. (b) Sketch of the axisymmetric simulation domain in its initial state. (c) Zoomed-in view showing the adaptive mesh refinement, achieving spatial resolutions down to 0.5 $\mu \mathrm{m}$.

Figure 2

Comparisons between an experiment and direct numerical simulation for a 2.56-mm diameter drop of a 5 cP water-glycerol solution impacting a FC-40 pool at 0.502 ${\mathrm{ms}}^{-1}$ with ${\mathrm{Re}}_d=274.1(\overline{\mathrm{Re}}=285.8),{\mathrm{We}}_d=9.4,{\mathrm{Fr}}_d=3.2$ at $-0.079$, 0.171, 0.393, and $0.837\mathrm{ms}$ after impact from left to right corresponding to dimensionless times $-0.015$, 0.034, 0.078, and 0.166, respectively. Top row: View from underside of the pool surface; bottom row: left-half side, view above pool; and right-half side, simulation results showing the magnitude of the velocity field. Interface contours are extracted from the numerical data.

Figure 3

Displacement of the center of the drop-pool interface for experiments and numerical simulations for three cases: $•$ drop and pool are of the same liquid [${\mathrm{Re}}_d=197.7(\overline{\mathrm{Re}}=197.7),{\mathrm{We}}_d=26.6,{\mathrm{Fr}}_d=4.5, \overline{V}=0.503\pm 0.002$], $•$ liquids have approximately the same density but different viscosities (${\mu }_r=0.24$ and ${\mathrm{Re}}_d=58.4(\overline{\mathrm{Re}}=243.3),{\mathrm{We}}_d=29.7,{\mathrm{Fr}}_d=3.9, \overline{V}=0.537\pm 0.003$), and $•$ liquids have approximately the same viscosity but different densities (${\rho }_r=1.76$ and ${\mathrm{Re}}_d=274.1(\overline{\mathrm{Re}}=285.8),{\mathrm{We}}_d=9.4,{\mathrm{Fr}}_d=3.2, \overline{V}=0.469\pm 0.005$). $\blacksquare$ represent experimental data from Ref. [9], for conditions similar to $•$. Different lines show the numerical results.

Figure 4

Penetration velocity including simulations, experiments, and models for (a) fixed viscosity ratio and varying density ratio and (b) fixed density ratio and varying viscosity ratio. In each case the droplet corresponds to a $D=1.9\mathrm{mm}$ 5 cSt SO drop impacting at either $1.1,0.55,$ or $0.275{\mathrm{ms}}^{-1}$, with the pool density or viscosity varied to produce the correct ratio. Note the abscissa is logarithmic in the main figure and linear in the inset. Where error bars are not visible they are smaller than the symbol itself. The numbering of the experimental results ($•$) is consistent between (a) and (b) and are as follows: (1) 5 cSt SO drop on 5 cSt SO pool ${\mathrm{Re}}_d=197.7(\overline{\mathrm{Re}}=197.7),{\mathrm{We}}_d=26.6,{\mathrm{Fr}}_d=4.5$; (2) water drop on water pool ${\mathrm{Re}}_d=1110.2(\overline{\mathrm{Re}}=1110.2),{\mathrm{We}}_d=6.6,{\mathrm{Fr}}_d=2.7$; (3) 5 cP water-glycerol solution drop on FC-40 pool ${\mathrm{Re}}_d=274.1(\overline{\mathrm{Re}}=285.8),{\mathrm{We}}_d=9.4,{\mathrm{Fr}}_d=3.2$; (4) 20-cSt SO drop on 5 cSt SO pool ${\mathrm{Re}}_d=58.4(\overline{\mathrm{Re}}=243.3),{\mathrm{We}}_d=29.7,{\mathrm{Fr}}_d=3.9$; and (5) 5 cSt SO drop on 20-cSt SO pool ${\mathrm{Re}}_d=201.9(\overline{\mathrm{Re}}=48.5),{\mathrm{We}}_d=27.0,{\mathrm{Fr}}_d=4.4$. The range $3.27\le \overline{\mathrm{Re}}\le 3344$ is explored in panel (b). $•$ represents DNS run with same conditions of experimental point denoted by $•$. Representative videos are presented as part of Appendix pp4.

Figure 5

The dataset from Fig. 4 showing the penetration velocity as a function of the composite Reynolds number $\overline{\mathrm{Re}}$ for density ratio ${\rho }_r=1$. The inset represents the same plot with linear scale for the abscissa. While in this case the droplet-based Reynolds number ${\mathrm{Re}}_d$ is not relevant, the distinction between the points is retained in order to facilitate comparisons to Fig. 4.

Figure 6

(a) The front displacement and (b) velocity against simulation time for four different measurement techniques. In the case of tracking either the top or bottom of the bubble (corresponding the underside of the drop or the top of the pool, respectively) the contraction of the air film into a bubble produces large variations in the displacement and thus velocity. Following the middle of the bubble lessens this effect, but it is still visible in the velocity. By fitting a quadratic function to the profile of the underside of the drop (as demonstrated in Fig. 7), an accurate measure of the front motion can be found. In this case the simulation corresponds to the impact of a 5 cSt SO drop onto an identical pool with ${\mathrm{Re}}_d=104.5(\overline{\mathrm{Re}}=104.5),{\mathrm{We}}_d=26.724,$ and ${\mathrm{Fr}}_d=4.0$ at resolution level 12.

Figure 7

Comparison of the different methods for tracking the interface position for (a) ${\rho }_r=1, {\mu }_r=1, t=0.116$; (b) ${\rho }_r=4, {\mu }_r=1, t=0.216$; (c) ${\rho }_r=1, {\mu }_r=0.5, t=0.216$; and (d) ${\rho }_r=1, {\mu }_r=1$ at early times ($t=0.016$) just after the rupture of the air film. In all cases the impact conditions are ${\mathrm{Re}}_d=104.5[\overline{\mathrm{Re}}=104.5\mathrm{except}\left(\mathrm{c}\right)\mathrm{where}\overline{\mathrm{Re}}=209.0],{\mathrm{We}}_{\mathrm{d}}=26.724,$ and ${\mathrm{Fr}}_d=4.0$ at resolution level 12. In each case the inset shows the bubble region in detail with the maroon square, triangle, and circle showing the bubble top, middle, and bottom, respectively and the cyan triangle the position of the interface by fitting a quadratic to the drop underside. The quadratic function is fitted to the points marked in green on the underside of the drop a sufficient distance away from the central bubble. The black line then shows the result of this procedure.

Figure 8

Dimensionless viscous energy dissipation per phase against viscosity ratio for the intermediate impact conditions of ${\mathrm{Re}}_d=104.5(6.53\le \overline{\mathrm{Re}}\le 1672),{\mathrm{We}}_d=26.724,$ and ${\mathrm{Fr}}_d=4.0$ at resolution level 12. In each case the density ratio is 1 and the viscosity ratio varies. The values of the viscous dissipation are taken at $t=0.6$. The inset illustrates the same data as a percentage of the total energy dissipated.

Figure 9

Temporal evolution of the viscous energy dissipated per phase (by color) for three different impact velocities (solid line ${\mathrm{Re}}_d=104.5(\overline{\mathrm{Re}}=104.5),{\mathrm{We}}_d=26.724,$ and ${\mathrm{Fr}}_d=4.0$, dashed line ${\mathrm{Re}}_d=52.25(\overline{\mathrm{Re}}=52.25),{\mathrm{We}}_d=3.341,$ and ${\mathrm{Fr}}_d=2.9$ and dotted line ${\mathrm{Re}}_d=26.125(\overline{\mathrm{Re}}=26.125),{\mathrm{We}}_d=0.835,$ and ${\mathrm{Fr}}_d=1.4$) for the impact of a 5 cSt SO drop onto an identical pool at resolution level 12. The three inset plots show the energy dissipation for each impact velocity separately, up until the point that the air film ruptures in each case. The time at which this occurs is indicated by the vertical black lines in the main plot, with the impact velocities again indicated by the line style.