Height of the Deforming Drop at the Symmetry Axis 78

If the impact Weber and Reynolds numbers are high, the flow far from the wall is determined mainly by inertia. Therefore, the dimensionless velocity of the lamella and its dimensionless shape in the central region of the deforming drop should not depend on the impact conditions. The recent study of Bakshi et al. [4] on the ax-isymmetric drop impact onto a dry spherical target confirms this assumption. The convex shape of the target geometry in this study allows one to observe the devel-opment of the lamella during the entire drop spreading process. Besides the other

parameters, the height of the deforming drop at the symmetry axis is measured. It was shown that in the initial phase of drop deformation the drop height first reduces with an almost constant velocity and the rear part of the drop moves almost as a rigid body. Then, at the time instant t ≈ 1/2 the process of drop deformation switches to a new regime and the drop height follows the inverse square dependence of time predicted by the remote asymptotic solution given in Eqs. (3.41–3.42). The evolution of the drop height depends neither on the impact Weber nor Reynolds number but is determined only by the ratio of the drop and target radii. Finally, when h is small enough, the viscous stresses govern the flow in the lamella. These viscous stresses, which become significant at time t = t_visc, lead to a damping of the flow. The residual film thickness is therefore a function of the Reynolds number only.

There is no reason to believe that drop impact onto a flat rigid substrate behaves differently. In the left graph in Fig. 3.28 the experimental data of Bakshi et al. [4]

and the results of numerical predictions from Fukai et al. [28], ˇSikalo et al. [107]

and Mukherjee and Abraham [60] for the evolution of the drop height h_C at the symmetry axis are shown as a function of time for various impact parameters. The corresponding results of the present numerical simulations of drop impact onto a symmetry plane are shown in the right graph in Fig. 3.28.

0 1 2 3 4 5 6

t 0.0

0.2 0.4 0.6 0.8 1.0

h c

Re=1068 We=144 (experiment) Re=210 We=30

Re=450 We=137 Re=4010 We=90 Re=6020 We=117 Re=6260 We=128 1 - t

0.39 (t + 0.25)²

0 1 2 3 4 5 6 7 8

t 0.0

0.2 0.4 0.6 0.8 1.0

h c

Re=61, We=397 Re=83, We=761 Re=104, We=1165 1 - t

0.39 (t + 0.25)²

Figure 3.28: Drop impact onto a dry substrate (left): the experimental data from Bakshi et al. [4] and the results of existing numerical predictions from Fukai et al. [28], ˇSikalo et al. [107] and Mukherjee and Abraham [60] for the evolution of the lamella thickness at the impact axis as a function of dimensionless time. Drop impact onto a symmetry plane (right):

the results of the present numerical predictions for the evolution of the lamella thickness at the impact axis as a function of dimensionless time.

During the first two regimes all the results lie approximately on a single curve for all the impact parameters. In the first and second non-viscous regimes, the height

of the drop at the symmetry axis can be approximated by the following expressions

h_C ≈ 1−t, at t <0.4, (3.48)

h_C ≈ 0.39

(t+ 0.25)², at 0.7< t < t_visc, (3.49) which are also plotted in graphs in Fig. 3.28. Equation (3.48) is chosen from the assumption that the rear part of the drop moves initially almost as a rigid body, while Eq. (3.49) is taken in the form of the remote solution, Eq. (3.42). Equations (3.49–

3.42) represent universal expressions for the thickness of the lamella generated by drop impact onto a dry flat substrate and onto a symmetry plane valid for all the impact parameters when both the Reynolds number and the Weber number are much larger than the unity.

In the left graph in Fig. 3.28 an indication of the third, viscous regime of the flow in the lamella at timest ≥t_visc can be observed, since the Eq. (3.49) underestimates slightly the numerically predicted values. However, the precise estimation of the residual film thickness and the time instant t_visc is difficult, since in the results of Fukai et al. [28] the lamella thickness at the latest stages of drop impact is comparable to the thickness of a line plotted in the graph. From the obtained results shown in the right graph in Fig. 3.28 it is obvious that the development of the near-wall boundary layer is not relevant in the case for drop impact onto a symmetry plane and therefore t_visc → ∞ in this case.

3.5.3.3 Flow in the Lamella at Later Times

Since the heighth_C of the lamella can be described well by Eq. (3.49), it can be con-cluded that the velocity in the close proximity of the drop axis can be approximated by the asymptotic solution, Eqs. (3.41).

In Fig. 3.29 the results of the numerical simulations of the average radial ve-locity and the veve-locity gradient at t = 1 in the lamella generated by drop impact onto a symmetry plane are shown as a function of the radius for various impact conditions. Surprisingly, the velocity is linear over a relatively wide range of the radius. Moreover, the velocity distribution almost does not depend on the impact parameters except at the edge region, where the lamella is compressed due to the viscous and capillary forces. Far from the edge the velocity gradient is nearly uni-form ∂U_r/∂r≈0.8 at t = 1 which agrees well with the remote asymptotic solution of Eqs. (3.41) with τ = 0.25. This value of τ is used in Eq. (3.49) to fit the data for the lamella thickness h_C at the impact axis.

Fig. 3.30 shows the shape of the central part of the spreading lamella. It is seen that also the lamella shape almost does not depend on the impact conditions except for the edge region associated with the rim formation. Therefore, the thickness of the lamella and its velocity distribution at very high Weber and Reynolds numbers are self-similar.

The exact shape of the lamella is not modeled analytically, but it is determined using the numerical simulations during the initial stage of drop deformation. The

0 0.5 1 1.5 2 r

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

U r

We=397, Re=61 We=761, Re=83 We=1165, Re=104

0.0 0.5 1.0 1.5 2.0

r -0.8

0.0 0.8 1.6 2.4

∂U r / ∂r

We=397, Re=61 We=761, Re=83 We=1165, Re=104

Figure 3.29: Drop impact onto a symmetry plane: the results of numerical predic-tions of the dimensionless average radial velocity (left) and the dimen-sionless gradient of the radial velocity (right) at the time instant t= 1 as a function of the dimensionless radius.

0.0 0.5 1.0 1.5 2.0

r 0.00

0.05 0.10 0.15 0.20 0.25 0.30

h

Re=6020 We=117 Re=6260, We=128 Re=61, We=397 Re=83, We=761 Re=104, We=1165 solid wall, Fukai et al. [28]:

symmetry plane:

Figure 3.30: Drop impact onto a symmetry plane: the results of numerical predic-tions for the dimensionless lamella thickness at the time instant t = 1 as a function of the dimensionless radius (lines) compared with the numerical simulations of drop impact onto a flat rigid substrate from Fukai et al. [28] (symbols).

lamella thickness distribution is approximated by the Gaussian function in Eq. (3.47) using τ = 0.25 determined from the numerical simulations of drop impact onto a symmetry plane. The corresponding function satisfying the condition h_L = h_C at r= 0 has the following form

hL= 0.39

(t+ 0.25)² exp [

− 2.34r² (t+ 0.25)2

]

. (3.50)

In Fig. 3.31 the approximate shape determined by Eq. (3.50) is compared with the numerical simulations of Fukai et al. [28] of drop impact with Re = 1565 and

0.0 0.5 1.0 1.5 2.0 r

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

h L

t = 0.5 t = 1 t = 2

t = 0.5 t = 1 t = 2

numerical, Fukai et al. [28]:

approximation:

Figure 3.31: Drop impact onto a dry substrate at Re = 1565 and We = 32: numer-ical predictions from Fukai et al. [28] of the lamella shape at various time instants compared with the approximate shape, Eq. (3.50).

We = 32. Att= 1 andt= 2 the agreement is rather good far from the lamella edge where the lamella is thicker than predicted (indicating that the velocity gradient here is smaller then near the axis r = 0) and where the rim formation becomes visible.

This agreement indicates that the inviscid remote asymptotic solution of Yarin and Weiss [135] given by Eqs. (3.41) can predict well the velocity field in the central part of the deforming drop far enough from the rim. At t = 0.5 the approximated solution overpredicts the results of numerical simulations. At this relatively early stage of spreading the velocity field is still two-dimensional and cannot be described by the remote asymptotic solution.