(a)Experimental results of a sliding drop on a tilted plate for a variety of constant inclination angles. (pictures from and experiments by Podgorski (2000)).
0 2 4 6 8 10
xin[mm]
(b)Simulation results using an increasing tilting an-gleβat the rate ofβ˙=5◦/s. The contact line shape is shown for different time steps that correspond to increasing tilting angles.
Figure 4.5:Shape of a PMMA drop (VD=6µl;ν=10.68×10−6m2/s) sliding down a glass plate (θA,R=55◦, 45◦). For an increasing Capillary number, from left to right, the contact line shape changes from spherical to oval and, then, to a rounded corner at the rear of the drop. The rounded corner forms to a cusp eventually emitting smaller droplets.
To conclude, the exceptional accordance of the simulation data with the experiments of Pierce et al. (2008) validates the here used framework for the incipient motion of drops on tilted plates. Considering the measurement uncertainties of Bommer et al. (2014), the simulation results are as well promising for different liquid-solid pairs. Post-processing of the adhesion force is verified and the definition of the critical tilting angle via the force balance of mass and adhesion forces is found to be valid.
0 0.5 1 1.5 Boβ
0 0.002 0.004 0.006 0.008 0.01
Ca
47V10
47V10 - Pearling Ca∝Boβ−Boc
Figure 4.6: Simulated drop velocity over the inclination Bond number similar to the experiments by Le Grand-Piteira et al.
(2005). The simulation is performed with increasing tilting an-gleβ˙=5◦/sversus different but constant tilting angles in exper-iments. The drop acceleration apparently plays a significant role in the simulations.
0 0.005 0.01 0.015
Ca 0
2 4 6 8 10 12
w,l,hinmm
Oval Corner Cusp/Pearling w- Sim.
w- Exp.
l- Exp.
l- Exp.
h- Sim.
h- Exp.
Figure 4.7:The drop shape measured in terms of height, width and length for an increasing drop velocity. The experimental and simulation results are of similar character.
Single experiments are performed at a constant inclination angle whereas in the simulation a tilting rate ofβ˙=5◦/sis used to cover the wholeC arange in one simulation. The drop is refined in three levels down to∆x=50µm. Note that in this section the Capillary number is calculated using the global drop velocity instead of the local contact line velocity.
The drop velocity and the plate inclination can be expressed in terms of the Capillary number and the Bond number.
Three forces act on the drop in quasi-static motion: a viscous drag on the glass plate, the interfacial forces and the weight of the drop. They scale with−ηU V1/3,−σV1/3∆and%V gsinβrespectively. The Bond number based on the component of gravity parallel to the tilted plate is defined asBoβ =Bosinβ=V2/3ρg/σ. Figure 4.6 shows the Capillary number over the inclination Bond numberBoβ. Le Grand-Piteira et al. suggest a scaling law as a result of a force balance
C a∝Boβ−Boc, (4.8)
whereBocis a constant depending on the wetting hysteresis∆θgiven by Dussan V. (1985) as
Boc=24 π
1/3(cosθR−cosθA)(1+cosθA)1/2
(2+cosθA)1/3(1−cosθA)1/6 . (4.9) The comparison of the scaling law with the simulation results indicates that the drop acceleration due to the continuous tilting rate ofβ˙=5◦/sstill has a strong influence. Consequently, a much lower tilting rate would be recommended for a quantitative comparison to the experiments.
Despite the differences in the case setup, a comparison of the drop geometry is still insightful. In Figure 4.7, the lengthL, widthW, and maximum drop heightHare compared with experimental results. The drop height stays almost constant for increasing Capillary numbers and the drop width decreases slightly, whereas the length of the drop almost doubles reaching a maximum before the breakup of the droplet at the cusp.
-0.01 -0.005 0 0.005 0.01 0.015 Ca
0 10 20 30 40 50 60 70 80 90
θin[◦]
Sim.
Kistler Exp.
Figure 4.8: Comparison of dynamic contact angle distribution using the Kistler model (see equation 3.43) in simulations and experimental measurements.
The dynamic contact angle model by Kistler (1993) used in simulations leads to contact angles at the receding contact line down to zero for high receding Capillary numbers as described in Section 2.4. The contact angles are shown in Figure 4.8. This stands in contrast to a finite contact angle found in experiments by Le Grand-Piteira et al. ofθmin=21◦. The formation of the corner at the rear end of the drop was first explained theoretically by Blake and Ruschak (1979), where they postulate that a maximum receding contact line speedC acr exists. WhenC acris exceeded, the contact line inclines to maintain the speed normal to the contact line constant at the maximum contact line speed
C a= C acr
sinφ, (4.10)
whereinφis defined between the drop motion direction and the tangent of the contact line. This speed normal to the contact line increases with the drop speed and plateaus at an almost constant level while forming a corner. This behavior is recovered in simulations shown in Figure 4.9 as well as in experiments.
In simulations, the tangent of the contact line is estimated by two points, of which one is defined at the rear tip of the drop. The tip serves as the center of a circle with radius0.5 mmwhich cuts the contact line in the second point, as indi-cated in Figure 4.5. To define the beginning of the corner regime, a threshold ofsinφ >0.98is set which coincides well with the transition shown in Figure 4.9 as well as Figure 4.10. In the corner regime, the dependence ofφis proportional to1/C alaw as already suggested by Podgorski et al. (2001).
A transition between the corner and the cusp regime is naturally set atφ=45◦which is in line with the saddle point corner tip description by Amar et al. (2001). Differently to Podgorski et al., Le Grand-Piteira et al. suggested a separate cusp and pearling regime distinguishable not only by the emitted small drops but also by the Capillary number. In simu-lations, however, the drop slows down when emitting droplets which is in accordance with the reduced mass of the main drop. In contrary to the simulations, the experiments are fed with new drops at a frequency of 0.5 Hz, such that the main drop collects the droplets from the predecessor drop.
To summarize, in our simulations all regimes shown in experiments by Le Grand-Piteira et al. (2005) are recovered.
The simulation setup expands the experimental results by an increasing tilting angle to cover a wide range of Capillary numbers versus many quasi-static experiments with fixed tilting angles. Because of an additional drop acceleration in the simulation, the transitions from an oval to a corner and from a corner to a cusp-shaped contact line appear for higher
0 0.002 0.004 0.006 0.008 0.01 Ca
0 1 2 3 4 5 6 7
Casinφ
×10−3
Oval Corner Cusp/Pearling
47V10
47V10 - pearling Casinφ= const.
Figure 4.9:The simulated contact line velocity in drop motion direction remains almost constant in the corner regime.
0 0.005 0.01 0.015
Ca 0.2
0.4 0.6 0.8 1 1.2
sinφ
Oval Corner Cusp/Pearling
47V10
47V10 - pearling
∝1/Ca
Figure 4.10:The contact line corner opening angleφexperiences a sharp transition in the corner regime in order to keep the Capil-lary number of the receding contact line constant.
Capillary numbers. Despite the setup difference, the constant plateau ofC asinφis comparable and in line with the the-oretical maximum receding speed postulated by Blake and Ruschak (1979). Also, the aspect ratio of the drop is similar to experiments. These observations lead to the conclusion that the receding motion of the contact line, in particular, is well described by this computational framework. Unlike shown in experiments, the pearling regime is not distinguishable from the cusps regime via the Capillary number, because of the differences in the setup.
As this study focuses on the drop motion in turbulent shear flow, the insight on the validity of the receding contact line motion is sufficient for this study. However, a further detailed investigation by simulations of the transition from corners to cusps and the break-up of droplets might be as well insightful. Moreover, it would help to understand the formation of rivulets from large drops.