10. The time evolution of the fingering pattern in a Newtonian oil
10.3. The number of fingers in comparison to linear 2D theory
10.3. The number of fingers in comparison to linear 2D theory
@T = 25◦C η[Pas] b0[µm] v0[µm/s] η[Pas] b0[µm] v0[µm/s]
τ0 '9.6×10−6 97.2 100 66 11.4 75 238
97.2 75 28 11.4 50 71
97.2 50 8 11.4 35 24
97.2 25 1 11.4 25 9
Table 10.2.: Different realizations of the control parameterτ= 9.6×10−6.
ti=Ti−Tfirst move+Tfirst move−Tlast still
2 . (10.6)
Ti is the time stamp of image i. Tfirst move and Tlast still denote the time stamps of the two images between which the oil started to move first. In this way, t0 lies between the last image on which the oil did not move and the first image on which the dust particles changed their position. This method yields an uncertainty in the time of the order of the inverse frame acquisition rate of the camera.
10.3. The number of fingers in comparison to linear 2D
Chapter 10. The fingering pattern in a Newtonian oil
100 80 60 40 20 0
N
12 10 8 6 4 2 0
t'
Figure 10.5.: N versust0 forτ0= 9.6×10−6. The solid line corresponds to the theoretical prediction from linear stability analysis equation10.9.
space is more and more restricted, their growth is hindered, and shielding effects are expected.
We concentrate now on the effect of the changing control parameter. To obtainτ as a function of t0, one simply replaces all quantities in τ0 by their time dependent analogue, see also section 4.2.2. The equation for the time dependent τ then reads [104]
τ(t0) = γb(t0)3
12v(t0)ηR(t0)3 . (10.8)
Hence follows a time dependence of the number of fingers N(t0) =
s 1 3
µ
1 + 1
2τ0(1 +t0)9/2
¶
. (10.9)
Note that for higher numbers of fingers, one can drop the 1 in equation 10.9, and the number of fingers depends on the time as a power law,
N2∝(1 +t0)−9/2 . (10.10)
This equation based on two dimensional linear stability analysis is the simplest way to approach the change in the number of fingers. One assumes a freshly starting experiment at each moment, albeit with an interface that is disturbed by the fingers that have evolved previously. The straight line in figure 10.5 corresponds to the theoretical prediction from equation 10.9forτ0= 9.6×10−6. At the beginning, the average number of fingers is of the same order of magnitude in experiments and the- ory. Subsequently, the number of fingers decreases much slower in the experiments than calculated and reaches zero at later times, so that the theoretical prediction
10.3. The number of fingers in comparison to linear 2D theory
Figure 10.6.: Overlay of two images at different times. The lighter contact line corresponds tot2> t1 (darker contact line).
always underestimates the actual number of fingers. This result has been described before. Lindneret al. compared the number of fingers from numerical experiments, laboratory experiments, and the linear theory [75]. They found that the number of fingers from experiments and simulations agreed, whereas they were always higher than the most unstable number of fingers from linear theory. In the following, we will discuss in more detail the origin of this deviation.
Growing and dying fingers
A closer look at the images of the experiments reveals that the decrease in the number of fingers is caused by adying out of fingers. We understand thereby that a certain finger stops growing inwards at a certain time and stays fixed at its current position. From that moment on, only the surrounding fingers continue to move inwards and they outrun the fixed finger. The material bridges that are left between the air fingers, see figure 10.6, contract to the middle of the cell, so that a fixed finger finally disappears. The retraction of these bridges is clearly visible in figure 10.4. Figure 10.6 shows an overlay of two images at different times t01 and t02 > t01. The lighter, more transparent image has been taken at a later time. The finger labeled as “growing finger” is advancing between t01 and t02. On the contrary, the finger labeled as “dying finger” does not change its position. We call a finger on an imagei dying (or stagnant), if its tip does not move towards the center of the cell between imageiand imagei+1. A dying finger can also be observed on the last two images of figure10.3. This coarsening starts very early in the debonding process.
Figure 10.7 shows the number of growing fingers Ngrow as a function of time. It is of course smaller than the total number of fingersNall. At the very beginning, where the fingers are harder to distinguish, it is difficult to judge whether each finger is growing. Comparing the number of growing fingers to the linear prediction from equation10.9, we find that the experimental results and the linear prediction agree
Chapter 10. The fingering pattern in a Newtonian oil
surprisingly well. This is a startling result, as it implies that the system chooses the number of fingers corresponding to the linearly most unstable wavelength at every moment in time and makes them grow.
As the number of growing fingers is governed by the changing control parameter, we conclude that geometrical shielding effect play only a secondary role. To separate geometrical effects, it is necessary to perform an experiment at constant control parameter. The lifting speed has to be changed in such way that the variable gap distance is compensated andτ(t0) =τ0. A decrease in the finger number would then be caused by geometric constraints. Such an experiment could unfortunately not be realized in our setup.
100 80 60 40 20 0 Ngrow
12 10 8 6 4 2 0
t'
Figure 10.7.: The number of growing fingers versust0in comparison to the linear theoretical prediction from equation10.9forτ0= 9.6×10−6.
To conclude so far, the total number of fingers as a function of time exceeds the number of fingers predicted from linear theory. This result has been observed as well in a laboratory experiment (this thesis and reference [75]) as in numerical simulations based on the 2D Navier–Stokes equations [75]. The experimental situation can be depicted in the following way. Consider a number of fingers N1 at t0 = t01. At t02 =t01+∆t, a number of fingersN2is growing. However,N1−N2“additional” fingers are present in the experiment. Even if they are not growing, they are contributing to the total number of fingers. The oil that remains between the air fingers retracts to the cell center subsequently and the stagnant fingers therefore die out.
The total number of fingers is imperatively higher than linear theory predicts, unless the material bridges contract very fast. In this case, the advancing finger base
“keeps up” with the advancing finger tips and the finger amplitudes are vanishing.
We have shown that the number of growing fingers is described by a time dependent control parameter derived from linear stability analysis.